A P.D.E. approach to Asian options: analytical and numerical evidence

Journalof BANKING & ELSEVIER

Journal of Banking & Finance 21 (1997) 613-640

FINANCE

A P.D.E. approach to Asian options: analytical and numerical evidence Brnrdicte Alziary a, Jean-Paul Drcamps b,*, Pierre-Franqois Koehl c a CEREMATH, Universit£ de Toulouse 1, Manufacture des Tabaes, Bat. F., 21 All~e de Brienne, F-31000 Toulouse, France b GREMAQ, Universit£ de Toulouse L Manufacture des Tabacs, Bttt. F., 21 All~e de Brienne, F-31000 Toulouse, France CREST-ENSAE, Timbre J 120, 3 Avenue Pierre Larousse, F-92241 Malakoff Cedex, France

Received 15 January 1996; accepted 28 September 1996

Abstract We first derive a one-state-variable partial differential equation, easy to implement, which characterizes the price of a European type Asian option. This result is explained and related to previous literature. We then derive new results on the hedging of an Asian option and propose analytical and numerical analysis on the comparison between Asian and European options. Our methodology which applies to "fixed-strike" Asian options as well as to "floating-strike" Asian options completes and clarifies various results in the literature. In this paper we focus on "backward-starting" Asian options. Our approach is

* Corresponding author. Tel.: +33.5.61.12.85.54 - - Fax: +33.5.61.22.55.63 - - email: decamps @gremaq.univ-tlse 1.fr The authors would like to thank Nicole E1 Karoui, Helyette Geman, Monique Jeanblanc-Picqu~, Ali Lazrak, Boris Leblanc, Eric Renault, Jean-Charles Rochet, Nizar Touzi, and an anonymous referee for helpful comments. We also thank the participants at the CREST and GREMAQ Finance seminar, at the 7th WCES, Tokyo, August 1995 and at the 14th LAMES, Rio, August 1996. The usual disclaimer applies. 0378-4266/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S03 7 8 - 4 2 6 6 ( 9 6 ) 0 0 0 5 7 - X

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quite general however, and we explain how to adapt our main results to the case of "forward-starting" Asian options. JEL classification: GI3 Keywords: Options; Asian options; Pricing; Hedging

1. Introduction

Derivative securities with more complicated payoffs than standard European or American options are becoming increasingly important in the field of finance. Some of the new products referred to as exotic options bring solutions to complex financial problems. Among exotic options, Asian options have been particularly considered by practitioners. The purpose of the present paper is to propose new analytical and numerical results on valuation and hedging of Asian options. Asian options are options in which the payoff depends on the arithmetical average price of the underlying asset during some part of the life of the option. See, for instance, Vorst (1996) for a review of the literature on Asian options. No general explicit pricing formulae are available for Asian options. This is because the distribution of the arithmetic average of a set of lognormal distributions is not explicit. We can roughly consider that there are two groups of studies on the valuation of Asian options. A first group of papers is centered around the idea that the distribution of the geometric average of a set of lognormal distribution is also lognormal. Therefore, Asian options on geometric average (uncommon in practice) are analytically valuable and may provide some information on arithmetical average options. Along this line Kemna and Vorst (1990) and Conze and Viswanathan (1991) obtain an explicit pricing formula for Asian options on geometric average. Bouaziz et al. (1994) use a simple linearization procedure and propose an approximate closed-form solution to the pricing of "floating-strike" Asian options. LEvy (1992) approximates the distribution of an arithmetic average by a lognormal distribution. A second group of papers deals directly with the arithmetic average. Caverhill and Clewlow (1990) use the fast Fourier transform to obtain numerical approximations of the price of Asian options. Kemna and Vorst (1990) [eq. (16), p.117] provide an explicit pricing formula [the same result lies in Geman and Yor (1993); eq. (3.7), p.361] in the particular case where the probability to exercise the Asian call is equal to one. Then, these authors propose, in the general case, Monte Carlo simulations to study Asian option prices. Geman and Yor (1993) propose an analytical study of Asian options. In particular, they characterize the case where an Asian call option price is higher than a standard European call option price. The present paper belongs to the second group: We directly deal with the arithmetic average, and thus work on an exact pricing formula. We propose a one-state-variable partial differential equation (P.D.E.) which characterizes the

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price of an Asian option and we provide quantitative and qualitative new results on the hedging of an Asian option. One important contribution of our P.D.E. approach, with respect to the previous literature, is to allow a detailed comparison between Asian and European options. Moreover our methodology is quite general and applies to all the various types of Asian options. Our paper is organized as follows: Section 2 defines the general framework and specifies the various types of Asian options that we consider. This section ends with a first result showing the relationship between "fixed-strike" Asian options, "floating-strike" Asian options and standard European options. Section 3 presents our P.D.E. approach and its application to the hedging: (i) we derive a one-statevariable P.D.E. which characterizes the price of an Asian option (this result is explained and related to the literature); and (ii) we prove general static results as monotonicity in the various parameters of the pricing model and derive results on the hedging of Asian options. Section 4 focuses on the comparison between Asian and European options in an analytical point of view. We deal with the prices as well as with the delta and the elasticity of Asian and European options. Section 5 is devoted to the numerical implementation of our approach. In particular, we explain how to adapt the classical finite-difference method to our problem. Section 6 presents the results of the numerical implementation. We focus on the differences between Asian and European options and propose an analysis of the various effects which interact in the pricing of an Asian option. Our conclusions are in Section 7.

2. The general framework We consider in this paper a continuous time economy on a bounded time interval [0,T]. Uncertainty is represented by a probability space ( ~ ' , J , ~ ) . The flow of information is represented by a complete continuous filtration ( ~ ) 0 _
where Wt is a Q Brownian motion.

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(ii) The price of any option is the discounted expectation of its terminal payoff (at date T) under Q. Using this general framework, we consider two different types of Asian options written on the stock S and exercisable at date T, the time interval over which the average value of the stock is calculated is defined by the interval [t0,T] with

O<_to
where K is the given exercise price. (ii) The "floating-strike" 3 Asian call option of which payoff at date T is equal to:

(

l

+ to

]

Thus along the Harrison and Kreps (1979) line, we focus our interest on the following expectations: def

[

1

and

where Ele represents the conditional expectation with respect to the filtration 9- t under the risk neutral probability Q. Remarks: (1) These expectations are clearly difficult to evaluate since the distribution, under Q, of ftroS(u)du is not explicit (as a mix of lognormal random variables). (2) We consider "European type" Asian options, i.e. options that may only be exercised at the expiration date. (3) We distinguish for each of the two types of Asian options two cases. (i) The time to maturity is less than or equal to the length of the averaging period (0 < t o < t < T) (these options are sometimes called in the literature "backward-starting" or "plain-vanilla" Asian options).

