# Questions tagged [jump-diffusion]

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### Second variation of a Brownian motion under jump-diffusion process

I am trying to solve exercise 15.3 from the book The concepts and practice of mathematical finance where it is asked Suppose the $\log S_t$ follows a Brownian motion over the period $[0, 1]$ except ...
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### Euler Scheme for Jump-Diffusion models

Jump-diffusion models (as Merton) have following SDE: $$dS_t=\mu S_tdt+\sigma S_t dW_t+S_tdJ_t$$ where $$J_t=\sum_{i=1}^{N_t}(\xi_i - 1)$$ $\xi_i$ - i.i.dn $N_t$ - Poisson process Do we in Euler ...
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### Vanilla Call Option Priced Using Jump Diffusion Model

I'm reading a book called Quant Job Interview Questions and Answers and came across the following question and its answer, but cannot make sense of it, so I really appreciate your advice: Question 2....
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### What is the intuition behind “jumps” causing volatility skew?

Some models use jumps as a way to explain volatility skew. I understand that if jumps exist, then you are "mishedged" as you no longer can continuously hedge. Options have a gamma component and ...
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### B-S derivative with another boundary condition

I want to use the derivation of BS for another type of derivative, not an option. Known the derivation of the Black-Scholes differential equation, is it possible to use in the same equation when my ...
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### Pricing barrier option under Levy process: Biased estimate?

I want to price a down and out call, barrier option, with the underlying asset following a Levy process. I am interest on the Kou double exponential model or the NIG process, to capture asymmetric ...
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### Why can't we create a “magic” basket of options to sell for no-arbitrage pricing in SVJ model?

I am learning how to price SVJ options and am reading some stuff on no-arbitrage pricing for SVJ model using the typical approach you would use (like in BSM option pricing) of creating a risk free ...
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### Predicting time series using Jump Diffusion model and Neural Networks

I am trying to understand the difference between using Jump diffusion model and Neural Networks or more precisely LSTM to predict time series data regardless what that data contains for example a ...
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### Jump Diffusion Model - Volatility and Mean of Jumps

I am trying to understand the concept of jump diffusion model. So far what I've understood is that by adding a Jump parameter to a GBM (Geometric Brownian Motion) we can generate a Jump diffusion ...
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### Poisson parameter in Merton's Jump-Diffusion Model to price call option

I've been taught the following European call valuation formula under jump-diffusion model: \begin{equation} price = E[e^{-rT}max(S_T-K,0)] =\sum_{j = 0}^\infty e^{-rT}P_j(\lambda)E[max(S_T-K,0)|J=j] \...
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### Basic book on stochastic calculus, Itô and jump processes and Brownian Motion

I was looking for a good book that explains at beginner-level the basic of stochastic calculus, probability and random variables, Itô and jump processes as well as Brownian Motion. At university we ...
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### Formal proof market incompleteness under jump diffusion

Does anyone have formal proof of markets incompleteness under jump diffusion ? I am familiar with the intuitive approach as mentioned in Tankov (delta), yet I am looking for a formal approach and ...
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### Understanding and simulating the jumps in Merton's Jump-Diffusion SDE?

I found this great post deriving the solution to the Merton Jump-Diffusion SDE $$S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right)\prod_{j=0}^{N_t}V_j$$ The first part of ...
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### Simulating compound Poisson jump-diffusion process with time-changed jump frequency

I want to simulate a jump-diffusion process with compound Poisson jumps and a deterministic jump frequency function $\lambda(t)$. The function should follow the following stochastic differential ...
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### Levy process and random measure

I am wondering if random measures are used under a Levy process and how this connects to finance (particularly pricing). Any paper or books for suggestions is welcomed.
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### Hedging jump models with a infinite number of derivatives

First of all, I inform you that I am not a financial mathematician and have vague knowledge about an incomplete market. Stochastic volatility models are incomplete so derivatives cannot be ...
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### Bond PDE under an Affine Jump Diffusion model

Under the Jump extended Vasicek model, the dynamics of the short rate are as follow : $$dr_t=\kappa(\theta-r_t)dt+\sigma\sqrt{r_t}\,dW_t+d\left(\sum\limits_{i=1}^{N_t}\,J_i\right)$$ where $N_t$ ...
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### Euler discretization with jumps

There is a process $B_t = B_0\prod_{i=1}^{N_t}(1-Z_n)$, where $Z_n=e^{-ξ_n}$ for i.i.d exponentially distributed random variables $(ξn)_{n≥1}$ with rate $ρ=20$. ${N_t}$ is a counting process ...
698 views