- Email: [email protected]
The American put under transactions costs
Journal of Economic Dynamics & Control 28 (2004) 915 – 935 www.elsevier.com/locate/econbase
The American put under transactions costs Stylianos Perrakisa;∗ , Jean Lefollb a Department
of nance, The John Molson School of business, Concordia University, Montr eal, Qu ebec, Canada, H3G 1M8 b HEC and International Finance Laboratory, University of Geneva, 1211 Gen) eve 4, Switzerland
Abstract This paper examines the optimal super-replication of American put options with physical delivery of the underlying asset, such as stock options, by means of a stock-plus-riskless asset portfolio. The framework of the analysis is the binomial model with proportional transactions costs on stock transactions. The paper extends the model for European options, originally presented in Merton (Geneva Papers Risk Insurance 14 (1989) 225) and Boyle and Vorst (J. of Finance 47 (1992) 271), and generalized in Bensaid et al. (Math. Finance 2 (1992) 63). The optimizing framework of this latter study is adapted to put options held by investors and perfectly hedged by a market maker, and to put options written by investors and both held and hedged perfectly by a market maker. It is shown that a unique optimal super-replicating portfolio exists at every node of the binomial tree for the long option, as well as for the short option when transactions costs are low. ? 2003 Elsevier B.V. All rights reserved. JEL classi cation: G13 Keywords: Pricing; Transactions costs; American options; Put
1. Introduction This paper examines the pricing of American put options by a perfectly-hedged market maker when there are transactions costs to be paid on the underlying stock. Thus, it extends European stock option pricing under perfect hedging, transactions costs, and binomial stock returns, formulated by Merton (1989), and extended by Boyle and Vorst (1992) BV. A similar extension, to American options on dividend-paying stocks under transaction costs, was done in an earlier study (Perrakis and Lefoll, 2000). ∗
Corresponding author. E-mail address: [email protected] (S. Perrakis).
0165-1889/03/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0165-1889(03)00099-X
916
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
While the perfect-hedging assumption may appear extreme for organized option markets, it does oBer a useful benchmark case for the derivation of option bounds, within which the option bid/ask prices must lie. Further, this assumption makes our results also useful to situations where there is a need for option replication such as, for instance, in portfolio insurance. Last, it allows the creation of options in cases in which no organized options market exists, as in most emergent Cnancial markets. Thus, while most results are expressed in terms of option pricing by a perfectly-hedged market maker, they also extend to these other important cases. In the option pricing models of the classic studies of Black and Scholes (1973) and Merton (1973), the call option is perfectly and continuously replicated by a stock-plus-riskless-asset portfolio. The introduction of Cxed transactions costs every time this portfolio is being rebalanced makes such a policy infeasible in a continuous time model. For this reason a number of papers have tackled the problem of portfolio selection and/or option pricing under transactions costs, both in continuous time 1 and on a binomial lattice. Since perfect option replication is infeasible in the continuous time models, those studies that dealt with option pricing speciCed either approximate replication at predetermined and exogenously given times, or expected utility-based portfolio selection under transactions costs. By contrast, the Merton–BV approach replicates both long and short call options at every node of the binomial lattice. While the replication of the long option is feasible in all cases, the replication of the short option requires some restrictions on parameter values. These restrictions are satisCed when transactions costs are ‘small’ for the chosen number of lattice steps, in a sense that will become more precise in Section 4 of this paper. Merton solved the replication problem when the option has only two periods to expiration; BV extended the Merton model to any number of periods. An important study by BensaFGd et al. (1992) BLPS, derived an algorithm to compute optimal perfect-hedging policies for an intermediary that issues long or short options (which they named super-replication), for several types of European options under binomial returns without necessarily replicating the option at every node. 2 The BLPS study found contrasting results for the important cases of physical delivery and cash settlement options: while the intermediary Cnds it optimal to replicate everywhere physical delivery long options, such a policy is suboptimal for cash settlement options, unless transactions costs are ‘small’, in the same sense as in the Merton–BV studies. In spite of its generality and powerful theoretical insights, the BLPS algorithm is rather diHcult to apply as stated for a large number of periods to expiration, since 1 See Leland (1985), Constantinides (1986), Hodges and Neuberger (1989) and Dumas and Luciano (1991). 2 Edirisinghe et al. (1993) also presented a two-stage dynamic programming algorithm for minimum cost hedging of an option without necessarily replicating it; see also Boyle and Tan (1994) for more on their method. The two-stage algorithm, however, was developed for options whose method of settlement is up to the seller, and it is not clear how it could be extended to cover the more realistic physical delivery and cash-settlement options. Further, for options with settlement up to the seller BLPS (1992) presented a closed form solution in their Theorem 4.
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
917
it requires the determination of the intersection points of two piecewise-linear convex functions at every node of the binomial tree. Further, it is based on the assumption that the option holder’s actions that are being hedged are predetermined at option expiration; such an assumption is not appropriate for hedging American options, where the option holder must decide whether to exercise or defer at every node. In this paper it is shown that in the presence of transactions costs there are cases where the option holder’s optimal action (exercise or not) depends on preferences or other holdings. Nonetheless, perfect hedging requires that the stock-plus-riskless-asset super-replicating portfolio be able to hedge all possible holder actions. This remark allows the extension of the BLPS algorithm, to cover the case of American put options with physical delivery of the underlying asset. European put options under transactions costs can be valued through put-call parity 3 from the above-cited studies’ call option prices. For American put options, however, it is well-known since Merton (1973) that early exercise may be proCtable even in the absence of dividends, implying that their value exceeds the one given by put-call parity. Although closed-form expressions do not exist for American puts, recursive or analytical methods have been derived, among others, by Parkinson (1977), Brennan and Schwartz (1977), Geske and Johnson (1984), MacMillan (1986), and Barone-Adesi and Whaley (1987). The binomial model of Cox et al. (1979) and Rendleman and Bartter (1979) can be used to value an American put in the absence of transactions costs. 4 The key element in such a valuation is the derivation of the early exercise boundary, the stock price separating early from deferred exercise at every time period prior to expiration. Here we show that transactions costs makes this boundary more complex, since the holder’s optimal action is ambiguous. The main weakness of the binomial model in handling option pricing under transactions costs is that no guidelines are given to determine the appropriate number of steps in the lattice. This is crucial because, for a xed transactions cost parameter, the total transactions costs increase as the number of steps rises. At the limit the value of the portfolio replicating the long European call option (without dividends) tends to the stock price, while the portfolio replicating the short option becomes equal to the well-known Merton (1973) European lower bound. 5 For this reason some studies such as Henrotte (1993) and Flesaker and Hughston (1994) have proposed replacing the Cxed transactions cost parameter with one that declines in proportion to the square root of the number of steps. This assumption provides reasonable non-trivial option prices under transactions costs, both in continuous time and on binomial lattices. Further, as 3 Put-call parity does not necessarily hold in the presence of transaction costs. Nonetheless, a put-call parity relation can be derived for portfolios replicating call and put options; see the comments in BV (1992, p. 285). None of the results in this paper depend on put-call parity. 4 See Cox and Rubinstein (1985, pp. 245–252). 5 For the long call it was shown in the BV study that, for a large number of binomial steps, the value of the replicating portfolio tends to a Black–Scholes option price with variance increasing proportionately with the square root of the number of steps; this becomes equal to the underlying stock price for inCnite steps. For the short call it was shown in Perrakis and Lefoll (1997) that the value of the super-replicating portfolio becomes equal to the Merton bound as soon as the number of steps rises suHciently to render the Cxed transactions costs parameter ‘large’ in comparison with the binomial parameters.
918
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
will be discussed in Section 5, it also satisCes the restriction of ‘small’ transactions costs in relation to the binomial parameters for most reasonable cases. In the next section we present the problem along the lines of the earlier Merton (1989), BV and BLPS studies. The long put is presented in Section 3, while Section 4 examines the short put for ‘small’ transactions costs. Some numerical examples and a discussion of the extension of the results to ‘large’ transactions costs are presented at the end. 2. The general model Let u, d and R denote, respectively, the size of the up and down moves and one plus the riskless rate of interest, with d ¡ R ¡ u. k denotes the transactions cost parameter, implying that the purchase (sale) of $1 of shares costs $(1 + k) (yields $(1 − k)). DeCne uL ≡ u(1 + k); dL ≡ d(1 − k); u ≡ u(1 − k) and d ≡ d(1 + k). There are no transactions costs in the riskless asset. The strike price of the option is X; n is the expiration time of the option in number of periods, and S0 is the initial stock price. We shall derive the ask and bid prices of American put options, respectively, Pa (S0 ; n) and Pb (S0 ; n), under the assumption, originally formulated by Merton (1989), that these prices are oBered by perfectly-hedged market makers. These prices are equal to the values of portfolios (Nj Sj + Bj ) containing Nj shares and Bj invested in the riskless asset at any time j 6 n, with Sj denoting the stock price. 6 The portfolios must be rebalanced to keep the market maker perfectly-hedged. Hence, Pa (S0 ; n) = N0 S0 + B0 and Pb (S0 ; n) = −(N0 S0 + B0 ) if the option is not going to be exercised immediately, and the portfolio (N0 ; B0 ) super-replicates the put option held long or written by the investor, respectively. We model the transactions costs by means of the convex function 7 ’(y), which is equal to (1 + k)y for y ¿ 0 and to (1 − k)y for y 6 0. Rebalancing the American put’s super-replicating portfolio at any time j ∈ [0; n] implies that one should hedge against both immediate and deferred exercise. We consider physical delivery options only, like the options on individual stocks. The basic feature of the BLPS approach, that will also be adopted in this paper, is that for any time j ∈ [0; n] the hedging portfolio must contain enough cash in the riskless asset to cover the subsequent position if the option is not exercised, including the cost of rebalancing the portfolio. Let !j denote a particular path from 0 to j, i.e. a particular sequence of up and down moves, and Sj (!j ) the corresponding stock price. The optimal hedging portfolio (Nj ; Bj ) may depend on !j ; this dependence will be suppressed for notational convenience, and will eventually be shown not to exist. For every !j there are two successor paths !j+1 , corresponding to uSj and dSj . For perfect hedging, therefore, we must have, if the option is not exercised: RBj ¿ Bj+1 + ’(Nj+1 − Nj )Sj+1 :
(1)
6 Although we use the same notation, the portfolios (N ; B ) are clearly not the same for long and short j j options. 7 The function ’(y) was deCned by BLPS, but our speciCcation of the transactions costs conforms to Merton–BV.
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
919
For European options the BLPS approach computes sequentially the optimal pathdependent portfolios, by evaluating the following function Qj for each path: 8 Qj (Nj−1 ; !j ) = Min{Bj + ’(Nj − Nj−1 )Sj }; Nj
j = 0; : : : ; n − 1;
(2)
subject to (Nj ; u!j ); Qj+1 (Nj ; d!j )}: RBj ¿ Max{Qj+1
(3)
Qj represents the minimal cash needed to hedge the option and cover the transactions costs if there is no immediate exercise. While (2) and (3) describe the optimal hedging for deferred exercise at j, the function Qj in (2) must be modiCed in the American put to account for possible early exercise; this new function will be denoted by Qj . Since this function diBers for long and short put options, it is left for subsequent sections. The formulation is completed by specifying the objective function at time zero, which is 9 Pa (S0 ; n) = Max{X − S0 ; Min[N0 S0 + B0 ]}; Pb (S0 ; n) = Max{X − S0 ; −Min[N0 S0 + B0 ]};
(4)
where the minimization within braces is with respect to N0 and subject to (3) in both cases. In (4), as in all the other nodes, the values derived by the minimization of the value of the hedging portfolio is compared to the value of immediate exercise at 0. 3. Ask price and super-replication of put options held long by investors At expiration the put-holding investor will always exercise when X ¿ (1 + k)Sn and will always let the option expire if X ¡ (1 − k)Sn . In-between these two values of Sn , though, rational investors may take either action. Thus, the holder of a naked put will let the option expire if Sn ¿ X=(1 + k); the holder of a fully-covered put who wants cash at n will always exercise if Sn ¡ X=(1 − k). Hence, the holder of a partly-covered put, holding a fraction ∈ (0; 1) of a share for each put option in his portfolio, will exercise (let expire) when Sn is less than (greater than) some value in the interval [X=(1 + k); X=(1 − k)], depending on . Perfect hedging, though, implies that the market maker must be hedged against either action for Sn in that interval. 10 8 The function Q is determined by the option’s settlement conditions at expiration time, as will become n clear in the next section. 9 In (4) we have ignored the transactions costs in establishing the initial position, as it was done in all the other studies. This omission does not aBect the solution when there is a unique replicating portfolio, but it needs to be reconsidered otherwise. 10 We have assumed, for simplicity, that the respective transactions cost parameters k and k of the market 1 2 maker and the put holder are the same. There is no generality lost by this assumption, since in reality it is expected that k1 6 k2 , and the region [X=(1 + k1 ); X=(1 − k1 )] is contained within the ambiguous exercise region [X=(1 + k2 ); X=(1 − k2 )]. The latter, therefore, plays no role whatsoever in the arguments that follow. It is the hedger’s parameter that determines the put’s early exercise boundary.
920
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
A 1.
---X/(1-k) .
A 2. A 3. Sn-1
. ---X/(1+k) . Sn
Fig. 1. The region of indeterminacy of option exercise.
