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Comparing Calibration Methods for Merton Jump-Diffusion Models

Article Quant Q&A · Author: Cavents
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Summary

The document describes a proposed comparison of calibration methods for Merton’s jump-diffusion model using S&P 500 option quotes. Its baseline method minimizes root mean squared error between market prices and prices computed with the Carr–Madan formula, fitting the jump mean, jump-size dispersion, jump intensity, and diffusion volatility separately for each maturity. The author also plans to compare model-implied moments and examine how calibrated parameters change across quote dates.

The question asks what other measures could help assess competing calibration procedures and whether one can be judged better overall. It provides a dataset spanning multiple years but contains no answers or empirical comparison results. The setup highlights that price fit alone may not capture differences in stability or economic usefulness, but it does not specify alternative metrics, out-of-sample tests, or a selection criterion. Any conclusion would depend on the chosen objective and evaluation context.

Key ideas

  • The baseline calibration minimizes pricing RMSE between model and market option prices.
  • The Merton model is fitted by maturity using jump and diffusion parameters.
  • The proposed comparison includes model moments and parameter evolution across quote dates.
  • The document poses evaluation questions but provides no results or prescribed measure for choosing a method.

Tags

OptionsVolatilityDerivatives pricingStatistics
Full text
# Calibration of Merton's jump diffusion model


# 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!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.