Analytical pricing of discretely monitored Asian-style options: Theory and application to commodity markets

Abstract

We compute an analytical expression for the moment generating function of the joint random vector consisting of a spot price and its discretely monitored average for a large class of square-root price dynamics. This result, combined with the Fourier transform pricing method proposed by Carr and Madan [Carr, P., Madan D., 1999. Option valuation using the fast Fourier transform. Journal of Computational Finance 2(4), Summer, 61-73] allows us to derive a closed-form formula for the fair value of discretely monitored Asian-style options. Our analysis encompasses the case of commodity price dynamics displaying mean reversion and jointly fitting a quoted futures curve and the seasonal structure of spot price volatility. Four tests are conducted to assess the relative performance of the pricing procedure stemming from our formulae. Empirical results based on natural gas data from NYMEX and corn data from CBOT show a remarkable improvement over the main alternative techniques developed for pricing Asian-style options within the market standard framework of geometric Brownian motion.

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

“…(i) The proof of (20) follows that of Proposition 2 in Fusai et al [39] given the expression (9) and using iterated expectations.…”

“…where ∂Φ (u; δ, X 0 ) /∂X 0 is computed using (37) and (14) for Lévy models; (37) and (16) for ASV models; (39) and (20)- (21) for the CEV model. In addition, ∂X 0 /∂S 0 = exp(−X 0 ) for Lévy and ASV models;…”

“…We mention, amongst others, contributed works by Andreasen [3] and Večeř [69] based on PDE approaches under lognormal asset price dynamics and the Fourier transform-based recursive convolution of Carverhill and Clewlow [20] on a reduced state space, whereas enhanced variations of the latter under general exponential Lévy dynamics appear, for example, in Benhamou [10], Fusai and Meucci [40],Černý and Kyriakou [21] and Zhang and Oosterlee [73]. Beyond the Lévy framework, Dassios and Nagaradjasarma [29] and Fusai et al [39] obtain explicit prices for Asian options under the square root asset price dynamics ( [55] consider a modified version with independent jumps added), whereas Cai et al [15] and Sesana et al [64] develop, respectively, an efficient asymptotic expansion and a quadrature method applicable to the generalized constant elasticity of variance (CEV) diffusion.…”

“…The special case of γ = 1 corresponds to the well known square root process of Cox and Ross [25], whereas when γ = 2 we obtain the lognormal model. The CEV diffusion with γ = 1 has been applied to the pricing of arithmetic Asian options in Dassios and Nagaradjasarma [29], Fusai et al [39] and, more recently, in its general form in Cai et al [15], Sesana et al [64] and Cai et al [16].…”

The Pricing of Freight Options with Stochastic Volatilities

“…Today few of the studies deal with the pricing of Asian-style options traded in freight derivatives market. But several approaches have been attempted to obtain pricing formulas for the pricing Asian-style options in financial market, assuming the underlying under the geometric Brownian motion hypothesis (Fusai and Roncoroni, 2008). …”