Pricing Discretely Monitored Asian Options by Maturity Randomization

Abstract

We present methodologies to price discretely monitored Asian options when the underlying evolves according to a generic Lévy process. For geometric Asian options we provide closed-form solutions in terms of the Fourier transform and we study in particular these formulas in the Lévy-stable case. For arithmetic Asian options we solve the valuation problem by recursive integration and derive a recursive theoretical formula for the moments to check the accuracy of the results. We compare the implementation of our method to Monte Carlo simulation implemented with control variates and using di¤erent parametric Lévy processes. We also discuss model-risk issues.JEL Classi…cation: G13, C63

Hedging of Asian options under exponential Lévy models: computation and performance

“…This powerful approach allows us to gauge explicitly the precision of the scheme and, by further exploiting its smooth convergence in the number of discretization points of the Fourier transforms, achieve highly accurate results as we show in the following section. Alternatively, one may consider using the maturity randomization algorithm presented in Fusai et al (2011) or the parallel wavelet-based procedure of Corsaro et al (2015) for potential speed-up gains with increasing monitoring frequency. Richardson extrapolation techniques also provide substantial CPU power saving when dealing with a large number of monitoring dates.…”

“…) Without loss of generality, we focus here on the call option with fixed strike price satisfying (2). Then, given the price of the fixed strike call, the price of the floating strike put option can be obtained using a symmetry relationship derived in Eberlein and Papapantoleon (2005) for underlyings driven by exponential Lévy models, while the prices of the fixed strike put and floating strike call options can be obtained via standard put-call parity (e.g., see Fusai et al, 2011).…”

“…Over the last few years fast and accurate algorithms emerged for pricing European discrete arithmetic Asian options beyond the traditional lognormal asset price model; we mention, amongst others,Černý and Kyriakou (2011), Fusai and Meucci (2008), , Fusai et al (2011), Marena et al (2013) and Zhang and Oosterlee (2013). Whereas the pricing of Asian options is well-understood, an additional interesting direction to explore is that of the computation of the price sensitivities, which poses the main challenge of proving validity of the interchange of differentiation and the integral price representation.…”

Efficient Pricing of European-Style Asian Options under Exponential Lévy Processes Based on Fourier Cosine Expansions

“…Advanced pricing methods for options on the arithmetic average are based on a recursive integration procedure in which the transitional probability density function of the log-return of the sum of asset prices is approximated; see [7,3,8,17,15,14]. In [7,3] an FFT and inverse FFT have been incorporated into the procedure to approximate the governing densities.…”

“…The total computational complexity in [15] was O(Mn 2 ), with M the number of monitoring dates and n the number of points used in the quadrature. The method in [15] is improved in [14], in which it is shown that the Asian option value can be derived by a price recursion or density recursion procedure. It is transformed into a complexvalued frequency-domain representation via the z-transform.…”