2 See, for instance, Vorst (1996) for a review of the literature on Asian options. 3 We use here the terminology of Geman and Yor (1993).

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(ii) The time to maturity is greater or equal to the length of the averaging period (0 < t < t o < T) (these options are sometimes called "forward-starting" Asian options in the literature). This paper essentially focuses on the case where the time to maturity is less than or equal to the length of the averaging period (that is the "backward-starting" Asian option case). Nevertheless we explain in the Appendix how to adapt our main results to the "forward-starting" Asian option case. (4) Options considered in this work are Asian call options. The case of Asian put options is easily deduced from the exact put-call parity. This put-call parity is derived from the first moment of ffoS(U)du. As a matter of fact, under our assumptions, using Fubini's theorem, we obtain Proposition l, which is derived in Bouaziz et al. (1994) for the "floating-strike" Asian option case. To simplify notation we suppose that t o --- 0 in the following.

Proposition 1. (i) let C b (resp. Ptb) denote the price at date t of a "floatingstrike" Asian call option (resp. "floating-strike" Asian put option), then: pth=Cb_S,

l_~r(l_e-r(r

t)) + e

~(r-t)_~

S(u)du.

(ii) let C~ (resp. P~") denote the price at date t of a "fixed-strike" Asian call option (resp. the price at date t of a "fixed-strike" Asian put option), then: Lr St Pt~ = C; - -~--(1 - e -r(r o)

+ e

r(T-t)( g - - T1 fjotS( u) du).

Let us remark that from part (ii) of Proposition 1, it is very easy to deduce the explicit pricing formula proposed by Kemna and Vorst (1990) [eq. (16), p. 117]. As a matter of fact, the probability to exercise the Asian call is equal to 1 if and only if, at the pricing date, the known part of the average is greater than the exercise price ( K - ( 1 / r ) f d S ( u ) d u ) < 0. In such a case the Asian put is worthless and part (ii) of Proposition 1 immediately gives the Kemna and Vorst formula: C ; = ~Str r ( l - e

r(T-,)) - e - r(r-,)( K - TI fotS(u)du ) .

Moreover, we easily deduce from Proposition 1 a deterministic relationship between "fixed-strike" Asian options, "floating-strike" Asian options and European options. Indeed, from the well-known put call parity for European options, with our notation, we have the following corollary:

Corollas. The following relation holds: pta + p b _ p,e = C~ + C~ - C;, where C 7 (resp. Pte) denotes the price at date t of a European call option (resp. a European put option) written on S with exercise price K and exercise date T.

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Following Geman and Yor (1993), we focus in the sequel on "fixed-strike" Asian call because the "floating-strike" case is less used in practice. Nevertheless, we point out in this paper some important differences in the analysis of "fixed-strike" and "floating-strike" options.

3. The P.D.E. approach The P.D.E. approach is the historical way of understanding options (Black and Scholes, 1973; Merton, 1973). The use of this approach to value Asian options can be summarized as follows. Kemna and Vorst (1990) establish a P.D.E. with two state variables (the stock price and the average) which characterize the price of a "fixed-strike" Asian option. These two state variables complicate the valuation problem and the hedging analysis. In particular, extra boundary conditions have to be specified and the numerical methods to solve the P.D.E. are computationally expensive. We suggest here to use a one-state-variable P.D.E., easy to implement and giving additional information on the hedging strategy. Our result is based on a homogeneity property of the price of the Asian call. Such a transformation, based on a homogeneity property of the price of the option, is very general and well known in the literature. This argument is first used by Ingersoll (1987) for studying "floating-strike" Asian options. Ingersoll (1987) derives a two-statevariable P.D.E., then, using a change of variables, he obtains a one-state-variable P.D.E. which characterizes the price of a "floating-strike" Asian option. His result is presented as a boundary value problem (where a full set of boundary conditions must be specified). Ingersoll (1987) claims without proof that his P.D.E. has a closed-form solution. We will see later why such an assertion is clearly difficult to verify. In the framework of the martingale approach, Kramkov and Mordecki (1993) use a similar homogeneity property to study American type Asian options with infinite maturity 4. This allows them to reduce an optimal stopping problem for a two-dimensional Markov process to an optimal stopping problem for a one-dimensional Markov process. Then, they define implicitly the optimal stopping time which characterizes the early exercise of an American type Asian option and propose a quasi explicit pricing formula. Of course, the assumption of infinite maturity is crucial in order to establish their result. Rogers and Shi (1995), independently of this work, propose the same approach as ours to value an Asian option, give a lower bound and an upper bound for the price, and investigate the computational aspects. However, they do not consider the hedging problem and the comparison between Asian and European call. Moreover they only consider the case of "backward-starting fixed-strike" Asian options. In this paper, using an appropriate change of numeraire, we directly obtain a one-state-

4 The exercise takes place at any date.

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variable P.D.E. which characterizes the price of fixed-strike Asian option 5. We apply our methodology to the hedging problem. We explain how to adapt our results to the different types of Asian options and we propose new results on the comparison between Asian and European options. This section is organized as follows. Proposition 2 presents the change of probability under which it is more convenient to study Asian options. We establish in Proposition 3 the key result of this section: a one-state-variable P.D.E. which characterizes the price of a "fixed-strike" Asian option. This result is related to literature and its financial meaning is explained. Then, we focus on the hedging problem. We derive exact formulas, not computationally expensive for the delta, the gamma and the elasticity of a "fixed-strike" Asian option and deduce more analytical results. 3.1. Pricing an Asian option Let us now define the probability associated to a numeraire change under which we will derive our main result. Proposition 2.

The following relation holds:

I/1 ) K - ~ £ r S ( u ) du

C 7 = St E y

Sr

where the measure QS is defined on 0 by its Radon-Nikodym derivative with respect to the risk neutral measure Q: dQ s dQ Proof

S(T) EQ[S(T)] "

Now, we have:

+] =e-r(r-t)EQ

Sr

Sr

5 This result is also derived in DEcampsand Koehl (1994).

"

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Let QS be defined by its Radon-Nikodym derivative with respect to the risk neutral measure Q:

dQ s

S(T)

dO

EQ[S(T)] '

Then, using the change of probability formula for conditional expectations, we obtain:

c; = s,e?"