Fig. 1 shows the relevant region of the binomial triangle at expiration time. Terminal values of Sn above and below the interval represent, respectively, the regions of expiration and unequivocal exercise of the put. Hence, the market maker must deliver to the option holder portfolios (0; 0) at the top and (−1; X ) at the bottom. The two points in-between these two regions, 11 though, correspond to indeterminacy of the option holder’s policy. This indeterminacy is resolved by examining the optimal hedging of the unexercised put option at n − 1 at the three points Ai ; i = 1; 2; 3, where the early exercise boundary will be located at n − 1. For any point Sn we have Nn = 0, Bn = ’(−Nn−1 )Sn if the option is not exercised, and Nn = −1; Bn = X + ’(−1 − Nn−1 )Sn if it is. Hence, applying the deCnitions (1)–(3) and omitting the path !n−1 in the arguments it is easy to see that at n − 1 the unexercised put will be hedged optimally by Nn−1 = Bn−1 = 0 for all points above A1 , (Nn−2 ) = ’(−Nn−2 )Sn−1 . Similarly, for all points below A3 we have implying that Qn−1 Nn−1 =−1, Bn−1 =X=R, and Qn−1 (Nn−2 )=X=R+’(−1−Nn−2 )Sn−1 . If now we consider the possibility of early exercise at n − 1 then we have, for these same regions, that (Nn−2 ) for points above A1 . Immediate exercise, however, requires Qn−1 (Nn−2 ) = Qn−1 cash equal to X + ’(−1 − Nn−2 )Sn−1 , greater than Qn−1 (Nn−2 ) for all Nn−2 for points below A3 , implying that Qn−1 (Nn−2 ) = X + ’(−1 − Nn−2 )Sn−1 for all points in that region. It remains to consider the points Ai ; i = 1; 2; 3. For these points the logic of perfect hedging implies that, for deferred exercise, the hedging portfolio must contain enough cash to hedge the maximum requirement of its two successor nodes. Thus, for points A2 (3) becomes RBn−1 ¿ Max{’(−Nn−1 )uSn−1 ; ’(−Nn−1 )dSn−1 ; X + ’(−1 − Nn−1 )uSn−1 ; X + ’(−1 − Nn−1 )dSn−1 }:
(5)
This relation represents the constraint set for the minimization problem (2). Similarly, for points A1 and A3 the set is like (5), with the terms X + ’(−1 − Nn−1 )uSn−1 and 11
The existence of such points depends on the parameters u; d and k. If u=d 6 (1 + k)=(1 − k) a point such as A2 , with both successor nodes in the interval [X=(1 + k); X=(1 − k)], cannot exist, while one or more such points may exist otherwise, depending on the relative sizes of u; d and k.
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
921
RBn-1 -(1+k)uSn-1 -(1-k)dSn-1 X -(1+k)dSn-1 -(1-k)uSn-1 -(1+k)uSn-1 -(1+k)dSn-1 -1
N'
0
Nn-1 -(1-k)dSn-1
-(1-k)uSn-1
Fig. 2. The constraint set at the early exercise boundary at n − 1.
’(−Nn−1 )dSn−1 missing from the braces in the right-hand side (RHS). Eq. (5) is sketched in Fig. 2. Applying (2), it can be seen from the constraint slopes that the optimal hedging portfolio for the unexercised put is, for all points Ai ; i = 1; 2; 3, at point 12 N of L n−1 and Fig. 2, at the intersection of X + ’(−1 − Nn−1 )dSn−1 = X − (1 + Nn−1 )dS ’(−Nn−1 )uSn−1 =−uN L n−1 Sn−1 . Further, it is clear from the deCnition of the intersection L n−1 − X ]=[uL − d]S L n−1 and denoted also by N ) point N that its abscissa (equal to [dS increases, while its ordinate decreases as Sn−1 increases, from A3 to A2 to A1 and when there are more than one points similar to A2 . Replacing Bn−1 by the ordinate of N in (2), we have for points Ai ; i = 1; 2; 3: L n−1 =R] + ’(N − Nn−2 )Sn−1 ; Qn−1 (Nn−2 ) = (X=R) − [(1 + N )dS
(6)
L n−1 = −uN where N is deCned by X − (1 + N )dS L n−1 Sn−1 . Up till here we have only considered deferred exercise at n−1. For immediate exercise at n−1 the optimal hedging portfolio must have at least X + ’(−1 − Nn−2 )Sn−1 in cash. This implies that Qn−1 (Nn−2 ) = Max{X + ’(−1 − Nn−2 )Sn−1 ; RHS of (6)}:
(7)
Observe that (7) is a convex function of Nn−2 , sketched in Figs. 3(a) – (c). Figs. 3(a) and (c) show the dominance of one of the two constraints within the braces in the RHS of (7). In Fig. 3(a) (Fig. 3(c)) the Crst (second) term within braces dominates for all Nn−2 , implying that the optimal hedging portfolio must hedge immediate (deferred) exercise for all inherited stockholdings Nn−2 . By contrast, in Fig. 3(b) the constraints intersect at a value N˜ ∈ (−1; N ), and the optimal portfolio must hedge an immediate (deferred) exercise for Nn−2 ¿ (6)N˜ . The hedging portfolio is thus set at 12 In (2) the coeHcients of N L ¿ 0, respectively, to L ¡ 0 and Sn−1 [1 − k − d=R] n−1 are Sn−1 [1 + k − u=R] the left and right of N .
922
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
Qn-1
Qn-1
-(1+k)S n-2
-(1+k)S n-2 -(1-k)Sn-2
X
-(1-k)Sn-2
X
(a)
-1 N'
0
Nn-2
-1
(b)
~ N N'
0
Nn-2
Qn-1 -(1+k)S n-2 -(1-k)Sn-2
-1
N'
0
Nn-2
(c) Fig. 3. (a) Immediate exercise, (b) in-between case, and (c) deferred exercise.
(−1; X ), at N˜ and its ordinate, and at N and its ordinate in Figs. 3(a)–(c), respectively. Removing the Max operator in (7), therefore, we Cnd that the corresponding expressions for the cash requirements function become Qn−1 (Nn−2 ) = X + ’(−1 − Nn−2 )Sn−1 if X (1 − 1=R) L ¿ [1 + k − d=R](1 + N )Sn−1 ; Qn−1 (Nn−2 ) = X − (1 − k)(1 + N˜ )Sn−1 + Sn−1 ’(N˜ − Nn−1 )
(8a) if
L L (1 − k − d=R)(1 + N )Sn−1 ¡ X (1 − 1=R) ¡ [1 + k − d=R]
×(1 + N )Sn−1 ;
(8b)
L n−1 =R + (1 + k) where N˜ is deCned by X − (1 − k)(1 + N˜ )Sn−1 = X=R − (1 + N )dS × (N − N˜ )Sn−1 , L n−1 =R + ’(N − Nn−2 )Sn−1 Qn−1 (Nn−2 ) = X=R − (1 + N )dS L if X (1 − 1=R) 6 (1 − k − d=R)(1 + N )Sn−1 :
(8c)
The functions given in (8a)–(8c) are all convex, decreasing and piecewise linear, with a single kink located within [ − 1; 0], at the value of Nn−2 that sets the arguments of ’ equal to 0; the same properties also apply to Qn−1 (Nn−2 ) for the points below A3 , with kinks at (−1; X ), and above A1 , with kinks at (0; 0). Further, the location
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
923
(i.e. stockholdings) of the kink is non-decreasing 13 in Sn−1 within the interval [ − 1; 0] for all Sn−1 . The form of the cash requirements function Qn−1 (Nn−2 ) depends only on N and N˜ , and is independent of the value of its argument. Hence, if the path !n−1 is re-introduced as an argument within Qn−1 , the function depends only on the stock price Sn−1 and not on the sequence of moves from 0 to n−1. At n−1, therefore, a unique super-replicating portfolio exists at every node of the binomial tree. Its composition varies, depending on the location of Sn−1 . Above A1 (below A3 ) in Fig. 1 it corresponds to immediate (deferred) exercise always. At points L n−1 =R) if (8c) Ai ; i = 1; 2; 3, it can be (−1; X ) if (8a) holds, (N ; X=R − (1 + N )dS holds, with N given in (6), and (N˜ ; X (1 − k)(1 + N˜ )Sn−1 ) if (8b) holds, where N˜ is given in (8b). Which one of (8a), (8b) or (8c) holds depends on X; k and the values of the binomial parameters. The following result extends these properties to any node till the origin. Theorem 1. For any time j ∈ [1; n−1] the optimal hedging portfolio (Nj∗ ; Bj∗ ) for a market maker facing investors holding the American put is unique at each node for all values of the inherited amount of stock Nj−1 . Its composition depends on the node, i.e. the stock price Sj , as follows: there exist three (possibly empty) subintervals of values of Sj , 1 ≡ [S0 dj ; S1j ], 2 ≡ [S1j ; S2j ], 3 ≡ [S2j ; S0 uj ], such that Nj∗ = −1, Bj∗ = X for Sj ∈ 1 , Nj∗ (Sj ) = N (Sj ), Bj∗ (Sj ) = Qj+1 (N ; uSj )=R for Sj ∈ 3 , while Nj∗ (Sj ) = N˜ (Sj ) and Bj∗ (Sj ) = X − (1 − k)(1 + N˜ )Sj for Sj ∈ 2 . Further, Nj∗ (Sj ) is non-decreasing, with values in the interval [ − 1; 0], while Bj∗ (Sj ) is non-increasing, with values in the interval [X; 0]. The values N (Sj ) and N˜ (Sj ) are found at each node from the solutions, respectively, of Eqs. (9a) and (9b), and N˜ 6 N : Qj+1 (N ; uSj ) = Qj+1 (N ; dSj );
(9a)
X − (1 − k)(1 + N˜ )Sj = Qj+1 (N ; dSj )=R + Sj (1 + k)(N − N˜ ):
∗ ∗ (Sj ) ∈ [Nj+1 (dSj ); Nj+1 (uSj )]
(9b)
solving (9a) exists for all Sj , and, for B (Sj )= A value N ˜ Qj+1 (N (Sj ); dSj )=R, a value N (Sj ) ∈ (−1; N (Sj )) solving (9b) exists if and only if X −B (Sj ) ∈ ((1−k)(1+N )Sj ), ((1+k)(1+N )Sj ). The value of the hedging portfolio, Sj Nj∗ + Bj∗ is decreasing and convex in Sj , and the same properties are also true for the functions (1+k)Sj Nj∗ +Bj∗ and (1−k)Sj Nj∗ +Bj∗ ; the slopes of these three functions lie, respectively, within the sets [ − 1; 0]; [ − (1 + k); 0] and [ − (1 − k); 0]. Last, for all nodes j ∈ [1; n − 1] we have Qj (Nj−1 ; Sj ) = Bj∗ (Sj ) + Sj ’(Nj∗ − Nj−1 ). Proof. See Appendix A. Corollary 1. The optimal super-replicating portfolio of an American long put is either (N (S0 ); B (S0 )) as in Theorem 1 or (−1; X ), depending on whether N (S0 )S0 + B (S0 ) or X − S0 is the largest. 13 This is derived from Figs. 3(a)–(c). N is non-decreasing and N˜ 6 N ; hence, we only need to show that the abscissas of two successive values of N˜ are non-decreasing in Sn−1 . However, Fig. 3(b) shows that N˜ increases as N increases, and the slope of the line from N becomes Tatter while that of the one from X becomes steeper as Sn−1 increases.
924
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
The implementation of the algorithm in Theorem 1 is, in spite of its apparent complexity, fairly straightforward. We Crst Cnd the super-replicating portfolio at n − 1 from (8a)–(8c) and the subsequent discussion. Then for each time j ∈ [0; n − 1] and for each node with stock price S we evaluate the replicating portfolio (N ; B ) from the successor nodes’ (Ni ; Bi ); i = 1; 2 and S1 ¿ S2 , through the solution of the Merton–BV equations, which are repeated here for convenience: N S uL + B R = N1 S uL + B1 ; N S dL + B R = N2 S dL + B2 : (10) Once (N ; B ) has been found, and starting from S = S0 dj , we examine whether B + (1 + k)S(1 + N ) 6 X ; if yes then we set N ∗ = −1; B∗ = X . Otherwise, we examine whether B + (1 − k)S(1 + N ) ¿ X ; if yes then we set N ∗ = N ; B∗ = B , for this node as well as for all nodes with a larger stock price. Last, if X lies within the two values we set N ∗ = N˜ ; B∗ = X − (1 − k)(1 + N˜ )S, with N˜ given by (9b). Transactions costs, therefore, complicate the derivation of the early exercise boundary for the American put. At any time j ∈ [0; n − 1] the transition from early to deferred exercise may pass through a set of nodes Sj ∈ 2 , where the super-replicating portfolio hedges immediate exercise for a subsequent down move, and deferred exercise for a subsequent up move. This set depends on X; k and the sizes of the binomial parameters. 4. Bid price and super-replication of options when investors are put writers As noted in the introduction, results for the replication of European short call options under transactions costs exist only when transactions costs are ‘small’. In our notation this condition, originally derived by Merton (1989), takes the form d 6 R(1 − k);
u ¿ R(1 + k):
(11)
We shall assume in this section that (11) holds, and discuss its relaxation in the following section. If (11) holds then put-call parity implies that perfect replication is also optimal for the European put. 14 The key diBerence in the hedging of long and short investor positions is that in the latter case the hedger (market maker) holds both the hedging portfolio and the option; hence, he also decides whether to exercise or not the option. At expiration date he will clearly exercise (let expire) if Sn 6 X=(1 + k)(Sn ¿ X=(1 − k)). For the in-between points, note from Fig. 1, that when (11) holds only one point Sn may lie in the region (X=(1 + k); X=(1 − k)). Point A2 , therefore, is missing and only points A1 and A3 may exist. In such a case the optimal super-replication policy 15 for the short call is to assume exercise at A1 and expiration at A3 . Similarly, for a European put the optimal portfolio can be shown by an argument similar to Lemma 2 in Perrakis and Lefoll (1997) to be Nn−1 = 1, Bn−1 = −X=R at A3 or below, and Nn−1 = Bn−1 = 0 at A1 or above. Otherwise, if no Sn lies within (X=(1 + k); X=(1 − k)), this latter interval is bracketed by the successor nodes of point A2 . Then, while points at or below A3 (at or above A1 ) replicate exercise (expiration without exercise), the replicating portfolio 14 15
See note 3. See Lemma 2 in Perrakis and Lefoll (1997).
