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Pricing of foreign exchange options with transaction costs: The choice of trading interval
Pricing of Foreign exchange Options with Transaction Costs: The Choice of Trading Interval
SHMUEL HAUSER AND AZRIEL LEVY
In this paper, we develop an arbitrage-free option valuation model in the presence of transaction costs. The model considers the trade-off between the choice of lowering costs by trading less often and the choice of reducing the hedging errors by trading more often. This trade-off allows derivation of a trading interval policy. We illustrate the model with actual transactions data and offer a procedure for the estimation of a trading interval that minimizes hedging errors and transaction costs. The findings suggest that currency options trading is most active in options with short time to expiration, especially if they are at-and out-ofthe-money options.
I. INTRODUCTION In the absence of transaction costs, a single-option equilibrium price may be derived through the creation of a perfectly hedged portfolio replication the option. The ability to create a perfect hedge, the basis for risk-neutral valuation models, including that of Black and Scholes, breaks down in the presence of transaction costs. This breakdown occurs because the construction of a perfect hedge requires continuous adjustments of the portfolio replicating the option, thus increasing total transaction costs to infinity. Several authors (e.g., Glister & Lee, 1984; Leland, 1985; Edirisinghe, Naik, & Uppal, 1993) argue that in order to reduce transaction costs, traders would avoid frequent trading, and that infrequent trading makes it impossible to create a perfectly hedged replicating portfolio, leading to hedging errors. One possible way to deal with these hedging errors is to use a preference-dependent hedging strategy, which raises questions regarding investors’ risk aversion. A second way is Leland’s (1985) approach, in which the hedging strategy depends on transaction costs and on the length of the trading interval. This approach relies on hedging errors’ being uncorrelated with the market and therefore requires assumptions on the pricing of securities in the market. In this paper, we develop an arbitrage model for the valuation of foreign currency options in the presence of transaction costs, which can be used to derive optimal trading interval policy. The model can be accommodated to other contingent claims as well.1 One advantage of the model is that it does not depend on investor preferences. A second advantage is that the Schmuel Hauser Ben Gurion University of the Negev, Beersheva, versity of Jerusalem, Mount Scopus, Jerusalem. l
Israel 84105; Azriel Levy
International Review of Financial Analysis, Vol. 5, No. 2,1996, pp. 145-160 Coptyright 0 1996 by JAI PRESS Inc., All Rights of reproduction in any form reserved.
l
The Hebrew Uni-
ISSN: 10573219
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model does not require any assumptions on the pricing of securities in the market. A third advantage is that the trading interval is an endogenous decision variable. The model extends the models of Boyle and Vorst (1992) and Merton (1990) by allowing the arbitrager to avoid trading every binomial period. Contrary to these models, we argue that when a portfolio replicating the option is not adjusted every binomial period, and as the trading interval becomes longer (i.e., the trading interval is more than one binomial period), to save on costs, hedging errors incur, and the underlying distribution of returns becomes multinomial rather than binomial (see Edirishinghe, Naik, & Uppal, 1993; Toft, 1994). In this case, two counteracting effects will occur. On one hand, lengthening the trading interval reduces transaction costs and thus reduces the range between the upper and lower option bounds. On the other hand, lengthening the trading interval increases hedging errors and therefore increases the range between the upper and lower option bounds.2 Note that given some level of transaction costs, arbitrage profits will be possible when the option price is higher than its upper bound or lower than its lower bound. In this regard, note also that the model developed by Toft evaluates a similar trade-off between transaction costs and the variance of the replicating portfolio. In his model, the hedger is concerned with the distribution of his cash flow at the end of each hedging period. The inflow at each rebalancing point is the hedging error minus the transaction costs. But, whereas Toft’s model is designed to provide an approximation to the evaluation of this inflow, the model suggested in this paper allows one to develop a procedure that evaluates the upper and the lower bounds which minimize hedging errors given the transaction costs. The paper is organized as follows. In Section II, we device the option price bounds in the presence of transaction costs, assuming a binomial distribution and assuming that trading takes place every binomial period. Then, the model is extended to the case where investors may choose not to trade every binomial period, and the distribution is multinomial. In Section III, we illustrate the model. In Section IV, we portray the choice of the optimal trading interval policy. Section V concludes the paper.
II. THE MODEL
Complete Market Call Option Valuation with Transaction Costs We assume that the rate of the underlying currency is binomial in each trading period, and that there are IZtrading periods prior to expiration. For an institutional investor operating in the inter-bank market, transaction costs are made up of the following three components: (1) bid/ask foreign exchange spread; (2) borrowing/lending interest rate spread; and (3) bid/ask option price spread. We employ the following notations: S,, S,, and S, are the ask, bid and middle (geometric average) exchange rate in period t (t = 0, 1, . . .), dollars per foreign currency; rL and rB are one plus local currency lending and borrowing interest rates, respectively; r*L and r*B are one plus foreign currency lending and borrowing interest rates, respectively; X is the strike price of the call option; C,, CtB, and C, are the ask, bid and middle (geometric average) price of the call option in period t(t = 0, 1, . . . ). With these notations a2 = SJS, and p2 = C,/C,, where (a2 - 1) and (p2 - 1) is the percentage difference between the bid and ask prices of the underlying currency and the option, respectively. Thus, the middle price is defined as the geometric average of the bid and ask prices:
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Options
s,=(S,Bs,,Y’* c, = cc,,cl,&*
(1) (2)
where bid and ask prices are: SrB = St/a; s,,
= S,a
(3)
CtB = C,/p; C,, = C,p. We further assume that the rate of return of the underlying distributed binomial and that cx and p are constant.
(4) asset, middle exchange rate S,, is a
Proposition 1: Binomial Upper Bound An arbitrage transaction in which an investor purchases 6 units of foreign currency at the ask rate, sells short one unit of the option at the bid price, and finances the transaction by borrowing dollars, yields the following option price upper bound:
c
i
[;( )e’(1 -
(5)
Q)“-j C(U, j)]
r B j=O
where C(u,j) is the value of the option at expiration, j times, and
when u occurs j times, d occurs n -
Q = ([a2rB-r*Ld]/[r*L(u-d)])whereOSQI
Proof: See Appendix
1.3
A.
Proposition 2: Binomial Lower Bound An arbitrage transaction by an investor who sells 6’ units of the foreign currency at the bid rate, buys one unit of the call option at the option ask price and lends dollars at the option ask price and lends dollars at the dollar lending rate, yields the following option price lower bound: c2-
,’ 2n
j f0
[(y)Q(l
-Q)“-’
C(u,j)]
(6)
rL p
where C(u, j) is the value of the option in period n, when u occurs j times and d occurs 12-j times, and Q’ = [ rL - r*Bdcx2]/[r*B(u Proof: See Appendix
- d)a*]
where 0 $ Q I 1.
B.
Propositions 1 and 2 indicate that in the presence of transaction costs, the efficient market price is bounded; whereas, in the absence of transaction costs, the upper and lower bounds coincide and a single equilibrium price is derived. In both cases, in equalities (5) and (6) represent preference-free strategies derived on the assumption that arbitrage opportunities are not available. For a further explanation, we elaborate on the upper bound. Suppose there are two market makers that compete in the market by selling (writing) a call option on the same underlying asset and that both market makers are subject to transaction costs. Suppose further that the transaction costs of market maker No. 1 are lower than those of market maker No. 2. Accord-
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ing to our model, market maker No. 1 will find it optimal to trade more often than market maker No. 2, and his/her option upper bound will be lower than the upper bound of market maker No. 2. Since the two market makers compete with each other, the equilibrium ask price in the market will be set by market maker No. 1 somewhere in the range between his/ her upper bound and the upper bound of market maker No. 2. If market maker No. 2 sells the option for a price lower than his/her upper bound he/she will incur a loss where as market maker 1 will profit as long as he sells the option for a price higher than his/her upper bound. If there are many market makers competing in the market, the ask price will be forced to come down to a level close (or even equal) to the upper bound of market maker No. 1. This equilibrium price is preference free since, in its derivation, we use only arbitrage considerations and do not make any assumption on the preferences of market participants.
