Pricing American Options under Stochastic Volatility: A New Method Using Chebyshev Polynomials to Approximate the Early Exercise Boundary

This paper presents a new numerical method for pricing American call options when the volatility of the price of the underlying stock is stochastic. By exploiting a log-linear relationship of the optimal exercise boundary with respect to volatility changes, we derive an integral representation of an American call price and the early exercise premium which holds under stochastic volatility. This representation is used to develop a numerical method for pricing the American options based on an approximation of the optimal exercise boundary by Chebyshev polynomials. Numerical results show that our numerical approach can quickly and accurately price American call options both under stochastic and/or constant volatility.American call option, Stochastic volatility, Early exercise boundary, Chebyshev polynomials

American options under stochastic volatility

Tese de mestrado em Matemática Financeira, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2016An option is a contract that gives the holder the right to buy, in the case of a call, or sell, in the case of a put, an underlying asset at a pre-determined strike price. A European option allows the holder to exercise the option only on a pre-determined expiration date, while with an American option the holder can exercise the option at any point in time until the maturity date. Options can incorporate dividends, which are a portion of a company's earning distributed to its shareholders, that can be issued as cash payments, as shares of stock or other property. Black and Scholes (1973) derived a closed form solution for the value of European options with constant volatility, while Heston (1993) provides a solution for European options with stochastic volatility. It was proved that assuming constant volatility leads to considerable mispricing. Bakshi, Cao and Chen (1997) did a series of tests comparing the Black and Scholes (1973) with three models which allow for stochastic volatility. They showed that incorporating stochastic volatility reduces the absolute pricing error by 20% to 70%. For example a call option with the price $1:68, under the Black and Scholes model has an error of$0:78, while with a model with stochastic volatility the error is reduced to $0:42. Hence, models that allow the volatility of the underlying asset to be stochastic better describe the market behavior. Unlike European options, American options do not have a closed form solution for its value with constant or stochastic volatility, due to the fact that the price depends on the optimal exercise policy. The models on American options under stochastic volatility can be separated in two approaches: the Partial Differential Equation, PDE, based and the non PDE based. There are various numerical methods to price American options. For example, Brennan and Schwartz (1977) introduced finite difference methods; the least squares Monte Carlo is a model developed by Longstaff and Schwartz (2001), where the model uses simulations of cash flows generated by the option and compare them to the value of immediate exercise to calculate the price. In Beliaeva and Nawalkha (2010) a bivariate tree is used where two independent trees are created for the stock price and for the variance. Broadie and Detemple (1996) developed a method for lower and upper bounds on the prices of American options based on regression coefficients. In the Clarke and Parrott (1999) model they use the Heston PDE, transformed into a non dimensional form, with a multigrid iteration method to solve the problem of option pricing. Detemple and Tian (2002) determine the exercise region by a single exercise boundary under general conditions on the interest rate and the dividend yield and derive a recursive integral equation for the exercise boundary. In this work, we will develop an implementation based on the Heston model with the explicit method. First, we will derive the Heston PDE, showing how it is used in the method described. Then we will test the accuracy of the results, randomly creating options and using the various methods to price them and calculate the errors of each method

Pricing American/Bermudan-style Options under Stochastic Volatility

A method to price American options under a stochastic volatility framework is introduced which is based on Rambharat and Brockwell (2010). We price American options under the Heston and Bates stochastic volatility models where volatility is assumed to be a latent process. The pricing algorithm is based on the least-squares Monte Carlo approach made popular by Longstaff and Schwartz (2001). Information about the volatility of the underlying asset is used to assist in solving the pricing problem. Since volatility is assumed to be a latent, a particle filter is used to estimate the filtering distribution of volatility. A summary vector is constructed which captures the essential features of the filtering distribution. At each time step before maturity, the elements of the summary vector and the current share price are used as explanatory variables in a regression function which estimates the continuation value of the option. Estimating the continuation value assists in finding the optimal time to exercise the option. This pricing approach is benchmarked against a method which assumes volatility is observable. Furthermore, our pricing approach is compared to simpler methods which do not use particle filtering. Results from our numerical experiments suggest the proposed approach produces accurate option prices

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

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

Pricing Options under Heston’s Stochastic Volatility Model via Accelerated Explicit Finite Differencing Methods

We present an acceleration technique, effective for explicit finite difference schemes describing diffusive processes with nearly symmetric operators, called Super-Time-Stepping (STS). The technique is applied to the two-factor problem of option pricing under stochastic volatility. It is shown to significantly reduce the severity of the stability constraint known as the Courant-Friedrichs-Lewy condition whilst retaining the simplicity of the chosen underlying explicit method. For European and American put options under Heston’s stochastic volatility model we demonstrate degrees of acceleration over standard explicit methods sufficient to achieve comparable, or superior, efficiencies to a benchmark implicit scheme. We conclude that STS is a powerful tool for the numerical pricing of options and propose them as the method-of-choice for exotic financial instruments in two and multi-factor models.