-;-

[]

C~/S t is thus the relative price of an Asian option with respect to the numeraire S. The probability associated to our change of numeraire is QS. Geman et al. (1995) propose a general framework to apply change of numeraire to study options. From Proposition 2, it is easy to remark the homogeneity in (S t, K - (1/T)fdS(u) d u) of the price of a "fixed-strike" Asian option. Let us recall that for standard European option the homogeneity is in (St,K). This result enlightens the fact that the "effective" strike price of our option is K minus the known part of the average. We deduce from this that the appropriate state variable x t to studying our option is the ratio effective strike price over the price of the underlying (x t = [ K - ( l / T ) f o ' S ( u ) d u ] / S t ) . As a consequence, the probability QS allows us to define the relative price of an Asian option with respect to the numeraire S as a function of the random realisation at date T of our state variable: C~/S, = E°S[ x T ]. The change of numeraire we consider and the previous remarks are quite general and can be used to study a wide class of derivative securities and in particular options with random strike. Using the Feyman Kac theorem, we are now in position to establish the main result of this section: to characterize E°S[x~] (and thus to characterize the price of an Asian option) by a one-state-variable P.D.E. Proposition 3. given by:

The price C~ at date t of a "fixed-strike" Asian call option is

ca'= stCa( xt,t) where 1

K Xt~

t

~foS(U) du S,

(1)

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and Ca(x t,t) is the solution of the partial differential equation: C]( x,t) (

lo.2x2+~(x,t)= rx ) + C ~ ( x,t) -~

T1

0

(2)

which holds in the domain D = {(x,t): x ~ R, 0 <_t < T} Proof From Ito's lemma, the dynamics under the probability QS of the state variable x t = [ K - ( 1/ T) fdS( u) d u ] / S t is defined by the stochastic differential equation: dx,=

(') -~-rx

t a t - o - x t d l , V t,

(3)

where ~ = Wt - ~rt is a QS Brownian motion (from Girsanov's theorem). From L e m m a 1, stated and proved in Appendix A, by a direct application of the Feyman Kac theorem (Karatzas and Shreve, 1988, p.360):

7 S~

is the solution of the P.D.E. (2). This ends the proof.

[]

The dynamics of our state variable (analyzed in L e m m a 1) can be seen as a generalization of the one used by Courtadon (1982). As a matter of fact, Courtadon (1982) 6 modelizes the term structure of interest rates as follows:

d x t= 3/( c ~ - x t ) dt + ~ x tdw t x0=x>0 Then, Courtadon shows that his process presents two natural boundaries at 0 and ~. Our case is a little more complex since we allow y and ~ and the initial value x 0 to be negative. The financial meaning of the stochastic differential Eq. (3) is the following: The state variable that we consider is x t = [ K - ( l / T ) f¢{S(u)du]/S t (with K > 0 ) and thus x o = K / ( S o ) ' is strictly positive. As a consequence, if there exists t ' > 0 such that ( l / T ) f ~ S ( u ) d u = K then, for all t greater than t', we have ( 1 / T ) f d S ( u ) d u > K, and finally for all t greater than t' we have x, < 0. L e m m a 1 captures the fact that if the process x t crosses 0 it stays

6 See also Courtadon (1985).

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in ]-o*,0[ forever. Therefore, if, at date t (1/T)fo'S(u)du > K (or equivalently x t _< 0) we are sure (from the date t) to exercise the option. Let us remark that the solution of Eq. (2) is explicit if x < 0, namely: 1

Ct = ~F (1 - e-r(T t)) _ e - r ( T t)xt.

(4)

Thus, again we find the result derived by Kemna and Vorst (1990) and by Geman and Yor (1993): In the case K - ( 1 / T ) f d S ( u ) d u <_O, it is easy to obtain a closed-form expression of the Asian option price. Moreover, Eq. (4) will be useful in the numerical implementation of our P.D.E. approach (Section 5): Our P.D.E. will be implemented on R + with Eq. (4) as boundary condition at x t = 0. Finally, let us remark that, from the proof of Proposition 3, the derivation of a closed-form solution of our P.D.E. is clearly related to the knowledge of the distribution of the arithmetic average of a set of lognormal distribution. In the case x < 0, we only require the first moment of f~S(u)du in order to obtain a closed-form solution of our P.D.E. In the case x ~ R we need the distribution of foTS(u)du which is not explicit. This is basically the reason why no one has published a closed-form solution for this type of P.D.E. Thus, in this section, we have established a partial differential equation which characterizes the price of a "fixed-strike" Asian option (an analogous result for a "floating-strike" Asian option is available). Such a P.D.E. is equivalent to the P.D.E. derived in Ingersoll (1987)). As far as we know the partial differential Eq. (2) does not admit explicit solutions. However, the very simple form of Eq. (2) permits the use of classical numerical techniques such as finite-difference methods to approximate the price of an Asian option (this is explained in Section 5). The P.D.E. approach we present here has three main advantages: (i) Firstly, we approximate the solution of the exact problem of the pricing of an Asian option. As opposed to an important part of the literature where the approximations consist of modifying geometric average option prices (see, e.g., Bouaziz et al., 1994), or to approximate the distribution of an arithmetic average (by Edgeworth series expansions, see, e.g., Turnbull and Wakeman, 1991). (ii) Secondly, Eq. (2) is a one-state-variable partial differential equation where the time parameter does not appear in the coefficients. We are typically in the case where the finite-difference approach requires in total fewer computations than the Monte Carlo approach in order to obtain similar accuracy. (iii) Thirdly, contrary to the papers mentioned above, our P.D.E. approach gives information on the hedging portfolio. This is discussed in the second part of this section.