Incomplete Market Call Option Valuation with Transaction Costs Continuous adjustment of the portfolio replicating the option can be very costly in the presence of transaction costs. In order to reduce transaction costs, an arbitrager will trade in discrete time intervals, rather than continuously (see Leland, 1985). In this section, we use the binomial model as analogous to the continuous model (e.g., Wiener process) and derive a trading policy and an arbitrage-free option price in the presence of transaction costs. It is well known4 that, in an incomplete market, investors cannot construct a perfect hedge, and the equilibrium arbitrage-free price of the option is bounded within some range, depending on the assumptions on the distribution of the underlying asset. In the presence of transaction costs, there are two basic approaches to the pricing of potions. The first is that taken by Merton (1990, Chapter 14 p. 432). Merton uses the most conservative estimate of transaction costs, by considering an intermediary in the market that liquidates his position at the end of each trading period and then opens up a new position. Thus, the option prices obtained are arbitrage-free bounds of the bid-ask option prices, that take into account the worst possible scenario from the point of view of the intermediary. In the second approach, Boyle and Vorst (1992) and Edirisinghe, Naik, and Uppal (1993) and Toft (1994) reduce transaction costs by avoiding total liquidation of the trader’s position. Both approaches are an extension of the binomial model, in the sense that, in each trading interval, there are only two possible states of nature. In this paper, we follow Merton’s approach, by considering the maximum transaction costs in each trading interval and by assuming that initially the return distribution of the underlying asset is binomial. Specifically, we assume that the distribution of the underlying asset is binomial and that each binomial period is known in advance. Figure 1 gives the distribution of returns if the trader trades every binomial period. However, if the trader refrains from trading every binomial period, but decides to trade every two binomial periods (periods 0,2, and 4 in Figure l), the distribution of returns in these periods is presented in Figure 2, and is derived directly from the binomial distribution, which is the “true” distribution of the underlying security. Note that the assumption made in this paper differs from Ritchken (1988) and Levy (1985) since they assume that the underlying distribution in each period is multinomial. In our model, the underlying distribution is binomial, but it becomes multinomial if the trader avoids trading every binomial period to reduce his/her transaction costs. Thus, in contrast with Merton (1990), Boyle and Vorst (1992) and Toft (1994), we do not constrain the trader to trade every binomial period, and recognize that, by increasing the trading travel, the distribution of the underlying asset returns becomes multinomial.5 As we have seen, for multinomial distributions, even without transaction costs, there will not exist a sin-
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gle arbitrage-free option price, but rather the price will be bounded within a range. Thus, if the trading interval is prolonged, two opposing effects will take place: (1) the upper and lower bounds will approach each other, as a result of the reduction in transaction costs, and (2) the upper and lower bounds will deviate from each other, as a result of the increase in the number of states of the distribution, leading to hedging errors. Given this trade-off we derive a trading policy in which the investor chooses a trading interval that enables him/her to exploit available arbitrage opportunities in the presence of transaction costs. At the end of every trading interval, the investor may reassess the position and choose a different trading strategy, depending on the trade-off between transaction costs and hedging errors. We consider an optimal trading interval as one that minimizes the upper bound, or maximizes the lower bound, since this strategy minimizes the cost of writing the option or maximizes the proceeds that finance the purchase of the option. The optimal trading interval of the upper bound need not be equal to that of the lower bound. To proceed now to the derivation of the option bounds in the presence of transaction costs, it is assumed that there are n binomial trading periods until the option expires. In order to reduce transaction costs, the investor may choose to increase the trading interval; i.e., to refrain from trading every binomial period. Denote by h the length of the actual trading period chosen by the investor. Thus, the number of actual trading periods prior to expiration, denoted HZ,will be: m = nlh
(7)
Figure 1 illustrates the case of n = 4 binomial periods. By choosing to trade every two (h = 2) binomial periods, the investor trades m = n/h = 2 times, and the distribution of returns in each trading period will be trinomial, as shown in Figure 2. In general, the distribution of returns
U4
u3d
u2d2
ud
d4
Binomial
Period
I
I
I
1
I
0
1
2
3
4
Figure 1.
Binomial
Distribution
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“‘d2
ud
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d4
I Multinomial
Period
I
I
I
2
1
0
Figure 2.
Multinomial
I
Distribution
in each trading period is multinomial, with 2h states (some of the states coincide and the number of unique states is only h + 1). To derive the multinomial upper and lower option bounds, we use the,following notations: d,, = dh
(8)
“,, = uh
(9)
Uh and dh are the maximum and minimum returns of the underlying asset, over trading interval h. ‘Lh, rBh, reLh, r*Bh, are on plus the lending and borrowing interest rates on local and foreign currencies, respectively, in the trading interval h. Thus, rLh = rLh; rBh = r& r*Lh = r*Lh; r*gh = r*#.
(10)
Denote by Ri, h the multinomial distribution return, when each trading interval is composed of h binomial periods (i = 1, . . . . h +l). Define U*h, d*,, such that, d*h = Rj, h
(11)
U*/,=R. I+ l> h
(12)
and, r*,,Rj,ha2
5 r,h < r*Bh Rj + ,, ha’.
(13)
Proposition 3: Multinomial Upper Bound When the length of the trading interval is of size h, the option price upper bound is given by,
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(14)
where Qh = [a*r,h - r*Lhdh]l[r*lh(uh expiration if Uh occurs j times and dh Proof: See Appendix
- dh)] and c(u*, j) is the value of the call option at occurs
m
-j
times.
C.
To elaborate on the argument that the lower the transaction costs, the lower the option upper bound, recall that the transaction costs are embedded in the terms a and p. For example, the option upper bound presented in equation (14) is a function of both a and p. It can be readily seen that a decrease in p results in a decrease in the upper bound. The effect of CCon this upper bound is a bit more complicated. Technically, this effect is similar to the effect of a decrease in the riskless interest rate in the binomial option valuation formula. To see this, note that the right side of equation (14) is analogous to the standard binomial option valuation formula. Thus, in this formula, an increase in the riskless interest rate or an increase in U. increases the option price. An increase of the same magnitude in (a*) will have a greater effect on the option price since it affects only Qh, but has no impact on the discounting term ll(r,,)m. Note that in our simulations the effect of p was found to be negligible in comparison with the effect of a. Proposition 4: Multinomial Lower Bound When the length of the trading interval is of size h, the option price lower bound is given by: (15)
where Qw = [rLh - r*Bhd*ha2]/[r*Bh(u*h -d*,)cc2], C(u*h. j) is the value of the option at expiration if U*h occurs j times and d*h occurs m -j times and m is the number of trading intervals to expiration.6 Proof: See appendix D. In the upper bound case, each trader with given transaction costs computes the upper bound in equation (14) for various trading intervals h. Given transaction costs of size a and p, the trader chooses the trading interval that minimizes the upper bound. The trading interval that minimizes the upper bound, as a function of cc and p, is denoted by /~,(a, p). Even if the option price is only slightly higher than this upper bound, one may construct a portfolio generating arbitrage profits. These profits will be realized only when trading is conducted every h,(a, B) periods. To prevent arbitrage profits, the option price must be below this upper bound. The lower the transaction costs, the lower this upper bound (see Figure 3). Likewise, there will be arbitrage profits if the option price is below the lower bound, which is the maximum lower bound for all trading intervals h. The trading interval that maximizes the lower bound [inequality (15)], as a function of a and p, is denoted by h,(ol, p). If the option price is below the lower bound obtained by trading every !~,(a, p) binomial periods, one may construct a portfolio that will generate arbitrage profits. The lower the transaction costs, the
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Price Bounds
____----
I :
1.0000
n
II : = 1.0001 Ill : \r 2
_______----------__-Ill t
n
1.0002
Black $ Scholes .___________________________,
-
I
7
8
9
higher the lower bound. The equilibrium option upper and lower bounds are determined by tradershaving est transaction costs and the ability to trade most often. Thus, a necessary condition librium is that the option price is within these upper and lower bounds.
the lowfor equi-
1
I
I
I
2
3
4
I
I
1
I
OL
5 6 Trading Interval
Figure 3. Upper and Lower Price Bounds (S=X=100;Sig=.15;r~r*=.1)
III. ILLUSTRATION
OF THE MODEL
The purpose of this section is to illustrate the valuation model presented in Section II, allowing for transaction costs and allowing the arbitrager to choose the appropriate trading interval. For simplicity, we assume that all transaction costs are embedded in the bid-ask spread of buying the underlying currency (i.e., p = 1). We also define the upper bound as the maximum option price that prevents arbitrage, given transaction costs a and trading interval h, and the lower bound as the minimum option price that prevents arbitrage, given transaction costs of level o? and trading interval h. The model is illustrated using the following parameters: S = 100; X = 100; T = .25; n = 180; h= 1,2, . ..) 9; and annualized domestic and foreign interest rates are 10% (rB = rL and Fan = r*& In order to compare our model with the Black-Scholes model, we follow Cox and Rubinstein’s approach (1985, p. 200) by setting u = exp(oddt), d =1/u, and s = .15. We calculate the upper and lower bounds of the currency call option price for different transaction costs levels a and different trading intervals, where h = 1 implies that the arbitrager trades every binomial period. Note that, if portfolio revision takes place every binomial period, the assumed underlying distribution may serve as a proxy to a Wiener process. In this case, in the absence of transaction costs (a = l.OOOO), our model converges to the BlackScholes option valuation model. Note also that the binomial option valuation model is
Pricing of Foreign Exchange Options
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viewed in the literature as a proxy of the Black-Scholes model, and does not depend on the number of binomial trading periods n (for relatively large n). With the introduction of transaction costs, the assumption regarding n becomes crucial. Assuming that the arbitrager trades every binomial period, and given some level of transaction costs ~1, the spread between the upper and lower option bound will be greater the higher is n. Figure 3 portrays the upper and lower bounds as a function of the trading interval h for various levels of transaction costs, denoted a. As expected, the higher the transaction cost, the higher the upper bound and the lower the lower bound, for every trading interval of size h. When a = 1.0000 (zero transaction costs) and h = 1 (trading is conducted every binomial period), the upper and lower bounds coincide (they are approximately equal to the BlackScholes price), and the trader can create a perfect hedge by revising the option replicating portfolio, every binomial period. also, when a = 1.OOOOand h > 1, the efficient price according to our model becomes bounded. Thus, as h increases, the upper bound increases and the lower bound decreases. Namely, the less often the trader trades, the higher the upper bound and the lower the lower bound. Evidently, in the absence of transaction costs, it is optimal to trade every binomial period, and the Black-Scholes price is the equilibrium option price. For a > 1, trading every binomial period incurs the greatest amount of transaction costs, and it may be worthwhile to trade less often. The trading interval that minimizes the upper bound or maximizes the lower bound may be greater than 1.O. For example, it is seen in Table 1 that when a - 1.0002 (bid-ask spread of 0.04%), the minimum upper bound is attained approximately at h = 3, namely when trading takes place every three binomial periods. The transaction costs effect outweighs the hedging ability effect up to a trading interval of size h = 3. If h > 3, the hedging effect becomes more crucial, and it does not pay to trade less often than every three binomial periods. As cx increases, the trading interval h that minimizes the upper bound increases, and the trading interval that maximizes the lower bound increases as well. However, for a given level of transaction costs, the upper-bound trading interval need not be equal to the lower bound trading interval.