Sequential Monte Carlo pricing of American-style options under stochastic volatility models

We introduce a new method to price American-style options on underlying investments governed by stochastic volatility (SV) models. The method does not require the volatility process to be observed. Instead, it exploits the fact that the optimal decision functions in the corresponding dynamic programming problem can be expressed as functions of conditional distributions of volatility, given observed data. By constructing statistics summarizing information about these conditional distributions, one can obtain high quality approximate solutions. Although the required conditional distributions are in general intractable, they can be arbitrarily precisely approximated using sequential Monte Carlo schemes. The drawback, as with many Monte Carlo schemes, is potentially heavy computational demand. We present two variants of the algorithm, one closely related to the well-known least-squares Monte Carlo algorithm of Longstaff and Schwartz [The Review of Financial Studies 14 (2001) 113-147], and the other solving the same problem using a "brute force" gridding approach. We estimate an illustrative SV model using Markov chain Monte Carlo (MCMC) methods for three equities. We also demonstrate the use of our algorithm by estimating the posterior distribution of the market price of volatility risk for each of the three equities.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS286 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

American options under stochastic volatility via a transformation procedure

Tese de mestrado em Matemática Financeira, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2017Nesta tese explora-se o pricing the opções Americanas através de um Transformation Procedure, tendo por base o modelo de volatilidade estocástica de Heston. Dado que a informação empírica mostra que o preço das ações contém variações na sua volatilidade, principalmente devido ao denominado efeito de alavancagem, esta tese incorpora um processo estocástico para volatilidade além de para o processo do ativo subjacente, estando estes correlacionados, sendo que nos modelos mais simples é típico a volatilidade ser determinística. Para resolver a equação de derivadas parciais associada ao modelo de Heston um método de diferenças finitas é utilizado complementado por condições de fronteira apropriadas para uma opção de venda. A utilização do método de diferenças finitas é instrumental para posteriormente através das suas partições no tempo conseguir que o preço seja solução para uma opção Americana, estando este sujeito a uma barreira-de-exercício, obtida através de um Transformation Procedure baseado na derivada da opção em relação ao seu preço, operando ao longo de várias iterações, contanto a tese com a prova de funcionalidade e uma ilustração deste Transformation Procedure. Esta tese também explora as condições de estabilidade numérica de acordo com a relação entre os parâmetros e as partições e também a precisão do método para diferentes partições das variáveis ao compará-lo com a solução de Heston para opções Europeias. Finalmente também é explorada a sensibilidade do preço das opções a diferentes variáveis, o efeito do preço quando introduzida a volatilidade estocástica face ao modelo determinístico e é explorada a eficiência face á precisão com a alteração de diferentes parâmetros.Empirical data shows that volatility of asset prices is not constant, although the basic derivative pricing settings do not take this into account, and so stochastic volatility models are more capable of providing reliable asset prices. Pricing assets under stochastic volatility in American option setting provides a bigger challenge when compared to European option setting. This thesis attempts to provide prices for American options under stochastic volatility by first constructing a optimal exercise boundary followed by an asset price through a transformation procedure. First, the baseline European pricing model is constructed and tested for accuracy and numerical stability. Then the procedure is described, its guarantees for convergence are elaborated and the method is desiccated through an illustration. Lastly, the method is explored to give insights on how the option behaves when its parameters are changed, and its speed is tested in different computational settings

Reduced Order Models for Pricing European and American Options under Stochastic Volatility and Jump-Diffusion Models

European options can be priced by solving parabolic partial(-integro) differential equations under stochastic volatility and jump-diffusion models like Heston, Merton, and Bates models. American option prices can be obtained by solving linear complementary problems (LCPs) with the same operators. A finite difference discretization leads to a so-called full order model (FOM). Reduced order models (ROMs) are derived employing proper orthogonal decomposition (POD). The early exercise constraint of American options is enforced by a penalty on subset of grid points. The presented numerical experiments demonstrate that pricing with ROMs can be orders of magnitude faster within a given model parameter variation range