The notation we follow in this article was created referencing to this Wolfram post on the topic, as the notation is more accessible. In this article we won't go into too much detail regarding proofs etc, we highly recommend reading Merton's original paper available in pdf here. In this article we will investigate the following:ġ) How to simulate a jump diffusion processĢ) Python implementation of Merton's formula to see if we can produce a volatility smile from artificial data.ģ) Model calibration to market prices to find optimal parameters using least squares. Jumps are often one of the explanations for the presence of this smile. The Merton jump diffusion model is also interesting due to the fact that it is able to produce the volatility smile which is observed in all options markets. Merton's 1979 paper Option Pricing When Underlying Stock Returns Are Discountious. The main idea regarding this paper was to extend the Black-Scholes model to incorporate more realistic assumptions and that deal with the fact that empirical studies of market returns, do not follow a constant variance log-normal distribution. The Merton Jump diffusion model is a result of Robert C.
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