IV. THE CHOICE OF TRADING INTERVAL An interesting issue that has not been addressed in the literature is how often has trading actually been conducted, given transaction costs. In this section, we suggest a policy of trading frequency. The choice of trading interval is illustrated via estimation of implied trading interval. The data used for the illustration are observations on German marks currency options and exchange rates compiled by the Philadelphia exchange, from August 1983 to December 1984. Euro-dollar and Euro-mark interest rates were collected from the London Financial Zmes.In the sample period, interest rates in Germany were much lower than interest rates in the U.S., permitting the use of European call option valuation models to price american call options. This is because, in such cases the probability of early exercise is practically zero, and the pricing errors of European currency call option models are insignificant.7 Volatility of the $/DM exchange rate is estimated by historical annualized logarithmic returns and is used as an input in estimating the binomial parameters u and d (see Cox & Rubinstein, 1985). For each observation, we compute the theoretical Black-Scholes option price (without transaction costs) and compare it with the observed market price. If the observed market price is higher than the Black-Scholes price, we equate the upper bound with the market price in order to derive the implied trading interval (hereinafter, ITI):
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(. , ITI),
where C,,,b is the observed option market price, CIJB,,d is the theoretical minimum upper bound call option price, as a function of the spot rate, exercise price, domestic and foreign interest rates, u and d, transaction costs, and implied trading interval (ITI). Likewise, if the observed price is lower than the Black-&holes price, we equate the lower bound with the market price in order to derive the implied trading interval, = CLB,,,
c mkt
(. , ITI),
where CLB,,,~~is the theoretical maximum lower bound call option price. Figure 4 illustrates the estimation procedure with the option upperbound. We first calculate the upper bound option price for a given level of transaction costs, for various binomial trading intervals (e.g., curve I in Figure 4). Second, we identify the trading interval that minimizes this upper bound (e.g., point F in Figure 4). Third, we repeat this process for different levels of transaction costs and draw a curve connecting these minimum points (curve UU in Figure 4). This curves the option upper bound (CUB,,d), given transaction costs a. Finally, we solve for the implied trading interval by equating the observed market call price (C,,& with CUB,,d (the intersection point of UU and Cmkt in Figure 4). Note that with this procedure the actual transaction costs [are not required because we simutaneously estimate IT1 and the implied transaction costs] (hereafter, ITC).* Assuming there are no arbitrage opportunities, these implied transaction costs cannot be higher than the transaction costs of any market participant. A similar procedure is used to derive the IT1 when Cmkr is lower than the Black-Scholes call price. Because the upper and lower bounds depend on the assumption of the number of binomial periods prior to expiration, we estimate ITC and IT1 for various 11’s (i.e., for various Call upper
bound
11 I II III IV v
uu
: Iyn1.0000 : Mn1.00005 :cI=1.0001 : c(n1.0002 : a n1.00025
Call Market J 3 1
Black 8 Scholes
I
I
I
I
2
3
4
5
Trading
interval
Figure 4. An Illustration of Estimation (S=X=100;T=0.25;r=r*=0.1;n=180)
Price
Procedure
6
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Exchange
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155
length of binomial periods). Recall that the assumption on n is tantamount to the assumption as to how often the arbitrager should trade in the continuous model in order to create a perfect hedge in the absence of transaction costs. Given that there are D trading days to expiration, II will depend on the assumption regarding the number of binomial periods per day, denoted I (n = D . I).
Table 1. Average Trading Interval and Implied Transaction Costs in German Marks Currency Optionsa Number of binomial periods (n) assuming I daily revisionsb I = 4 times a day ITI All
1001
By Boundary In
Status
517
out
383
90
ITC
ITILTCC
0.482
1.163
0.685
0.241
0.581
0.685
(0.019)
(0.03 1)
(0.004)
(0.010)
(0.03 1)
0.278
0.484
1.664
0.139
0.242
1.664
(0.008)
(0.046)
(0.243)
(0.004)
(0.023)
(0.243)
0.311
0.744
0.730
0.155
0.377
0.730
(0.005)
(0.018)
(0.046)
(0.003)
(0.009)
(0.046)
0.757
1.885
0.403
0.379
0.943
0.403
(0.016)
(0.004)
(0.008)
(0.008)
(0.002)
(0.008)
54.7
By Maturity
45
ITI
(0.009)
Fd
T<45
ITUITCC
(In-, At-, and Out-of-the-Money) 88
At
ITC
I = 8 times a day
54.7
(in Days) 63 262 676
0.219
1.471
0.549
0.109
0.736
0.549
(0.014)
(0.078)
(0.177)
(0.007)
(0.039)
(0.177)
0.437
1.476
0.474
0.219
0.738
0.474
(0.011)
(0.041)
(0.057)
(0.005)
(0.021)
(0.057)
0.518
1.016
0.779
0.259
0.508
0.779
(0.009)
(0.020)
(0.039)
(0.005)
(0.010)
(0.039)
8.3
Fd
8.3
By Upper and Lower Bounds Upper Lower
Fd Notes:
103 898
0.274
0.447
1.909
0.137
0.224
1.909
(0.007)
(0.041)
(0.249)
(0.004)
(0.021)
(0.249)
0.506
1.245
0.545
0.253
0.623
0.545
(0.010)
(0.02 1)
(0.021)
(0.005)
(0.01 1)
(0.011)
173.9
173.9
a Numbers in parentheses are standard errors. b n = D * I is the assumed number of binomial periods until maturity, where D is the number of days to maturity and 1 is the number of binomial periods per one day. c ITI is the implied trading interval in days. ITC is the implied bid-ask spread (~(CX- 1)) of exchange rate as a proportion of the level of the exchange rate, in basis points. ITUITC is implied trading interval per one unit of transaction costs measured by the bid-ask spread. d One way analysis of variance is used to test the hypothesis H,: I.lt = pL2= = pi vs. H,: at least two means are not equal.
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Table 1 presents ITC and IT1 for I = 4, 8. The first notable result is that as n increases (or I increases) ITC and IT1 decrease. The explanation of this result is that, for some level of transaction costs a, the lower is n (or 1), the lower is the upper bound and the higher is the lower bound.9 Thus, given the observed option price, which is taken as the upper or lower bound in our model, IT1 must be higher the smaller is the number of binomial periods, in order to induce the same option bounds. The problem with the estimation of IT1 and ITC is that they depend on IZ, the exact number of binomial trading intervals that approximate the continuous Wiener process, and we do not know a priori the value of II. However, both ITC and IT1 decrease proportionally as the number of assumed daily binomial trades increases; i.e., the ratio ITI/ITC is constant and is independent of 1 (or of n, given D). Thus, we observe in Table 1 that when 1 = 4, both ITC and IT1 are twice as high as when 1 = 8. This finding allows us to circumvent the difficulty of finding the correct length of the binomial trading interval that serves as a proxy for a continuous process. The ratio ITI/ITC, is the trading interval in terms of a trading day per ne unit of transaction costs; i.e., its reciprocal is the implied number of trades per day. For example, if ITI/ITC = S, the implied trading strategy is to trade twice a day, per one unit of transaction costs, and if transaction costs are actually twice as high, our model predicts that arbitragers will actually trade once a day. Note that, give ITI/ITC and the correct proportional transaction costs, one could determine not only the implied trading interval, but also the number of binomial trades per day, denoted 1, that approximate the continuous process. The results in Table 1 also indicate that the shorter the time to maturity and the more outof-the-money the option, the smaller the implied trading interval per one unit of implied transaction costs (ITUITC). These results suggest that options trading is most active in options with short time to expiration (about two portfolio revisions per one day in short-term options compared with only a little more than one portfolio revision in long-term option), especially if they are at-or out-of-the-money options.
V. CONCLUSIONS
In the presence of transaction costs, traders will try to avoid trading very often to minimize their costs. However, in doing so, it becomes impossible for them to create a perfectly hedged portfolio replicating the option, and there will be hedging errors. Thus, the arbitrager faces a trade-off between the choice of lowering costs by trading less often and the choice of reducing the hedging errors by trading more often. We developed an arbitragefree model for the valuation of foreign currency options (or any other contingent claim) in which this trade-off prevails. The model enables us to derive a trading interval policy. We illustrated the model with actual transactions data and offered a procedure for the estimation of a trading interval that minimizes hedging errors and transaction costs. The findings suggest that options trading is most active in options with short time to expiration, especially if they are at- or out-of-the-money options.
ACKNOWLEDGMENT This paper was partially supported by the Krueger Fund.
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APPENDIX A: BINOMIAL UPPER BOUND Proof of Proposition 1 Assume that an arbitrager who purchases an amount 6 of the foreign currency at the ask rate, sells short one unit of the option at the bid price, and finances the transaction by borrowing dollars. The hedge ratio 6, will yield the same return in the two states of nature (U and 6> if the following condition is met: GSur*,/a
- CUP = GSdr*,/a
- C&3,
(Al)
Where C, and Cd are the value of the middle call option price next period, in states u and d, respectively. Solving for 6 we get: 6 = ([aP(C, - Cd)l/[Sr*L(u
- d)l).
