# Option Pricing with Expansion Methods: New Approaches to Advanced Stochastic Volatility Models and American Options

##### Hu, Jun (2016)

Hu, Jun

Tampere University of Technology

2016

Tieto- ja sähkötekniikan tiedekunta - Faculty of Computing and Electrical Engineering

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http://urn.fi/URN:ISBN:978-952-15-3844-5

##### Tiivistelmä

In the thesis, we aim to develop a new framework for pricing advanced options quickly and accurately. Speciﬁcally, we price European options under stochastic volatility models and American options under the Black – Scholes model using expansion methods, which are widely used in physical sciences.

Efficient and accurate pricing of option contracts has long been the central problem of mathematical ﬁnance. Apart from the classic Black – Scholes model, which has a closedform solution, there is no universally accepted method for pricing of options. The Monte Carlo simulation is general but slow, while ﬁnite-difference and numerical integration (Fourier transform) methods are comparably accurate but are not always applicable to exotic options and non-affine models. Furthermore, although those popular numerical methods can provide decent estimates of option prices, their discrete nature makes it difficult for them to achieve efficiency and accuracy simultaneously. While considerable amount of research has been devoted to these methods, we believe that the potential of expansion methods are underestimated.

The expansion methodology, which is widely used in mathematics and physics, divides an unknown quantity into an inﬁnite and converging series whose neighbouring terms are related by algebraic or differential equations. Therefore, starting from the known leading terms, we can work out the iteration equations and obtain any number of terms in the series, as long as the iteration equations are exact and explicitly solvable. In the context of ﬁnance, the option price can be expanded as an inﬁnite series of analytical functions, which are related by the pricing partial differential equation (PDE). Besides the ﬂexibility to deal with different models and options, the biggest advantage of expansion methods is that, users can evaluate the formulae derived by the author by plugging in parameter values, which greatly reduces computational intensity.

First, we show that European options under stochastic volatility models can be expanded with various pairs of parameters in the volatility process, such as initial volatility, speed of mean-reversion, volatility of volatility and long-term volatility. The methods use powers of parameters as basis functions, and work with small parameter values. To achieve better performance, a modiﬁed version of expansion with initial volatility and volatility of volatility is proposed to reduce the pricing error when the parameters are large. The new method uses bounded basis functions, rather than the unbounded power series, and the numerical results conﬁrm that the promotion from unbounded to bounded greatly improves the ability of expansion methods to approximate option prices. Moreover, symmetry considerations are also helpful for expansion methods. When the scale invariance is broken, we are equipped with one more degree of freedom to ﬁne-tune the convergence, which is not proven or guaranteed.

Then, we show that the non-linear problem of American options under the Black – Scholes model can be solved as a series of special functions that we deﬁned earlier. Such special functions remain in the same family under many operations, making explicit expression of the option prices possible. We formally demonstrate two of the many ways of expansion, which work numerically except in the case of low volatility and high interest rate. Thus, an improved version, is proposed. It treats the Black – Scholes model as an advanced model with an additional operator. The improved method is able to deal with reasonable values of volatility, interest rate, moneyness and maturity. Finally, we outline the possibility of combining advanced models with advanced option types. American options can be treated similarly under many popular models, as long as the extra operators preserve the closedness of the special functions.

The main contribution of the thesis is the demonstration that expansion methods can be used efficiently with non-affine stochastic volatility models and American options, which have no closed-form solutions. Additionally, explicit formulae, instead of formal relations in terms of integrals, are derived and available for reproduction.

Efficient and accurate pricing of option contracts has long been the central problem of mathematical ﬁnance. Apart from the classic Black – Scholes model, which has a closedform solution, there is no universally accepted method for pricing of options. The Monte Carlo simulation is general but slow, while ﬁnite-difference and numerical integration (Fourier transform) methods are comparably accurate but are not always applicable to exotic options and non-affine models. Furthermore, although those popular numerical methods can provide decent estimates of option prices, their discrete nature makes it difficult for them to achieve efficiency and accuracy simultaneously. While considerable amount of research has been devoted to these methods, we believe that the potential of expansion methods are underestimated.

The expansion methodology, which is widely used in mathematics and physics, divides an unknown quantity into an inﬁnite and converging series whose neighbouring terms are related by algebraic or differential equations. Therefore, starting from the known leading terms, we can work out the iteration equations and obtain any number of terms in the series, as long as the iteration equations are exact and explicitly solvable. In the context of ﬁnance, the option price can be expanded as an inﬁnite series of analytical functions, which are related by the pricing partial differential equation (PDE). Besides the ﬂexibility to deal with different models and options, the biggest advantage of expansion methods is that, users can evaluate the formulae derived by the author by plugging in parameter values, which greatly reduces computational intensity.

First, we show that European options under stochastic volatility models can be expanded with various pairs of parameters in the volatility process, such as initial volatility, speed of mean-reversion, volatility of volatility and long-term volatility. The methods use powers of parameters as basis functions, and work with small parameter values. To achieve better performance, a modiﬁed version of expansion with initial volatility and volatility of volatility is proposed to reduce the pricing error when the parameters are large. The new method uses bounded basis functions, rather than the unbounded power series, and the numerical results conﬁrm that the promotion from unbounded to bounded greatly improves the ability of expansion methods to approximate option prices. Moreover, symmetry considerations are also helpful for expansion methods. When the scale invariance is broken, we are equipped with one more degree of freedom to ﬁne-tune the convergence, which is not proven or guaranteed.

Then, we show that the non-linear problem of American options under the Black – Scholes model can be solved as a series of special functions that we deﬁned earlier. Such special functions remain in the same family under many operations, making explicit expression of the option prices possible. We formally demonstrate two of the many ways of expansion, which work numerically except in the case of low volatility and high interest rate. Thus, an improved version, is proposed. It treats the Black – Scholes model as an advanced model with an additional operator. The improved method is able to deal with reasonable values of volatility, interest rate, moneyness and maturity. Finally, we outline the possibility of combining advanced models with advanced option types. American options can be treated similarly under many popular models, as long as the extra operators preserve the closedness of the special functions.

The main contribution of the thesis is the demonstration that expansion methods can be used efficiently with non-affine stochastic volatility models and American options, which have no closed-form solutions. Additionally, explicit formulae, instead of formal relations in terms of integrals, are derived and available for reproduction.

##### Kokoelmat

- Väitöskirjat [4135]