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
925
Qn-1
B'n-1
-X N'n-1
Ñn-1
1
Nn-2
Fig. 4. Minimum cash requirements at point A2 .
for the European put at A2 is found, again by the same technique as in Lemma 2 of Perrakis and Lefoll (1997), to be equal to 16 Nn−1 = [X − Sn−1 d]=[Sn−1 (u − d)];
Bn−1 = −Sn−1 uNn−1 =R:
(12)
We shall use this case as a benchmark for our analysis at n − 1, since the other case can be considered as a subcase of it. 17 Consider now the market maker’s decision whether to exercise or not the put at any node at n−1. Such early exercise is optimal for all points at or below A3 , and will never be considered for points at or above A1 . The interesting case is at A2 , where (Nn−2 ) = Bn−1 + Sn−1 ’(Nn−1 − Nn−2 ). This minimum cash requirements function Qn−1 hedges the option if it is not exercised. It should be compared to −X + Sn−1 ’(1 − Nn−2 ), the corresponding function in the event of early exercise, in order to choose the smallest 18 of the two, denoted by Qn−1 (Nn−2 ). Aside from the fact that this comparison depends on the inherited stock Nn−2 , the resulting function is not necessarily convex, since it is the minimum of two convex functions. The function Qn−1 (Nn−2 ) is shown in Fig. 4 for the case of non-convexity. The form of Qn−1 (Nn−2 ) depends on what happens at Nn−2 = 0 and Nn−2 = 1. The point N˜ n−1 of Fig. 4, deCned by the intersection of −X + Sn−1 (1 + k)(1 − Nn−2 ) and Bn−1 + Sn−1 (1 − k)(Nn−1 − Nn−2 ), exists if and only if Qn−1 (0) ¡ − X + Sn−1 (1 + k) 16 This portfolio is the solution of the program (2) subject to the constraint RB n−1 ¿ Max{uNn−1 Sn−1 ; −X + dSn−1 (1 − Nn−1 )}. 17 At n−2 there is a node of the binomial tree whose successor nodes are A and A , corresponding to 1 3 expiry and immediate exercise at n−1; hence, it behaves exactly like point A2 when the latter exists. 18 At any time j the market maker’s wealth is X − S ’(1 − N j j−1 ) if it exercises, and Bj + Sj ’(Nj − Nj−1 ) if it defers, where the portfolio (Nj ; Bj ) hedges optimally the subsequent positions. Hence, the market maker will choose the maximum of these two functions, implying that its cash requirements equal the minimum of the negative of these functions.
926
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
and −X ¡ Qn−1 (1); otherwise, one of the two functions is less than the other for all Nn−2 . Hence, we have at point A2 : Qn−1 (Nn−2 ) = Bn−1 + Sn−1 ’(Nn−1 − Nn−2 ) if − X ¿ Bn−1 +Sn−1 (1 − k)(Nn−1 − 1);
(13a)
Qn−1 (Nn−2 ) = −X + Sn−1 ’(1 − Nn−2 ) if − X + Sn−1 (1 + k) 6 Bn−1 + Sn−1 (1 + k)Nn−1 ;
(13b)
Qn−1 (Nn−2 ) = {RHS of (13a) for Nn−2 6 N˜ n−1 ; RHS of (13b) for Nn−2 ¿ N˜ n−1 } otherwise; Bn−1 − N˜ n−1 ) = −X + Sn−1 (1 − k)(Nn−1
+ Sn−1 (1 + k)(1 − N˜ n−1 );
(13c)
and (Nn−1 ; Bn−1 ) are given by (12). (13a) and (13b) apply, respectively, to all points at or above A1 and at or below A3 . We observe from the analysis at n−1 that the non-convexity case of Fig. 4 may appear at most at one node. Further, the replicating portfolio (Nn−1 ; Bn−1 ) given by (12) is found at the intersection of two convex functions, −uSn−1 ’(Nn−1 ) and −X + dSn−1 ’(1 − Nn−1 ). These functions have a single link each at Nn−1 = 0 and 1, respectively, and the Tattest branch of the u-node intersects with the steepest branch of the ∗ ∗ d-node. Last, it can be easily veriCed that the quantities Bn−1 + (1 − k)Nn−1 Sn−1 and ∗ ∗ Bn−1 + (1 + k)Nn−1 Sn−1 are increasing and concave as functions of Sn−1 , with derivatives that are 6 (1 − k) and 6 (1 + k), respectively. These properties are extended to any node by the following result.
Theorem 2. When (11) holds then for any time j ∈ [1; n−1] the optimal hedging portfolio (Nj∗ ; Bj∗ ) and exercise policy for a market maker facing investors who are put writers is unique at each node for all but possibly one node for all val∗ ∗ (m); Bj+1 (m)); m = u; d, ues of the inherited amount of stock Nj−1 . Further, if (Nj+1 ∗ ∗ ∗ denote the hedging portfolios of the successor nodes, we have Nj ∈ [Nj+1 (u); Nj+1 (d)], ∗ ∗ ∗ ∗ ∗ Bj ∈ [Bj+1 (d); Bj+1 (u)] and Nj ∈ [0; 1], Bj ∈ [−X; 0]. For every j there may be a node Sjl , with immediate exercise for all Sj ¡ Sjl and deferred exercise for Sj ¿ Sj; l+1 , while for Sjl the optimal policy may depend on Nj−1 , as in Fig. 4. The portfolio (Nj ; Bj ) hedging the unexercised put at any node is given by the solution of ∗ ∗ ∗ ∗ Bj+1 (u) + uSj (Nj+1 (u) − Nj ) = Bj+1 (d) + dSj (Nj+1 (d) − Nj ) = RBj ;
(14)
and immediate (deferred) exercise is optimal if (15a) (15b) holds: − X + Sj (1 + k) 6 Bj + Sj (1 + k)Nj ;
(15a)
Bj + Sj (1 − k)Nj 6 − X + Sj (1 − k):
(15b)
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
927
If neither (15a) nor (15b) hold then there is a value N˜ j ∈ (Nj ; 1) solving Bj + Sj (1 − k)(Nj − N˜ j ) = −X + Sj (1 + k)(1 − N˜ j ), with immediate (deferred) exercise optimal for Nj−1 ¿ (6)N˜ j . Proof. See Appendix A. Corollary 2. The optimal super-replicating portfolio of an American short put when (1) holds is either −(N (S0 ); B (S0 )) as in Theorem 2 or (−1; X ), depending on whether −(S0 N (S0 ) + B (S0 )) or X − S0 is largest. Hence, this case is very similar to that of the previous section. For each node the optimal exercise policy and hedging portfolio are found by evaluating the replicating portfolio (N ; B ) from the successor nodes on the basis of (14). Once this portfolio is found, we examine whether immediate exercise is proCtable by applying (15a) and (15b). At any time between origin and expiration the early exercise boundary may include at most a single point, where the optimal exercise policy is ambiguous. When that point exists the cash requirements function is non-convex and has two kinks, whose locations are, however, independent from the inherited stock. As with the long put the implementation of the algorithm described in this section is quite simple. We Crst determine whether at option expiration there is a point within ∗ ∗ ∗ ∗ (X=(1 + k); X=(1 − k)). If yes, then at n − 1 set Nn−1 = 1; Bn−1 = −X (Nn−1 = Bn−1 = 0) for nodes below (above) that point and start the algorithm as if the option expired at n−1. If no, then Cnd (Nn−1 ; Bn−1 ) from (12) and verify from (13a)–(13c) whether ˜ N n−1 exists. For all nodes at all times j ∈ [0; n−1) Cnd recursively the portfolio (Nj ; Bj ) from (14) and determine the early exercise boundary, including any possible points N˜ j , by applying (15a) and (15b); set Nj∗ = 1; Bj∗ = −X (Nj∗ = Nj ; Bj∗ = Bj ) below (above) the boundary. 5. Discussion and numerical results As noted in the introduction, the key issue in applying the results of this paper (as well as of the earlier Merton (1989), BV (1992) and BLPS (1992) studies) is the discrete time to expiration, the number n of subdivisions in the binomial lattice. This number also determines the binomial parameters, since in most applications following Cox et al. (1979) these parameters are evaluated from the expressions used to Cnd the convergence of the binomial model to continuous time as n increases. If t denotes the (continuous) time to option expiration, r the riskless rate, and the volatility of √ the associated stock return process, we have u = e t=n ; d = u−1 and R = ert=n . The √ choice of n also determines the validity of the key relation (11). Letting t=n ≡ t and t = x, we observe that (11) is satisCed iB x − rx2 ¿ ln[(1 + k)=(1 − k)]. For xed parameter values this relation is eventually violated as n increases. Increasing n also increases the upper bound for both European puts and calls, while, if (11) is violated so that u ¡ d (or 2 x ¡ ln[(1 + k)=(1 − k)]) the European call lower bound becomes equal to the Merton (1973) bound, Max{0; S0 − Xe−rt }. A similar result also
928
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
Table 1 k = 0:005; n = 20 and n = 40 X
85 90 95 100 105 110 115
UPa
Pa
Pb
UPb
UP
n = 20
n = 40
n = 20
n = 40
n = 20
n = 40
n = 20
n = 40
n = 20
n = 40
0.295 0.849 1.954 3.867 6.689 10.42 15.00
0.374 1.009 2.218 4.155 6.964 10.60 15.03
0.010 0.032 0.096 0.248 0.539 1.001 1.726
0.012 0.040 0.110 0.244 0.489 0.901 1.509
0.015 0.122 0.606 2.092 5.210 10.00 15.00
0.002 0.035 0.329 1.598 5.00 10.00 15.00
0.005 0.006 0.056 0.266 0.888 2.024 2.640
0.000 0.002 0.032 0.253 1.140 2.331 2.759
0.280 0.727 1.348 1.776 1.479 0.416 0.00
0.372 0.973 1.889 2.558 1.964 0.601 0.028
holds for the American put lower bound. While these results cannot be easily dismissed for all option markets, 19 they clearly cannot represent option bid and ask prices for highly liquid markets as in the US. √ √ By contrast, (11) holds if k depends inversely on n (is proportional to t) for reasonable values of the other parameters, as suggested by Henrotte (1993) and Flesaker and Hughston (1994). Setting k = !x, expanding ln[(1 + k)=(1 − k)] in Taylor series around k = 1, and replacing into (11), we observe that (11) is satisCed if ¿ rx + 2! + o(x). This holds for x → 0 as long as ¿ 2!, as is true for most underlying assets. The Henrotte–Flesaker and Hughston assumption also has implications for the asymptotic behaviour of the option’s long (short) hedging portfolio. These authors showed (in continuous time) √ that the value of a Black–Scholes expres√ this portfolio tends to √ sion with volatility (1 + 2!= )( (1 − 2!= )). For k = ! t, however, these are also the asymptotic values of the replicating portfolios derived within the binomial model of BV (1992). Hence, our American put results should also converge to the corresponding results of a binomial costs but with implied √ model without transactions √ volatilites adjusted by the factors (1 + 2!= ) and (1 − 2!= ). In the tables, we provide values of the super-replicating portfolios for American long and short puts when (11) holds for the following values: = 0:2; r = 0:1; t = 0:25; S0 = $100; X varying from $85 to $115 in increments of $5. Table 1 assumes √ constant √ transactions costs with k = 0:5% and n = 20 or 40. Table 2 assumes k = ! t = ! t=n, with ! = 0:005 and n = 64 or 100. In both tables we show the values Pa and Pb of the portfolios super-replicating the long and short put options, respectively; these can also be the ask and bid values for a perfectly-hedged market maker. We also show the values UPa and UPb , representing the corresponding early exercise premium, the excess of the American over European put values; the latter were computed from the put-call parity relation, with the European call value computed as in BV. Last, we show the diBerence UP, representing the bid/ask spread of the American put in absolute terms under the perfect hedging assumption. 19 For instance, in the empirical study of the Swiss option market of Lefoll and Perrakis (1995) a third of the sample had call bid prices violating the Merton (1973) lower bound.
