Pricing Step Options under the CEV and other Solvable Diffusion Models

We consider a special family of occupation-time derivatives, namely proportional step options introduced by Linetsky in [Math. Finance, 9, 55--96 (1999)]. We develop new closed-form spectral expansions for pricing such options under a class of nonlinear volatility diffusion processes which includes the constant-elasticity-of-variance (CEV) model as an example. In particular, we derive a general analytically exact expression for the resolvent kernel (i.e. Green's function) of such processes with killing at an exponential stopping time (independent of the process) of occupation above or below a fixed level. Moreover, we succeed in Laplace inverting the resolvent kernel and thereby derive newly closed-form spectral expansion formulae for the transition probability density of such processes with killing. The spectral expansion formulae are rapidly convergent and easy-to-implement as they are based simply on knowledge of a pair of fundamental solutions for an underlying solvable diffusion process. We apply the spectral expansion formulae to the pricing of proportional step options for four specific families of solvable nonlinear diffusion asset price models that include the CEV diffusion model and three other multi-parameter state-dependent local volatility confluent hypergeometric diffusion processes.Comment: 30 pages, 16 figures, submitted to IJTA

Modelling FX smile : from stochastic volatility to skewness

Imperial Users onl

Recursive marginal quantization: extensions and applications in finance

Quantization techniques have been used in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and the efficient calibration of large derivative books. Recursive marginal quantization of an Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This algorithm is generalized and it is shown that it is possible to perform recursive marginal quantization for two higher-order schemes: the Milstein scheme and a simplified weak-order 2.0 scheme. Furthermore, the recursive marginal quantization algorithm is extended by showing how absorption and reflection at the zero boundary may be incorporated. Numerical evidence is provided of the improved weak-order convergence and computational efficiency for the geometric Brownian motion and constant elasticity of variance models by pricing European, Bermudan and barrier options. The current theoretical error bound is extended to apply to the proposed higher-order methods. When applied to two-factor models, recursive marginal quantization becomes computationally inefficient as the optimization problem usually requires stochastic methods, for example, the randomized Lloyd’s algorithm or Competitive Learning Vector Quantization. To address this, a new algorithm is proposed that allows recursive marginal quantization to be applied to two-factor stochastic volatility models while retaining the efficiency of the original Newton-Raphson gradientdescent technique. The proposed method is illustrated for European options on the Heston and Stein-Stein models and for various exotic options on the popular SABR model. Finally, the recursive marginal quantization algorithm, and improvements, are applied outside the traditional risk-neutral pricing framework by pricing long-dated contracts using the benchmark approach. The growth-optimal portfolio, the central object of the benchmark approach, is modelled using the time-dependent constant elasticity of variance model. Analytic European option prices are derived that generalize the current formulae in the literature. The time-dependent constant elasticity of variance model is then combined with a 3/2 stochastic short rate model to price zerocoupon bonds and zero-coupon bond options, thereby showing the departure from risk-neutral pricing

Essays on option pricing, with applications on interest rates, equities and credit derivatives

