Spectral Expansions for Asian (Average Price) Options

Abstract

Arithmetic Asian or average price options deliver payoffs based on the average underlying price over a prespecified time period. Asian options are an important family of derivative contracts with a wide variety of applications in currency, equity, interest rate, commodity, energy, and insurance markets. We derive two analytical formulas for the value of the continuously sampled arithmetic Asian option when the underlying asset price follows geometric Brownian motion. We use an identity in law between the integral of geometric Brownian motion over a finite time interval [0, t] and the state at time t of a one-dimensional diffusion process with affine drift and linear diffusion and express Asian option values in terms of spectral expansions associated with the diffusion infinitesimal generator. The first formula is an infinite series of terms involving Whittaker functions M and W. The second formula is a single real integral of an expression involving Whittaker function W plus (for some parameter values) a finite number of additional terms involving incomplete gamma functions and Laguerre polynomials. The two formulas allow accurate computation of continuously sampled arithmetic Asian option prices.

General Optimized Lower and Upper Bounds for Discrete and Continuous Arithmetic Asian Options

“…To check the accuracy of our MLB, we adhere to the test cases considered in Cai and Kou [14] and compare against existing methods devoted to the Gaussian dynamics and the continuous average case. Table 10 shows that in most of the cases of moderate to high volatility, 0.1 ≤ σ ≤ 0.5, our MLB results agree to 4 decimal places with the prices obtained from the methods of Geman and Yor [42], Linetsky [53], Cai and Kou [14] (accurate to 10 decimal places), and Večeř [69], Zhang [74] (accurate to 6 decimal places). Although the MLB seems less accurate than the other approaches (still sufficiently accurate for practical applications), its performance improves substantially with decreasing volatility to extremely low levels in which case many numerical methods for Asian options perform poorly.…”

“…where the expectations in (53) and (54) are taken under the P and P measures, respectively. (Note that for CEV elasticity γ > 2, the inequality sign in the indicator functions in (51)- (54) is reversed.…”

“…Without being exhaustive, we mention Rogers and Shi [61], who are the first to provide the Asian option price by solving a partial differential equation (PDE) in one space dimension, but also Večeř [69] and Zhang [74] with improved PDE-based solutions in low-volatility settings; Linetsky [53] derives an elegant spectral expansion for the option price; Geman and Yor [42] are the first to give a solution in terms of a single-Laplace transform, which is re-derived in Dewynne and Shaw [31] where it is shown how to treat the low-volatility case more effectively. An also accurate double-transform at low volatility levels is suggested in Fusai [38], whereas Cai and Kou [14] generalize to a double-Laplace transform under the hyperexponential jump diffusion model, encompassing the Gaussian model and Kou's double exponential jump diffusion as special cases.…”

Pricing of Discretely Sampled Asian Options Under Levy Processes

“…Later, the approach has been extensively studied, for example, by Fu et al [35], Carr and Schröder [24,25], Shaw [59]. Linetsky [45] takes a new direction, and derives analytical formulas using spectral expansion for the value of the continuously sampled arithmetic Asian option.…”

Pricing Discretely Monitored Asian Options by Maturity Randomization

“…Among analytical approaches, we mention the Laplace transform approach in Geman and Yor (1993), the spectral expansion derived by Linetsky (2004), and the approximation of the average distribution by …tting integer moments in Turnbull and Wakeman (1991), Lévy (1992), Milevsky and Posner (1998) or logarithmic moments as in Fusai and Tagliani (2002). Another approach uses binomial trees, such as Gaudenzi et al (2007).…”