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
929
Table 2 √ k = 0:005; t=n; n = 64 and 100 X
85 90 95 100 105 110 115
Pa
UPa
Pb
UP
UPb
n = 64
n = 100
n = 64
n = 100
n = 64
n = 100
n = 64
n = 100
n = 64
n = 100
0.130 0.488 1.39 3.16 6.04 10.07 15.00
0.127 0.487 1.40 3.16 6.04 10.07 15.00
0.005 0.023 0.09 0.25 0.60 1.23 2.12
0.005 0.023 0.09 0.25 0.58 1.24 2.13
0.096 0.401 1.23 2.96 5.86 10.02 15.00
0.095 0.404 1.25 2.97 5.88 10.02 15.00
0.004 0.019 0.08 0.24 0.62 1.35 2.23
0.004 0.020 0.08 0.25 0.62 1.35 2.24
0.034 0.087 0.14 0.20 0.18 0.05 0.00
0.032 0.083 0.15 0.19 0.16 0.05 0.00
A comparison of Table 1 with the corresponding results for k = 0 (not shown here) illustrates the eBect of transactions costs on the early exercise premium. Thus, for the long put price Pa transactions costs tend to reduce the importance of early exercise, decreasing its premium in relative terms (as a proportion of the European put value), but also in absolute terms as the exercise price increases. This was to be expected, given our super-replication results. The super-replicating portfolio is unaBected by transactions costs at the nodes of the binomial tree where early exercise takes place, while the value of the corresponding portfolio for the European put rises at these same nodes. Further, transactions costs would tend to reduce the region where early exercise takes place: perfect hedging implies that the portfolio should hedge the largest of the early or deferred exercise policy, and it is only the latter that rises. Nonetheless, the early exercise premium continues to be appreciable in most cases, especially for deep-in-the-money options. The other results derived from Table 1 were more or less to be expected from equivalent results for European options. The bid/ask spread in the last two columns rises sharply in both absolute and percentage terms with the number of periods to expiration of the option, with the ask price increasing and the bid price decreasing. Further, the eBect of possible early exercise on the bid/ask spread depends very much on the striking price X : the spread rises for low values of X but decreases for high X s. The results shown in Table 2 were compared to American put √ values derived from a binomial model without transactions costs but with = 0:2 1:05 for Pa and = √ 0:2 0:95 for Pb ; these are the asymptotic volatilities of the Black–Scholes model that the European options would converge to as n rises. For n = 100 the results of the binomial model were in all cases within less than $0:01 from the values of both Pa and Pb derived from our algorithms and shown in the table. Note also that for n = 64 both Pa and Pb are already very close to their asymptotic values. Thus, our algorithms converge to the binomial evaluation of the American put under adjusted volatility if √ we assume k = ! t. 6. Conclusions We have presented algorithms evaluating, at each node of the binomial tree, portfolios super-replicating American put options in the presence of transactions costs. Such
930
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
portfolios also represent the tightest possible bounds on such option prices when they are computed under the assumption that the market maker is perfectly hedged. The expressions are valid for all options held long by the investors and for short options when transactions costs are ‘small’ (i.e. when (11) holds). In general, the shape of the optimal exercise policy found by our results is fairly similar to the no-transaction-costs case. It is optimal to exercise the option immediately for low values of the stock price, and to defer for high values. An important diBerence, however, is that under transactions costs there exists an intermediate region of stock prices where the exercise policy is ambiguous. Nonetheless, the algorithms developed in the theorems presented in this paper are easy to implement, and allow the quick computation of the optimal super-replicating portfolios for any number of periods. The approach followed in this paper can be extended to other relevant cases of American options under transactions costs. In Perrakis and Lefoll (PL, forthcoming) super-replication algorithms are presented for a physical delivery option with a single constant dividend prior to expiration. Options on assets with a constant payout rate, such as index options, can also be handled in principle in the same way. Such options, however, are cash-settled and may be exercised once a day. As BLPS showed, the optimal hedging policy for both long and short European cash-settled options does not correspond to simple replication when (11) does not hold. The matter is currently under study. Acknowledgements The authors wish to thank the Canadian Social Sciences and Humanities Research Council, the Swiss Fonds National pour la Recherche ScientiCque (under grant 1214040687.94) and the CERESSEC for Cnancial support, and IrWene Collin for research assistance. A preliminary version of this paper was presented at the Northern Finance Association meetings, September 1995, at the JournWees Internationales de l’Association FranXcaise de Finance, June 1996, at the Second International Conference on Computing in Economics and Finance, June 1996, and at the Annual Meeting of the European Financial Management Association, June 1996. We wish to thank our discussants, Yisong Tian and Stanley Pliska. We also acknowledge the helpful comments from MichaFel Selby. Appendix A. All proof are basically geometric demonstrations of the desired results. This preserves the original format of the BLPS (1992) approach, on which this paper is largely based. In addition, the intuition is clearer than alternative algebraic proofs. Proof of Theorem 1. Apart from the convexity properties, the rest of the theorem was already shown to be true at n−1. For the convexity properties, we Crst observe that at n−1 they hold for the unexercised put, on the basis of the put-call parity; this
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
931
2
- d (1+k) R - ud(1+k) R
Qj+2
-u(1+k)
. . D. .D . D .D D1
D11
2
D22
. D. 2
12
D12
2
22
- d (1-k) R
. .D . D . D
- ud(1-k) R
N’(Sj)
- u (1+k) R -1
Nj+1
11
2
(a)
-d(1-k)
1
2
(b)
- u (1-k) R
Fig. 5. (a) A portion of the binomial tree. (b) The portfolio replicating the unexercised option when both successor nodes are in 3 .
convexity is also preserved under early exercise. 20 Now we use induction and consider the nodes at j; j + 1 and j + 2, shown in Fig. 5(a), assuming that the theorem holds ∗ ∗ at j + 1 and j + 2. The existence of N (Sj ) ∈ [Nj+1 (dSj ); Nj+1 (uSj )] is demonstrated in Fig. 5(b), which shows the functions Qj+2 (Nj+1 ; Sj+2 ) corresponding to the points D11 , D12 and D22 of Fig. 5(a). If both successor nodes D1 and D2 are in 3 then it follows, from the slopes of the broken lines emanating from them that deCne D, that ∗ ∗ (dSj ); Nj+1 (uSj )), QED. Similarly, if both Di ∈ 1 ; i = 1; 2, then D ∈ 1 N (Sj ) ∈ (Nj+1 ∗ ∗ (uSj ) = Nj+1 (dSj ), QED. The relative positions of as well, in which case N (Sj ) = Nj+1 ∗ ∗ N (Sj ); Nj+1 (uSj ) and Nj+1 (dSj ) are also shown in Figs. 6(a)–(c), respectively, for the cases Di ∈ 2 ; {D2 ∈ 1 ; D1 ∈ 2 } and {D2 ∈ 2 ; D1 ∈ 3 }. We have thus shown that a unique replicating portfolio (N (Sj ); B (Sj )) exists for the unexercised put at time j; with N non-decreasing and B non-increasing, within the sets of values [ − 1; 0] and [0; X=R], respectively. Consider now early exercise. We must show that, for any two successive nodes Sj1 and Sj2 ¡ Sj1 , Sj2 ∈ 2 implies that Sj1 ∈ 1 , while Sj2 ∈ 3 implies Sj1 ∈ 3 . If Sj2 ∈ 2 then the point X on the vertical axis lies below the intersection of that axis with the steeper line emanating from point D2 in Fig. 6(a), implying that it also lies below the intersection of the axis with the steeper line from D1 , QED. If Sj2 ∈ 3 then this implies that X on the vertical axis lies below the intersection of that axis with the Tatter line emanating from D2 in Fig. 7 (point 2 ), which lies below point 1 where the Tatter line from D1 intersects the axis. This last statement is proven by the ∗ ∗ convexity in S of the function Bj+1 (S) + (1 − k)SNj+1 (S), which holds by the induction hypothesis, QED. 20
∗ ∗ S For instance, Bn−1 + (1 − k)Nn−1 n−1 is equal to X − (1 − k)Sn−1 within 1 and 2 , and to B (Sn−1 ) + −1 L L (1 − k)N (Sn−1 )Sn−1 = [R(1 − k) − u]N L (Sn−1 )Sn−1 =R within 3 . Since @(N S)=@S = d[uL − d] , the slope of the function increases as we pass from 2 to 3 . Similar results also hold for the other two functions.
932
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935 Q
Q j+1 D D2 X
j+1
X,D2
-d(1-k) D1
.
D
.
D1
-d(1+k) -u(1+k) -u(1-k) Nj
(a) N*(dSj) N'(Sj) N*(uSj)
N’(Sj)
(b)
Nj
Q j+1
X
.. D2
D
. (c)
D1
N’(Sj)
Nj
Fig. 6. (a) The replicating portfolio when both successor nodes are in 2 . (b) The replicating portfolio when the ‘down’ successor nodes is in 1 and the ‘up’ node is in 2 . (c) The replicating portfolio when the ‘down’ successor node is in 2 and the ‘up’ node is in 3 .
Q j+1
-d(1+k) -u(1+k)
α1 α2 X
.
D2
.
D
.
D1
-d(1-k) -u(1-k) Nj
Fig. 7. Demonstration that early exercise is unproCtable in the ‘up’ successor node when the ‘down’ node is in 3 .
We have thus shown that a unique replicating portfolio exists at j, whose composition depends on its location in the three regions i ; i=1; 2; 3. The only part of the proof that remains concerns the convexity of the three functions. This is proven Crst by showing that convexity holds for SN (S) + B (S); (1 − k)SN (S) + B (S), and (1 + k)SN (S) +
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
933
Q j+1 ,Q j+2 2
-u (1-k) R
.
-ud(1+k) R
D11
.
-d(1+k)
D1
-u(1-k)
..
-ud(1-k) R
D 12
D
. . D2
D22
N'(uS j) N'j
2
-d (1+k) R
2
-d (1-k) R
N'(dS j)
N j,N j+1
Fig. 8. The portfolio hedging the unexercised short put.
B (S), i.e. if early exercise is ignored; the proof by induction, which is identical to the one used in Lemma 3 of PL (forthcoming), is available from the authors on request. Since convexity implies that the slope of (1 − k)SN (S) + B (S) is non-decreasing and ¿ − (1 − k), it is also preserved by adding early exercise, since it would simply take the maximum of it and the line X − (1 − k)S, QED. The proof for the other two functions is virtually identical. This completes the proof of the theorem. Proof of Theorem 2. The theorem clearly holds at n and n − 1. We use induction and consider the three nodes of the binomial tree depicted in Fig. 5(a). If both successor nodes at j + 1 correspond to immediate exercise then Nj∗ = 1; Bj∗ = −X , QED; likewise, Nj∗ = Bj∗ = 0 if these are also the optimal portfolios for both successor nodes. Now suppose that at j + 2 the theorem holds and that at nodes D11 and D22 the optimal ∗ ∗ ∗ portfolio has Nj+2 ¿ 0 and Nj+2 6 1, respectively, while at D12 we have 0 ¡ Nj+2 ¡ 1. Then the situation is as depicted in Fig. 8, which shows that if points D1 and D2 correspond to non-exercise then point D must lie at the intersection of the dotted lines, implying that (Nj ; Bj ) satisCes (14), QED. If node D1 corresponds to immediate exercise then nodes D2 and D also correspond to immediate exercise, QED; (Nj ; Bj ) ∗ also satisfy (14) if two of the three nodes at j + 2 have the same Nj+1 , at 0 or at 1. Next we consider what happens if one point at j + 1 has optimal exercise policies as in Fig. 4. Such a case is depicted in Fig. 9, which shows that it is impossible for two successive nodes to have such optimal exercise policies. If D1 and D2 are found from the solutions of equations (14) for the two nodes at j + 1 before early exercise is considered, then if a point N˜ exists at the u-node then (15a) must hold at the d-node, implying that we should have immediate exercise at that node; the node D can then easily be seen to correspond to immediate exercise as well. Hence, the situation depicted in Fig. 4 can occur at only one node at most for every j ∈ [1; n − 1],
934
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
Q
j+1
,Q
j+2
-u(1+k)Sj -u(1-k)S j
.
D1
-d(1+k)Sj -u(1+k)Sj
.
D'2
-d(1-k)Sj D,D2
-X N'(uSj)
N'(dSj)
Ñ(uSj)
N j,Nj+1
1=N*(dSj)
Fig. 9. Demonstration that only one node may have optimal exercise policies as in Fig. 4: ‘up’ successor node as in Fig. 4, immediate exercise in ‘down’ node.
Q
j+1
,Q
j+2
-u(1+k)Sj
.
-u(1-k)S j
D1
.
D
-d(1-k)Sj
.
D2
-d(1+k)Sj
-X N'(uSj)=N*(uSj) N'(dSj) Ñ(dSj )
1
N j,Nj+1
Fig. 10. Demonstration that only one node may have optimal exercise policies as in Fig. 4: ‘down’ successor node as in Fig. 4, deferred exercise in the ‘up’ node.
QED. By contrast, Fig. 10 shows that, if it is the d-node that has policies as in Fig. 4, or corresponds to deferred exercise for all Nj , then the u-node must correspond to deferred exercise as well, QED. This completes the proof of the Theorem. References Barone-Adesi, G., Whaley, R.E., 1987. EHcient analytic approximation of American option values. Journal of Finance 42, 301–320. BensaFGd, B., Lesne, J.P., Pag[es, H., Scheinkman, J., 1992. Derivative asset pricing with transaction costs. Mathematical Finance 2, 63–86.
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
935
Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–659. Boyle, P.P., Tan, K.S., 1994. Lure of the linear. Risk 7, 43–46. Boyle, P.P., Vorst, T., 1992. Option replication in discrete time with transactions costs. Journal of Finance 47, 271–293. Brennan, M., Schwartz, E., 1977. The valuation of American put options. Journal of Finance 32, 449–462. Constantinides, G., 1986. Capital market equilibrium with transactions costs. Journal of Political Economy 94, 842–862. Cox, J.C., Rubinstein, M., 1985. Options Markets. Prentice-Hall, Englewood CliBs, NJ. Cox, J.C., Ross, S., Rubinstein, M., 1979. Option pricing: a simpliCed approach. Journal of Financial Economics 7, 229–263. Dumas, B., Luciano, E., 1991. An exact solution to a dynamic portfolio choice problem under transactions costs. Journal of Finance 46, 577–595. Edirisinghe, C., Naik, V., Uppal, R., 1993. Optimal replication of options with transaction costs and trading restrictions. Journal of Financial and Quantitative Analysis 28, 117–138. Flesaker, B., Hughston, L.P., 1994. Contingent claim replication in continuous time with transaction costs. Working paper, Merrill Lynch. Geske, R., Johnson, H.E., 1984. The American put valued analytically. Journal of Finance 39, 1511–1524. Henrotte, P., 1993. Transactions costs and duplication strategies. Working paper, Graduate School of Business, Stanford University. Hodges, S., Neuberger, A., 1989. Optimal replication of contingent claims under transactions costs. Review of Futures Markets 8, 222–242. Lefoll, J., Perrakis, S., 1995. Transactions costs and option bid/ask spread on the Swiss Options and Financial Futures Exchange (SOFFEX). Canadian Journal of Administrative Sciences 12, 276–289. Leland, H., 1985. Option pricing and replication with transactions costs. Journal of Finance 40, 1283–1301. MacMillan, L.W., 1986. Analytic approximation for the American put option. Advances in Futures and Options Research 1, 119–139. Merton, R.C., 1973. Theory of rational option pricing. Bell Journal of Economics and Management Science 4, 141–183. Merton, R.C., 1989. On the application of the continuous-time theory of Cnance to Cnancial intermediation and insurance. The Geneva Papers on Risk and Insurance 14, 225–261. Parkinson, M., 1977. Option pricing: the American put. Journal of Business 50, 1–36. Perrakis, S., Lefoll, J., 1997. Derivative asset pricing with transaction costs: an extension. Computational Economics 10, 359–376. Perrakis, S., Lefoll, J., 2000. Option pricing and replication with transaction costs and dividends. Journal of Economic Dynamics and Control 24, 1527–1561. Rendleman, R., Bartter, B., 1979. Two-state option pricing. Journal of Finance 34, 1093–1110.
