The COS method for option valuation under the SABR dynamics

In this paper, we consider the COS method for pricing European and Bermudan options under the stochastic alpha beta rho (SABR) model. In the COS pricing method, we make use of the characteristic function of the discrete forward process. We observe second-order convergence by using a second-order Taylor scheme in the discretization, or by using Richardson extrapolation in combination with a Euler–Maruyama discretization on the forward process. We also consider backward stochastic differential equations under the SABR model, using the discretized forward process and Fourier-cosine expansion for the occurring expectations. For this purpose, we extend the so-called BCOS method from one to two dimensions

On the data-driven COS method

In this paper, we present the data-driven COS method, ddCOS. It is a Fourier-based finan- cial option valuation method which assumes the availability of asset data samples: a char- acteristic function of the underlying asset probability density function is not required. As such, the presented technique represents a generalization of the well-known COS method [1]. The convergence of the proposed method is O(1 / √ n ) , in line with Monte Carlo meth- ods for pricing financial derivatives. The ddCOS method is then particularly interesting for density recovery and also for the efficient computation of the option’s sensitivities Delta and Gamma. These are often used in risk management, and can be obtained at a higher accuracy with ddCOS than with plain Monte Carlo methods

On the data-driven COS method

In this paper, we present the data-driven COS method, ddCOS. It is a Fourier-based financial option valuation method which assumes the availability of asset data samples: a characteristic function of the underlying asset probability density function is not required. As such, the presented technique represents a generalization of the well-known COS method [1]. The convergence of the proposed method is in line with Monte Carlo methods for pricing financial derivatives. The ddCOS method is then particularly interesting for density recovery and also for the efficient computation of the option's sensitivities Delta and Gamma. These are often used in risk management, and can be obtained at a higher accuracy with ddCOS than with plain Monte Carlo methods

Option data and modeling BSM implied volatility

This contribution to the Handbook of Computational Finance, Springer-Verlag, gives an overview on modeling implied volatility data. After introducing the concept of Black-Scholes-Merton implied volatility (IV), the empirical stylized facts of IV data are reviewed. We then discuss recent results on IV surface dynamics and the computational aspects of IV. The main focus is on various parametric, semi- and nonparametric modeling strategies for IV data, including ones which respect no-arbitrage bounds.Implied volatility

A Non-Gaussian Option Pricing Model with Skew

Closed form option pricing formulae explaining skew and smile are obtained within a parsimonious non-Gaussian framework. We extend the non-Gaussian option pricing model of L. Borland (Quantitative Finance, {\bf 2}, 415-431, 2002) to include volatility-stock correlations consistent with the leverage effect. A generalized Black-Scholes partial differential equation for this model is obtained, together with closed-form approximate solutions for the fair price of a European call option. In certain limits, the standard Black-Scholes model is recovered, as is the Constant Elasticity of Variance (CEV) model of Cox and Ross. Alternative methods of solution to that model are thereby also discussed. The model parameters are partially fit from empirical observations of the distribution of the underlying. The option pricing model then predicts European call prices which fit well to empirical market data over several maturities.Comment: 37 pages, 11 ps figures, minor changes, typos corrected, to appear in Quantitative Financ

Spectral Decomposition of Option Prices in Fast Mean-Reverting Stochastic Volatility Models

Using spectral decomposition techniques and singular perturbation theory, we develop a systematic method to approximate the prices of a variety of options in a fast mean-reverting stochastic volatility setting. Four examples are provided in order to demonstrate the versatility of our method. These include: European options, up-and-out options, double-barrier knock-out options, and options which pay a rebate upon hitting a boundary. For European options, our method is shown to produce option price approximations which are equivalent to those developed in [5]. [5] Jean-Pierre Fouque, George Papanicolaou, and Sircar Ronnie. Derivatives in Financial Markets with Stochas- tic Volatility. Cambridge University Press, 2000

