# Tag Info

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### Solution of Merton's Jump-Diffusion SDE

Let $$dS_t = \mu S_t dt + \sigma S_t dW_t + S_{t^-} dJ_t$$ where $$J_t = \sum_{j=1}^{N_t} (V_j - 1)$$ is a compound Poisson process, with $V_j$ i.i.d. jump sizes (positive random variables) whose ...
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### Simple question on jump-diffusion

Could it be that your problem is only due to the $t^-$ notation convention? Think of it that way, it is only worth distinguishing $S_{t^-}$ from $S_t$ at a jump time. Elsewhere, knowing that Brownian ...

### Price of Call Option with or without jumps

You can check out those discussions in Merton paper when introducing jumps "Option pricing when underlying stock returns are discontinuous". In the very last part he discusses the influence ...
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### Modelling considerations for a jump model

Note that, \begin{align*} d\big( e^{-\mu t}S_t \big) &= -\mu e^{-\mu t}S_t dt + e^{-\mu t}S_{t-}(\mu dt + Y_t dN_t)\\ &=e^{-\mu t}S_{t-}Y_t dN_t. \end{align*} From the Doleans-Dade exponential ...
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### What is the purest way to get exposure to Jump risk premia, is there a jump swap

The closest contract to this is gap risk which does trade, either as OTC swap (client looking for a hedge) or embedded inside a structured note (bank looking to recycle risk). Basic starting point ...
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### Characteristic function of CGMY model

Y in the CGMY model is not defined for negative integer values due to divergence of the gamma function at those values, and implicitly the characteristic function. However, in the case of negative non-...
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### Crash cliquet price

Defining $\tilde{S}_n = S_n/S_{n-1}$ (which is well defined, assuming $S_n > 0$ for all $n$), the problem becomes that of barrier option pricing. In particular, you're looking to price a down-and-...
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### Simulate double exponential process with correlated jumps?

I don't think you can correlate discrete processes in the traditional sense. Instead, I would make the two Poisson intensities time-varying through which a degree of "jump similarity" can be ...
1 vote
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### How to estimate lambda for Jump-Diffusion Process from Empirical data?

TLDR: The jump frequency depends on how you specify the jump size distribution. If you want the $\lambda$ to actually represent the jump frequency under a certain jump-diffusion model, then you ...
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### Marked poisson process vs compounded

The compound Poisson process isn't technically a marked process because we formulate the process with respect to $\sum_i D_i$ instead of $(\tau_i, D_i)$. However the compound process is constructed ...
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### Hawkes process intensity solution

Set $\tilde{\lambda}_t = e^{\kappa t} \lambda_t$ and solve for $\tilde{\lambda}_t$
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### Numerical Methods for Merton Model

You should take a look at the BENCHOP project. There we benchmarked around 15 different numerical methods against 6 option pricing problems. One of the problems was the Merton model. The methods were ...
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### How to price jumps in payoffs

There is nothing to model in the payoff. A payoff is a collection of cash flows. A cash flow is a function of market observables. Your function just happens to be discontinuous. From a risk point of ...
1 vote

### Where to find pricing formulas for affine stochastic volatility jump-diffusion models?

One of these two books may help you: A Guide to Quantitative Finance Option Pricing via Quadrature They are both from the same author. Both price European vanilla options under various stochastic ...

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