The American put under transactions costs Stylianos Perrakisa;∗ , Jean Lefollb a Department
of nance, The John Molson School of business, Concordia University, Montr eal, Qu ebec, Canada, H3G 1M8 b HEC and International Finance Laboratory, University of Geneva, 1211 Gen) eve 4, Switzerland
Abstract This paper examines the optimal super-replication of American put options with physical delivery of the underlying asset, such as stock options, by means of a stock-plus-riskless asset portfolio. The framework of the analysis is the binomial model with proportional transactions costs on stock transactions. The paper extends the model for European options, originally presented in Merton (Geneva Papers Risk Insurance 14 (1989) 225) and Boyle and Vorst (J. of Finance 47 (1992) 271), and generalized in Bensaid et al. (Math. Finance 2 (1992) 63). The optimizing framework of this latter study is adapted to put options held by investors and perfectly hedged by a market maker, and to put options written by investors and both held and hedged perfectly by a market maker. It is shown that a unique optimal super-replicating portfolio exists at every node of the binomial tree for the long option, as well as for the short option when transactions costs are low. ? 2003 Elsevier B.V. All rights reserved. JEL classi cation: G13 Keywords: Pricing; Transactions costs; American options; Put
1. Introduction This paper examines the pricing of American put options by a perfectly-hedged market maker when there are transactions costs to be paid on the underlying stock. Thus, it extends European stock option pricing under perfect hedging, transactions costs, and binomial stock returns, formulated by Merton (1989), and extended by Boyle and Vorst (1992) BV. A similar extension, to American options on dividend-paying stocks under transaction costs, was done in an earlier study (Perrakis and Lefoll, 2000). ∗
Corresponding author. E-mail address: [email protected] (S. Perrakis).
0165-1889/03/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0165-1889(03)00099-X
916
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
While the perfect-hedging assumption may appear extreme for organized option markets, it does oBer a useful benchmark case for the derivation of option bounds, within which the option bid/ask prices must lie. Further, this assumption makes our results also useful to situations where there is a need for option replication such as, for instance, in portfolio insurance. Last, it allows the creation of options in cases in which no organized options market exists, as in most emergent Cnancial markets. Thus, while most results are expressed in terms of option pricing by a perfectly-hedged market maker, they also extend to these other important cases. In the option pricing models of the classic studies of Black and Scholes (1973) and Merton (1973), the call option is perfectly and continuously replicated by a stock-plus-riskless-asset portfolio. The introduction of Cxed transactions costs every time this portfolio is being rebalanced makes such a policy infeasible in a continuous time model. For this reason a number of papers have tackled the problem of portfolio selection and/or option pricing under transactions costs, both in continuous time 1 and on a binomial lattice. Since perfect option replication is infeasible in the continuous time models, those studies that dealt with option pricing speciCed either approximate replication at predetermined and exogenously given times, or expected utility-based portfolio selection under transactions costs. By contrast, the Merton–BV approach replicates both long and short call options at every node of the binomial lattice. While the replication of the long option is feasible in all cases, the replication of the short option requires some restrictions on parameter values. These restrictions are satisCed when transactions costs are ‘small’ for the chosen number of lattice steps, in a sense that will become more precise in Section 4 of this paper. Merton solved the replication problem when the option has only two periods to expiration; BV extended the Merton model to any number of periods. An important study by BensaFGd et al. (1992) BLPS, derived an algorithm to compute optimal perfect-hedging policies for an intermediary that issues long or short options (which they named super-replication), for several types of European options under binomial returns without necessarily replicating the option at every node. 2 The BLPS study found contrasting results for the important cases of physical delivery and cash settlement options: while the intermediary Cnds it optimal to replicate everywhere physical delivery long options, such a policy is suboptimal for cash settlement options, unless transactions costs are ‘small’, in the same sense as in the Merton–BV studies. In spite of its generality and powerful theoretical insights, the BLPS algorithm is rather diHcult to apply as stated for a large number of periods to expiration, since 1 See Leland (1985), Constantinides (1986), Hodges and Neuberger (1989) and Dumas and Luciano (1991). 2 Edirisinghe et al. (1993) also presented a two-stage dynamic programming algorithm for minimum cost hedging of an option without necessarily replicating it; see also Boyle and Tan (1994) for more on their method. The two-stage algorithm, however, was developed for options whose method of settlement is up to the seller, and it is not clear how it could be extended to cover the more realistic physical delivery and cash-settlement options. Further, for options with settlement up to the seller BLPS (1992) presented a closed form solution in their Theorem 4.
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
917
it requires the determination of the intersection points of two piecewise-linear convex functions at every node of the binomial tree. Further, it is based on the assumption that the option holder’s actions that are being hedged are predetermined at option expiration; such an assumption is not appropriate for hedging American options, where the option holder must decide whether to exercise or defer at every node. In this paper it is shown that in the presence of transactions costs there are cases where the option holder’s optimal action (exercise or not) depends on preferences or other holdings. Nonetheless, perfect hedging requires that the stock-plus-riskless-asset super-replicating portfolio be able to hedge all possible holder actions. This remark allows the extension of the BLPS algorithm, to cover the case of American put options with physical delivery of the underlying asset. European put options under transactions costs can be valued through put-call parity 3 from the above-cited studies’ call option prices. For American put options, however, it is well-known since Merton (1973) that early exercise may be proCtable even in the absence of dividends, implying that their value exceeds the one given by put-call parity. Although closed-form expressions do not exist for American puts, recursive or analytical methods have been derived, among others, by Parkinson (1977), Brennan and Schwartz (1977), Geske and Johnson (1984), MacMillan (1986), and Barone-Adesi and Whaley (1987). The binomial model of Cox et al. (1979) and Rendleman and Bartter (1979) can be used to value an American put in the absence of transactions costs. 4 The key element in such a valuation is the derivation of the early exercise boundary, the stock price separating early from deferred exercise at every time period prior to expiration. Here we show that transactions costs makes this boundary more complex, since the holder’s optimal action is ambiguous. The main weakness of the binomial model in handling option pricing under transactions costs is that no guidelines are given to determine the appropriate number of steps in the lattice. This is crucial because, for a xed transactions cost parameter, the total transactions costs increase as the number of steps rises. At the limit the value of the portfolio replicating the long European call option (without dividends) tends to the stock price, while the portfolio replicating the short option becomes equal to the well-known Merton (1973) European lower bound. 5 For this reason some studies such as Henrotte (1993) and Flesaker and Hughston (1994) have proposed replacing the Cxed transactions cost parameter with one that declines in proportion to the square root of the number of steps. This assumption provides reasonable non-trivial option prices under transactions costs, both in continuous time and on binomial lattices. Further, as 3 Put-call parity does not necessarily hold in the presence of transaction costs. Nonetheless, a put-call parity relation can be derived for portfolios replicating call and put options; see the comments in BV (1992, p. 285). None of the results in this paper depend on put-call parity. 4 See Cox and Rubinstein (1985, pp. 245–252). 5 For the long call it was shown in the BV study that, for a large number of binomial steps, the value of the replicating portfolio tends to a Black–Scholes option price with variance increasing proportionately with the square root of the number of steps; this becomes equal to the underlying stock price for inCnite steps. For the short call it was shown in Perrakis and Lefoll (1997) that the value of the super-replicating portfolio becomes equal to the Merton bound as soon as the number of steps rises suHciently to render the Cxed transactions costs parameter ‘large’ in comparison with the binomial parameters.
918
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
will be discussed in Section 5, it also satisCes the restriction of ‘small’ transactions costs in relation to the binomial parameters for most reasonable cases. In the next section we present the problem along the lines of the earlier Merton (1989), BV and BLPS studies. The long put is presented in Section 3, while Section 4 examines the short put for ‘small’ transactions costs. Some numerical examples and a discussion of the extension of the results to ‘large’ transactions costs are presented at the end. 2. The general model Let u, d and R denote, respectively, the size of the up and down moves and one plus the riskless rate of interest, with d ¡ R ¡ u. k denotes the transactions cost parameter, implying that the purchase (sale) of $1 of shares costs $(1 + k) (yields $(1 − k)). DeCne uL ≡ u(1 + k); dL ≡ d(1 − k); u ≡ u(1 − k) and d ≡ d(1 + k). There are no transactions costs in the riskless asset. The strike price of the option is X; n is the expiration time of the option in number of periods, and S0 is the initial stock price. We shall derive the ask and bid prices of American put options, respectively, Pa (S0 ; n) and Pb (S0 ; n), under the assumption, originally formulated by Merton (1989), that these prices are oBered by perfectly-hedged market makers. These prices are equal to the values of portfolios (Nj Sj + Bj ) containing Nj shares and Bj invested in the riskless asset at any time j 6 n, with Sj denoting the stock price. 6 The portfolios must be rebalanced to keep the market maker perfectly-hedged. Hence, Pa (S0 ; n) = N0 S0 + B0 and Pb (S0 ; n) = −(N0 S0 + B0 ) if the option is not going to be exercised immediately, and the portfolio (N0 ; B0 ) super-replicates the put option held long or written by the investor, respectively. We model the transactions costs by means of the convex function 7 ’(y), which is equal to (1 + k)y for y ¿ 0 and to (1 − k)y for y 6 0. Rebalancing the American put’s super-replicating portfolio at any time j ∈ [0; n] implies that one should hedge against both immediate and deferred exercise. We consider physical delivery options only, like the options on individual stocks. The basic feature of the BLPS approach, that will also be adopted in this paper, is that for any time j ∈ [0; n] the hedging portfolio must contain enough cash in the riskless asset to cover the subsequent position if the option is not exercised, including the cost of rebalancing the portfolio. Let !j denote a particular path from 0 to j, i.e. a particular sequence of up and down moves, and Sj (!j ) the corresponding stock price. The optimal hedging portfolio (Nj ; Bj ) may depend on !j ; this dependence will be suppressed for notational convenience, and will eventually be shown not to exist. For every !j there are two successor paths !j+1 , corresponding to uSj and dSj . For perfect hedging, therefore, we must have, if the option is not exercised: RBj ¿ Bj+1 + ’(Nj+1 − Nj )Sj+1 :
(1)
6 Although we use the same notation, the portfolios (N ; B ) are clearly not the same for long and short j j options. 7 The function ’(y) was deCned by BLPS, but our speciCcation of the transactions costs conforms to Merton–BV.
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
919
For European options the BLPS approach computes sequentially the optimal pathdependent portfolios, by evaluating the following function Qj for each path: 8 Qj (Nj−1 ; !j ) = Min{Bj + ’(Nj − Nj−1 )Sj }; Nj
j = 0; : : : ; n − 1;
(2)
subject to (Nj ; u!j ); Qj+1 (Nj ; d!j )}: RBj ¿ Max{Qj+1
(3)
Qj represents the minimal cash needed to hedge the option and cover the transactions costs if there is no immediate exercise. While (2) and (3) describe the optimal hedging for deferred exercise at j, the function Qj in (2) must be modiCed in the American put to account for possible early exercise; this new function will be denoted by Qj . Since this function diBers for long and short put options, it is left for subsequent sections. The formulation is completed by specifying the objective function at time zero, which is 9 Pa (S0 ; n) = Max{X − S0 ; Min[N0 S0 + B0 ]}; Pb (S0 ; n) = Max{X − S0 ; −Min[N0 S0 + B0 ]};
(4)
where the minimization within braces is with respect to N0 and subject to (3) in both cases. In (4), as in all the other nodes, the values derived by the minimization of the value of the hedging portfolio is compared to the value of immediate exercise at 0. 3. Ask price and super-replication of put options held long by investors At expiration the put-holding investor will always exercise when X ¿ (1 + k)Sn and will always let the option expire if X ¡ (1 − k)Sn . In-between these two values of Sn , though, rational investors may take either action. Thus, the holder of a naked put will let the option expire if Sn ¿ X=(1 + k); the holder of a fully-covered put who wants cash at n will always exercise if Sn ¡ X=(1 − k). Hence, the holder of a partly-covered put, holding a fraction ∈ (0; 1) of a share for each put option in his portfolio, will exercise (let expire) when Sn is less than (greater than) some value in the interval [X=(1 + k); X=(1 − k)], depending on . Perfect hedging, though, implies that the market maker must be hedged against either action for Sn in that interval. 10 8 The function Q is determined by the option’s settlement conditions at expiration time, as will become n clear in the next section. 9 In (4) we have ignored the transactions costs in establishing the initial position, as it was done in all the other studies. This omission does not aBect the solution when there is a unique replicating portfolio, but it needs to be reconsidered otherwise. 10 We have assumed, for simplicity, that the respective transactions cost parameters k and k of the market 1 2 maker and the put holder are the same. There is no generality lost by this assumption, since in reality it is expected that k1 6 k2 , and the region [X=(1 + k1 ); X=(1 − k1 )] is contained within the ambiguous exercise region [X=(1 + k2 ); X=(1 − k2 )]. The latter, therefore, plays no role whatsoever in the arguments that follow. It is the hedger’s parameter that determines the put’s early exercise boundary.
920
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
A 1.
---X/(1-k) .
A 2. A 3. Sn-1
. ---X/(1+k) . Sn
Fig. 1. The region of indeterminacy of option exercise.