3.2. Hedging an Asian option Very few results have been derived on the Asian option hedging problem. Turnbull and Wakeman (1991) give a rough estimate of the delta based on an Edgeworth series expansion. Geman and Yor (1993) prove that when K -

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( I / T ) f d S ( u ) d u < 0 then the gamma is 0 while the delta is not constant. The meaning of this result is clear since in the case K - ( 1 / T ) f d S ( u ) d u < 0 we are sure to exercise the option. A more constructive result on the hedging problem is proposed by E1 Karoui and Jeanblanc-Picqu6 (1993). These authors show that the price of an Asian option on a stock is equal to the price of a European option on a fictitious asset which has a random volatility. This random volatility admits for the upper bound the constant volatility o- of the underlying stock. Then they propose a "super hedging strategy" built on the Black and Scholes formula with volatility equal to the upper bound o-, Finally, Bouaziz et al. (1994) prove on an approximate pricing formula that the price of a "forward-starting floating-strike" Asian option is a linear function in S t. As a consequence, the gamma of the option is just 0. We have proved that their result can be easily extended to the exact pricing formula. Such a property does not apply to "backward-starting floating strike" and "forward-starting fixed-strike" Asian options. In the sequel, we focus on "backward-starting fixed-strike" Asian options. We give exact formulae easy to implement, for the delta, the gamma and the elasticity of an Asian option (Proposition 4). Then we derive analytical results as the monotonicity of the price of the option in the various parameters of the model. In particular, this allows us to prove some classical intuitive results on the hedging strategy. Under the notation of Proposition 3, by straightforward derivation we have: Proposition 4. (i) The delta, the gamma and the elastici~ of a "fixed-strike" Asian call option is giuen by: aef aCf

^a

Aa~- - -

= C ( xt,t)

aSt

a2

^

r"=

st c " , , ( x t , t ) ,

s, ,Q -

- XlCa( xt,t),

~

Q(xt,t) = 1 -Xt~x,,t)

These relations are very general. By simple algorithm, such as finite-difference methods, Proposition 4 can be used by practitioners to obtain numerical information to hedge Asian options. From Proposition 4, we can now deduce more analytical results on the hedging strategy. With the notations of Proposition 3 we prove the following 7.

7 The proof is available from the authors upon request.

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Proposition 5. (i) The price at date t of a "fixed-strike" Asian call option is an increasing function in S t and a decreasing function in K.

oc;'

(ii) OK = - - e - r ( T - t ) O ( X T < O ) '

oc;

OS

1

Tr~_le-r(-O)aaxlT
x

c

x

where Qa is the probability defined by its Radon-Nikodym derivative with respect to the risk neutral probability Q: dQa

ftTS(u) du

(iii) The price C t at date t of a "backward-starting" Asian option is given by the formula: 1

C~ = St-T7 ( 1 - e-r(r-,)) Q~( Xr < O) - e - r ( r - t ) ( K - -~ l fos(u)du)Q(xT
).

(5)

Proposition 5 deserves some comment: First notice that, as for the European call option, the derivative of the price of a "fixed-strike" Asian call with respect to the exercise price is equal, in absolute value, to the discounted risk neutral probability to exercise the call. Moreover, we obtain a general expression for the delta: 1

Aa = --(1

- e--r(r-t))Qa(xT < 0).

Tr As a consequence: (i) The probability QO highlights the average nature of Asian options. (ii) We obtain an upper bound for the delta: 1

~o < T r ( 1 - e - " > ' ~ ) . (iii) When the known part of the average is greater than K, we have Q(x r < 0 ) - 1 (we are sure from the valuation date t to exercise the option). Thus, we meet again the Geman and Yor (1993) formula for the delta of an Asian option with a probability of exercise equal to one: 1

Aa-- - - ( 1

Tr

--e-r(T-t)).

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(iv) When t tends to T, the delta tends to 0. Therefore we have proved a very simple result which is intuitively explained in Hull (1992): Near the exercise date T, the change in the average value with respect to the stock asset price is small and the option becomes progressively easier to hedge. (v) Finally, notice that Eq. (5) has the same structure as the Black and Scholes formula: We have proved that the price of an Asian option, as the price of a standard European option, is equal to the price of the underlying asset times the delta of the considered option minus the discounted effective strike price times the risk neutral probability to exercise the option. The same analysis holds for a "floating-strike" Asian option (the analogous formula is given in Appendix B). Of course, Eq. (5) is not explicit, however our P.D.E. approach allows us to compute numerically the probabilities Q a ( x T < 0) and Q ( x r < 0) and thus to investigate the various effect which interact in the pricing of an Asian option. Moreover such an analysis allows us, as we will see in the following sections, to obtain new results on the comparison between Asian and European options.

4. Comparison between Asian and European options: analytical evidence As we have noted, most of the papers on Asian options consider either a modifying pricing problem (Bouaziz et al., 1994) or a numerical approximation of the distribution of an arithmetic average (Turnbull and Wakeman (1991)). As a consequence, these papers cannot propose a detailed description of the comparison between Asian and European options. Our methodology is based on the exact pricing problem and thus permits such a detailed description. Section 4 completes and extends to the hedging tools the analytical results of Geman and Yor (1993) on the comparison between Asian and European options while Section 6, (through the implementation of our P.D.E, approach explained in Section 5), derives new numerical results on this subject. This section is organized as follows: We first state in Propositions 6 and 7 analogous results of Propositions 3 and 4 for European options. This allows us to compare the results of the previous section with the case of European options and to define the way we use to compare the two types of options. Then, we extend a result of Geman and Yor (1993) (Proposition 8) and we provide some elements of comparison between the delta and the elasticity of Asian and European options.

Proposition 6.

The price C 7 at date t o f a European call option is given by."

K C;=StCe(y~,t)

with

Yt=--,

St

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and Ce( yt,t) is solution of the partial differential equation: l

ry) + C ~ ( y , t ) - ~ o ' 2 y 2 + C ; ( y , t ) = 0

Cy(y,t)(-

Ce(y,T)=(1-y)

(6)

+

which holds in the domain D = {(y,t): 0 _< y < 0% 0 _< t _< T}. Proposition 6 is proved in the same way as Proposition 3. We just have to consider the well-known homogeneity in (S t, K ) of the price of a European option. As a consequence, the relevant state variable in the European case is Yt = ( K ) / ( S t ) . The dynamics under QS of the state variable y, is here: dy, = - r y , d t -

~r y, dW,

(7)

Eq. (6) has of course an explicit solution:

de( Yt, t)

= Et[(1 - YT) + ] = N ( d l ) - Yte-r(r-ON( d a - o ~ -

t ),

with 1

dI = ~

1

1

In yte_r(r_t ) + - ~ o ' ~ - t ,

where N is the standard cumulative normal function. Through the relation C[ =StCe(y,,t), we meet again the Black and Scholes (1973) formula, The difference between Eqs. (2) and (6) (Eqs. (3) and (7), respectively) lies in the coefficient I / T which is due to the "arithmetic average nature" of Asian options 8. From the last proposition, we obtain again the well-known explicit results:

Proposition 7. are given by."

