# Calibration of Merton's jump diffusion model

Setting

In my financial engineering project I'm working on a new calibration formalism for jump-diffusion models and in particular Merton's jump diffusion model. A jump diffusion process $\{X(t), t \geq 0\}$ is a mix of a diffusion process with an additional jump part, that is, $$X(t) = \sigma W(t) + \sum_{i=1}^{N(t)} Y_i,$$ where $\sigma$ is the diffusion volatility, $N(t)$ is a Poisson process with jump-intensity $\lambda$ independent of $W(t)$ and $Y_i \sim N(\mu, \delta^2)$ represent the jump sizes which are i.i.d and independent of $W(t)$ and $N(t)$.

To compare the new calibration method, I use a standard calibration method where the RMSE between model prices (computed by Carr-Madan option pricing formula) and market prices is minimized. Given a quote date with options quoted for, say $15$, different maturities. The result of the calibration is then an optimal parameter set $\{\mu,\delta, \lambda,\sigma\}$ for each maturity such that the distance between model prices and market prices is minimal.

To compare both calibration methods for a specific quoting date, for instance $29/10/2009$, I do the following:

(1) I compute the optimal parameter sets with corresponding RMSE's.

(2) I match the moments, that is, I compute variance, skewness, kurtosis and hyper skewness under the Merton jump diffusion model for the specific optimal parameter set.

(3) I plot the evolution of the different parameters $\mu,\delta, \lambda$ and $\sigma$ for a specific maturity for quoting dates ranging from $2008$ until $2009$.

Question

What are some other useful ways to compare two calibration methods? Is it possible to say if a calibration method is better than another? I know this is probably a hard question to answer since the performance depends on a lot of different factors. However, it would be great if I have some different ways to compare them both.

The dataset I'm using: options quoted on the S&P 500 with quoting dates ranging from 1993 until 2009.

• ... This link might help be useful: google.be/url?sa=t&source=web&rct=j&url=http://…. Anyway isn't it weird to use a fixed parameter set for each expiry? Let $T$ be a listed expiry, this means that you use one model for options expiring before $T$ (eg $T-1$day) and another one for option S expiring after $T$ (eg $T+1$day). How do you preclude arbitrage? May 20 '16 at 6:33