Fig. 1 shows the relevant region of the binomial triangle at expiration time. Terminal values of Sn above and below the interval represent, respectively, the regions of expiration and unequivocal exercise of the put. Hence, the market maker must deliver to the option holder portfolios (0; 0) at the top and (−1; X ) at the bottom. The two points in-between these two regions, 11 though, correspond to indeterminacy of the option holder’s policy. This indeterminacy is resolved by examining the optimal hedging of the unexercised put option at n − 1 at the three points Ai ; i = 1; 2; 3, where the early exercise boundary will be located at n − 1. For any point Sn we have Nn = 0, Bn = ’(−Nn−1 )Sn if the option is not exercised, and Nn = −1; Bn = X + ’(−1 − Nn−1 )Sn if it is. Hence, applying the deCnitions (1)–(3) and omitting the path !n−1 in the arguments it is easy to see that at n − 1 the unexercised put will be hedged optimally by Nn−1 = Bn−1 = 0 for all points above A1 , (Nn−2 ) = ’(−Nn−2 )Sn−1 . Similarly, for all points below A3 we have implying that Qn−1 Nn−1 =−1, Bn−1 =X=R, and Qn−1 (Nn−2 )=X=R+’(−1−Nn−2 )Sn−1 . If now we consider the possibility of early exercise at n − 1 then we have, for these same regions, that (Nn−2 ) for points above A1 . Immediate exercise, however, requires Qn−1 (Nn−2 ) = Qn−1 cash equal to X + ’(−1 − Nn−2 )Sn−1 , greater than Qn−1 (Nn−2 ) for all Nn−2 for points below A3 , implying that Qn−1 (Nn−2 ) = X + ’(−1 − Nn−2 )Sn−1 for all points in that region. It remains to consider the points Ai ; i = 1; 2; 3. For these points the logic of perfect hedging implies that, for deferred exercise, the hedging portfolio must contain enough cash to hedge the maximum requirement of its two successor nodes. Thus, for points A2 (3) becomes RBn−1 ¿ Max{’(−Nn−1 )uSn−1 ; ’(−Nn−1 )dSn−1 ; X + ’(−1 − Nn−1 )uSn−1 ; X + ’(−1 − Nn−1 )dSn−1 }:
(5)
This relation represents the constraint set for the minimization problem (2). Similarly, for points A1 and A3 the set is like (5), with the terms X + ’(−1 − Nn−1 )uSn−1 and 11
The existence of such points depends on the parameters u; d and k. If u=d 6 (1 + k)=(1 − k) a point such as A2 , with both successor nodes in the interval [X=(1 + k); X=(1 − k)], cannot exist, while one or more such points may exist otherwise, depending on the relative sizes of u; d and k.
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
921
RBn-1 -(1+k)uSn-1 -(1-k)dSn-1 X -(1+k)dSn-1 -(1-k)uSn-1 -(1+k)uSn-1 -(1+k)dSn-1 -1
N'
0
Nn-1 -(1-k)dSn-1
-(1-k)uSn-1
Fig. 2. The constraint set at the early exercise boundary at n − 1.
’(−Nn−1 )dSn−1 missing from the braces in the right-hand side (RHS). Eq. (5) is sketched in Fig. 2. Applying (2), it can be seen from the constraint slopes that the optimal hedging portfolio for the unexercised put is, for all points Ai ; i = 1; 2; 3, at point 12 N of L n−1 and Fig. 2, at the intersection of X + ’(−1 − Nn−1 )dSn−1 = X − (1 + Nn−1 )dS ’(−Nn−1 )uSn−1 =−uN L n−1 Sn−1 . Further, it is clear from the deCnition of the intersection L n−1 − X ]=[uL − d]S L n−1 and denoted also by N ) point N that its abscissa (equal to [dS increases, while its ordinate decreases as Sn−1 increases, from A3 to A2 to A1 and when there are more than one points similar to A2 . Replacing Bn−1 by the ordinate of N in (2), we have for points Ai ; i = 1; 2; 3: L n−1 =R] + ’(N − Nn−2 )Sn−1 ; Qn−1 (Nn−2 ) = (X=R) − [(1 + N )dS
(6)
L n−1 = −uN where N is deCned by X − (1 + N )dS L n−1 Sn−1 . Up till here we have only considered deferred exercise at n−1. For immediate exercise at n−1 the optimal hedging portfolio must have at least X + ’(−1 − Nn−2 )Sn−1 in cash. This implies that Qn−1 (Nn−2 ) = Max{X + ’(−1 − Nn−2 )Sn−1 ; RHS of (6)}:
(7)
Observe that (7) is a convex function of Nn−2 , sketched in Figs. 3(a) – (c). Figs. 3(a) and (c) show the dominance of one of the two constraints within the braces in the RHS of (7). In Fig. 3(a) (Fig. 3(c)) the Crst (second) term within braces dominates for all Nn−2 , implying that the optimal hedging portfolio must hedge immediate (deferred) exercise for all inherited stockholdings Nn−2 . By contrast, in Fig. 3(b) the constraints intersect at a value N˜ ∈ (−1; N ), and the optimal portfolio must hedge an immediate (deferred) exercise for Nn−2 ¿ (6)N˜ . The hedging portfolio is thus set at 12 In (2) the coeHcients of N L ¿ 0, respectively, to L ¡ 0 and Sn−1 [1 − k − d=R] n−1 are Sn−1 [1 + k − u=R] the left and right of N .
922
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
Qn-1
Qn-1
-(1+k)S n-2
-(1+k)S n-2 -(1-k)Sn-2
X
-(1-k)Sn-2
X
(a)
-1 N'
0
Nn-2
-1
(b)
~ N N'
0
Nn-2
Qn-1 -(1+k)S n-2 -(1-k)Sn-2
-1
N'
0
Nn-2
(c) Fig. 3. (a) Immediate exercise, (b) in-between case, and (c) deferred exercise.
(−1; X ), at N˜ and its ordinate, and at N and its ordinate in Figs. 3(a)–(c), respectively. Removing the Max operator in (7), therefore, we Cnd that the corresponding expressions for the cash requirements function become Qn−1 (Nn−2 ) = X + ’(−1 − Nn−2 )Sn−1 if X (1 − 1=R) L ¿ [1 + k − d=R](1 + N )Sn−1 ; Qn−1 (Nn−2 ) = X − (1 − k)(1 + N˜ )Sn−1 + Sn−1 ’(N˜ − Nn−1 )
(8a) if
L L (1 − k − d=R)(1 + N )Sn−1 ¡ X (1 − 1=R) ¡ [1 + k − d=R]
×(1 + N )Sn−1 ;
(8b)
L n−1 =R + (1 + k) where N˜ is deCned by X − (1 − k)(1 + N˜ )Sn−1 = X=R − (1 + N )dS × (N − N˜ )Sn−1 , L n−1 =R + ’(N − Nn−2 )Sn−1 Qn−1 (Nn−2 ) = X=R − (1 + N )dS L if X (1 − 1=R) 6 (1 − k − d=R)(1 + N )Sn−1 :
(8c)
The functions given in (8a)–(8c) are all convex, decreasing and piecewise linear, with a single kink located within [ − 1; 0], at the value of Nn−2 that sets the arguments of ’ equal to 0; the same properties also apply to Qn−1 (Nn−2 ) for the points below A3 , with kinks at (−1; X ), and above A1 , with kinks at (0; 0). Further, the location
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
923
(i.e. stockholdings) of the kink is non-decreasing 13 in Sn−1 within the interval [ − 1; 0] for all Sn−1 . The form of the cash requirements function Qn−1 (Nn−2 ) depends only on N and N˜ , and is independent of the value of its argument. Hence, if the path !n−1 is re-introduced as an argument within Qn−1 , the function depends only on the stock price Sn−1 and not on the sequence of moves from 0 to n−1. At n−1, therefore, a unique super-replicating portfolio exists at every node of the binomial tree. Its composition varies, depending on the location of Sn−1 . Above A1 (below A3 ) in Fig. 1 it corresponds to immediate (deferred) exercise always. At points L n−1 =R) if (8c) Ai ; i = 1; 2; 3, it can be (−1; X ) if (8a) holds, (N ; X=R − (1 + N )dS holds, with N given in (6), and (N˜ ; X (1 − k)(1 + N˜ )Sn−1 ) if (8b) holds, where N˜ is given in (8b). Which one of (8a), (8b) or (8c) holds depends on X; k and the values of the binomial parameters. The following result extends these properties to any node till the origin. Theorem 1. For any time j ∈ [1; n−1] the optimal hedging portfolio (Nj∗ ; Bj∗ ) for a market maker facing investors holding the American put is unique at each node for all values of the inherited amount of stock Nj−1 . Its composition depends on the node, i.e. the stock price Sj , as follows: there exist three (possibly empty) subintervals of values of Sj , 1 ≡ [S0 dj ; S1j ], 2 ≡ [S1j ; S2j ], 3 ≡ [S2j ; S0 uj ], such that Nj∗ = −1, Bj∗ = X for Sj ∈ 1 , Nj∗ (Sj ) = N (Sj ), Bj∗ (Sj ) = Qj+1 (N ; uSj )=R for Sj ∈ 3 , while Nj∗ (Sj ) = N˜ (Sj ) and Bj∗ (Sj ) = X − (1 − k)(1 + N˜ )Sj for Sj ∈ 2 . Further, Nj∗ (Sj ) is non-decreasing, with values in the interval [ − 1; 0], while Bj∗ (Sj ) is non-increasing, with values in the interval [X; 0]. The values N (Sj ) and N˜ (Sj ) are found at each node from the solutions, respectively, of Eqs. (9a) and (9b), and N˜ 6 N : Qj+1 (N ; uSj ) = Qj+1 (N ; dSj );
(9a)
X − (1 − k)(1 + N˜ )Sj = Qj+1 (N ; dSj )=R + Sj (1 + k)(N − N˜ ):
∗ ∗ (Sj ) ∈ [Nj+1 (dSj ); Nj+1 (uSj )]
(9b)
solving (9a) exists for all Sj , and, for B (Sj )= A value N ˜ Qj+1 (N (Sj ); dSj )=R, a value N (Sj ) ∈ (−1; N (Sj )) solving (9b) exists if and only if X −B (Sj ) ∈ ((1−k)(1+N )Sj ), ((1+k)(1+N )Sj ). The value of the hedging portfolio, Sj Nj∗ + Bj∗ is decreasing and convex in Sj , and the same properties are also true for the functions (1+k)Sj Nj∗ +Bj∗ and (1−k)Sj Nj∗ +Bj∗ ; the slopes of these three functions lie, respectively, within the sets [ − 1; 0]; [ − (1 + k); 0] and [ − (1 − k); 0]. Last, for all nodes j ∈ [1; n − 1] we have Qj (Nj−1 ; Sj ) = Bj∗ (Sj ) + Sj ’(Nj∗ − Nj−1 ). Proof. See Appendix A. Corollary 1. The optimal super-replicating portfolio of an American long put is either (N (S0 ); B (S0 )) as in Theorem 1 or (−1; X ), depending on whether N (S0 )S0 + B (S0 ) or X − S0 is the largest. 13 This is derived from Figs. 3(a)–(c). N is non-decreasing and N˜ 6 N ; hence, we only need to show that the abscissas of two successive values of N˜ are non-decreasing in Sn−1 . However, Fig. 3(b) shows that N˜ increases as N increases, and the slope of the line from N becomes Tatter while that of the one from X becomes steeper as Sn−1 increases.
924
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
The implementation of the algorithm in Theorem 1 is, in spite of its apparent complexity, fairly straightforward. We Crst Cnd the super-replicating portfolio at n − 1 from (8a)–(8c) and the subsequent discussion. Then for each time j ∈ [0; n − 1] and for each node with stock price S we evaluate the replicating portfolio (N ; B ) from the successor nodes’ (Ni ; Bi ); i = 1; 2 and S1 ¿ S2 , through the solution of the Merton–BV equations, which are repeated here for convenience: N S uL + B R = N1 S uL + B1 ; N S dL + B R = N2 S dL + B2 : (10) Once (N ; B ) has been found, and starting from S = S0 dj , we examine whether B + (1 + k)S(1 + N ) 6 X ; if yes then we set N ∗ = −1; B∗ = X . Otherwise, we examine whether B + (1 − k)S(1 + N ) ¿ X ; if yes then we set N ∗ = N ; B∗ = B , for this node as well as for all nodes with a larger stock price. Last, if X lies within the two values we set N ∗ = N˜ ; B∗ = X − (1 − k)(1 + N˜ )S, with N˜ given by (9b). Transactions costs, therefore, complicate the derivation of the early exercise boundary for the American put. At any time j ∈ [0; n − 1] the transition from early to deferred exercise may pass through a set of nodes Sj ∈ 2 , where the super-replicating portfolio hedges immediate exercise for a subsequent down move, and deferred exercise for a subsequent up move. This set depends on X; k and the sizes of the binomial parameters. 4. Bid price and super-replication of options when investors are put writers As noted in the introduction, results for the replication of European short call options under transactions costs exist only when transactions costs are ‘small’. In our notation this condition, originally derived by Merton (1989), takes the form d 6 R(1 − k);
u ¿ R(1 + k):
(11)
We shall assume in this section that (11) holds, and discuss its relaxation in the following section. If (11) holds then put-call parity implies that perfect replication is also optimal for the European put. 14 The key diBerence in the hedging of long and short investor positions is that in the latter case the hedger (market maker) holds both the hedging portfolio and the option; hence, he also decides whether to exercise or not the option. At expiration date he will clearly exercise (let expire) if Sn 6 X=(1 + k)(Sn ¿ X=(1 − k)). For the in-between points, note from Fig. 1, that when (11) holds only one point Sn may lie in the region (X=(1 + k); X=(1 − k)). Point A2 , therefore, is missing and only points A1 and A3 may exist. In such a case the optimal super-replication policy 15 for the short call is to assume exercise at A1 and expiration at A3 . Similarly, for a European put the optimal portfolio can be shown by an argument similar to Lemma 2 in Perrakis and Lefoll (1997) to be Nn−1 = 1, Bn−1 = −X=R at A3 or below, and Nn−1 = Bn−1 = 0 at A1 or above. Otherwise, if no Sn lies within (X=(1 + k); X=(1 − k)), this latter interval is bracketed by the successor nodes of point A2 . Then, while points at or below A3 (at or above A1 ) replicate exercise (expiration without exercise), the replicating portfolio 14 15
See note 3. See Lemma 2 in Perrakis and Lefoll (1997).