The delta, the gamma and the elasticity of a European call option

Aedef OCte =~e(yt,t) = - OSt -

__ ytCy ^e ( Yt ,t) = N ( d l ) ,

def 22 C e y2 --=-..f_t ^e Yt,t ) = F e= ~S 2 St Cyy( def St 1"2e = A ~ - - = 1 C[

Cy(yt,t) Y' d e ( y, ,t)

1

s,

r~U~-t

1

1

2q~ e- U ~, SN( d, )

SN( d, ) - K e - r ( r - t W ( d, - ~

) "

8 Notice that Proposition 6 allow us to explain how to adapt our P.D.E. approach to the case of "forward-starting" Asian option. This is done in Appendix C.

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627

We remark that the two state variables we consider for studying Asian and European options ( x t and yt ) are equal at the time t = 0 (x 0 = Y0 = K/So). Thus, for our comparison to be relevant, we consider in the sequel the pricing date t = 0. Therefore, comparing the prices and the hedging tools between Asian and European options is equivalent to comparing the functions Ca(x,0), C'~(x,0) and ^e C e (x,O), C~(x,O).

In the following, we first clarify and complete a result due to Geman and Yor (1993) (Proposition 8). Then we extend our analysis to the comparison of the delta between Asian and European options (Proposition 9). Proposition 10 focuses on elasticities. Geman and Yor (1993) prove that, ceteris paribus, for all values of the exercise price K in a neighborhood of 0, if r is negative 9, the Asian option price is greater than the standard European option price. We extend their result in two ways. Firstly, we prove that their result holds when x 0 = ( K ) / ( S o) is in the neighborhood of 0 (part (i) of Proposition 8). Secondly, we prove that for "small negative" value of r, the previous result cannot be extended to all x 0 = ( K ) / ( S o) (part (ii) of Proposition 8).

Proposition 8. (i) Forallr C e ( x , 0 ) . (ii) There exists r* < O, Vr ~ [r *,0], 3 x > 0, Ca(x,0) < C"(x,0). Proof If K = 0, it is straightforward, using Fubini's theorem, to derive a closed-form expression for the price of the Asian option and to check that, if r is negative, the price of the Asian option is greater than the price of a European option. As a consequence: Vr < 0

c a ( 0 , 0 ) > ce(0,0).

Now, part (i) of Proposition 8 comes from the continuity of da and d e in the variable x. Let us prove part (ii) of the proposition. From Geman and Yor (1993), if r = 0, then:

Vx >_ o

d°(x,0) _< de(x,0).

Moreover, direct calculus gives (for r = 0):

d°(o,o) = d2o,o). 9 As Geman and Yor (1993) notice it, the sign of r can be negative for currency Asian options or Asian options on oil spreads. When r is positive (and the time to maturity equal to to the length of the averaging period, which is here t = 0), the Asian call option price is smaller than the standard European call price (Geman and Yor (1993)).

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Let us now show that for r = 0, we have: 3x>0

Ca(x,0)
If not, we would have: Vx>0

ca(x,O)

VK> 0

V X 0 ~_~0

de(x,O),

=

then, ca(so,o)

= ce( so,o),

therefore, ~C a

aC e

VK>0 VSo>_0 TE ( s°'0)=-gE ( s°'0)' From Proposition 5, this can be written: VK>0

VS0>0

e

( u ) du

K

=Q(S(T)>_K)

In other words, the distribution of ( 1 / T ) [ r S ( u ) du is lognormal. This gives us the contradiction. Thus, we have proved, for r = 0:

3 x ~. O Ca( x,O) < de( x,O). The continuity of d a and ~e in r gives (ii).

[]

Now, let us apply Proposition 8 to compare the delta and Asian and European options. We deduce a result analogous is negative for all x 0 = ( K ) / ( S o) in the neighborhood of 0, call is greater than the delta of the European call. The result positive. More precisely, we have:

the elasticity between to Proposition 8: if r the delta of the Asian is reversed when r is

Proposition9. (i) For all r < O, 3x* > O, V ( K ) / ( S o ) <_x*, Aa > Aq (ii) There exists r > O, 3x* > O, V ( K ) / ( S o) <_x*, A a < A e. Proof

At t --- 0, we have from Propositions 4 and 7:

Aa=da(x,t)_xCa(x,t),

A e=d e_xde(x,t).

Thus, if x = 0 (i.e. K = 0), we have A a = Ca(x,t) and Proposition 8 gives clearly: ifr<0and ifr=0and ifr>0and

x=0then x=0then x=0then

Ae =

A a > A e, A a = A e, A a < A e.

Proposition 9 follows from the continuity of d a

d e

in x. []

Ce(x,t).

Therefore,

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629

Let us now make a more precise comparison between the elasticity of the Asian and European options. From Propositions 4 and 7, we have for x = 0 ( K = 0), for all values of r, Oa = ~ ' ~ e = 1. Using the previous result we can propose a little more.

Proposition 10. There exists x * * positive, there exists a neighborhood U of (O,x* *) such that, f o r all ( r , x ) ~ U, the elasticity of the Asian option is greater than the elasticity of the European option. Proof.

From Propositions 4 and 7, let us first remark that for x different from 0, ,Q, < ~Qe <=~

a C°(x,t) ax de(x,t)

>0.

Now, for r = 0, x = 0 we have C a ( 0 , 0 ) / c e ( 0 , 0 ) = 1 and, from Proposition 7, there exists x * such that C " ( x * , O ) / C e ( x *,0) < 1. As a consequence, there exists x* * ~ [0, x* ] such that, for r = 0, (a/Ox)(Ca(x * * , 0 ) ) / ( C ' ( x * *,0)) < 0. From the initial remark, this is equivalent to O a > D e. The standard continuity argument ends the proof. [] Of course, we just obtain locally analytical results. However, they prove the existence of interesting features. For instance, the elasticity of an Asian option can be greater than the elasticity of a European option. As a consequence, in such a case, it is costless to hedge a stock with an Asian option rather than with a European option. In the following section we focus on the numerical implementation of our P.D.E. approach. This allows us in Section 6, to make precise from a numerical point of view, the analytical results established in this section.

