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 m...
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| 主要作者: | |
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| 格式: | Thesis |
| 语言: | English |
| 出版: |
2021
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| 主题: | |
| 在线阅读: | http://psasir.upm.edu.my/id/eprint/104713/1/SIOW%20WOON%20JENG%20-IR.pdf |
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| 总结: | 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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