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
925
Qn-1
B'n-1
-X N'n-1
Ñn-1
1
Nn-2
Fig. 4. Minimum cash requirements at point A2 .
for the European put at A2 is found, again by the same technique as in Lemma 2 of Perrakis and Lefoll (1997), to be equal to 16 Nn−1 = [X − Sn−1 d]=[Sn−1 (u − d)];
Bn−1 = −Sn−1 uNn−1 =R:
(12)
We shall use this case as a benchmark for our analysis at n − 1, since the other case can be considered as a subcase of it. 17 Consider now the market maker’s decision whether to exercise or not the put at any node at n−1. Such early exercise is optimal for all points at or below A3 , and will never be considered for points at or above A1 . The interesting case is at A2 , where (Nn−2 ) = Bn−1 + Sn−1 ’(Nn−1 − Nn−2 ). This minimum cash requirements function Qn−1 hedges the option if it is not exercised. It should be compared to −X + Sn−1 ’(1 − Nn−2 ), the corresponding function in the event of early exercise, in order to choose the smallest 18 of the two, denoted by Qn−1 (Nn−2 ). Aside from the fact that this comparison depends on the inherited stock Nn−2 , the resulting function is not necessarily convex, since it is the minimum of two convex functions. The function Qn−1 (Nn−2 ) is shown in Fig. 4 for the case of non-convexity. The form of Qn−1 (Nn−2 ) depends on what happens at Nn−2 = 0 and Nn−2 = 1. The point N˜ n−1 of Fig. 4, deCned by the intersection of −X + Sn−1 (1 + k)(1 − Nn−2 ) and Bn−1 + Sn−1 (1 − k)(Nn−1 − Nn−2 ), exists if and only if Qn−1 (0) ¡ − X + Sn−1 (1 + k) 16 This portfolio is the solution of the program (2) subject to the constraint RB n−1 ¿ Max{uNn−1 Sn−1 ; −X + dSn−1 (1 − Nn−1 )}. 17 At n−2 there is a node of the binomial tree whose successor nodes are A and A , corresponding to 1 3 expiry and immediate exercise at n−1; hence, it behaves exactly like point A2 when the latter exists. 18 At any time j the market maker’s wealth is X − S ’(1 − N j j−1 ) if it exercises, and Bj + Sj ’(Nj − Nj−1 ) if it defers, where the portfolio (Nj ; Bj ) hedges optimally the subsequent positions. Hence, the market maker will choose the maximum of these two functions, implying that its cash requirements equal the minimum of the negative of these functions.
926
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
and −X ¡ Qn−1 (1); otherwise, one of the two functions is less than the other for all Nn−2 . Hence, we have at point A2 : Qn−1 (Nn−2 ) = Bn−1 + Sn−1 ’(Nn−1 − Nn−2 ) if − X ¿ Bn−1 +Sn−1 (1 − k)(Nn−1 − 1);
(13a)
Qn−1 (Nn−2 ) = −X + Sn−1 ’(1 − Nn−2 ) if − X + Sn−1 (1 + k) 6 Bn−1 + Sn−1 (1 + k)Nn−1 ;
(13b)
Qn−1 (Nn−2 ) = {RHS of (13a) for Nn−2 6 N˜ n−1 ; RHS of (13b) for Nn−2 ¿ N˜ n−1 } otherwise; Bn−1 − N˜ n−1 ) = −X + Sn−1 (1 − k)(Nn−1
+ Sn−1 (1 + k)(1 − N˜ n−1 );
(13c)
and (Nn−1 ; Bn−1 ) are given by (12). (13a) and (13b) apply, respectively, to all points at or above A1 and at or below A3 . We observe from the analysis at n−1 that the non-convexity case of Fig. 4 may appear at most at one node. Further, the replicating portfolio (Nn−1 ; Bn−1 ) given by (12) is found at the intersection of two convex functions, −uSn−1 ’(Nn−1 ) and −X + dSn−1 ’(1 − Nn−1 ). These functions have a single link each at Nn−1 = 0 and 1, respectively, and the Tattest branch of the u-node intersects with the steepest branch of the ∗ ∗ d-node. Last, it can be easily veriCed that the quantities Bn−1 + (1 − k)Nn−1 Sn−1 and ∗ ∗ Bn−1 + (1 + k)Nn−1 Sn−1 are increasing and concave as functions of Sn−1 , with derivatives that are 6 (1 − k) and 6 (1 + k), respectively. These properties are extended to any node by the following result.
Theorem 2. When (11) holds then for any time j ∈ [1; n−1] the optimal hedging portfolio (Nj∗ ; Bj∗ ) and exercise policy for a market maker facing investors who are put writers is unique at each node for all but possibly one node for all val∗ ∗ (m); Bj+1 (m)); m = u; d, ues of the inherited amount of stock Nj−1 . Further, if (Nj+1 ∗ ∗ ∗ denote the hedging portfolios of the successor nodes, we have Nj ∈ [Nj+1 (u); Nj+1 (d)], ∗ ∗ ∗ ∗ ∗ Bj ∈ [Bj+1 (d); Bj+1 (u)] and Nj ∈ [0; 1], Bj ∈ [−X; 0]. For every j there may be a node Sjl , with immediate exercise for all Sj ¡ Sjl and deferred exercise for Sj ¿ Sj; l+1 , while for Sjl the optimal policy may depend on Nj−1 , as in Fig. 4. The portfolio (Nj ; Bj ) hedging the unexercised put at any node is given by the solution of ∗ ∗ ∗ ∗ Bj+1 (u) + uSj (Nj+1 (u) − Nj ) = Bj+1 (d) + dSj (Nj+1 (d) − Nj ) = RBj ;
(14)
and immediate (deferred) exercise is optimal if (15a) (15b) holds: − X + Sj (1 + k) 6 Bj + Sj (1 + k)Nj ;
(15a)
Bj + Sj (1 − k)Nj 6 − X + Sj (1 − k):
(15b)
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
927
If neither (15a) nor (15b) hold then there is a value N˜ j ∈ (Nj ; 1) solving Bj + Sj (1 − k)(Nj − N˜ j ) = −X + Sj (1 + k)(1 − N˜ j ), with immediate (deferred) exercise optimal for Nj−1 ¿ (6)N˜ j . Proof. See Appendix A. Corollary 2. The optimal super-replicating portfolio of an American short put when (1) holds is either −(N (S0 ); B (S0 )) as in Theorem 2 or (−1; X ), depending on whether −(S0 N (S0 ) + B (S0 )) or X − S0 is largest. Hence, this case is very similar to that of the previous section. For each node the optimal exercise policy and hedging portfolio are found by evaluating the replicating portfolio (N ; B ) from the successor nodes on the basis of (14). Once this portfolio is found, we examine whether immediate exercise is proCtable by applying (15a) and (15b). At any time between origin and expiration the early exercise boundary may include at most a single point, where the optimal exercise policy is ambiguous. When that point exists the cash requirements function is non-convex and has two kinks, whose locations are, however, independent from the inherited stock. As with the long put the implementation of the algorithm described in this section is quite simple. We Crst determine whether at option expiration there is a point within ∗ ∗ ∗ ∗ (X=(1 + k); X=(1 − k)). If yes, then at n − 1 set Nn−1 = 1; Bn−1 = −X (Nn−1 = Bn−1 = 0) for nodes below (above) that point and start the algorithm as if the option expired at n−1. If no, then Cnd (Nn−1 ; Bn−1 ) from (12) and verify from (13a)–(13c) whether ˜ N n−1 exists. For all nodes at all times j ∈ [0; n−1) Cnd recursively the portfolio (Nj ; Bj ) from (14) and determine the early exercise boundary, including any possible points N˜ j , by applying (15a) and (15b); set Nj∗ = 1; Bj∗ = −X (Nj∗ = Nj ; Bj∗ = Bj ) below (above) the boundary. 5. Discussion and numerical results As noted in the introduction, the key issue in applying the results of this paper (as well as of the earlier Merton (1989), BV (1992) and BLPS (1992) studies) is the discrete time to expiration, the number n of subdivisions in the binomial lattice. This number also determines the binomial parameters, since in most applications following Cox et al. (1979) these parameters are evaluated from the expressions used to Cnd the convergence of the binomial model to continuous time as n increases. If t denotes the (continuous) time to option expiration, r the riskless rate, and the volatility of √ the associated stock return process, we have u = e t=n ; d = u−1 and R = ert=n . The √ choice of n also determines the validity of the key relation (11). Letting t=n ≡ t and t = x, we observe that (11) is satisCed iB x − rx2 ¿ ln[(1 + k)=(1 − k)]. For xed parameter values this relation is eventually violated as n increases. Increasing n also increases the upper bound for both European puts and calls, while, if (11) is violated so that u ¡ d (or 2 x ¡ ln[(1 + k)=(1 − k)]) the European call lower bound becomes equal to the Merton (1973) bound, Max{0; S0 − Xe−rt }. A similar result also
928
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
Table 1 k = 0:005; n = 20 and n = 40 X
85 90 95 100 105 110 115
UPa
Pa
Pb
UPb
UP
n = 20
n = 40
n = 20
n = 40
n = 20
n = 40
n = 20
n = 40
n = 20
n = 40
0.295 0.849 1.954 3.867 6.689 10.42 15.00
0.374 1.009 2.218 4.155 6.964 10.60 15.03
0.010 0.032 0.096 0.248 0.539 1.001 1.726
0.012 0.040 0.110 0.244 0.489 0.901 1.509
0.015 0.122 0.606 2.092 5.210 10.00 15.00
0.002 0.035 0.329 1.598 5.00 10.00 15.00
0.005 0.006 0.056 0.266 0.888 2.024 2.640
0.000 0.002 0.032 0.253 1.140 2.331 2.759
0.280 0.727 1.348 1.776 1.479 0.416 0.00
0.372 0.973 1.889 2.558 1.964 0.601 0.028
holds for the American put lower bound. While these results cannot be easily dismissed for all option markets, 19 they clearly cannot represent option bid and ask prices for highly liquid markets as in the US. √ √ By contrast, (11) holds if k depends inversely on n (is proportional to t) for reasonable values of the other parameters, as suggested by Henrotte (1993) and Flesaker and Hughston (1994). Setting k = !x, expanding ln[(1 + k)=(1 − k)] in Taylor series around k = 1, and replacing into (11), we observe that (11) is satisCed if ¿ rx + 2! + o(x). This holds for x → 0 as long as ¿ 2!, as is true for most underlying assets. The Henrotte–Flesaker and Hughston assumption also has implications for the asymptotic behaviour of the option’s long (short) hedging portfolio. These authors showed (in continuous time) √ that the value of a Black–Scholes expres√ this portfolio tends to √ sion with volatility (1 + 2!= )( (1 − 2!= )). For k = ! t, however, these are also the asymptotic values of the replicating portfolios derived within the binomial model of BV (1992). Hence, our American put results should also converge to the corresponding results of a binomial costs but with implied √ model without transactions √ volatilites adjusted by the factors (1 + 2!= ) and (1 − 2!= ). In the tables, we provide values of the super-replicating portfolios for American long and short puts when (11) holds for the following values: = 0:2; r = 0:1; t = 0:25; S0 = $100; X varying from $85 to $115 in increments of $5. Table 1 assumes √ constant √ transactions costs with k = 0:5% and n = 20 or 40. Table 2 assumes k = ! t = ! t=n, with ! = 0:005 and n = 64 or 100. In both tables we show the values Pa and Pb of the portfolios super-replicating the long and short put options, respectively; these can also be the ask and bid values for a perfectly-hedged market maker. We also show the values UPa and UPb , representing the corresponding early exercise premium, the excess of the American over European put values; the latter were computed from the put-call parity relation, with the European call value computed as in BV. Last, we show the diBerence UP, representing the bid/ask spread of the American put in absolute terms under the perfect hedging assumption. 19 For instance, in the empirical study of the Swiss option market of Lefoll and Perrakis (1995) a third of the sample had call bid prices violating the Merton (1973) lower bound.