5. Numerical

implementation

To compute the solution of the partial differential equation (2), we apply an explicit finite-difference method. Explicit methods have the advantage of being much simpler to implement than implicit methods. There are two principal difficulties to solve: -First the domain of the P.D.E. is unbounded, so we need to reduce it to a bounded domain to be able to discretise it. This is done with a change of variables. -Then, with an explicit method, the convergence of the calculated value to the correct solution is not necessarily ensured. Indeed a well-known disadvantage of explicit methods for parabolic equations is the stability condition: Thus the computation of the P.D.E. considered in this paper involves some modification of the classical explicit method. [The solution we use to overcome the problem of stability is quite similar to the approach of Hull and White (1990).]

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630

5.1. A change of variables First, using the fact that the solution is known in the case x < 0, we consider the P.D.E. (2) only for x > 0 with the following boundary conditions: 1

6a(0,t) = ~r(1

e-r(T-,)),

-

lira 6 ~ ( x , t ) = O. The boundary condition at infinity comes obviously from the definition of the Asian call. Then we define a new state variable y = e -x, so that, for x ~ [0,~], y ~ [0,1]. Finally, the P.D.E. (2) becomes: t 0 -2

0=

C~(y,t) + - ~ - y 2 ( l n __

+

T

__

rlny

y ) 2 C-ar y ( y , t )

y+--~--y(lny)

2

~a

C,(y,t) (a)

d~( y,T) = 0 1

C a ( l , / ) = -~r(1 - e - r ( r - o ) (~(0,t) = 0 and so the new P.D.E. holds on a bounded domain.

5.2. Modification of the classical explicit finite-difference method To implement the explicit finite-difference method, two increments A t, and A y of the variables t and y are chosen. A grid is then constructed for considering values of ~a when y is equal to: 0, Ay, 2 A y , . . . , n y A y = l, and time is equal to 0, At, 2At, - . . , n t A t = T We will denote iAt by ti, j a y b y yj, and the approximation of Ca(jAy,iAt) ~a by Cij. The partial derivatives of C a with respect to y at node ( i - 1,j) are approximated as follows: 0~

-a 1 - Ci,j -o Ci,j+

0y

Ay

&

i,j+ 1

0y 2 =

'

+&

~,J- 1

Ay2

(9)

+

-o 2Cij ,

(10)

B. Alziary et al. / Journal of Banking & Finance 21 (1997) 613-640

631

and the time derivative is approximated as Ci,j

Ot

t- I,j

(11)

At

Substituting Eqs. (9)-(1 I) into Eq. (8) gives: (~,'[,,~

=0

ca,-I,j

-__P j j - I C i . j -a -I

& i - 1,0

= 0

C"i_ ,,..

= ~r(1 _ e-r(r

VO
+Pj4+lCi,j+J ~a (12)

1

(i-I)At))

where 1 At PJ4-' = ~0-2(YJ)2( In yj)Z × --,(A y) 2

At PJ4 = 1 - ~r2( yj)2(ln yj)2 (AY) 2

--

[(1 ~,

]

rln yj yj + ~o-~-yj(lnyj) 2 S-yy' At

1 At ej,j+l = ~0-2(yj)2( In yj)Z ( A y)"

+ [(1~ - r l n 3) ) Yj+

10_ 2yj(ln yi) 2] A t •

Ay

.

Eqs. (9) and (10) are standard. Instead of using a centered estimate of the first partial derivative of C a with respect to y, we use a right estimate. Indeed, with the usual estimate it is not possible to satisfy a stability condition on the whole domain y ~ [0,1 ]. Proposition 11. Let 0- and r be such that o-z < r. The solution of the difference equation (12) approaches the solution of the P.D.E. (8) as A y ~ 0 and At--* O,

if.. 1

0-2 At e2 (Ay) 2

1 At T A~ > 0

(13)

Proof First, let us note that the explicit finite-difference approximation (12) is obviously consistent with the P.D.E. (8) and that Eq. (13) insures the stability.

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[The proof based on a theorem in Ames (1977), p.75, is available upon request.] Then, for this type of parabolic equation, consistency and stability imply convergence as it is shown in Lamberton and Lapeyre (1991). [] To check the accuracy our method, we test it on the European call option equation, since it is very similar to the Asian one, and since we can compare the computed solution and the exact one. Computations were carried out on a Alpha 800 Digital Workstation with FORTRAN, and a quite acceptable accuracy was obtained. Of course, for a fixed increment Ay, the maximum increment At required by the stability condition decreases as the volatility increases. But P.D.E. results improve as the volatility increases. Moreover, it is important to emphasize that with P.D.E. methods, we obtain with one computation the price at t = 0 for any strike. That is why we are able to approximate the derivative of the price function and so A and /2.

6. Comparison between Asian and European options: Numerical evidence When we consider Asian options we can have two basic intuitions: Due to the average nature of Asian options their price is less sensitive to volatility, the probability to exercise them is more sensitive to the strike when compared to European options. The numerical implementation of our P.D.E. approach allows to verify these intuitions, to quantify them, to explain how they are related and to analyse their consequences on the general behavior of Asian options. In the sequel we first analyse the price difference between European and Asian options (Fig. 1; Table 1). Then we compare the delta and the elasticity of Asian and European options. The effect of the level of the riskless rate is also analysed.

60,00 F

50,00 F

40,00 F

~iiiiiii. -~az- ..

-..

_

European

30,00 F "%.",

-.

.

.

____

20,00 F

10,00 F

o,ooF

I. . . . .

K

Fig. 1. European and Asian prices.

Asian

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Table i Relative price difference D in percent between European and Asian option with respect to the European option price: o = [(c; - c 7 ) / c , ] x 10o o-

K

D (%)

Aa

z~e

f~a

~e

~Oa-- .(2e

0.05 0.05 0.05 0.1 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.5

95 100 105 95 100 105 95 100 105 95 100 105 95 100 105 95 100 105

33.15 50.23 78.04 33.97 48.27 66.08 36.64 46.29 56.65 38.29 45.43 52.57 39.32 44.83 50.31 39.89 44.40 48.65

0.9557 0.8921 0.4391 0.9092 0.7488 0.4652 0.7736 0.6388 0.4848 0.7016 0.6015 0.4974 0.6639 0.5863 0.5081 0.6441 0.5806 0.5180

0.9978 0.9660 0.8021 0.9283 0.8289 0.6780 0.7900 0.7088 0.6202 0.7327 0.6736 0.6131 0.7099 0.6646 0.6191 0.7028 0.6664 0.6302

10.8500 20.6400 43.45011 10.1900 15.1200 21.9101/ 7.7250 9.3880 11.2000 6.0090 6.7940 7.6000 4.9060 5.3540 5.8010 4.1640 4.4470 4.7270

7.5710 11.1400 17.4300 6.8720 8.6650 10.8400 4.9990 5.5890 6.2100 3.8710 4.1530 4.4390 3.1840 3.3480 3.5110 2.7300 2.8370 2.9410

3.2790 9.500 26.0200 3.3180 6.4550 11.0700 2.7260 3.7990 4.9900 2.1380 2.6410 3.1610 1.7220 2.0060 2.2900 1.4340 1.6100 1.7860

Current price of the underlying asset: SO= 10O. Riskless interest rate: r = 0.09. Time to maturity: T = 1 year.

