# 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.

Thanks in advance!

• Your question is ill-posed. You did not even describe what the "new" calibration method is. A minor point is that "matching" does not just mean computing. "Matching" means pairing two things. What are your two things? – Hans May 19 '16 at 22:30
• Comparing the quality of the fit with respect to the vanilla market (hence RMSEs, or any other error metric) is one thing. But what about: calibration risk in terms of the pricing of second gen exotics (see "a perfect calibration! Now what?" article), stability of calibrated params over time (avoid spurious rebalancing costs), time required to run a single calibration. If you plan on comparing the usual way (find parameters to minimise an error metric) with a moment matching approach (find parameters that match the first 4 moments of the risk-neutral distributions for instance) ... – Quantuple May 20 '16 at 6:26
• ... 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? – Quantuple May 20 '16 at 6:33
• @Siron Where did you get that wrong idea? I mean, this would be OK if you are just looking for a parametrisation of the different IV slices on any given day (e.g. using SVI or SABR) but as soon as you are doing pricing/hedging, there can be only one model (because the underlying asset has only one dynamics). If not, how could you possibly manage books with several options (exotics/vanillas) written on the same underlying: How do you obtain consistent Greeks across your full portfolio? How to deal with instruments maturing on non listed expiries? ... – Quantuple May 20 '16 at 9:22
• ... I think you are confused, the model (e.g. Merton, Heston, Local Vol) is calibrated each day simultaneously on all expiries. Not one parameter set for each expiry each day. This is the difference between calibrating an arbitrage free pricing model vs. calibrating a simple representation of the future risk-neutral distributions using a parametric form. The first is explanatory, the second is purely descriptive. – Quantuple May 20 '16 at 9:24