Appl. J. Prob. 39, 391394 (2002) Printed in Israel Applied Probability Trust 2002
ON THE EQUIVALENCE OF FLOATINGAND FIXED-STRIKE ASIAN OPTIONS
VICKY HENDERSON, University of Oxford RAFA WOJAKOWSKI, Lancaster University Abstract There are two types of Asian options in the nancial markets which differ according to the role of the average price. We give a symmetry result between the oating- and xed-strike Asian options. The proof involves a change of numraire and time reversal of Brownian motion. Symmetries are very useful in option valuation, and in this case the result allows the use of more established xed-strike pricing methods to price oating-strike Asian options. Keywords: Asian option; oating-strike Asian option; put call symmetry; change of numraire; time reversal; Brownian motion AMS 2000 Subject Classication: Primary 60G44 Secondary 91B28
1. Introduction The purpose of this paper is to establish a useful symmetry result between oating- and xed-strike Asian options. A change of probability measure or numraire and a time-reversal argument are used to prove the result for models where the underlying asset follows exponential Brownian motion. There are many known symmetry results in nancial option pricing dating back to Kruizenga [11]. Bartels [2] derives relationships between European puts and calls with different strikes, under general assumptions on the stock price process. Vecer [13] discusses some more recent work on symmetries. Such results are useful for transferring knowledge about one type of option to another, and may be used to simplify coding of one type of option when the other is already coded. For example, Henderson and Hobson [10] derive an equivalence between a passport option and a xed-strike lookback option. However, care must be taken as the transformed option may not exist in the market or have a sensible economic interpretation. These tricks become very useful for exotic options, when perhaps no closed-form solution exists but an equivalence relation holds, together with an accurate computational procedure for the related class. This is of particular interest for the Asian option since much is known about the xed-strike case, but comparatively little work has been done for the oating-strike option. In contrast to the above symmetries, our results for Asian options relate two different types of arithmetic Asians, a oating-strike call (or put) and a xed-strike put (or call) option. This is interesting, since from the parity shown by Alziary et al. [1] involving all four Asians and a
Received 12 September 2001; revision received 6 December 2001. Postal address: Nomura Centre for Quantitative Finance, Mathematical Institute, 24-29 St Giles, Oxford, OX1 3LB, UK. Email address: [e-mail] Postal address: Department of Accounting and Finance, Lancaster University, Management School, Lancaster, LA1 4YX, UK.
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European put and call, it does not seem likely that there would be such a result. It is simple, however, to obtain a putcall parity between two Asians of the same type; see [1]. Asian options have a payoff which depends on the average price of the underlying asset during some part of its life. The average is usually arithmetic, and if the asset price is assumed to follow exponential Brownian motion, an explicit option price is not available as the arithmetic average of a set of log-normal distributions is not known explicitly. There are two types of Asian options: the xed-strike option, where the average relates to the underlying asset and the strike is xed; and the oating-strike option, where the average relates to the strike price. Pricing of the xed-strike Asian has been the subject of much research over the last ten years and academic interest in these options has experienced a recent revival; see [3], [5], which continue the earlier work of [9]. The reader is referred to the recent papers [7] and [6] for an overview of the literature on pricing xed-strike Asian options. The oating-strike Asian option has received much less attention in the literature, perhaps because the problem is more difcult in that the joint law of {St , At } is needed. Chung et al. [4] generalise earlier efforts which derive approximations using joint log-normality. A PDE approach can also be taken; see [12], and [13] for an excellent new method. Even so, pricing methods for oating-strike options are underdeveloped compared with the more established methods for the xed-strike option. 2. The model We consider the standard BlackScholes economy with a risky asset (stock) and a money market account. We take as given a complete probability space ( , F , P ) with a ltration (F )0t which is right-continuous and such that F0 contains all the P -null sets of F . We also assume the existence of a risk-neutral probability measure Q (equivalent to P ) under which discounted asset prices are martingales, implying no arbitrage. We denote expectation under measure Q by E, and under the Q stock price follows dSt = (r ) dt + dWt , St (1)
where r is the constant continuously compounded interest rate, is a continuous dividend yield, is the instantaneous volatility of asset return and W is a Q-Brownian motion. An Asian option contract is written at t = 0 and expires at T > 0. The arithmetic average A can be calculated at T given the price history from time t0 < T . Let D > 0 denote the duration of averaging D = T t0 . If T = D, the average is computed over the whole life of the option, which is termed the plain vanilla option. If T > D, the option is forward starting, and if T < D, the Asian option is in progress. The arithmetic average is dened by At = 1 D
t t0
Su du,
where t 0. The xed-strike Asian call option is dened to be the security which pays (AT K)+ at time T ; the oating-strike Asian call option pays (ST AT )+ . By arbitrage arguments, the price of a xed-strike call cx at time t = 0 is given by: cx (K, S0 , r, , 0, T ) = cx = erT E(AT K)+ ,
On the equivalence of oating- and xed-strike Asian options
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whilst the price of a oating-strike call cf at time t = 0 is given by cf (S0 , , r, , 0, T ) = cf = erT E(ST AT )+ , with = 1 being the important case in nancial option pricing. Asian put options are dened analogously with prices denoted by px (K, S0 , r, , 0, T ) and pf (S0 , , r, , 0, T ) for xed- and oating-strike calls respectively. 3. A symmetry between oating-strike and xed-strike Asian options Theorem 1. Under the assumption that S follows exponential Brownian motion in (1), the following symmetry results hold: (i) cf (S0 , , r, , 0, T ) = px (S0 , S0 , , r, 0, T ) and (ii) cx (K, S0 , r, , 0, T ) = pf (S0 , K/S0 , , r, 0, T ). Proof. We prove (i) rst. Taking t0 = 0, the oating-strike Asian call price expressed in units of stock as numraire is
cf
cf S0
=
erT ST erT (ST AT )+ . E (ST AT )+ = E S0 S0 ST dQ ST erT 2 = e( /2)T + WT = , S0 eT dQ
By changing numraire to S via
we dene the measure Q (see [8]). Under Q , Wt = Wt t is a Brownian motion, using the Girsanov theorem. Now the price becomes
cf = eT E ( A )+ , T
AT 1 T Su 1 T = du = S (T ) du, ST T 0 ST T 0 u where for u T we dene the FT -measurable random variable Su 1 = exp{(r + 2 2 )(u T ) + (Wu WT )}, Su (T ) ST A T using Wt , a Q Brownian motion. Now note that if, for all t, Wt Wt is a Q -Brownian law motion starting at zero, then Wu WT = WT u and 1 T WT u +(r+(1/2) 2 )(uT ) law A = AT e du. T T 0 Reversing time via the variable change s = T u gives 1 AT = T
T 0
with
e Ws (r+(1/2)
2 )s
ds.
Thus cf = eT E ( A )+ = eT E ( AT )+ and so the result (i) is proved. The result T (ii) follows directly from (i) using putcall parity.
Remark 1. The above results extend trivially to forward-start options where the averaging period begins at t0 > 0, and the option is priced at times up to t0 . They do not extend to in progress Asians due to the extra term created by the average to date At , at ...