American options under stochastic volatility: control variates, maturity randomization & multiscale asymptotics

Abstract

American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of American option price under stochastic volatility models. We achieve this by using the maturity randomization method known as Canadization. The volatility process is characterized by fast and slow scale fluctuating factors. In particular, we study the case of an American put with a single underlying asset and use perturbative expansion techniques to approximate its price as well as the optimal exercise boundary up to the first order. We then use the approximate optimal exercise boundary formula to price American put via Monte Carlo. We also develop efficient control variates for our simulation method using martingales resulting from the approximate price formula. A numerical study is conducted to demonstrate that the proposed method performs better than the least squares regression method popular in the financial industry, in typical settings where values of the scaling parameters are small. Further, it is empirically observed that in the regimes where scaling parameter value is equal to unity, fast and slow scale approximations are equally accurate.

Kriging metamodels and experimental design for Bermudan option pricing

“…Compare for the first parameter set a kriging model with LHS design D that uses N = 10, 000 (and estimates option value at 16.06) relative to the LSMC model with N = 128, 000 (and estimates option value of 16.05). For the set of examples from [1] we observe that the space-filling design does not perform as well, which most likely is due to having the inefficient rectangular LHS domain X = [30, 50] × [−4, 1]. Still, the sequential design method wins handily.…”

“…We consider the asset Put h(t, x 1 , x 2 ) = e −rt (K − x 1 ) + ; exercise opportunities are spaced out by ∆t δt. Related experiments have been carried out in [35], and recently in [1]. Table 5 lists three different parameter sets.…”

Discrete-type approximations for non-Markovian optimal stopping problems: Part I

“…In this case, one cannot interpret (X, V ) as an "augmented" finite-dimensional Markov process due to non-trivial correlations associated with FBM and hence a concrete solution of the related optimal stopping problem is very challenging. We stress even optimal stopping problems based on finite-dimensional "augmented" Markovian-type models (X, V ) (e.g CIR-type models driven by Brownian motion) are not easy to solve it (see e.g [31,1]). In practice, the volatility V is not directly observed so that it has to be approximated.…”

Pricing commodity futures options in the Schwartz multi factor model with stochastic volatility: An asymptotic method

“…The zero order term in this expansion corresponds to the classical Black-Scholes term, where the constant spot volatility is replaced by the long term average volatility. This technique has been extended to cover other volatility dynamics, including Heston volatility, in as well as for real options and optimal exercise timing in and Agarwal et al (2016). Recently, Barsotti and Pontier (2016) used the asymptotic approach to study the optimal capital structure in a Merton structural firm model with stochastic volatility.…”