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A new PDE approach for pricing arith- metic average Asian options∗ Jan Večeř† Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213. Email:
[email protected]. May 15, 2001 Abstract. In this paper, arithmetic average Asian options are studied. It is ob- served that the Asian option is a special case of the option on a traded account. The price of the Asian option is characterized by a simple one-dimensional par- tial differential equation which could be applied to both continuous and discrete average Asian option. The article also provides numerical implementation of the pricing equation. The implementation is fast and accurate even for low volatility and/or short maturity cases. Key words: Asian options, Options on a traded account, Brownian motion, fixed strike, floating strike. 1 Introduction Asian options are securities with payoff which depends on the average of the underlying stock price over certain time interval. Since no general analytical solution for the price of the Asian option is known, a variety of techniques have been developed to analyze arithmetic average Asian options. A number of ap- proximations that produce closed form expressions have appeared, see Turnbull and Wakeman [18], Vorst [19], Levy [13], Levy and Turnbull [14]. Geman and Yor [8] computed the Laplace transform of the Asian option price, but numerical inversion remains problematic for low volatility and/or short maturity cases (see Geman and Eydeland [6] or Fu, Madan and Wang [5]). Monte Carlo simulation works well, but it can be computationally expensive without the enhancement of variance reduction techniques and one must account for the inherent discretiza- tion bias resulting from the approximation of continuous time processes through discrete sampling (see Broadie and Glasserman [3], Broadie, Glasserman and Kou [4] and Kemma and Vorst [12]). ∗This work was supported by the National Science Foundation under grant DMS-98-02464. †I would like to thank Fredrik Åkesson, Julien Hugonnier, Steven Shreve, Dennis Wong and Mingxin Xu for helpful comments and suggestions on this paper. 1 In general, the price of an Asian option can be found by solving a PDE in two space dimensions (see Ingersoll [10]), which is prone to oscillatory solutions. Ingersoll [10] also observed that the two-dimensional PDE for a floating strike Asian option can be reduced to a one-dimensional PDE. Rogers and Shi [17] have formulated a one-dimensional PDE that can model both floating and fixed strike Asian options. They reduced the dimension of the problem by dividing K−S̄t (K is the strike, S̄t is the average stock price over [0, t]) by the stock price St. However this one-dimensional PDE is difficult to solve numerically since the diffusion term is very small for values of interest on the finite difference grid. The dirac delta function also appears as a coefficient of the PDE in the case of the floating strike option. Zvan, Forsyth and Vetzal [21] were able to improve the numerical accuracy of this method by using computational fluid dynamics tech- niques. Andreasen [2] applied Rogers and Shi’s reduction to discretely sampled Asian option. More recently, Lipton [15] noticed similarity of pricing equations for the passport and the Asian option, again using Rogers and Shi’s reduction. In this article, an alternative one-dimensional PDE is derived by a similar space reduction. It is noted that the arithmetic average Asian option (both floating and fixed strike) is a special case of an option on a traded account. See Shreve and Večeř [16] and [20] for a detailed discussion about options on a traded account. Options on a traded account generalize the concept of many options (passport, European, American, vacation) and the same pricing techniques could be applied to price the Asian option. The resulting one-dimensional PDE for the price of the Asian option is simple enough to be easily implemented to give very fast and accurate results. Section 2 of the article briefly describes options on a traded account. It is shown in section 3 that the Asian option is a special case of the option on a traded account. The one-dimensional PDE for the price of the Asian option is given. Section 4 describes the numerical implementation and compares results with results of other methods. Section 5 concludes the paper. 