Pricing early-exercise and discrete barrier options by fourier-cosine series expansions

Abstract

We present a pricing method based on Fourier-cosine expansions for early-exercise and discretely-monitored barrier options. The method works well for exponential Lévy asset price models. The error convergence is exponential for processes characterized by very smooth (C ∞ [a, b] ∈ R) transitional probability density functions. The computational complexity is O((M − 1)N log N ) with N a (small) number of terms from the series expansion, and M , the number of early-exercise/monitoring dates. This paper is the follow-up of [22] in which we presented the impressive performance of the Fourier-cosine series method for European options.

The Stochastic Grid Bundling Method: Efficient pricing of Bermudan options and their Greeks

“…Figure 3(a) and (b) show the convergence with an increasing number of bundles for the two methods. A highly accurate option reference price is computed using the COS method [10]. Figure 3(c) compares the total computational time, i.e.…”

“…As is well known, it is possible to reduce this problem to a one-dimensional problem, which can then be priced accurately using the COS method [10], thus providing a benchmark result for the algorithm. A geometric average put option on d assets has intrinsic value: The asset prices are assumed to follow correlated geometric Brownian motion processes, i.e.…”

Pricing of early-exercise Asian options under Lévy processes based on Fourier cosine expansions

“…In [9,10] the COS method was developed for the computation of continuation value c( y m−1 , t m−1 ) and option price v( y 0 , t 0 ), for Bermudan options, based on the insight that the characteristic function of the underlying Lévy asset price process is known.…”

“…Truncating the integration range, gives us approximationĉ as, [10], and the calculation of b 2 will be explained in Section 2.4. By applying the chain rule to the joint conditional density function in (5), we find…”

“…As pointed out in [10] At the final stage of the algorithm, we need to add up all k · p elements, i.e.…”