The Rough Hawkes Heston Stochastic Volatility Model
Copyright policy )The Rough Hawkes Heston Stochastic Volatility Model
AbstractWe study an extension of the Heston stochastic volatility model that incorporates rough volatility and jump clustering phenomena. In our model, named the rough Hawkes Heston stochastic volatility model, the spot variance is a rough Hawkes‐type process proportional to the intensity process of the jump component appearing in the dynamics of the spot variance itself and the log returns. The model belongs to the class of affine Volterra models. In particular, the Fourier‐Laplace transform of the log returns and the square of the volatility index can be computed explicitly in terms of solutions of deterministic Riccati‐Volterra equations, which can be efficiently approximated using a multi‐factor approximation technique. We calibrate a parsimonious specification of our model characterized by a power kernel and an exponential law for the jumps. We show that our parsimonious setup is able to simultaneously capture, with a high precision, the behavior of the implied volatility smile for both S&P 500 and VIX options. In particular, we observe that in our setting the usual shift in the implied volatility of VIX options is explained by a very low value of the power in the kernel. Our findings demonstrate the relevance, under an affine framework, of rough volatility and self‐exciting jumps in order to capture the joint evolution of the S&P 500 and VIX.
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR], [QFIN.PR]Quantitative Finance [q-fin]/Pricing of Securities [q-fin.PR], 330, Stochastic models in economics, leverage effect, rough volatility, [QFIN.PR] Quantitative Finance [q-fin]/Pricing of Securities [q-fin.PR], [QFIN.CP]Quantitative Finance [q-fin]/Computational Finance [q-fin.CP], 510, FOS: Economics and business, joint calibration of S&, JEL: C - Mathematical and Quantitative Methods/C.C6 - Mathematical Methods • Programming Models • Mathematical and Simulation Modeling/C.C6.C63 - Computational Techniques • Simulation Modeling, Derivative securities (option pricing, hedging, etc.), FOS: Mathematics, Stochastic volatility, Leverage effect, stochastic volatility, joint calibration of S\&P 500 and VIX smiles, P 500 and VIX smiles, affine Volterra processes, Rough volatility, jump clusters, [QFIN.CP] Quantitative Finance [q-fin]/Computational Finance [q-fin.CP], Jump processes on discrete state spaces, Probability (math.PR), Jump clusters, JEL: G - Financial Economics/G.G1 - General Financial Markets/G.G1.G13 - Contingent Pricing • Futures Pricing, Mathematical Finance (q-fin.MF), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], joint calibration of S&amp, affine Volterra processes, Hawkes processes, jump clusters, joint calibration of S&P 500 and VIX smiles, leverage effect, rough volatility, stochastic volatility, VIX, Quantitative Finance - Mathematical Finance, VIX, Point processes (e.g., Poisson, Cox, Hawkes processes), Hawkes processes, JEL: G - Financial Economics/G.G1 - General Financial Markets/G.G1.G12 - Asset Pricing • Trading Volume • Bond Interest Rates, Mathematics - Probability
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR], [QFIN.PR]Quantitative Finance [q-fin]/Pricing of Securities [q-fin.PR], 330, Stochastic models in economics, leverage effect, rough volatility, [QFIN.PR] Quantitative Finance [q-fin]/Pricing of Securities [q-fin.PR], [QFIN.CP]Quantitative Finance [q-fin]/Computational Finance [q-fin.CP], 510, FOS: Economics and business, joint calibration of S&, JEL: C - Mathematical and Quantitative Methods/C.C6 - Mathematical Methods • Programming Models • Mathematical and Simulation Modeling/C.C6.C63 - Computational Techniques • Simulation Modeling, Derivative securities (option pricing, hedging, etc.), FOS: Mathematics, Stochastic volatility, Leverage effect, stochastic volatility, joint calibration of S\&P 500 and VIX smiles, P 500 and VIX smiles, affine Volterra processes, Rough volatility, jump clusters, [QFIN.CP] Quantitative Finance [q-fin]/Computational Finance [q-fin.CP], Jump processes on discrete state spaces, Probability (math.PR), Jump clusters, JEL: G - Financial Economics/G.G1 - General Financial Markets/G.G1.G13 - Contingent Pricing • Futures Pricing, Mathematical Finance (q-fin.MF), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], joint calibration of S&amp, affine Volterra processes, Hawkes processes, jump clusters, joint calibration of S&P 500 and VIX smiles, leverage effect, rough volatility, stochastic volatility, VIX, Quantitative Finance - Mathematical Finance, VIX, Point processes (e.g., Poisson, Cox, Hawkes processes), Hawkes processes, JEL: G - Financial Economics/G.G1 - General Financial Markets/G.G1.G12 - Asset Pricing • Trading Volume • Bond Interest Rates, Mathematics - Probability
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