(A2)
To prevent arbitrage, the return in states u and d must be non-positive: GSur*L/a
- CUP - (Ma
- C/P)rB
IO,
where the second term on the left side of (A3) is the principal loan. Substitute (A2) into (A3) and solve for C,
(A3)
and interest on the financing
CI B21Qc, + ( l-Q)Cd]/rB where Q = [a2r,‘*,
d ]/ [ r*L (u - d)]
and 0 I Q 2 1. Equation
644)
(A4) enables us to the derive
the n period upper bound. In period 1, there are two possible states of nature: St, = US; St2 = dS
where S is the initial price of the underlying currency. The middle price of the call option in each of the states is denoted C,, Cd, respectively. For St 1 = US (and Cl = C,) we can repeat the arbitrage described above to find the upper bound of C,: CU I p2[Qc,,
(A-3
+ ( lmQ)’ ud]/rB.
Likewise, Cd2
I32 [QC,u + Cl-
(‘46)
Q)CdWrB.
where C,,, Cud, Cdd, or Cdd, is the option price in period 2 in each of the four possible states of nature. To find the two-period upper bound, we substitute (A.5) and (A6) into (A4): C5p4[Q2C,,+
2Q(l -Q)C,,+
2
(1 -Q)*C&rs
.
(A7)
This is a recursive process in which the n-period upper bound becomes:
cs<
&h-Q)n-jc(u,j)]
,
648)
'B j=O where C(u, j) is the value of the option at expiration when u occurs j times and d occurs n - j times. Recall that Q is a function of the stock transaction cost a and that the option will be exercised in period II if Sddn -j la - X 2 arbitrage process that leads to the upper selling the option. Exercising the option transaction will deliver the underlying S~dn-jla
(the bid exchange rate).
0.
The bound means asset,
first term is the bid currency price. requires purchasing the underlying that the writer who conducted the which could otherwise have been
Thus, the asset and arbitrage sold for
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Proof of Proposition 2 Assume an arbitrage transaction where the investor sells 6’ of the foreign currency at the bid rate, buys one unit of the call option at the option ask price, and lends dollars at the dollar lending rate. The hedge ratio 8, which equates the return in the two states of nature u and d, is derived from the following equation: - G’Sur*Ba + c/p
= - 8’sur*#X - c,/p.
(Bl)
Solving for 6’: 6’ = [(C,-C,)]/[Sr*,(u-d)@].
(B2)
To avoid arbitrage opportunities, the return in both states, u and d, must be non-positive. Thus, in solving this inequality, we derive the following lower bound. c1 [QC, + (1 - e’)cdl/p2rL,
(B3)
where Q’ = ([I~-T*~~cL~]/[~*~(u-&cL~])~~~O~Q’< 1. Continuing this recursive process, the n-period lower bound is derived:
1
C2-
np2n rL
i
[(J)Q’(l
-Q)“mjC(u,j)]
(B4)
j=O
where C(u, j) is the value of the option in period n, when u occurs j times and d occurs n - j times. Note that, in creating the arbitrage transaction that leads to the lower bound, the investor sells ‘short’ the underlying asset and buys ‘long’ the call option. Therefore, if he exercises his option in period 12,he must buy the underlying currency and pay the ask price, S,a. The option will be exercised if dd(” - )’ -‘ct - X 2 0.
APPENDIX
C: MULTINOMIAL
LOWER BOUND
Proof of Proposition 3 Let Rh be the return on the underlying asset in each trading period of length h. Ritchken and Kuo (1988) and Levy and Levy (1991) prove that, in the absence of transaction costs and dividends, the option upper bound will be the binomial formula, with the binomial parameters replaced by the maximum and minimum returns on the underlying asset. In the arbitrage process that leads to the derivation of the upper bound, the underlying asset is purchased, the option is shorted and dollars are borrowed. uh and dh are the maximum and minimum returns, respectively [see equations (8) and (9)]; rBh is the dollar borrowing rate; and r*u is the foreign lending interest rate [equation (lo)]. In addition, there are transaction costs in purchasing foreign currency and the call option [equations (3) and (4)]. Because the upper bound is given by the binomial formula with m trading intervals, we can substitute the maximum and minimum returns and relevant interest rates into inequality (5), which gives the multinomial upper bound presented inequality (14).
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APPENDIX
D: MULTINOMIAL
LOWER BOUND
Proof of Proposition 4 Ritchken and Kuo (1988) prove that, in the absence of transaction costs, the multinomial lower bound will be given by the binomial formula, with the binomial parameters replaced by the two returns of the underlying asset surrounding the risk-free rate (see footnote 6). In the arbitrage process that leads to the lower bound, the option is purchased, the underlying currency is shorted, and dollars are lent. Thus, the relevant dollar risk-free rate is the lending rate, whereas the relevant foreign currency total return should include the foreign borrowing rate and transaction costs. The two returns that surround the dollar lending rate are denoted d*, and Use [equations (ll), (12), (13)]. lo Because the lower bound is the binomial formula, we can substitute d*h and u*h and the relevant interest rates into inequality (6), and get the multinomial lower bound, presented in inequality (15).
NOTES
1. The model can be generalized to any commodity option by replacing the foreign currency interest rates, r*, by r - b, where r is the domestic interest rate and b is the cost of carrying the commodity (see Whaley, 1986, for example). 2. Our model relies on the pricing mechanism developed in Ritchken and Kuo (1988) and Levy and Levy (1991). These studies show that, when the underlying distribution is multinomial, even in the absence of transaction costs, the option value will be bounded within some range. 3. It can be shown by no-arbitrage arguments that @O. Although theoretically it is possible that Q 1 1, it will occur only in cases where transaction costs (a, rb - rl) are extremely high. 4. See Levy (1985), Ritchken (1985), Perrrakis (1986), Ritchken and Kuo (1988), and Levy and Levy (1991). 5. Edirisinghe, Naik, and Uppal (1993) employ a linear programming recursive model that allows the trader to avoid trading every period. 6. For example, assume that the return distribution is multinomial with R = uI, . . ., uk, and Uj 5 r < Uj + 1 where r is the risk-free interest rate. In the absence of dividends costs, the multinomial
and transaction
option lower bound is the binomial price where d*h = Uj and uXh = uj + l
are substituted for the binomial parameters u and d, respectively (Ritchken & Kuo, 1988). 7. Shastri and Tandon (1986) and others find insignificant pricing errors in using the European option model to price american calls on foreign currency. 8. The estimation of implied transaction costs is required since the marginal transaction cost is not observable. 9. Total transaction costs increase with the number of portfolio adjustments. As a result, other things being equal, the larger the number of trading intervals, the higher the upper bound and the lower the lower bound. Also, avoiding trading often increases the upper bound and decreases the lower bound. 10. When the currency (underlying asset) is shorted, it is sold at the bid price, and repurchased at the ask price, and the interest rate is the foreign borrowing rate. Therefore, the total return to the borrower is r*fij, ha*.
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REFERENCES
Boyle, P.P., & Vorst, T. (1992). Option replication in discrete time with transaction costs. Journal of Finance 47,271-294. COX, C.C., & Rubinstein, M. (1985). Options Markets, Englewood Cliffs, NJ: PrenticeHall, Inc. Edirisinghe, C., Naik, V., and Uppal, R. (1993). Optimal replication of options with transactions costs and trading restrictions. Journal of Financial and Quantitative Analysis 28, 117-138. Hodges, S.D., & Neuberger, A. (1990). Optimal replication of contingent claims under transaction costs. Working paper, London Business School. Galai, D. (1990). Comments on valuation and replication of options with transaction costs. Working paper, Hebrew University. Garman, M.S., & Kohlhagen, W.S. (1983). Foreign currency option values. Journal oflnternational Money and Finance 2,23 I-237. Gilster, J. & Lee, W. The effects of transactions costs and different borrowing lending rates on the option pricing model: a note. Journal of Finance 39, 10151222. Leland, H. (1985). Option pricing and replication with transaction costs. Journal of Finance 40,1283-1301. Levy, H. (1985). Upper and lower bounds on the put and call option value: stochastic dominance approach. Journal of Finance 40,1197-1217. Levy, H., & Levy, A. (1991). Option valuation: an extension of the binomial model. Advances in Futures and Options Research 5,49-69. Merton, R.C. (1990). Continuous-Time Finance. 1990 Cambridge, MA: Basil Blackwell. Perrakis, S. (1986). Option pricing bounds in discrete time: extensions and the pricing of the american put. Journal of Business 59, 119-141. Ritchken, P.H. (1985). On option pricing bounds. Journal of Finance 40, 1219-1233. Ritchken, P.H., & Kuo, S. (1988). Option bounds with finite revision opportunities. Journal of Finance 43,301-308. Rubinstein, M. (1976). The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics 7,407-425. Shastri, K., & Tandon, K. (1986). On the use of European models to price American options on foreign currency. Journal of Futures Markets 6,93-108. Toft, K. (1994). Exact formulas for expected hedging error and transaction costs in option replication. Working paper, University of California at Berkeley. Whaley, R.E. (1986). Valuation of American futures options: theory and empirical tests. Journal of Finance 41, 127-150.
