Closed-Form Formulae for European Options Under Three-Factor Models
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Closed-Form Formulae for European Options Under Three-Factor Models Joanna Goard1 Received: 15 June 2018 / Revised: 15 August 2018 / Accepted: 28 December 2018 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this paper, we derive new closed-form valuations to European options under threefactor hybrid models that include stochastic interest rates and stochastic volatility and incorporate a nonzero covariance structure between factors. We make novel use of the empirically proven 3/2 stochastic volatility model with a time-dependent drift in which we are free to choose the moving reversion target. This model has been shown by many authors to empirically outperform other volatility models in maximising model fit. We also improve the valuation of European options under the Heston volatility and Cox, Ingersoll, Ross interest rate model, recently published in the literature, by replacing open-form infinite series with closed-form analytic expressions. For completeness, we also add a fuller covariance structure in this setting and detail closed-form valuations for options. The inclusion of nonzero covariances amongst the factors can significantly improve option pricing by allowing for a wider variety of market behaviour. The solutions are derived by firstly formulating the price of a European call option in terms of the corresponding characteristic function of the underlying price and then determining a partial differential equation for the characteristic function. By including empirically proven models into our analysis, the options formulae could provide more realistic prices for investors and practitioners. Keywords European option valuation · Stochastic volatility · Stochastic interest rates · Heston model · 3/2 model Mathematics Subject Classification 91G20 · 35K10 · 35R60
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Joanna Goard [email protected] University of Wollongong, Wollongong, NSW, Australia
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J. Goard
1 Introduction Many of the assumptions of the famous Black–Scholes (BS) model by Black and Scholes [9] are not supported by empirical evidence. For example, the assumed asset price dynamics with constant volatility and interest rates are inadequate in describing features of asset returns such as skewness, leptokurtosis and pronounced conditional heteroskedasticity. For this reason, stochastic volatility models were introduced and have become particularly popular for derivative pricing and hedging over the last 25 years. Empirical evidence on underlying asset prices and on option prices strongly suggests that asset volatility is stochastic. Further, the research literature supports the use of stochastic volatility in an effort to reproduce the implied volatility smile observed in markets and to avoid many of the shortcomings of the constant variance diffusion assumed by the Black–Scholes model. Work on stochastic volatility models includes the early efforts of Hull and White [25], Scott [33] and Wiggins [36]. Assuming that volatility is u
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