JEL Classification: G13This thesis is devoted to option pricing, with applications on interest rates, equities and credit derivatives, and is comprised of three separate and self-contained essays: A Pricing Swaptions under Multifactor Gaussian HJM Models Several approximations have been proposed in the literature for the pricing of European-style swaptions under multifactor term structure models. However, none of them provides an estimate for the inherent approximation error. Until now, only the Edgeworth expansion technique of Collin-Dufresne and Goldstein (2002) is able to characterize the order of the approximation error. Under a multifactor Heath, Jarrow, and Morton (1992) Gaussian framework, this paper proposes a new approximation for European-style swaptions, which is able to set bounds on the magnitude of the approximation error and is based on the conditioning approach initiated by Curran (1994) and Rogers and Shi (1995). All the proposed pricing bounds will arise as a simple by-product of the Nielsen and Sandmann (2002) setup, and will be shown to provide a better accuracy-efficiency trade-off than all the approximations already proposed in the literature. B Pricing of European-style Barrier Options under Stochastic Interest Rates This paper offers an extremely fast and accurate novel methodology for the pricing of (long-term) European-style single barrier options on underlying spot prices driven by a geometric Brownian motion and under the stochastic interest rates framework of Vasiček (1977). The proposed valuation methodology extends the stopping time approach of Kuan and Webber (2003) to a more general setting, and expresses the price of a European-style barrier option in terms of the first passage time density of the underlying asset price to the barrier level. Using several model parameter constellations and option maturities, our numerical results show that the proposed pricing approach is much more accurate and faster than the two-dimensional extended Fortet method of Bernard et al. (2008). C Pricing Credit and Equity Default Swaps under the Jump to Default Extended CEV Model This paper offers a novel methodology for the pricing of credit and equity default swaps under the jump to default extended constant elasticity of variance (JDCEV) model of Carr and Linetsky (2006). The proposed method extends the stopping time approach of Kuan and Webber (2003), and expresses the value of the building blocks of both contracts in terms of the first passage time density of the underlying asset price to the contract triggering level. The numerical results show that the proposed pricing methodology is extremely accurate and much faster than the Laplace transform approach of Mendoza-Arriaga and Linetsky (2011).Esta tese dedica-se ao tema da avaliação de opções, com aplicações a taxas de juro, ações e derivados de crédito, e é composta por três artigos distintos: A Pricing Swaptions under Multifactor Gaussian HJM Models Várias aproximações foram já propostas na literatura para a avaliação de swaptions de estilo Europeu, no âmbito de modelos de taxa de juro multi-fator. Contudo, nenhuma delas fornece uma estimativa para o erro de aproximação subjacente. Até agora, apenas a Edgeworth expansion technique de Collin-Dufresne e Goldstein (2002) é capaz de caracterizar a ordem do erro de aproximação. No âmbito de um modelo Heath, Jarrow e Morton (1992) Gaussiano multi-fator, este artigo propõe uma nova aproximação para swaptions de estilo Europeu, que é capaz de estabelecer limites para a magnitude do erro de aproximação e é baseada na conditioning approach iniciada por Curran (1994) e Rogers e Shi (1995). Todos os limites de preço propostos surgirão como um simples sub-produto da estrutura de Nielsen e Sandmann (2002), e será demonstrado que estes proporcionam um melhor equilíbrio entre precisão e eficiência do que todas as aproximações já propostas na literatura. B Pricing of European-style Barrier Options under Stochastic Interest Rates Este artigo oferece uma nova metodologia, extremamente rápida e precisa, para a avaliação de opções de estilo Europeu com barreira sobre ativos subjacentes caracterizados por um geometric Brownian motion e no âmbito do modelo de taxas de juro estocásticas de Vasiček (1977). A metodologia de avaliação proposta estende a stopping time approach de Kuan e Webber (2003) a uma configuração mais geral, e expressa o preço de uma opção de estilo Europeu com barreira em termos da densidade de probabilidade do primeiro tempo de passagem do preço do ativo subjacente pelo nível da barreira. Utilizando várias configurações de parâmetros e maturidades de opções, os nossos resultados numéricos mostram que a metodologia de avaliação proposta é muito mais precisa e rápida do extended Fortet method bi-dimensional de Bernard et al. (2008). C Pricing Credit and Equity Default Swaps under the Jump to Default Extended CEV Model Este artigo oferece uma nova metodologia para a avaliação de credit e equity default swaps no âmbito do modelo jump to default extended constant elasticity of variance (JDCEV) de Carr e Linetsky (2006). A abordagem proposta estende a stopping time approach de Kuan e Webber (2003), e expressa o valor das componentes de ambos os contratos em termos da densidade de probabilidade do primeiro tempo de passagem do preço do ativo subjacente pelo nível de acionamento do contrato. Os resultados numéricos mostram que a abordagem de avaliação proposta é precisa e muito mais rápida do que a Laplace transform approach de Mendoza-Arriaga e Linetsky (2011)

Recursive marginal quantization of higher-order schemes

© 2018 Informa UK Limited, trading as Taylor & Francis Group. Quantization techniques have been applied in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and efficient calibration of large derivative books. Recursive marginal quantization (RMQ) of the Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This method involves recursively quantizing the conditional marginals of the discrete-time Euler approximation of the underlying process. By generalizing this approach, we show that it is possible to perform RMQ for two higher-order schemes: the Milstein scheme and a simplified weak order 2.0 scheme. We further extend the applicability of RMQ by showing how absorption and reflection at the zero boundary may be incorporated, when necessary. To illustrate the improved accuracy of the higher-order schemes, various computations are performed using geometric Brownian motion and the constant elasticity of variance model. For both models, we provide evidence of improved weak order convergence and computational efficiency. By pricing European, Bermudan and barrier options, further evidence of improved accuracy of the higher-order schemes is demonstrated

A Black-Scholes user's guide to the Bachelier model

To cope with the negative oil futures price caused by the COVID-19 recession, global commodity futures exchanges temporarily switched the option model from Black--Scholes to Bachelier in 2020. This study reviews the literature on Bachelier's pioneering option pricing model and summarizes the practical results on volatility conversion, risk management, stochastic volatility, and barrier options pricing to facilitate the model transition. In particular, using the displaced Black-Scholes model as a model family with the Black-Scholes and Bachelier models as special cases, we not only connect the two models but also present a continuous spectrum of model choices