Abstract
Jump-diffusion processes have been utilised to capture the leptokurtic nature of asset returns and to fit the market observed option volatility skew with great success. These models can be calibrated to historical share price data or forward-looking option market data. In this chapter, we infer South African equity option prices using the Bayesian inference framework. This approach allows one to attain uncertainties in the parameters of the calibrated models and confidence intervals with any predictions produced with the models. We calibrate the one-dimensional Merton jump-diffusion model to European put and call option data on the All-Share price index using MCMC methods: the MALA, HMC, and the NUTS. Our approach produces a distribution of the jump-diffusion model parameters, which can be used to build economic scenario generators and price exotic options such as those embedded in life insurance contracts. The empirical results show that our approach can, on test data, exactly price all put option prices regardless of their moneyness, with slight miss-pricing on very deep in the money calls.
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