Essays about Option Valuation under Stochastic Interest Rates

This thesis consists of three essays on the valuation of options under stochastic interest rates. In the first essay we examine multivariate models where the stochastic process of a log-normally distributed underlying depends on the evolution of correlated interest rate processes. There the correlation structure can change by constant factor volatilities, which influences not only the values of financial instruments but also their hedge strategies. In this model class we propose a unified framework for the pricing and hedging of chooser options under stochastic interest rates. Chooser options are exotic derivatives who give the owner the right to enter at their exercise date a call or a put option with the same underlying. The chosen multivariate framework allows to derive closed form solutions of the arbitrage price for different specifications of chooser options such as different strike prices or time to maturities. The second essay deals with the so called convexity correction of swap rates. A convexity correction needs to be computed in the case of constant maturity swaps (CMS) for example. The expectation is taken under a different measure than the assets martingale measure. Then, the expectation is not the forward value but the forward value plus a convexity correction. One approach in the literature suggests, that the convexity correction is the price of a static portfolio of plain-vanilla swaptions. This portfolio approach has the advantage that the volatility cube can be incorporated by using a stochastic or local volatility models, but it is the solution of an integral over an infinite number of strike prices. We propose an algorithm to approximate the replication portfolio with a finite number and therefore a discrete set of strike prices. The accuracy of the method is examined using numerical examples and different valuation models as well as different sets of strike prices. The modeling of multi-asset options within an interest rate model is the topic of the third essay. There, we consider the joint dynamic of a basket of n-assets with the application to CMS spread options in mind. Therefore we use a Swap Market Model (SMM) with deterministic volatility and the SABR model with stochastic volatility. Using the Markovian Projection methodology we approximate multivariate SMM/SABR dynamic with a univariate SMM/SABR dynamic to price caps and floors in closed form. This enables us to consider not only the asset correlation but, in the case of the SABR model, as well the skew, the cross-skew and the decorrelation in our approximation. If for example, spread options are considered the latter is not possible in alternative approximations. We illustrate the method by considering the example where the underlyings are two constant maturity swap rates. There we examine the influence of the swaption volatility cube on CMS spread options and compare our approximation formulae to results obtained by Monte Carlo simulation and a copula approach

Valuing options in Heston's stochastic volatility model: Another analytical approach

We are concerned with the valuation of European options in Heston's stochastic volatility model with correlation. Based on Mellin transforms we present new closed-form solutions for the price of European options and hedging parameters. In contrast to Fourier-based approaches where the transformation variable is usually the log-stock price at maturity, our framework focuses on transforming the current stock price. Our solution has the nice feature that similar to the approach of Carr and Madan (1999) it requires only a single integration. We make numerical tests to compare our results to Heston's solution based on Fourier inversion and investigate the accuracy of the derived pricing formulae. --Stochastic volatility,European option,Mellin transform

Foreign Exchange Option Valuation under Stochastic Volatility

>Magister Scientiae - MScThe case of pricing options under constant volatility has been common practise for decades. Yet market data proves that the volatility is a stochastic phenomenon, this is evident in longer duration instruments in which the volatility of underlying asset is dynamic and unpredictable. The methods of valuing options under stochastic volatility that have been extensively published focus mainly on stock markets and on options written on a single reference asset. This work probes the effect of valuing European call option written on a basket of currencies, under constant volatility and under stochastic volatility models. We apply a family of the stochastic models to investigate the relative performance of option prices. For the valuation of option under constant volatility, we derive a closed form analytic solution which relaxes some of the assumptions in the Black-Scholes model. The problem of two-dimensional random diffusion of exchange rates and volatilities is treated with present value scheme, mean reversion and non-mean reversion stochastic volatility models. A multi-factor Gaussian distribution function is applied on lognormal asset dynamics sampled from a normal distribution which we generate by the Box-Muller method and make inter dependent by Cholesky factor matrix decomposition. Furthermore, a Monte Carlo simulation method is adopted to approximate a general form of numeric solution The historic data considered dates from 31 December 1997 to 30 June 2008. The basket contains ZAR as base currency, USD, GBP, EUR and JPY are foreign currencies

Interest-rate models: an extension to the usage in the energy market and pricing exotic energy derivatives.

In this thesis, we review various popular pricing models in the interest-rate market. Among these pricing models, we choose the LIBOR Market model (LMM) as the benchmark model. Based on market practice experience, we also develop a pricing model named the “Market volatility model”. By pricing vanilla interest-rate options such as interest-rate caps and swaptions, we compare the performance of our Market volatility model to that of the LMM. It is proved that the Market Volatility model produce comparable results to the LMM, while its computing efficiency largely exceeds that of the LMM. Following the recent rapid development in the commodity market, in particular the energy market, we attempt to extend the use of our proposed Market volatility model from the interest-rate market to the energy market. We prove that the Market Volatility model is capable of pricing various energy derivative under the assumption of absence of the convenience yield. In addition, we propose a new type of exotic energy derivative which has a flexible option structure. This energy derivative is named as the Flex-Asian spread options (FASO). We give examples of different option structures within the FASO framework and use the Market volatility model to generate option prices and greeks for each structure. Although the Market volatility model can be used to price various energy derivatives based on oil/gas contracts, it is not compatible with the structure of one of the most advanced derivatives in the energy market, the storage option. We modify the existing pricing model for storage options and use our own 3D-binomial tree approach to price gas storage contracts. By doing these, we improve the performance of the traditional storage model