Calibration Merton Jump-Diffusion - Quantitative Finance Stack Exchange most recent 30 from quant.stackexchange.com 2019-08-23T10:53:40Z https://quant.stackexchange.com/feeds/question/18075 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://quant.stackexchange.com/q/18075 6 Calibration Merton Jump-Diffusion Vittorio Apicella https://quant.stackexchange.com/users/11943 2015-05-31T14:09:26Z 2016-11-22T22:59:58Z <p>Consider the following SDE $dV_t = rV_tdt +\sigma V_t dW_t + dJ_t$</p> <p>where $J_t$ is a Compound poisson process with log-Normal jump size $Y_i$.</p> <p>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...</p> <p>[EDIT] Well, what I'd need is in fact the distribution of the first hitting time, that is</p> <p>$\tau = \inf\{t&gt;0 : V_t = x\}$</p> <p>where x is some barrier $\in R$</p> <p>$Pr\left\{V_0 e^{(r-(1/2) \sigma^2)t + \sigma W_t + \sum_{i=0}^{N(t)} Y_i} = x \right\} =\\Pr \left\{(r-(1/2)\sigma^2)t + \sigma W_t + \sum_{i=0}^{N(t)}Y_i =\ln(x/V_0) \right\} = \\ Pr\left\{\sigma W_t + \sum_{i=0}^{N(t)}Y_i =\ln(x/V_0) - (r-(1/2)\sigma^2)t \right\}$</p> <p>The problem is here...I don't know which distribution comes out in the left hand side</p> https://quant.stackexchange.com/questions/18075/-/18099#18099 2 Answer by q.t.f. for Calibration Merton Jump-Diffusion q.t.f. https://quant.stackexchange.com/users/11936 2015-06-01T13:46:37Z 2015-06-01T13:46:37Z <p>There is no analytic solution. You have to solve numerically, either by monte carlo or PIDE. </p> https://quant.stackexchange.com/questions/18075/-/31133#31133 3 Answer by Mehness for Calibration Merton Jump-Diffusion Mehness https://quant.stackexchange.com/users/19239 2016-11-22T22:59:58Z 2016-11-22T22:59:58Z <p>Hi am having to write as an 'answer' as am new to forum. </p> <p>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.</p> <p>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:</p> <p><a href="https://www.amazon.co.uk/Interest-Rate-Models-Practice-Inflation/dp/3540221492/ref=sr_1_1?ie=UTF8&amp;qid=1479854412&amp;sr=8-1&amp;keywords=brigo" rel="nofollow noreferrer">https://www.amazon.co.uk/Interest-Rate-Models-Practice-Inflation/dp/3540221492/ref=sr_1_1?ie=UTF8&amp;qid=1479854412&amp;sr=8-1&amp;keywords=brigo</a></p> <p>Here's the SDE </p> <p>$$d\lambda_t=\kappa(\mu-\lambda_t)dt+\nu\sqrt{\lambda_t}dZ_t+dJ_t^{\alpha,\gamma}$$</p> <p>$\lambda_t$ 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!</p>