This section ends with two features of our approach: Firstly, our approach allows to explain and to precise from a numerical point of view when the price of a European option is lower than the price of an Asian option. Secondly, our approach applies to " b a c k w a r d - s t a r t i n g " Asian options as well as to " f o r w a r d starting" Asian options. Fig. 1 gives the prices of Asian and European options as a function of the exercise price K for various volatilities o- and for a given riskless interest rate r equal to 0.09. Asian and E u r o p e a n option prices are both decreasing convex functions in the exercise price K. Notice that these curves illustrate the property analytically proved by K e m n a and Vorst (1990) and G e m a n and Yor 11993): W h e n the time to maturity is equal to the length of the averaging period European option prices are greater than Asian option prices. More interesting are the observations of the behavior of Asian options as a function of the volatility (since until n o w no analytical result has been derived on this subject). The price of an Asian option is a decreasing function in the volatility of the u n d e r l y i n g stock. Moreover, A s i a n options are less sensitive to an increase in volatility than European options. The third c o l u m n of Table 1 specifies this point and focuses on

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the relative price difference D between the two types of options with respect to the European option price. Our numerical results are the following: For a given volatility, the relative price difference D is increasing in the strike K. Moreover, the higher the given volatility, the smaller the increase of D. Notice that for options out of the money (S o < K), the relative price difference D is "rapidly" decreasing in ~r. For options in the money (S o > K), D is "slowly" increasing in o- and for options at the money (S o = K), D is "slowly" decreasing in tr. Let us now focus on the comparison of the deltas and the elasticities between Asian and European options. By implementing the formulas of Propositions 4 and 7 we obtain Table 1. Let us first remark that the numerical results given in Table 1 seem to generalize the analytical results of Section 4: for values of parameters currently observed the value of European delta is greater than the value of the Asian one. The Asian elasticity is greater than the European one. More precisely, Table 1 indicates that, as in the case for European options, in the money Asian options have higher delta than out of the money Asian options. The behaviour of the elasticities are the same for Asian and European options. In particular, out of the money Asian options have higher elasticities than in the money Asian options. However, due to the average nature of Asian options, the difference in risk is smaller for Asian options than for European options. Moreover Table 1 indicates that the difference between the elasticities of Asian and European options is increasing in K. The higher the volatility the smaller the increase. Notice that the difference between the elasticities is decreasing in the volatility. The following proposition, easily deduced from assertion (iii) of Proposition 5, enlightens two effects which interact in these numerical results:

Proposition 12. The difference between the deltas and the elasticities of standard European and Asian options are given by: 1

Ae--

Aa=

-~o(( C e - C a) -[- K e - r r [ Q ( y r < 1 ) - Q ( x r < 0)1),

•a -- Oe = z~a--~a -- A e ~ e

=Ke-rr(

Q(xr<0)~2

Q(Yr
Two effects interact in these differences: (i) a price effect and (ii) a probability effect. The probability effect is easy to quantify. Using our P.D.E. approach, the implementation to exercise the risk neutral probability to exercise the Asian option comes from Proposition 5, part (ii). Our numerical implementation indicates, for instance, that Asian and European options have the same probability of exercise for value of the strike around the money. As a consequence for options around the

B. Alziary et al. / Journal of Banking & Finance 21 (1997) 613-640

635

money the difference in prices between Asian and European options is approximated by the value of the underlying asset times the difference of the delta. Let us now summarize how sensitive the Asian call is with respect to the riskless interest rate: We find that, as for European options, the risk neutral probability to exercise the Asian call is increasing in the riskless interest rate r. In the same manner, the price of an Asian call is increasing in the riskless interest rate. Moreover, the smaller the riskless interest rate the greater the relative price difference between Asian and European options. Our P.D.E. approach also specifies when the price of an Asian call is greater than the European one. As Turnbull and Wakeman (1991) and Geman and Yor (1993) notice, if the maturity of the option is less than the averaging period, the price of an Asian option may be greater than the price of a European option (priced at the same date with the same maturity and the same strike K). Such a result happens when the known part of the average at the pricing date is sufficiently high. Our numerical approach allows us to catch this point. As a matter of fact, we compute in a single step the prices of Asian or European calls for all positive strikes. Therefore, we are able to give the required amount of the known part of the average at the pricing date such that the price of an Asian call is

Table 2 Relative price difference D in percent between European Asian option with respect to the European option price: D = [(C; - Cf)/C~] tr

K

× lOO D

A"

Ae

/2"

.Q~

1 2 " - 12e

0.9552 0.9257 0.7327 0.9002 0.7966 0.6262 0.7672 0.6775 0.5767 0.7052 0.6386 0.5695 0.6769 0.6254 0.5733 0.6650 0.6232 0.5816

0.9987 0.9840 0.9054 0.9397 0.8678 0.7582 0.8063 0.7409 0.6695 0.7499 0.7015 0.6518 0.7285 0.6912 0.6537 0.7232 0.6933 0.6635

8.1470 12.5800 21.1200 7.5440 9.8840 12.8500 5.6220 6.4190 7.2640 4.3760 4.7580 5.1460 3.5970 3.8180 4.0380 3.0770 3.2190 3.3600

6.2040 8.4000 12.0200 5.7250 6.9410 8.3850 4.2610 4.6770 5.1100 3.3400 3.5410 3.7430 2.7720 2.8890 3.0050 2.3950 2.4710 2.5450

1.9430 4.1800 9.1000 1.8190 2.9430 4.4650 1.3610 1.7420 2.1540 1.0360 1.2170 1.4030 0.8250 0.9290 1.0330 0.6820 0.7480 0.8150

(%) 0.05 0.05 0.05 0.1 0.1 0.1 0.2 0.2 0.2 0.3 0.3 00.3 0.4 0.4 0.4 0.5 0.5 0.5

95 100 105 95 100 105 95 100 105 95 100 105 95 100 105 95 100 105

16.84 25.35 41.60 17.02 23.70 33.01 17.59 21.98 27.05 17.96 21.15 24.62 18.16 20.65 23.26 18.22 20.26 22.40

Current price of the underlying asset: SO= 100. Risldess interest rate: r = 0.09. Time to maturity: T = l year. to = 0.5 (i.e. the pricing date is 26 weeks prior to the averaging period).