SHMUEL HAUSER AND AZRIEL LEVY
In this paper, we develop an arbitrage-free option valuation model in the presence of transaction costs. The model considers the trade-off between the choice of lowering costs by trading less often and the choice of reducing the hedging errors by trading more often. This trade-off allows derivation of a trading interval policy. We illustrate the model with actual transactions data and offer a procedure for the estimation of a trading interval that minimizes hedging errors and transaction costs. The findings suggest that currency options trading is most active in options with short time to expiration, especially if they are at-and out-ofthe-money options.
I. INTRODUCTION In the absence of transaction costs, a single-option equilibrium price may be derived through the creation of a perfectly hedged portfolio replication the option. The ability to create a perfect hedge, the basis for risk-neutral valuation models, including that of Black and Scholes, breaks down in the presence of transaction costs. This breakdown occurs because the construction of a perfect hedge requires continuous adjustments of the portfolio replicating the option, thus increasing total transaction costs to infinity. Several authors (e.g., Glister & Lee, 1984; Leland, 1985; Edirisinghe, Naik, & Uppal, 1993) argue that in order to reduce transaction costs, traders would avoid frequent trading, and that infrequent trading makes it impossible to create a perfectly hedged replicating portfolio, leading to hedging errors. One possible way to deal with these hedging errors is to use a preference-dependent hedging strategy, which raises questions regarding investors’ risk aversion. A second way is Leland’s (1985) approach, in which the hedging strategy depends on transaction costs and on the length of the trading interval. This approach relies on hedging errors’ being uncorrelated with the market and therefore requires assumptions on the pricing of securities in the market. In this paper, we develop an arbitrage model for the valuation of foreign currency options in the presence of transaction costs, which can be used to derive optimal trading interval policy. The model can be accommodated to other contingent claims as well.1 One advantage of the model is that it does not depend on investor preferences. A second advantage is that the Schmuel Hauser Ben Gurion University of the Negev, Beersheva, versity of Jerusalem, Mount Scopus, Jerusalem. l
Israel 84105; Azriel Levy
International Review of Financial Analysis, Vol. 5, No. 2,1996, pp. 145-160 Coptyright 0 1996 by JAI PRESS Inc., All Rights of reproduction in any form reserved.
l
The Hebrew Uni-
ISSN: 10573219
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model does not require any assumptions on the pricing of securities in the market. A third advantage is that the trading interval is an endogenous decision variable. The model extends the models of Boyle and Vorst (1992) and Merton (1990) by allowing the arbitrager to avoid trading every binomial period. Contrary to these models, we argue that when a portfolio replicating the option is not adjusted every binomial period, and as the trading interval becomes longer (i.e., the trading interval is more than one binomial period), to save on costs, hedging errors incur, and the underlying distribution of returns becomes multinomial rather than binomial (see Edirishinghe, Naik, & Uppal, 1993; Toft, 1994). In this case, two counteracting effects will occur. On one hand, lengthening the trading interval reduces transaction costs and thus reduces the range between the upper and lower option bounds. On the other hand, lengthening the trading interval increases hedging errors and therefore increases the range between the upper and lower option bounds.2 Note that given some level of transaction costs, arbitrage profits will be possible when the option price is higher than its upper bound or lower than its lower bound. In this regard, note also that the model developed by Toft evaluates a similar trade-off between transaction costs and the variance of the replicating portfolio. In his model, the hedger is concerned with the distribution of his cash flow at the end of each hedging period. The inflow at each rebalancing point is the hedging error minus the transaction costs. But, whereas Toft’s model is designed to provide an approximation to the evaluation of this inflow, the model suggested in this paper allows one to develop a procedure that evaluates the upper and the lower bounds which minimize hedging errors given the transaction costs. The paper is organized as follows. In Section II, we device the option price bounds in the presence of transaction costs, assuming a binomial distribution and assuming that trading takes place every binomial period. Then, the model is extended to the case where investors may choose not to trade every binomial period, and the distribution is multinomial. In Section III, we illustrate the model. In Section IV, we portray the choice of the optimal trading interval policy. Section V concludes the paper.
II. THE MODEL
Complete Market Call Option Valuation with Transaction Costs We assume that the rate of the underlying currency is binomial in each trading period, and that there are IZtrading periods prior to expiration. For an institutional investor operating in the inter-bank market, transaction costs are made up of the following three components: (1) bid/ask foreign exchange spread; (2) borrowing/lending interest rate spread; and (3) bid/ask option price spread. We employ the following notations: S,, S,, and S, are the ask, bid and middle (geometric average) exchange rate in period t (t = 0, 1, . . .), dollars per foreign currency; rL and rB are one plus local currency lending and borrowing interest rates, respectively; r*L and r*B are one plus foreign currency lending and borrowing interest rates, respectively; X is the strike price of the call option; C,, CtB, and C, are the ask, bid and middle (geometric average) price of the call option in period t(t = 0, 1, . . . ). With these notations a2 = SJS, and p2 = C,/C,, where (a2 - 1) and (p2 - 1) is the percentage difference between the bid and ask prices of the underlying currency and the option, respectively. Thus, the middle price is defined as the geometric average of the bid and ask prices:
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147
Options
s,=(S,Bs,,Y’* c, = cc,,cl,&*
(1) (2)
where bid and ask prices are: SrB = St/a; s,,
= S,a
(3)
CtB = C,/p; C,, = C,p. We further assume that the rate of return of the underlying distributed binomial and that cx and p are constant.
(4) asset, middle exchange rate S,, is a
Proposition 1: Binomial Upper Bound An arbitrage transaction in which an investor purchases 6 units of foreign currency at the ask rate, sells short one unit of the option at the bid price, and finances the transaction by borrowing dollars, yields the following option price upper bound:
c
i
[;( )e’(1 -
(5)
Q)“-j C(U, j)]
r B j=O
where C(u,j) is the value of the option at expiration, j times, and
when u occurs j times, d occurs n -
Q = ([a2rB-r*Ld]/[r*L(u-d)])whereOSQI
Proof: See Appendix
1.3
A.
Proposition 2: Binomial Lower Bound An arbitrage transaction by an investor who sells 6’ units of the foreign currency at the bid rate, buys one unit of the call option at the option ask price and lends dollars at the option ask price and lends dollars at the dollar lending rate, yields the following option price lower bound: c2-
,’ 2n
j f0
[(y)Q(l
-Q)“-’
C(u,j)]
(6)
rL p
where C(u, j) is the value of the option in period n, when u occurs j times and d occurs 12-j times, and Q’ = [ rL - r*Bdcx2]/[r*B(u Proof: See Appendix
- d)a*]
where 0 $ Q I 1.
B.
Propositions 1 and 2 indicate that in the presence of transaction costs, the efficient market price is bounded; whereas, in the absence of transaction costs, the upper and lower bounds coincide and a single equilibrium price is derived. In both cases, in equalities (5) and (6) represent preference-free strategies derived on the assumption that arbitrage opportunities are not available. For a further explanation, we elaborate on the upper bound. Suppose there are two market makers that compete in the market by selling (writing) a call option on the same underlying asset and that both market makers are subject to transaction costs. Suppose further that the transaction costs of market maker No. 1 are lower than those of market maker No. 2. Accord-
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ing to our model, market maker No. 1 will find it optimal to trade more often than market maker No. 2, and his/her option upper bound will be lower than the upper bound of market maker No. 2. Since the two market makers compete with each other, the equilibrium ask price in the market will be set by market maker No. 1 somewhere in the range between his/ her upper bound and the upper bound of market maker No. 2. If market maker No. 2 sells the option for a price lower than his/her upper bound he/she will incur a loss where as market maker 1 will profit as long as he sells the option for a price higher than his/her upper bound. If there are many market makers competing in the market, the ask price will be forced to come down to a level close (or even equal) to the upper bound of market maker No. 1. This equilibrium price is preference free since, in its derivation, we use only arbitrage considerations and do not make any assumption on the preferences of market participants.
