# Calibration Merton Jump-Diffusion

Consider the following SDE $dV_t = rV_tdt +\sigma V_t dW_t + dJ_t$

where $J_t$ is a Compound poisson process with log-Normal jump size $Y_i$.

How am I supposed to calibrate this model to CDS spreads? The problem of course is there doesn't exist an analytical formula for the survival probability function...

[EDIT] Well, what I'd need is in fact the distribution of the first hitting time, that is

$\tau = \inf\{t>0 : V_t = x\}$

where x is some barrier $\in R$

$Pr\left\{V_0 e^{(r-(1/2) \sigma^2)t + \sigma W_t + \sum_{i=0}^{N(t)} Y_i} = x \right\} =\\Pr \left\{(r-(1/2)\sigma^2)t + \sigma W_t + \sum_{i=0}^{N(t)}Y_i =\ln(x/V_0) \right\} = \\ Pr\left\{\sigma W_t + \sum_{i=0}^{N(t)}Y_i =\ln(x/V_0) - (r-(1/2)\sigma^2)t \right\}$

The problem is here...I don't know which distribution comes out in the left hand side

• Could you please develop the last part of your question: "The problem of course is there doesn't exist an analytical formula for the survival probability function". It will help people to answer, and may be yourself to better understand your problem. – lehalle May 31 '15 at 14:24
• sure...actually my problem is I don't know how to compute the probability in the EDIT – Vittorio Apicella May 31 '15 at 14:41
• Just to be sure I understand your question: what is V? – lehalle May 31 '15 at 14:55
• Generically, you can fit any model numerically. No need a closed form. Usually when a model is widely used, there are known good practices. And in this case I do not know them. In your case (in dimension one), it seems tractable numerically. – lehalle May 31 '15 at 17:54
• there are formulas for barrier option prices when the jump takes a specific form. These may help. There is some work by Kou. – Mark Joshi Jun 3 '15 at 3:23

Hi am having to write as an 'answer' as am new to forum.

We used stochastic intensity models on desk from a while back. Generally Black-Karasinski to avoid negative hazard rates (and for useful features such as mean reversion). Now in your choice of structural approach with lognormal jumps as some respondents have pointed out you will have to simulate to calibrate your model params. which may be computationally onerous particularly when it comes to risk measures.

Forgive me if you have seen but an elegant alternative are the affine jump diffusions, in particular I like Brigo's 'JCIR++' based on a square root process with exponential jumps. This has analytical survival probabilities, AND the intensities are non-negative. See for example:

https://www.amazon.co.uk/Interest-Rate-Models-Practice-Inflation/dp/3540221492/ref=sr_1_1?ie=UTF8&qid=1479854412&sr=8-1&keywords=brigo

Here's the SDE

$$d\lambda_t=\kappa(\mu-\lambda_t)dt+\nu\sqrt{\lambda_t}dZ_t+dJ_t^{\alpha,\gamma}$$

$\lambda_t$ is the intensity, the jump arrives at rate $\alpha$ and has distribution $Exp(\gamma)$. We also are able to have mean reversion. p832 in the reference has the formula for the survivals. But maybe that's old news to you in which case sorry!

There is no analytic solution. You have to solve numerically, either by monte carlo or PIDE.