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B. AIziary et al./ Journal of Banking & Finance 21 (1997) 613-640

greater than the price of a European call. For instance, our numerical results indicate that for a riskless rate equal to 0.09, a volatility cr equal to 0.1 and a strike equal to 100 francs, if the known part of the average is greater than 6 francs then the price of the Asian call is greater than the price of the European call. Moreover, the higher the volatility the higher the required amount of the known part of the average. Finally, using Appendix C, we have computed in Table 2 the prices of a "forward-starting" Asian call characterized by a length of average equal to 1 year and a pricing date which is 26 weeks prior to the averaging period. The behavior of the relative price difference D between "forward-starting" call and standard European call is qualitatively the same that the relative price difference between "backward-starting" Asian call and standard European call. We notice nevertheless that this relative price difference is lower in the case of "forward-starting" Asian call. The intuition is clear: When the length of the averaging period tends to 0, the price of "forward-starting" Asian option tends to the price of a standard European option. Here again our P.D.E. methodology allows to quantify this point.

7. Summary and conclusions We derive in this paper a P.D.E. methodology to study Asian options ("floating-strike" as well as "fixed-strike", "forward-starting" as well as "backwardstarting" Asian options). We focus more precisely on the case of a "backwardstarting fixed-strike" Asian option. The price of such an option is characterized by a one-state-variable P.D.E. This is obtained in a very general manner using an appropriate change of numeraire. Such a methodology can be used to study a wide class of derivative securities and in particular options with random strike. Our analysis highlights the fact that the "effective" strike price of a "backward-starting fixed-strike" Asian option is the fixed strike K minus the known part of the average. As a consequence, the appropriate state variable for studying this Asian option is the ratio "effective" strike price over the price of the underlying asset. Using our methodology we derive exact formulae, easy to implement, for the delta, the gamma and the elasticity of a "fixed-strike backward-starting" Asian call. We prove some classical intuitions on the hedging of such an option. Furthermore, we specify some differences between the various types of Asian options. The second part of our work is devoted to the comparison between Asian and European options and to the numerical implementation of our methodology. For our comparison to be relevant we consider a "fixed-strike" Asian option with the time to maturity equal to the length of the averaging period. We first clarify and extend to the hedging tools some analytical results of Geman and Yor (1993). Along the Hull and White (1990) line, we then adapt the explicit finite-difference method to our problem. We propose a numerical comparison of the prices the deltas and the elasticities between Asian and European options.

B. Alziary et al./ Journal of Banking & Finance 21 (1997) 613-640

637

The P.D.E. approach we develop in this paper is an appropriate tool to understand and to quantify the differences between European options and the various types of Asian options. Besides its practical interest, our methodology can also be used to obtain more theoretical results. For instance, to answer the question: In a continuous time stochastic volatility model, does an Asian option complete the market? The problem has been solved for European options by Romano and Touzi (1993) and by Bajeux and Rochet (1996). The basic idea of Romano and Touzi (1993) is to deal simultaneously with martingales and partial differential equations to show that European options complete the market. It is long but straightforward to adapt our approach to this methodology and to prove that Asian options complete the market as well.

Appendix A L e m m a 1. Let us consider the f o l l o w i n g differential equation dxt=

(') T

rx t d t - o - x l d W

t

x o = x.

(14)

where t ~ [0,~[, W t is a B r o w n i a n motion. T, r and o" are constants and T > O. Then the stochastic differential equation (14) has a unique solution x t in [O,T]. Moreover, this solution ( xl) o ~_t ~_r verifies: Vt ~ R +

P ( x t < O / x o = O) = 1,

P( x, > O/x o <_O) = O, P ( x, = O/xo < O) = O.

Proof The proof is a direct application of Karatzas and Shreve's (1988) theorem 2.9, p.289. From Ito's lemma, the solution of Eq. (2) is given by the expression: 0"2

1

x t =Xoe(2 --r)'-¢wO)-

1

0"2

_ f e~T-r×t-s)-¢~w,-W,)ds. T Jo

The distribution of x t is not explicit (as a mix of lognormal variables); however, it is clear that we have: Vt ~ R +

P ( x , < O / x o = O) = 1,

P( x, > O/xo <- O) = O, P ( x~ = O/xo < O) = O.

This ends the proof of Lemma 1.

[]

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Appendix B Proposition. The price at date t of a "backward-starting floating-strike" Asian call is given by the following formula:

(

l

C~= S, Qs( E) - - ~ r ( 1 - e

r(T-t))Qa( (.) ) -- e - r ( T - t ) _ _l fo'S(u)duQ(e) T

where E is the event the call is exercised:

Moreover, we have the following expression for the delta: Ab -

8C~

1 = Q S ( e ) -- - - ( 1 - - e - r ( T - t ) ) Q a ( E ) .

OS

Tr

Appendix C Let us consider the price at date t of a "forward-starting" Asian option. The length of the averaging period is T - t o and we have 0 < t _< t 0 < T. Remark that, at the pricing date t < t 0, the price of our option is a function of the time t and of the current price S t of the underlying asset. (The average has not begun to run.) As a consequence the price of the "forward-starting" Asian option is characterized, on the interval [0,to], by the European P.D.E. of Proposition 6 and equal to the price of a "backward-starting" Asian option priced at t o and maturing at date T. To summarize we deduce from Propositions 3 and 6:

Proposition. The price C t at date t of a "forward-starting"2 Asian option is given by the relation: c, =

where C( xt, t) is defined by the following partial differential equations: 1 - rXCx(

,t) +

2

+

C(X,to) = C"( X,to) ce(o,t) --

1

(T-to)r

(1 -- e - rfr-t°))

=0

B. Alziary et al. / Journal of Banking & Finance 21 (1997) 613-640

639

and

1 T-

rx

)

to

^a

C (x,t) +

1 2 2^o x

2

^

=0

(°(x,v) 1 Ca(O,t)

(1 -- e - r ( T - ' / )

(r-to)r

There are no difficulties implementing an explicit finite-difference approximation for the previous P.D.E. We only have to check that both stability conditions for Asian and European equations are satisfied.

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