Incomplete Market Call Option Valuation with Transaction Costs Continuous adjustment of the portfolio replicating the option can be very costly in the presence of transaction costs. In order to reduce transaction costs, an arbitrager will trade in discrete time intervals, rather than continuously (see Leland, 1985). In this section, we use the binomial model as analogous to the continuous model (e.g., Wiener process) and derive a trading policy and an arbitrage-free option price in the presence of transaction costs. It is well known4 that, in an incomplete market, investors cannot construct a perfect hedge, and the equilibrium arbitrage-free price of the option is bounded within some range, depending on the assumptions on the distribution of the underlying asset. In the presence of transaction costs, there are two basic approaches to the pricing of potions. The first is that taken by Merton (1990, Chapter 14 p. 432). Merton uses the most conservative estimate of transaction costs, by considering an intermediary in the market that liquidates his position at the end of each trading period and then opens up a new position. Thus, the option prices obtained are arbitrage-free bounds of the bid-ask option prices, that take into account the worst possible scenario from the point of view of the intermediary. In the second approach, Boyle and Vorst (1992) and Edirisinghe, Naik, and Uppal (1993) and Toft (1994) reduce transaction costs by avoiding total liquidation of the trader’s position. Both approaches are an extension of the binomial model, in the sense that, in each trading interval, there are only two possible states of nature. In this paper, we follow Merton’s approach, by considering the maximum transaction costs in each trading interval and by assuming that initially the return distribution of the underlying asset is binomial. Specifically, we assume that the distribution of the underlying asset is binomial and that each binomial period is known in advance. Figure 1 gives the distribution of returns if the trader trades every binomial period. However, if the trader refrains from trading every binomial period, but decides to trade every two binomial periods (periods 0,2, and 4 in Figure l), the distribution of returns in these periods is presented in Figure 2, and is derived directly from the binomial distribution, which is the “true” distribution of the underlying security. Note that the assumption made in this paper differs from Ritchken (1988) and Levy (1985) since they assume that the underlying distribution in each period is multinomial. In our model, the underlying distribution is binomial, but it becomes multinomial if the trader avoids trading every binomial period to reduce his/her transaction costs. Thus, in contrast with Merton (1990), Boyle and Vorst (1992) and Toft (1994), we do not constrain the trader to trade every binomial period, and recognize that, by increasing the trading travel, the distribution of the underlying asset returns becomes multinomial.5 As we have seen, for multinomial distributions, even without transaction costs, there will not exist a sin-
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gle arbitrage-free option price, but rather the price will be bounded within a range. Thus, if the trading interval is prolonged, two opposing effects will take place: (1) the upper and lower bounds will approach each other, as a result of the reduction in transaction costs, and (2) the upper and lower bounds will deviate from each other, as a result of the increase in the number of states of the distribution, leading to hedging errors. Given this trade-off we derive a trading policy in which the investor chooses a trading interval that enables him/her to exploit available arbitrage opportunities in the presence of transaction costs. At the end of every trading interval, the investor may reassess the position and choose a different trading strategy, depending on the trade-off between transaction costs and hedging errors. We consider an optimal trading interval as one that minimizes the upper bound, or maximizes the lower bound, since this strategy minimizes the cost of writing the option or maximizes the proceeds that finance the purchase of the option. The optimal trading interval of the upper bound need not be equal to that of the lower bound. To proceed now to the derivation of the option bounds in the presence of transaction costs, it is assumed that there are n binomial trading periods until the option expires. In order to reduce transaction costs, the investor may choose to increase the trading interval; i.e., to refrain from trading every binomial period. Denote by h the length of the actual trading period chosen by the investor. Thus, the number of actual trading periods prior to expiration, denoted HZ,will be: m = nlh
(7)
Figure 1 illustrates the case of n = 4 binomial periods. By choosing to trade every two (h = 2) binomial periods, the investor trades m = n/h = 2 times, and the distribution of returns in each trading period will be trinomial, as shown in Figure 2. In general, the distribution of returns
U4
u3d
u2d2
ud
d4
Binomial
Period
I
I
I
1
I
0
1
2
3
4
Figure 1.
Binomial
Distribution
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“‘d2
ud
3
d4
I Multinomial
Period
I
I
I
2
1
0
Figure 2.
Multinomial
I
Distribution
in each trading period is multinomial, with 2h states (some of the states coincide and the number of unique states is only h + 1). To derive the multinomial upper and lower option bounds, we use the,following notations: d,, = dh
(8)
“,, = uh
(9)
Uh and dh are the maximum and minimum returns of the underlying asset, over trading interval h. ‘Lh, rBh, reLh, r*Bh, are on plus the lending and borrowing interest rates on local and foreign currencies, respectively, in the trading interval h. Thus, rLh = rLh; rBh = r& r*Lh = r*Lh; r*gh = r*#.
(10)
Denote by Ri, h the multinomial distribution return, when each trading interval is composed of h binomial periods (i = 1, . . . . h +l). Define U*h, d*,, such that, d*h = Rj, h
(11)
U*/,=R. I+ l> h
(12)
and, r*,,Rj,ha2
5 r,h < r*Bh Rj + ,, ha’.
(13)
Proposition 3: Multinomial Upper Bound When the length of the trading interval is of size h, the option price upper bound is given by,
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(14)
where Qh = [a*r,h - r*Lhdh]l[r*lh(uh expiration if Uh occurs j times and dh Proof: See Appendix
- dh)] and c(u*, j) is the value of the call option at occurs
m
-j
times.
C.
To elaborate on the argument that the lower the transaction costs, the lower the option upper bound, recall that the transaction costs are embedded in the terms a and p. For example, the option upper bound presented in equation (14) is a function of both a and p. It can be readily seen that a decrease in p results in a decrease in the upper bound. The effect of CCon this upper bound is a bit more complicated. Technically, this effect is similar to the effect of a decrease in the riskless interest rate in the binomial option valuation formula. To see this, note that the right side of equation (14) is analogous to the standard binomial option valuation formula. Thus, in this formula, an increase in the riskless interest rate or an increase in U. increases the option price. An increase of the same magnitude in (a*) will have a greater effect on the option price since it affects only Qh, but has no impact on the discounting term ll(r,,)m. Note that in our simulations the effect of p was found to be negligible in comparison with the effect of a. Proposition 4: Multinomial Lower Bound When the length of the trading interval is of size h, the option price lower bound is given by: (15)
where Qw = [rLh - r*Bhd*ha2]/[r*Bh(u*h -d*,)cc2], C(u*h. j) is the value of the option at expiration if U*h occurs j times and d*h occurs m -j times and m is the number of trading intervals to expiration.6 Proof: See appendix D. In the upper bound case, each trader with given transaction costs computes the upper bound in equation (14) for various trading intervals h. Given transaction costs of size a and p, the trader chooses the trading interval that minimizes the upper bound. The trading interval that minimizes the upper bound, as a function of cc and p, is denoted by /~,(a, p). Even if the option price is only slightly higher than this upper bound, one may construct a portfolio generating arbitrage profits. These profits will be realized only when trading is conducted every h,(a, B) periods. To prevent arbitrage profits, the option price must be below this upper bound. The lower the transaction costs, the lower this upper bound (see Figure 3). Likewise, there will be arbitrage profits if the option price is below the lower bound, which is the maximum lower bound for all trading intervals h. The trading interval that maximizes the lower bound [inequality (15)], as a function of a and p, is denoted by h,(ol, p). If the option price is below the lower bound obtained by trading every !~,(a, p) binomial periods, one may construct a portfolio that will generate arbitrage profits. The lower the transaction costs, the
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____----
I :
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II : = 1.0001 Ill : \r 2
_______----------__-Ill t
n
1.0002
Black $ Scholes .___________________________,
-
I
7
8
9
higher the lower bound. The equilibrium option upper and lower bounds are determined by tradershaving est transaction costs and the ability to trade most often. Thus, a necessary condition librium is that the option price is within these upper and lower bounds.
the lowfor equi-
1
I
I
I
2
3
4
I
I
1
I
OL
5 6 Trading Interval
Figure 3. Upper and Lower Price Bounds (S=X=100;Sig=.15;r~r*=.1)
III. ILLUSTRATION
OF THE MODEL
The purpose of this section is to illustrate the valuation model presented in Section II, allowing for transaction costs and allowing the arbitrager to choose the appropriate trading interval. For simplicity, we assume that all transaction costs are embedded in the bid-ask spread of buying the underlying currency (i.e., p = 1). We also define the upper bound as the maximum option price that prevents arbitrage, given transaction costs a and trading interval h, and the lower bound as the minimum option price that prevents arbitrage, given transaction costs of level o? and trading interval h. The model is illustrated using the following parameters: S = 100; X = 100; T = .25; n = 180; h= 1,2, . ..) 9; and annualized domestic and foreign interest rates are 10% (rB = rL and Fan = r*& In order to compare our model with the Black-Scholes model, we follow Cox and Rubinstein’s approach (1985, p. 200) by setting u = exp(oddt), d =1/u, and s = .15. We calculate the upper and lower bounds of the currency call option price for different transaction costs levels a and different trading intervals, where h = 1 implies that the arbitrager trades every binomial period. Note that, if portfolio revision takes place every binomial period, the assumed underlying distribution may serve as a proxy to a Wiener process. In this case, in the absence of transaction costs (a = l.OOOO), our model converges to the BlackScholes option valuation model. Note also that the binomial option valuation model is
Pricing of Foreign Exchange Options
153
viewed in the literature as a proxy of the Black-Scholes model, and does not depend on the number of binomial trading periods n (for relatively large n). With the introduction of transaction costs, the assumption regarding n becomes crucial. Assuming that the arbitrager trades every binomial period, and given some level of transaction costs ~1, the spread between the upper and lower option bound will be greater the higher is n. Figure 3 portrays the upper and lower bounds as a function of the trading interval h for various levels of transaction costs, denoted a. As expected, the higher the transaction cost, the higher the upper bound and the lower the lower bound, for every trading interval of size h. When a = 1.0000 (zero transaction costs) and h = 1 (trading is conducted every binomial period), the upper and lower bounds coincide (they are approximately equal to the BlackScholes price), and the trader can create a perfect hedge by revising the option replicating portfolio, every binomial period. also, when a = 1.OOOOand h > 1, the efficient price according to our model becomes bounded. Thus, as h increases, the upper bound increases and the lower bound decreases. Namely, the less often the trader trades, the higher the upper bound and the lower the lower bound. Evidently, in the absence of transaction costs, it is optimal to trade every binomial period, and the Black-Scholes price is the equilibrium option price. For a > 1, trading every binomial period incurs the greatest amount of transaction costs, and it may be worthwhile to trade less often. The trading interval that minimizes the upper bound or maximizes the lower bound may be greater than 1.O. For example, it is seen in Table 1 that when a - 1.0002 (bid-ask spread of 0.04%), the minimum upper bound is attained approximately at h = 3, namely when trading takes place every three binomial periods. The transaction costs effect outweighs the hedging ability effect up to a trading interval of size h = 3. If h > 3, the hedging effect becomes more crucial, and it does not pay to trade less often than every three binomial periods. As cx increases, the trading interval h that minimizes the upper bound increases, and the trading interval that maximizes the lower bound increases as well. However, for a given level of transaction costs, the upper-bound trading interval need not be equal to the lower bound trading interval.
