15 votes
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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 ...
Quantuple's user avatar
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6 votes
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Price of Call Option with or without jumps

The call for the stock that can jump downward will be more valuable due to put-call parity. Suppose you have two stocks, both with a price of $100 and the same diffusive volatility. Stock A does not ...
Robert McDonald's user avatar
4 votes
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Stochastic integral involving Poisson Process

Let \begin{align*} X_t=\left(\int_0^t f(s,N_{s-}) d\tilde{N}_s\right)^2 \end{align*} and \begin{align*} Y_t = \int_0^t f(s,N_{s-}) d\tilde{N}_s. \end{align*} By It$\hat{\text{o}}$'s product rule for ...
Gordon's user avatar
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4 votes
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Hawkes process intensity solution

Let us define the auxiliary process $\Lambda_t=e^{\kappa t}\lambda_t$. Note that: $$ \Lambda_t = \kappa e^{\kappa t} \int_0^t(\rho_s-\lambda_s)ds+\delta e^{\kappa t}\int_0^tdN_t$$ Hence after a jump ...
Daneel Olivaw's user avatar
4 votes

Price of Call Option with or without jumps

There's a lot left unspecified in this question, since it is stated without precision, but the effective idea of the answer given here is that those jumps introduce extra variation into the forward ...
Brian B's user avatar
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3 votes
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Geometric Brownian Motion unable to model / predict jumps

Assume that the value of the sample path of the geometric Brownian motion equals $10$ at time $t_0$ and equals 100 at time $t_0 + \Delta t$. For the value to change from $10$ to $100$, the path should ...
Holden's user avatar
  • 165
3 votes

Solution of Merton's Jump-Diffusion SDE

Use Ito for jumps $$ dS_t = \frac{\partial S_t}{\partial t} dt + \frac{\partial S_t}{\partial W_t}dW_t + \frac{1}{2}\frac{\partial^2 S_t}{\partial W_t^2} dt + \frac{\partial S_t}{\partial N_t}d N_t $$...
Phun's user avatar
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2 votes
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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 ...
Quantuple's user avatar
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2 votes

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 ...
mchen's user avatar
  • 121
2 votes

How to compute the conditional variance of this jump process?

Alternatively, let $\{\tau_i\}_{i=1}^{\infty}$ be the jump time of the Poisson process $N$. Moreover, let \begin{align*} X_t = \int_0^t (J_s-1)dN_s. \end{align*} Here, we assume that the jump sizes $...
Gordon's user avatar
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2 votes

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 ...
Kupoc's user avatar
  • 98
1 vote

Kou model — solving PIDE for European and American options in Python

The issue I described in my initial question is linked to the integral term. In the paper, this term is multiply by $ \theta \Delta \text{t} $ but this is only the "implicit" part of the ...
pierrot's user avatar
  • 96
1 vote

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 ...
Gordon's user avatar
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1 vote

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 ...
James Spencer-Lavan's user avatar
1 vote

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-...
Forgottenscience's user avatar
1 vote

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-...
torbonde's user avatar
  • 181
1 vote

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 ...
James Spencer-Lavan's user avatar
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 ...
LocalVolatility's user avatar
1 vote

Hawkes process intensity solution

Set $\tilde{\lambda}_t = e^{\kappa t} \lambda_t$ and solve for $\tilde{\lambda}_t$
Antoine Conze's user avatar
1 vote
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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 ...
millovanovic's user avatar
1 vote
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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 ...
AFK's user avatar
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