Abstract
We propose an extension of the Gamma -OU Barndorff-Nielsen and Shephard model taking into account jump clustering phenomena. We assume that the intensity process of the Hawkes driver coincides, up to a constant, with the variance process. By applying the theory of continuous-state branching processes with immigration, we prove existence and uniqueness of strong solutions of the SDE governing the asset price dynamics. We propose a measure change of self-exciting Esscher type in order to describe the relation between the risk-neutral and the historical dynamics, showing that the Gamma -OU Hawkes framework is stable under this probability change. By exploiting the affine features of the model we provide an explicit form for the Laplace transform of the asset log-return, for its quadratic variation and for the ergodic distribution of the variance process. We show that the proposed model exhibits a larger flexibility in comparison with the Gamma -OU model, in spite of the same number of parameters required. We calibrate the model on market vanilla option prices via characteristic function inversion techniques, we study the price sensitivities and propose an exact simulation scheme. The main financial achievement is that implied volatility of options written on VIX is upward shaped due to the self-exciting property of Hawkes processes, in contrast with the usual downward slope exhibited by the Gamma -OU Barndorff-Nielsen and Shephard model.
Highlights
In recent years, the implied volatility indices, such as VIX in the US and V2X in Europe, proved themselves to be key financial instruments for investment strategies, hedging and indicators of the “stress” on the market [35]
In their celebrated papers Barndorff-Nielsen and Shephard [5,6] propose a stochastic volatility model of Ornstein–Uhlenbeck type driven by a subordinator; by considering as a concrete specification for the subordinator a compound Poisson process with exponentially distributed jumps size, they obtain a model where both the variance and the log-returns are driven by the same jump process
From this plot is possible to see that the proposed method exhibits the optimal convergence rate of 0.5 typical of the exact methods, contrarily to the Euler scheme, whose convergence rate is smaller. In this concluding section we want to provide a systematic comparison of the properties of the proposed -OU Hawkes volatility model and the two most similar stochastic volatility models available in the literature, i.e. the Barndorff-Nielsen and Shephard (BNS) and Heston models
Summary
The implied volatility indices, such as VIX in the US and V2X in Europe, proved themselves to be key financial instruments for investment strategies, hedging (see [9,55]) and indicators of the “stress” on the market [35]. Duffie et al [22] generalize the jump-diffusion framework by including a stochastic intensity for the jump processes both in the asset returns and the volatility dynamics, no self-exciting example is proposed in their paper. Our idea is to build up our model as an extension of the Barndorff-Nielsen and Shephard (BNS) model in order to include jumps clusters in both the volatility and the stock return dynamics In their celebrated papers Barndorff-Nielsen and Shephard [5,6] propose a stochastic volatility model of Ornstein–Uhlenbeck type driven by a subordinator; by considering as a concrete specification for the subordinator a compound Poisson process with exponentially distributed jumps size, they obtain a model where both the variance and the log-returns are driven by the same jump process.
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