IV. THE CHOICE OF TRADING INTERVAL An interesting issue that has not been addressed in the literature is how often has trading actually been conducted, given transaction costs. In this section, we suggest a policy of trading frequency. The choice of trading interval is illustrated via estimation of implied trading interval. The data used for the illustration are observations on German marks currency options and exchange rates compiled by the Philadelphia exchange, from August 1983 to December 1984. Euro-dollar and Euro-mark interest rates were collected from the London Financial Zmes.In the sample period, interest rates in Germany were much lower than interest rates in the U.S., permitting the use of European call option valuation models to price american call options. This is because, in such cases the probability of early exercise is practically zero, and the pricing errors of European currency call option models are insignificant.7 Volatility of the $/DM exchange rate is estimated by historical annualized logarithmic returns and is used as an input in estimating the binomial parameters u and d (see Cox & Rubinstein, 1985). For each observation, we compute the theoretical Black-Scholes option price (without transaction costs) and compare it with the observed market price. If the observed market price is higher than the Black-Scholes price, we equate the upper bound with the market price in order to derive the implied trading interval (hereinafter, ITI):
154
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C mkt = CUB,,,
OF FINANCIAL
ANALYSIS
/Vol.
5(2)
(. , ITI),
where C,,,b is the observed option market price, CIJB,,d is the theoretical minimum upper bound call option price, as a function of the spot rate, exercise price, domestic and foreign interest rates, u and d, transaction costs, and implied trading interval (ITI). Likewise, if the observed price is lower than the Black-&holes price, we equate the lower bound with the market price in order to derive the implied trading interval, = CLB,,,
c mkt
(. , ITI),
where CLB,,,~~is the theoretical maximum lower bound call option price. Figure 4 illustrates the estimation procedure with the option upperbound. We first calculate the upper bound option price for a given level of transaction costs, for various binomial trading intervals (e.g., curve I in Figure 4). Second, we identify the trading interval that minimizes this upper bound (e.g., point F in Figure 4). Third, we repeat this process for different levels of transaction costs and draw a curve connecting these minimum points (curve UU in Figure 4). This curves the option upper bound (CUB,,d), given transaction costs a. Finally, we solve for the implied trading interval by equating the observed market call price (C,,& with CUB,,d (the intersection point of UU and Cmkt in Figure 4). Note that with this procedure the actual transaction costs [are not required because we simutaneously estimate IT1 and the implied transaction costs] (hereafter, ITC).* Assuming there are no arbitrage opportunities, these implied transaction costs cannot be higher than the transaction costs of any market participant. A similar procedure is used to derive the IT1 when Cmkr is lower than the Black-Scholes call price. Because the upper and lower bounds depend on the assumption of the number of binomial periods prior to expiration, we estimate ITC and IT1 for various 11’s (i.e., for various Call upper
bound
11 I II III IV v
uu
: Iyn1.0000 : Mn1.00005 :cI=1.0001 : c(n1.0002 : a n1.00025
Call Market J 3 1
Black 8 Scholes
I
I
I
I
2
3
4
5
Trading
interval
Figure 4. An Illustration of Estimation (S=X=100;T=0.25;r=r*=0.1;n=180)
Price
Procedure
6
Pricing
of Foreign
Exchange
Options
155
length of binomial periods). Recall that the assumption on n is tantamount to the assumption as to how often the arbitrager should trade in the continuous model in order to create a perfect hedge in the absence of transaction costs. Given that there are D trading days to expiration, II will depend on the assumption regarding the number of binomial periods per day, denoted I (n = D . I).
Table 1. Average Trading Interval and Implied Transaction Costs in German Marks Currency Optionsa Number of binomial periods (n) assuming I daily revisionsb I = 4 times a day ITI All
1001
By Boundary In
Status
517
out
383
90
ITC
ITILTCC
0.482
1.163
0.685
0.241
0.581
0.685
(0.019)
(0.03 1)
(0.004)
(0.010)
(0.03 1)
0.278
0.484
1.664
0.139
0.242
1.664
(0.008)
(0.046)
(0.243)
(0.004)
(0.023)
(0.243)
0.311
0.744
0.730
0.155
0.377
0.730
(0.005)
(0.018)
(0.046)
(0.003)
(0.009)
(0.046)
0.757
1.885
0.403
0.379
0.943
0.403
(0.016)
(0.004)
(0.008)
(0.008)
(0.002)
(0.008)
54.7
By Maturity
45
ITI
(0.009)
Fd
T<45
ITUITCC
(In-, At-, and Out-of-the-Money) 88
At
ITC
I = 8 times a day
54.7
(in Days) 63 262 676
0.219
1.471
0.549
0.109
0.736
0.549
(0.014)
(0.078)
(0.177)
(0.007)
(0.039)
(0.177)
0.437
1.476
0.474
0.219
0.738
0.474
(0.011)
(0.041)
(0.057)
(0.005)
(0.021)
(0.057)
0.518
1.016
0.779
0.259
0.508
0.779
(0.009)
(0.020)
(0.039)
(0.005)
(0.010)
(0.039)
8.3
Fd
8.3
By Upper and Lower Bounds Upper Lower
Fd Notes:
103 898
0.274
0.447
1.909
0.137
0.224
1.909
(0.007)
(0.041)
(0.249)
(0.004)
(0.021)
(0.249)
0.506
1.245
0.545
0.253
0.623
0.545
(0.010)
(0.02 1)
(0.021)
(0.005)
(0.01 1)
(0.011)
173.9
173.9
a Numbers in parentheses are standard errors. b n = D * I is the assumed number of binomial periods until maturity, where D is the number of days to maturity and 1 is the number of binomial periods per one day. c ITI is the implied trading interval in days. ITC is the implied bid-ask spread (~(CX- 1)) of exchange rate as a proportion of the level of the exchange rate, in basis points. ITUITC is implied trading interval per one unit of transaction costs measured by the bid-ask spread. d One way analysis of variance is used to test the hypothesis H,: I.lt = pL2= = pi vs. H,: at least two means are not equal.
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Table 1 presents ITC and IT1 for I = 4, 8. The first notable result is that as n increases (or I increases) ITC and IT1 decrease. The explanation of this result is that, for some level of transaction costs a, the lower is n (or 1), the lower is the upper bound and the higher is the lower bound.9 Thus, given the observed option price, which is taken as the upper or lower bound in our model, IT1 must be higher the smaller is the number of binomial periods, in order to induce the same option bounds. The problem with the estimation of IT1 and ITC is that they depend on IZ, the exact number of binomial trading intervals that approximate the continuous Wiener process, and we do not know a priori the value of II. However, both ITC and IT1 decrease proportionally as the number of assumed daily binomial trades increases; i.e., the ratio ITI/ITC is constant and is independent of 1 (or of n, given D). Thus, we observe in Table 1 that when 1 = 4, both ITC and IT1 are twice as high as when 1 = 8. This finding allows us to circumvent the difficulty of finding the correct length of the binomial trading interval that serves as a proxy for a continuous process. The ratio ITI/ITC, is the trading interval in terms of a trading day per ne unit of transaction costs; i.e., its reciprocal is the implied number of trades per day. For example, if ITI/ITC = S, the implied trading strategy is to trade twice a day, per one unit of transaction costs, and if transaction costs are actually twice as high, our model predicts that arbitragers will actually trade once a day. Note that, give ITI/ITC and the correct proportional transaction costs, one could determine not only the implied trading interval, but also the number of binomial trades per day, denoted 1, that approximate the continuous process. The results in Table 1 also indicate that the shorter the time to maturity and the more outof-the-money the option, the smaller the implied trading interval per one unit of implied transaction costs (ITUITC). These results suggest that options trading is most active in options with short time to expiration (about two portfolio revisions per one day in short-term options compared with only a little more than one portfolio revision in long-term option), especially if they are at-or out-of-the-money options.
V. CONCLUSIONS
In the presence of transaction costs, traders will try to avoid trading very often to minimize their costs. However, in doing so, it becomes impossible for them to create a perfectly hedged portfolio replicating the option, and there will be hedging errors. Thus, the arbitrager faces a trade-off between the choice of lowering costs by trading less often and the choice of reducing the hedging errors by trading more often. We developed an arbitragefree model for the valuation of foreign currency options (or any other contingent claim) in which this trade-off prevails. The model enables us to derive a trading interval policy. We illustrated the model with actual transactions data and offered a procedure for the estimation of a trading interval that minimizes hedging errors and transaction costs. The findings suggest that options trading is most active in options with short time to expiration, especially if they are at- or out-of-the-money options.
ACKNOWLEDGMENT This paper was partially supported by the Krueger Fund.
Pricing
157
of Foreign Exchange Options
APPENDIX A: BINOMIAL UPPER BOUND Proof of Proposition 1 Assume that an arbitrager who purchases an amount 6 of the foreign currency at the ask rate, sells short one unit of the option at the bid price, and finances the transaction by borrowing dollars. The hedge ratio 6, will yield the same return in the two states of nature (U and 6> if the following condition is met: GSur*,/a
- CUP = GSdr*,/a
- C&3,
(Al)
Where C, and Cd are the value of the middle call option price next period, in states u and d, respectively. Solving for 6 we get: 6 = ([aP(C, - Cd)l/[Sr*L(u
- d)l).
