Calibrating Jump Models to Credit Spreads and Survival Probabilities
Summary
The document considers calibrating a structural Merton jump-diffusion model to credit default swap spreads. The modeled firm value combines diffusion with compound Poisson jumps, and default is defined by the first time firm value reaches a barrier. The questioner observes that the distribution needed for survival probabilities is not readily available in closed form. The responses state that calibration in this setup requires numerical methods, such as Monte Carlo simulation or solving a partial integro-differential equation, which may be costly, especially for risk calculations.
As an alternative, one answer describes affine jump-diffusion intensity models, particularly a mean-reverting square-root intensity with exponential jumps. This approach keeps hazard rates nonnegative and can provide analytical survival probabilities. The document does not provide calibration steps, equations for the survival curve, or a comparison of empirical fit and computational performance, so the suggested alternative is a model choice rather than a demonstrated result.
Key ideas
- The Merton jump-diffusion default barrier leads to a first-passage survival problem.
- The stated structural model has no analytic solution for the required distribution.
- Monte Carlo or partial integro-differential equation methods can support numerical calibration.
- A mean-reverting square-root intensity model with exponential jumps can yield analytical survival probabilities.
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# Calibration Merton Jump-Diffusion
# 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
## Answer by Mehness (score 3, accepted)
https://quant.stackexchange.com/a/31133
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!
## Answer by q.t.f. (score 2)
https://quant.stackexchange.com/a/18099
There is no analytic solution. You have to solve numerically, either by monte carlo or PIDE.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.