Option pricing for rough Heston model using numerical methods

The value of an option is largely affected by the underlying assumptions or models, such as the modelling of the volatility process. Fractional Brownian motion has been shown to be able to accurately model and forecast volatility processes displayed in the financial market. The key attribute of modelling the empirical volatility using the fractional Brownian motion is its rough movement nature which is governed by a parameter called Hurst parameter H with the valid range of H ∈ (0,0.5) to display the roughness effect. In response to the development, we study the option pricing methods of rough volatility model to price derivatives such as the widely acceptable option–S&P 500 (SPX) option. This thesis will focus on the option pricing methods under a particular rough volatility model called rough Heston model. The main problem of this study is that the characteristic function of the rough Heston model contains a fractional Riccati equation which has no closed-form solution. Solving the fractional Riccati equation using the standard iterative method (fractional Adams-Bashforth-Moulton method) would require O(N2) time complexity where N is the number of steps of the standard method. If Nc is the number of steps in the numerical integration of the Fourier inversion method, the computational cost would further increase to O(N2Nc) time complexity when fractional Adams-Bashforth-Moulton is used as the medium to price option under rough Heston model. The huge computational cost on the computation of option price under the rough Heston model would undoubtedly be a barrier to most practitioners. The main objectives of this study are to improve an existing approximation method called Pad´e approximant to approximate fractional Riccati equation’s solution and construct an approximation formula for option price without involving the characteristic function of rough Heston model. The main contribution of our study is that we have modified and improved an existing Pad´e approximant such that it can accurately approximate the solutions of fractional Riccati equation on the Hurst parameter range of H ∈ (0,0.5) unlike the Pad´e approximant from previous study where its accuracy will increasingly deteriorate when the Hurst parameter H increases up to 0.5. The time complexity of modified Pad´e approximant is kept at O(1) time complexity. In addition, we have also constructed an approximation option pricing formula under rough Heston model. Specifically, the method utilises the decomposition formula of option price under certain stochastic volatility, and depending on the structure of forward variance curve used, the approximation formula would require O(1) or O(nf ) time complexity to compute the option value where nf is the number of integration steps. The result of the numerical experiment has shown that the methods are capable of matching the SPX options very accurately in a non-extreme market condition and moderately accurate in an extreme market condition, and most importantly, the option pricing method can be computed in a time-efficient manner.

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