(A2)
To prevent arbitrage, the return in states u and d must be non-positive: GSur*L/a
- CUP - (Ma
- C/P)rB
IO,
where the second term on the left side of (A3) is the principal loan. Substitute (A2) into (A3) and solve for C,
(A3)
and interest on the financing
CI B21Qc, + ( l-Q)Cd]/rB where Q = [a2r,‘*,
d ]/ [ r*L (u - d)]
and 0 I Q 2 1. Equation
644)
(A4) enables us to the derive
the n period upper bound. In period 1, there are two possible states of nature: St, = US; St2 = dS
where S is the initial price of the underlying currency. The middle price of the call option in each of the states is denoted C,, Cd, respectively. For St 1 = US (and Cl = C,) we can repeat the arbitrage described above to find the upper bound of C,: CU I p2[Qc,,
(A-3
+ ( lmQ)’ ud]/rB.
Likewise, Cd2
I32 [QC,u + Cl-
(‘46)
Q)CdWrB.
where C,,, Cud, Cdd, or Cdd, is the option price in period 2 in each of the four possible states of nature. To find the two-period upper bound, we substitute (A.5) and (A6) into (A4): C5p4[Q2C,,+
2Q(l -Q)C,,+
2
(1 -Q)*C&rs
.
(A7)
This is a recursive process in which the n-period upper bound becomes:
cs<
&h-Q)n-jc(u,j)]
,
648)
'B j=O where C(u, j) is the value of the option at expiration when u occurs j times and d occurs n - j times. Recall that Q is a function of the stock transaction cost a and that the option will be exercised in period II if Sddn -j la - X 2 arbitrage process that leads to the upper selling the option. Exercising the option transaction will deliver the underlying S~dn-jla
(the bid exchange rate).
0.
The bound means asset,
first term is the bid currency price. requires purchasing the underlying that the writer who conducted the which could otherwise have been
Thus, the asset and arbitrage sold for
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ANALYSIS
/ Vol. S(2)
LOWER BOUND
Proof of Proposition 2 Assume an arbitrage transaction where the investor sells 6’ of the foreign currency at the bid rate, buys one unit of the call option at the option ask price, and lends dollars at the dollar lending rate. The hedge ratio 8, which equates the return in the two states of nature u and d, is derived from the following equation: - G’Sur*Ba + c/p
= - 8’sur*#X - c,/p.
(Bl)
Solving for 6’: 6’ = [(C,-C,)]/[Sr*,(u-d)@].
(B2)
To avoid arbitrage opportunities, the return in both states, u and d, must be non-positive. Thus, in solving this inequality, we derive the following lower bound. c1 [QC, + (1 - e’)cdl/p2rL,
(B3)
where Q’ = ([I~-T*~~cL~]/[~*~(u-&cL~])~~~O~Q’< 1. Continuing this recursive process, the n-period lower bound is derived:
1
C2-
np2n rL
i
[(J)Q’(l
-Q)“mjC(u,j)]
(B4)
j=O
where C(u, j) is the value of the option in period n, when u occurs j times and d occurs n - j times. Note that, in creating the arbitrage transaction that leads to the lower bound, the investor sells ‘short’ the underlying asset and buys ‘long’ the call option. Therefore, if he exercises his option in period 12,he must buy the underlying currency and pay the ask price, S,a. The option will be exercised if dd(” - )’ -‘ct - X 2 0.
APPENDIX
C: MULTINOMIAL
LOWER BOUND
Proof of Proposition 3 Let Rh be the return on the underlying asset in each trading period of length h. Ritchken and Kuo (1988) and Levy and Levy (1991) prove that, in the absence of transaction costs and dividends, the option upper bound will be the binomial formula, with the binomial parameters replaced by the maximum and minimum returns on the underlying asset. In the arbitrage process that leads to the derivation of the upper bound, the underlying asset is purchased, the option is shorted and dollars are borrowed. uh and dh are the maximum and minimum returns, respectively [see equations (8) and (9)]; rBh is the dollar borrowing rate; and r*u is the foreign lending interest rate [equation (lo)]. In addition, there are transaction costs in purchasing foreign currency and the call option [equations (3) and (4)]. Because the upper bound is given by the binomial formula with m trading intervals, we can substitute the maximum and minimum returns and relevant interest rates into inequality (5), which gives the multinomial upper bound presented inequality (14).
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Pricing of Foreign Exchange Options
APPENDIX
D: MULTINOMIAL
LOWER BOUND
Proof of Proposition 4 Ritchken and Kuo (1988) prove that, in the absence of transaction costs, the multinomial lower bound will be given by the binomial formula, with the binomial parameters replaced by the two returns of the underlying asset surrounding the risk-free rate (see footnote 6). In the arbitrage process that leads to the lower bound, the option is purchased, the underlying currency is shorted, and dollars are lent. Thus, the relevant dollar risk-free rate is the lending rate, whereas the relevant foreign currency total return should include the foreign borrowing rate and transaction costs. The two returns that surround the dollar lending rate are denoted d*, and Use [equations (ll), (12), (13)]. lo Because the lower bound is the binomial formula, we can substitute d*h and u*h and the relevant interest rates into inequality (6), and get the multinomial lower bound, presented in inequality (15).
NOTES
1. The model can be generalized to any commodity option by replacing the foreign currency interest rates, r*, by r - b, where r is the domestic interest rate and b is the cost of carrying the commodity (see Whaley, 1986, for example). 2. Our model relies on the pricing mechanism developed in Ritchken and Kuo (1988) and Levy and Levy (1991). These studies show that, when the underlying distribution is multinomial, even in the absence of transaction costs, the option value will be bounded within some range. 3. It can be shown by no-arbitrage arguments that @O. Although theoretically it is possible that Q 1 1, it will occur only in cases where transaction costs (a, rb - rl) are extremely high. 4. See Levy (1985), Ritchken (1985), Perrrakis (1986), Ritchken and Kuo (1988), and Levy and Levy (1991). 5. Edirisinghe, Naik, and Uppal (1993) employ a linear programming recursive model that allows the trader to avoid trading every period. 6. For example, assume that the return distribution is multinomial with R = uI, . . ., uk, and Uj 5 r < Uj + 1 where r is the risk-free interest rate. In the absence of dividends costs, the multinomial
and transaction
option lower bound is the binomial price where d*h = Uj and uXh = uj + l
are substituted for the binomial parameters u and d, respectively (Ritchken & Kuo, 1988). 7. Shastri and Tandon (1986) and others find insignificant pricing errors in using the European option model to price american calls on foreign currency. 8. The estimation of implied transaction costs is required since the marginal transaction cost is not observable. 9. Total transaction costs increase with the number of portfolio adjustments. As a result, other things being equal, the larger the number of trading intervals, the higher the upper bound and the lower the lower bound. Also, avoiding trading often increases the upper bound and decreases the lower bound. 10. When the currency (underlying asset) is shorted, it is sold at the bid price, and repurchased at the ask price, and the interest rate is the foreign borrowing rate. Therefore, the total return to the borrower is r*fij, ha*.
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/Vol. S(2)
REFERENCES
Boyle, P.P., & Vorst, T. (1992). Option replication in discrete time with transaction costs. Journal of Finance 47,271-294. COX, C.C., & Rubinstein, M. (1985). Options Markets, Englewood Cliffs, NJ: PrenticeHall, Inc. Edirisinghe, C., Naik, V., and Uppal, R. (1993). Optimal replication of options with transactions costs and trading restrictions. Journal of Financial and Quantitative Analysis 28, 117-138. Hodges, S.D., & Neuberger, A. (1990). Optimal replication of contingent claims under transaction costs. Working paper, London Business School. Galai, D. (1990). Comments on valuation and replication of options with transaction costs. Working paper, Hebrew University. Garman, M.S., & Kohlhagen, W.S. (1983). Foreign currency option values. Journal oflnternational Money and Finance 2,23 I-237. Gilster, J. & Lee, W. The effects of transactions costs and different borrowing lending rates on the option pricing model: a note. Journal of Finance 39, 10151222. Leland, H. (1985). Option pricing and replication with transaction costs. Journal of Finance 40,1283-1301. Levy, H. (1985). Upper and lower bounds on the put and call option value: stochastic dominance approach. Journal of Finance 40,1197-1217. Levy, H., & Levy, A. (1991). Option valuation: an extension of the binomial model. Advances in Futures and Options Research 5,49-69. Merton, R.C. (1990). Continuous-Time Finance. 1990 Cambridge, MA: Basil Blackwell. Perrakis, S. (1986). Option pricing bounds in discrete time: extensions and the pricing of the american put. Journal of Business 59, 119-141. Ritchken, P.H. (1985). On option pricing bounds. Journal of Finance 40, 1219-1233. Ritchken, P.H., & Kuo, S. (1988). Option bounds with finite revision opportunities. Journal of Finance 43,301-308. Rubinstein, M. (1976). The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics 7,407-425. Shastri, K., & Tandon, K. (1986). On the use of European models to price American options on foreign currency. Journal of Futures Markets 6,93-108. Toft, K. (1994). Exact formulas for expected hedging error and transaction costs in option replication. Working paper, University of California at Berkeley. Whaley, R.E. (1986). Valuation of American futures options: theory and empirical tests. Journal of Finance 41, 127-150.