2 Options on a traded account An option on a traded account is a contract which allows the holder of the option to switch during the life of the option among various positions in an underlying asset (stock). The holder accumulates gains and losses resulting from this trading, and at the expiration of the option he gets the call option payoff with strike 0 on his final account value, i.e., he keeps any gain from trading and is forgiven any loss. Suppose that the stock evolves under the risk neutral measure according to the equation dSt = St(rdt + σdWt), (2.1) where r is the interest rate and σ is the volatility of the stock. Denote the option holder’s trading strategy by qt, the number of shares held at time t. The strategy qt is subject to the contractual constraint qt ∈ [αt, βt], where αt ≤ βt. It turns out that the holder of the option should never take an intermediate position, i.e, 2 at any time he should hold either αt shares of stock or βt shares. In the case of Asian options, αt = βt, so option holder’s trading strategy is a priori given to him. In our model the value of the option holder’s account corresponding to the strategy qt satisfies dXqt = qtdSt + µ(X q t − qtSt)dt (2.2) Xq0 = X0. This represents a trading strategy in the money market and the underlying asset, where X0 is the initial wealth and µ is the interest rate corresponding to reinvesting the cash position Xqt − qtSt (possibly different from the risk-neutral interest rate r). The trading strategy is self-financing when µ = r. The holder of the option will receive at time T the payoff [XqT ] +. The objective of the seller of the option, who makes this payment, is to be prepared to hedge against all possible strategies of the holder of the option. Therefore the price of this contract at time t should be the maximum over all possible strategies qu of the discounted expected value under the risk-neutral probability P of the payoff of the option, i.e., V [α,β](t, St, Xt) = max qu∈[α,β] e−r(T−t)E[[XqT ] +|Ft], t ∈ [0, T ]. (2.3) Computation of the expression in (2.3) is a problem of stochastic optimal control, and the function V [α,β](t, s, x) is characterized by the corresponding Hamilton– Jacobi–Bellman (HJB) equation − rV + Vt + rsVs + max q∈[α,β] [(µx + q(r − µ))Vx + 12σ 2s2(Vss + 2qVsx + q2Vxx)] = 0 (2.4) with the boundary condition V (T, s, x) = x+. (2.5) The maximum in (2.4) is attained by the optimal strategy qoptt . The case αt = βt = 1 reduces to the European call, the case αt = βt = −1 reduces to the European put. The American call and put give the holder of the option the right to switch at most once during the life of the option to zero position (i.e., exercise the option), but it does not pay interest on the traded account while the holder has a position in the stock market. These can be modelled by setting µ = 0 in (2.2) and allowing only one switch in qt, either from 1 to 0 (American call) or from −1 to 0 (American put). The passport option has contractual conditions αt = −1, βt = 1, the so-called vacation call has αt = 0, βt = 1 and the so-called vacation put has αt = −1, βt = 0. By the change of variable Zqt = Xqt St , (2.6) 3 we can reduce the dimensionality of the problem (2.3), as we show below. The same change of variable was used in Hyer, Lipton-Lifschitz and Pugachevsky [9] and in Andersen, Andreasen and Brotherton-Ratcliffe [1] to price passport options and in Shreve and Večeř [16] to price options on a traded account. Applying Itô’s formula to the process Zqt , we get dZqt = (qt − Z q t ) (r − µ− σ2)dt + (qt − Z q t ) σdWt. (2.7) We next define a new probability measure P̃ by P̃(A) = ∫ A DT dP, A ∈ F , where DT = e−rT · STS0 = exp ( σWT − 12σ 2T ) . (2.8) Under P̃, W̃t = −σt+Wt is a Brownian motion, according to Girsanov’s theorem. Notice that e−rT E[XqT ] + = e−rT Ẽ [ XqT DT ]+ = S0 · Ẽ [ XqT ST ]+ = S0 · Ẽ [ZqT ] + (2.9) and dZqt = (qt − Z q t ) (r − µ)dt + (qt − Z q t ) σdW̃t. (2.10) The corresponding reduced HJB equation becomes ut + max q∈[α,β] ( (r − µ)(q − z)uz + 12 (q − z) 2σ2uzz ) = 0 (2.11) with the boundary condition u(T, z) = z+. (2.12) The relationship between V and u is V (0, S0, X0) = S0 · u ( 0, X0S0 ) . (2.13) Closed form solutions and optimal strategies are provided in Shreve and Večeř [16] for the prices of the option on a traded account for any general constraints of the type αt ≡ α and βt ≡ β when µ = r. 3 Asian option as an option on a traded account Options on a traded account also represent Asian options. Notice that d(tSt) = tdSt + Stdt, or equivalently, TST = ∫ T 0 tdSt + ∫ T 0 Stdt. (3.1) After dividing by the maturity time T and rearranging the terms we get 1 T ∫ T 0 Stdt = ∫ T 0 ( 1− tT ) dSt + S0. (3.2) 4 In the terminology of the option on a traded account, the Asian fixed strike call payoff (S̄T −K)+ is achieved by taking qt = 1− tT and X0 = S0−K and where the traded account evolves according to the equation dXt = ( 1− tT ) dSt, (3.3) i.e., when µ = 0 so no interest is added or charged to the traded account. We have then XT = ∫ T 0 (1− tT )dSt + S0 −K = S̄T −K. (3.4) Thus the average of the stock price could be achieved by a selling off one share of stock at the constant rate 1T shares per unit time. Similarly, the Asian fixed strike put payoff (K − S̄T )+ is achieved by taking qt = tT − 1 and X0 = K − S0. For the Asian floating strike call with payoff (KST − S̄T )+ we take simply qt = tT − 1 + K and X0 = S0(K