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
929
Table 2 √ k = 0:005; t=n; n = 64 and 100 X
85 90 95 100 105 110 115
Pa
UPa
Pb
UP
UPb
n = 64
n = 100
n = 64
n = 100
n = 64
n = 100
n = 64
n = 100
n = 64
n = 100
0.130 0.488 1.39 3.16 6.04 10.07 15.00
0.127 0.487 1.40 3.16 6.04 10.07 15.00
0.005 0.023 0.09 0.25 0.60 1.23 2.12
0.005 0.023 0.09 0.25 0.58 1.24 2.13
0.096 0.401 1.23 2.96 5.86 10.02 15.00
0.095 0.404 1.25 2.97 5.88 10.02 15.00
0.004 0.019 0.08 0.24 0.62 1.35 2.23
0.004 0.020 0.08 0.25 0.62 1.35 2.24
0.034 0.087 0.14 0.20 0.18 0.05 0.00
0.032 0.083 0.15 0.19 0.16 0.05 0.00
A comparison of Table 1 with the corresponding results for k = 0 (not shown here) illustrates the eBect of transactions costs on the early exercise premium. Thus, for the long put price Pa transactions costs tend to reduce the importance of early exercise, decreasing its premium in relative terms (as a proportion of the European put value), but also in absolute terms as the exercise price increases. This was to be expected, given our super-replication results. The super-replicating portfolio is unaBected by transactions costs at the nodes of the binomial tree where early exercise takes place, while the value of the corresponding portfolio for the European put rises at these same nodes. Further, transactions costs would tend to reduce the region where early exercise takes place: perfect hedging implies that the portfolio should hedge the largest of the early or deferred exercise policy, and it is only the latter that rises. Nonetheless, the early exercise premium continues to be appreciable in most cases, especially for deep-in-the-money options. The other results derived from Table 1 were more or less to be expected from equivalent results for European options. The bid/ask spread in the last two columns rises sharply in both absolute and percentage terms with the number of periods to expiration of the option, with the ask price increasing and the bid price decreasing. Further, the eBect of possible early exercise on the bid/ask spread depends very much on the striking price X : the spread rises for low values of X but decreases for high X s. The results shown in Table 2 were compared to American put √ values derived from a binomial model without transactions costs but with = 0:2 1:05 for Pa and = √ 0:2 0:95 for Pb ; these are the asymptotic volatilities of the Black–Scholes model that the European options would converge to as n rises. For n = 100 the results of the binomial model were in all cases within less than $0:01 from the values of both Pa and Pb derived from our algorithms and shown in the table. Note also that for n = 64 both Pa and Pb are already very close to their asymptotic values. Thus, our algorithms converge to the binomial evaluation of the American put under adjusted volatility if √ we assume k = ! t. 6. Conclusions We have presented algorithms evaluating, at each node of the binomial tree, portfolios super-replicating American put options in the presence of transactions costs. Such
930
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
portfolios also represent the tightest possible bounds on such option prices when they are computed under the assumption that the market maker is perfectly hedged. The expressions are valid for all options held long by the investors and for short options when transactions costs are ‘small’ (i.e. when (11) holds). In general, the shape of the optimal exercise policy found by our results is fairly similar to the no-transaction-costs case. It is optimal to exercise the option immediately for low values of the stock price, and to defer for high values. An important diBerence, however, is that under transactions costs there exists an intermediate region of stock prices where the exercise policy is ambiguous. Nonetheless, the algorithms developed in the theorems presented in this paper are easy to implement, and allow the quick computation of the optimal super-replicating portfolios for any number of periods. The approach followed in this paper can be extended to other relevant cases of American options under transactions costs. In Perrakis and Lefoll (PL, forthcoming) super-replication algorithms are presented for a physical delivery option with a single constant dividend prior to expiration. Options on assets with a constant payout rate, such as index options, can also be handled in principle in the same way. Such options, however, are cash-settled and may be exercised once a day. As BLPS showed, the optimal hedging policy for both long and short European cash-settled options does not correspond to simple replication when (11) does not hold. The matter is currently under study. Acknowledgements The authors wish to thank the Canadian Social Sciences and Humanities Research Council, the Swiss Fonds National pour la Recherche ScientiCque (under grant 1214040687.94) and the CERESSEC for Cnancial support, and IrWene Collin for research assistance. A preliminary version of this paper was presented at the Northern Finance Association meetings, September 1995, at the JournWees Internationales de l’Association FranXcaise de Finance, June 1996, at the Second International Conference on Computing in Economics and Finance, June 1996, and at the Annual Meeting of the European Financial Management Association, June 1996. We wish to thank our discussants, Yisong Tian and Stanley Pliska. We also acknowledge the helpful comments from MichaFel Selby. Appendix A. All proof are basically geometric demonstrations of the desired results. This preserves the original format of the BLPS (1992) approach, on which this paper is largely based. In addition, the intuition is clearer than alternative algebraic proofs. Proof of Theorem 1. Apart from the convexity properties, the rest of the theorem was already shown to be true at n−1. For the convexity properties, we Crst observe that at n−1 they hold for the unexercised put, on the basis of the put-call parity; this
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
931
2
- d (1+k) R - ud(1+k) R
Qj+2
-u(1+k)
. . D. .D . D .D D1
D11
2
D22
. D. 2
12
D12
2
22
- d (1-k) R
. .D . D . D
- ud(1-k) R
N’(Sj)
- u (1+k) R -1
Nj+1
11
2
(a)
-d(1-k)
1
2
(b)
- u (1-k) R
Fig. 5. (a) A portion of the binomial tree. (b) The portfolio replicating the unexercised option when both successor nodes are in 3 .
convexity is also preserved under early exercise. 20 Now we use induction and consider the nodes at j; j + 1 and j + 2, shown in Fig. 5(a), assuming that the theorem holds ∗ ∗ at j + 1 and j + 2. The existence of N (Sj ) ∈ [Nj+1 (dSj ); Nj+1 (uSj )] is demonstrated in Fig. 5(b), which shows the functions Qj+2 (Nj+1 ; Sj+2 ) corresponding to the points D11 , D12 and D22 of Fig. 5(a). If both successor nodes D1 and D2 are in 3 then it follows, from the slopes of the broken lines emanating from them that deCne D, that ∗ ∗ (dSj ); Nj+1 (uSj )), QED. Similarly, if both Di ∈ 1 ; i = 1; 2, then D ∈ 1 N (Sj ) ∈ (Nj+1 ∗ ∗ (uSj ) = Nj+1 (dSj ), QED. The relative positions of as well, in which case N (Sj ) = Nj+1 ∗ ∗ N (Sj ); Nj+1 (uSj ) and Nj+1 (dSj ) are also shown in Figs. 6(a)–(c), respectively, for the cases Di ∈ 2 ; {D2 ∈ 1 ; D1 ∈ 2 } and {D2 ∈ 2 ; D1 ∈ 3 }. We have thus shown that a unique replicating portfolio (N (Sj ); B (Sj )) exists for the unexercised put at time j; with N non-decreasing and B non-increasing, within the sets of values [ − 1; 0] and [0; X=R], respectively. Consider now early exercise. We must show that, for any two successive nodes Sj1 and Sj2 ¡ Sj1 , Sj2 ∈ 2 implies that Sj1 ∈ 1 , while Sj2 ∈ 3 implies Sj1 ∈ 3 . If Sj2 ∈ 2 then the point X on the vertical axis lies below the intersection of that axis with the steeper line emanating from point D2 in Fig. 6(a), implying that it also lies below the intersection of the axis with the steeper line from D1 , QED. If Sj2 ∈ 3 then this implies that X on the vertical axis lies below the intersection of that axis with the Tatter line emanating from D2 in Fig. 7 (point 2 ), which lies below point 1 where the Tatter line from D1 intersects the axis. This last statement is proven by the ∗ ∗ convexity in S of the function Bj+1 (S) + (1 − k)SNj+1 (S), which holds by the induction hypothesis, QED. 20
∗ ∗ S For instance, Bn−1 + (1 − k)Nn−1 n−1 is equal to X − (1 − k)Sn−1 within 1 and 2 , and to B (Sn−1 ) + −1 L L (1 − k)N (Sn−1 )Sn−1 = [R(1 − k) − u]N L (Sn−1 )Sn−1 =R within 3 . Since @(N S)=@S = d[uL − d] , the slope of the function increases as we pass from 2 to 3 . Similar results also hold for the other two functions.
932
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935 Q
Q j+1 D D2 X
j+1
X,D2
-d(1-k) D1
.
D
.
D1
-d(1+k) -u(1+k) -u(1-k) Nj
(a) N*(dSj) N'(Sj) N*(uSj)
N’(Sj)
(b)
Nj
Q j+1
X
.. D2
D
. (c)
D1
N’(Sj)
Nj
Fig. 6. (a) The replicating portfolio when both successor nodes are in 2 . (b) The replicating portfolio when the ‘down’ successor nodes is in 1 and the ‘up’ node is in 2 . (c) The replicating portfolio when the ‘down’ successor node is in 2 and the ‘up’ node is in 3 .
Q j+1
-d(1+k) -u(1+k)
α1 α2 X
.
D2
.
D
.
D1
-d(1-k) -u(1-k) Nj
Fig. 7. Demonstration that early exercise is unproCtable in the ‘up’ successor node when the ‘down’ node is in 3 .
We have thus shown that a unique replicating portfolio exists at j, whose composition depends on its location in the three regions i ; i=1; 2; 3. The only part of the proof that remains concerns the convexity of the three functions. This is proven Crst by showing that convexity holds for SN (S) + B (S); (1 − k)SN (S) + B (S), and (1 + k)SN (S) +
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
933
Q j+1 ,Q j+2 2
-u (1-k) R
.
-ud(1+k) R
D11
.
-d(1+k)
D1
-u(1-k)
..
-ud(1-k) R
D 12
D
. . D2
D22
N'(uS j) N'j
2
-d (1+k) R
2
-d (1-k) R
N'(dS j)
N j,N j+1
Fig. 8. The portfolio hedging the unexercised short put.
B (S), i.e. if early exercise is ignored; the proof by induction, which is identical to the one used in Lemma 3 of PL (forthcoming), is available from the authors on request. Since convexity implies that the slope of (1 − k)SN (S) + B (S) is non-decreasing and ¿ − (1 − k), it is also preserved by adding early exercise, since it would simply take the maximum of it and the line X − (1 − k)S, QED. The proof for the other two functions is virtually identical. This completes the proof of the theorem. Proof of Theorem 2. The theorem clearly holds at n and n − 1. We use induction and consider the three nodes of the binomial tree depicted in Fig. 5(a). If both successor nodes at j + 1 correspond to immediate exercise then Nj∗ = 1; Bj∗ = −X , QED; likewise, Nj∗ = Bj∗ = 0 if these are also the optimal portfolios for both successor nodes. Now suppose that at j + 2 the theorem holds and that at nodes D11 and D22 the optimal ∗ ∗ ∗ portfolio has Nj+2 ¿ 0 and Nj+2 6 1, respectively, while at D12 we have 0 ¡ Nj+2 ¡ 1. Then the situation is as depicted in Fig. 8, which shows that if points D1 and D2 correspond to non-exercise then point D must lie at the intersection of the dotted lines, implying that (Nj ; Bj ) satisCes (14), QED. If node D1 corresponds to immediate exercise then nodes D2 and D also correspond to immediate exercise, QED; (Nj ; Bj ) ∗ also satisfy (14) if two of the three nodes at j + 2 have the same Nj+1 , at 0 or at 1. Next we consider what happens if one point at j + 1 has optimal exercise policies as in Fig. 4. Such a case is depicted in Fig. 9, which shows that it is impossible for two successive nodes to have such optimal exercise policies. If D1 and D2 are found from the solutions of equations (14) for the two nodes at j + 1 before early exercise is considered, then if a point N˜ exists at the u-node then (15a) must hold at the d-node, implying that we should have immediate exercise at that node; the node D can then easily be seen to correspond to immediate exercise as well. Hence, the situation depicted in Fig. 4 can occur at only one node at most for every j ∈ [1; n − 1],
934
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
Q
j+1
,Q
j+2
-u(1+k)Sj -u(1-k)S j
.
D1
-d(1+k)Sj -u(1+k)Sj
.
D'2
-d(1-k)Sj D,D2
-X N'(uSj)
N'(dSj)
Ñ(uSj)
N j,Nj+1
1=N*(dSj)
Fig. 9. Demonstration that only one node may have optimal exercise policies as in Fig. 4: ‘up’ successor node as in Fig. 4, immediate exercise in ‘down’ node.
Q
j+1
,Q
j+2
-u(1+k)Sj
.
-u(1-k)S j
D1
.
D
-d(1-k)Sj
.
D2
-d(1+k)Sj
-X N'(uSj)=N*(uSj) N'(dSj) Ñ(dSj )
1
N j,Nj+1
Fig. 10. Demonstration that only one node may have optimal exercise policies as in Fig. 4: ‘down’ successor node as in Fig. 4, deferred exercise in the ‘up’ node.
QED. By contrast, Fig. 10 shows that, if it is the d-node that has policies as in Fig. 4, or corresponds to deferred exercise for all Nj , then the u-node must correspond to deferred exercise as well, QED. This completes the proof of the Theorem. References Barone-Adesi, G., Whaley, R.E., 1987. EHcient analytic approximation of American option values. Journal of Finance 42, 301–320. BensaFGd, B., Lesne, J.P., Pag[es, H., Scheinkman, J., 1992. Derivative asset pricing with transaction costs. Mathematical Finance 2, 63–86.
S. Perrakis, J. Lefoll / Journal of Economic Dynamics & Control 28 (2004) 915 – 935
935
Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–659. Boyle, P.P., Tan, K.S., 1994. Lure of the linear. Risk 7, 43–46. Boyle, P.P., Vorst, T., 1992. Option replication in discrete time with transactions costs. Journal of Finance 47, 271–293. Brennan, M., Schwartz, E., 1977. The valuation of American put options. Journal of Finance 32, 449–462. Constantinides, G., 1986. Capital market equilibrium with transactions costs. Journal of Political Economy 94, 842–862. Cox, J.C., Rubinstein, M., 1985. Options Markets. Prentice-Hall, Englewood CliBs, NJ. Cox, J.C., Ross, S., Rubinstein, M., 1979. Option pricing: a simpliCed approach. Journal of Financial Economics 7, 229–263. Dumas, B., Luciano, E., 1991. An exact solution to a dynamic portfolio choice problem under transactions costs. Journal of Finance 46, 577–595. Edirisinghe, C., Naik, V., Uppal, R., 1993. Optimal replication of options with transaction costs and trading restrictions. Journal of Financial and Quantitative Analysis 28, 117–138. Flesaker, B., Hughston, L.P., 1994. Contingent claim replication in continuous time with transaction costs. Working paper, Merrill Lynch. Geske, R., Johnson, H.E., 1984. The American put valued analytically. Journal of Finance 39, 1511–1524. Henrotte, P., 1993. Transactions costs and duplication strategies. Working paper, Graduate School of Business, Stanford University. Hodges, S., Neuberger, A., 1989. Optimal replication of contingent claims under transactions costs. Review of Futures Markets 8, 222–242. Lefoll, J., Perrakis, S., 1995. Transactions costs and option bid/ask spread on the Swiss Options and Financial Futures Exchange (SOFFEX). Canadian Journal of Administrative Sciences 12, 276–289. Leland, H., 1985. Option pricing and replication with transactions costs. Journal of Finance 40, 1283–1301. MacMillan, L.W., 1986. Analytic approximation for the American put option. Advances in Futures and Options Research 1, 119–139. Merton, R.C., 1973. Theory of rational option pricing. Bell Journal of Economics and Management Science 4, 141–183. Merton, R.C., 1989. On the application of the continuous-time theory of Cnance to Cnancial intermediation and insurance. The Geneva Papers on Risk and Insurance 14, 225–261. Parkinson, M., 1977. Option pricing: the American put. Journal of Business 50, 1–36. Perrakis, S., Lefoll, J., 1997. Derivative asset pricing with transaction costs: an extension. Computational Economics 10, 359–376. Perrakis, S., Lefoll, J., 2000. Option pricing and replication with transaction costs and dividends. Journal of Economic Dynamics and Control 24, 1527–1561. Rendleman, R., Bartter, B., 1979. Two-state option pricing. Journal of Finance 34, 1093–1110.