The major problem in pricing Asian option is that we do not know much about the distribution of the underlying asset’s average price A(T ) (see Eq. (2.1) and (2.5)).
A(T ) can be viewed as the sum of log-normal random variables; and the density function of a sum of log normal random variables is currently unavailable. That is why there is no simple and exact closed-form solutions for pricing Asian options.
Approximation methods suggested in the academic literature can be grouped into
r σ T S0 GE Shaw Euler PW TW MC10 MC100 SE 5.0% 50% 1 1.9 .195 .193 .194 .194 .195 .192 .196 .004 5.0% 50% 1 2.0 .248 .246 .247 .247 .250 .245 .249 .004 5.0% 50% 1 2.1 .308 .306 .307 .307 .311 .305 .309 .005 2.0% 10% 1 2.0 .058 .520 .056 .0624 .0568 .0559 .0565 .0008 18.0% 30% 1 2.0 .227 .217 .219 .219 .220 .219 .220 .003 12.5% 25% 2 2.0 .172 .172 .172 .172 .173 .173 .172 .003 5.0% 50% 2 2.0 .351 .350 .352 .352 .359 .351 .348 .007
Table 2.1: Stress Tests.
The exercise price X is 2.0. The approximation methods for comparison are from Geman and Eydeland [22] (GE), Shaw [39], Euler, Post-Widder method (PW) [1], and Turnbull-Wakeman [41] (TW). The benchmark values (MC10 and MC100) and the approximation values are from [21]. MC10 uses 10 time steps per day, whereas MC100 uses 100. Both are based on 100,000 trials. SE stands for standard error, also from [21].
three different categories: approximation analytical formulae, (quasi-)Monte Carlo simulations, and the lattice (and the related PDE) approach.
2.4.1 Approximation Analytical Formulae
This approach denotes that the value of the option is approximated by (semi-)closed form formulae. Some related academic works in this category try to approximate the probability density function of A. Turnbull and Wakeman [41] and Levy [32] try to approximate the density function of A by Edgeworth series expansion. Milevsky and Posner [37] approximate it by the reciprocal gamma distribution. Carverhill and Clewlow [12] and Benhamou [6] use Fourier transform to approximate the payoff func-tion at maturity. Geman and Yor [23] derive an analytical expression for the Laplace transform of the continuous Asian calls, and numerical inversion of this transform is considered by Geman and Eydeland [22] and Shaw [39]. Some inversion algorithms based on the Euler and Post-Widder methods are suggested in Abate and Whitt [1].
The forward starting Asian options are approximated with Taylor’s series expansion as in [8, 40]. Zhang [42] approximates the option value by combining an analytical closed form with a numerical adjustment (computed by finite difference method).
The major problem of the approximation analytical formulae approach is that most suggested methods from this approach lack error control [21]. Table 2.1 illustrates that some well-known approximation methods fail in extreme cases. Besides, the American-style Asian option’s value cannot be approximated by this approach easily.
2.4.2 Monte Carlo Simulation
The Monte Carlo simulation approach denotes a pricing procedure that values a derivative by randomly sampling changes in economic variables. To value a European-style Asian option, a typical Monte Carlo simulation can divide the time interval between the option initial date and the maturity date into n time steps. Then it simulate the price path of the underlying asset by the following formula:
Si = Si−1exp[(r − 1/2σ2)∆t + σ√
∆tω],
where Si denotes the price of the underlying asset at time i, ∆t is equal to T /n, and ω denotes the a standard normal random variable. Note that the distribution of this n + 1-dimensional random price vector (S0, S1, S2, · · · , Sn) is the same as the distribution of random vector (S(0), S(∆t), S(2∆t), · · · , S(n∆t)) which is governed by Eq. (2.1). The average price of each price path is computed by Eq. (2.6). Thus the payoff for each price path for the European-style Asian option can be computed by Eq. (2.7). The output option value is obtained by averaging the discounted payoffs.
The Monte Carlo simulation approach suffers from the following problems:
1. The pricing result is only probabilistic.
2. The number of simulated price paths should be large enough to obtain satisfac-tory pricing results. Thus the algorithm is not efficient enough.
3. The pricing results are significantly influenced by the random sources used to obtain the random variable ω. Biased results are produced if the random sources are unreliable.
4. The American-style option can not be handled by this approach easily.
Some related works that address the first three problems are listed below: Boyle, Broadie, and Glasserman [9], Broadie and Glasserman [10], Broadie, Glasserman, and Kou [11], Kemna and Vorst [28], and Lapeyre and Temam [31]. Briefly speaking, they try to reduce the variance of the pricing results and refine the quality of the random variable (ω) they use. Four simple methods used by them are
• Antithetic variates:
Assume that the normal random variables ˆω1, ˆω2, · · · , ˆωn are sampled. A simu-lated price path ( ˆS0, ˆS1, . . . , ˆSn) is constructed by defining ˆSi as
Sˆi = ˆS0e(r−σ2/2)i∆t+σ√∆t(ˆω1+ˆω2+...ˆωi).
A dual price path ( ˆS0, ˆS0, . . . , ˆSn) can be constructed by defining ˆS(i) as Sˆ(i) = ˆS0e(r−σ2/2)i∆t+σ√∆t(−ˆω1−ˆω2−...−ˆωi)
.
• Moment matching:
This is done by tuning the sampled random variables so that the first few moments of the tuned sampled random variables match the moments of the distribution where the random variables are sampled from. For example, if we sample l random variables (ω1, ω2, · · · , ωl) from a normal distribution with mean µ and standard deviation σ. The sample mean and standard error of these sampled random variables are µl and σl, respectively. Then the tuned variable ωi is
ωi ≡ (ωi− µl)σ σl
+ µ.
• Latin hypercube sampling:
This is a stratified sampling method that forces the cumulative probabilities of the empirical distribution (determined by the sampled random variables) to match the cumulative probabilities of the theoretical distribution (where the variables are sampled from). For example, assume that ω1, ω2, . . . , ω100 are sampled from a normal distribution, then each observation ωi is forced to lie between (i − 1)th and ith percentile.
• Control variates:
This method is widely applicable to reduce the variance of the output results.
This is done by replacing the evaluation of an unknown expectation with the evaluation of the difference between the unknown quantity and another expec-tation whose value is known [9]. This is used by Kemna and Vorst [28] for pricing Asian options.
For pricing American-style Asian options, Longstaff and Schwartz (2001) develop a least-squares Monte Carlo approach to tackle the problem.