# Questions tagged [jump-diffusion]

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### Jump-Diffusion Model for pricing Convertible Bonds

I am looking for research papers on pricing convertible bonds using jump-diffusion model. Most of the material I am able to obtain so far is related to binomial tree methodology for pricing the bonds. ...
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### What can the area under a GBM jump curve tell you

So I used matlab and simulated stock prices with the Merton diffusion model. Now I want to take the integral of the area. Now would there be any financial insight by taking the integral of a stock ...
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### I just got Matlab, what are some options that I should model in a jump diffusion

Don't worry I understand mathematics: ito's calc, martingales, etc. I am just curious what options I should test, and from what indices. Is there stuff I can test from the 2008 crash to measure their ...
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### Solution for a SDE for a Bond found in Bugard & Kjaer

I'm going over the paper -Partial Differential Equation Representation of Derivatives with Bilateral Counterparty Risk and Funding Costs- from Burgard and Kjaer. There the following SDE is given for ...
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### Unique risk neutral measure for jumps or incomplete markets for jumps

I wanted to understand why the market is incomplete in jump-diffusion models. whereas if we have a model following geometric Brownian motion then we can get a risk-neutral measure and hence a complete ...
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### SDE Jump-Diffusion

If you combine the compound Poisson process with the Brownian motion you obtain the simplest case of a Jump-diffusion. Let’s define $$X_t = \mu t + \sigma W_t + J_t$$ where $W_t$ is a Wiener process ...
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### Binomial tree with jumps

I am struggling with developing a binomial tree with jumps. although there are models such as CRR, could you suggest a book or have any idea to proceed? Thanks, Amir
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### Exact Simulation algorithm SVCJ (Broadie Kaja)

I'm trying to write the code for Exact simulation algorithm SVCJ http://www.columbia.edu/~mnb2/broadie/Assets/broadie_kaya_WSC2004.pdf The code seems to be working but fluctuates a lot. Could anyone ...
I cant see the link between my method of calculation and the method done in the book Cartea and Jaimungal (Algorithmic and High Frequency Trading, page 220.) We have a mean-reverting process d\mu_t=-... 0answers 108 views +50 ### Relating two equations in a jump-diffusion process I am trying to understand an argument involving the pricing kernel \xi_t in the context of a simple jump diffusion model for the price of an asset S_t: \begin{align} \xi_t = \exp \left[ -\theta ... 0answers 62 views ### Greeks for Pricing Convertible Bond Using Jump Diffusion Model I'm learning the jump diffusion model used to price a convertible bond, and got the following stochastic differential equation under risk neutral measure:dS = (r+\lambda*p)Sdt + \sigma*SdW+Sdq$$... 1answer 102 views ### 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.... 4answers 202 views ### 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 ... 0answers 41 views ### 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 ... 0answers 27 views ### 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 ... 0answers 48 views ### 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 ... 1answer 217 views ### 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 ... 0answers 106 views ### 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 ... 0answers 112 views ### 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] \... 4answers 364 views ### 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 ... 0answers 62 views ### 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 ... 0answers 1k views ### 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$$The first part of ... 0answers 262 views ### Simulating compound Poisson jump-diffusion process with time-changed jump frequency I want to simulate a jump-diffusion process with compound Poisson jumps and a deterministic jump frequency function \lambda(t). The function should follow the following stochastic differential ... 1answer 80 views ### Levy process and random measure I am wondering if random measures are used under a Levy process and how this connects to finance (particularly pricing). Any paper or books for suggestions is welcomed. 0answers 69 views ### Hedging jump models with a infinite number of derivatives First of all, I inform you that I am not a financial mathematician and have vague knowledge about an incomplete market. Stochastic volatility models are incomplete so derivatives cannot be ... 1answer 257 views ### Bond PDE under an Affine Jump Diffusion model Under the Jump extended Vasicek model, the dynamics of the short rate are as follow :$$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 ... 0answers 66 views ### 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 ... 1answer 483 views ### Crash cliquet price Denote by n the n-th trading day in a year and by S_n the stock price on that day. An instrument expirying in 1 year pays \max(0,1-\frac{S_n}{S_{n-1}}) and early terminates if \frac{S_n}{S_{n-1}... 1answer 201 views ### Barrier Option under Jump Diffusion I am trying to price a Barrier Option under a model with jumps. I am using a brownian bridge approach but struggle with the jumps around these bridges and don't know how to handle this. My main ... 1answer 115 views ### stochastic vol modelling not enough for smile It seems in practice models that include Stochastic Volatility alone do not have enough power to produce actual observed implied vol surfaces. Is there recent empirical literature documenting this? 1answer 189 views ### Simulate double exponential process with correlated jumps? So, I'm trying to simulate a correlated double exponential jump process for two assets, and I understand the pure exponential jump process (\eta_1 and \eta_2, the probability of an upward jump ... 1answer 820 views ### How to estimate lambda for Jump-Diffusion Process from Empirical data? So, I have really no idea how to go about this, but how would I go about choosing sensible parameter values for a basic jump-diffusion simulation, namely \lambda ? For example, getting the average ... 1answer 471 views ### exercise on multivariate Ito's lemma + jumps (Poisson) Given the two jump-diffusions: \begin{equation} \begin{aligned} dX_{1,t} &= a_1 dt + b_1 dW_t + c_1 dN_t(\lambda) \\ dX_{2,t} &= a_2 dt + b_2 dW'_t + c_2 dN_t(\lambda) \\ corr(dW,dW') &= \... 1answer 644 views ### Merton's jump diffusion Can someone help me finding the expected value of the solution to Merton's jump diffusion model: \begin{align} S_t &= S_0 \exp \left( \left(r - \frac{\sigma^2}{2} - \lambda k \right) t + \sigma ... 1answer 148 views ### Cadlag Property of Jump Proccesses I've recently started studying Cont & Tankov's "financial modelling with jump processes". I'm curious as to why that this assumption of the cadlag property (also called RCLL "right continuous ... 1answer 449 views ### Trouble understanding jump part in Kou double exponential jump diffusion model I am trying to work with Kou's double exponential Jump-diffusion model and simulate a price path in a programming language. So the dynamics of the asset price in Kou's model follow: \begin{equation} ... 1answer 969 views ### Black-Scholes formula for Poisson jumps For underlying assetd S = r S dt + \sigma S d W + (J-1)Sd N$$here W is a Brownian motion, N(t) is Poisson process with intensity \lambda. Suppose J is log-normal with standard deviation \... 1answer 56 views ### influence of exponential-Lévy on a call price Thank you all for answering my question. I wanted to know what influence has the exponential-Lévy model on a call price (how the curve changes). If we add Merton jumps, we get an EDPID like this one: ... 0answers 82 views ### How are Levy driven SDE simulated? Do you just use an Euler scheme as before? E.g. take this process, OU process with a Levy driver. \begin{equation} \text{d}V_t = -\lambda V_t\text{d}t + dZ_t \end{equation} Do you just have V_{... 1answer 436 views ### Pricing the discount zero-coupon bond under a jump-diffusion model I am going to get the price of a zero coupon bond in a jump-diffusion model. The dynamic of interest rate as follow$$dr_t=\kappa(\theta-r_t)dt+\sigma\sqrt{r_t}\,dW_t+d\left(\sum\limits_{i=1}^{N_t}\,...
Assume we model the dynamics of a tradable asset as follows $$S_t = S_0 \exp\left[\sigma W_t +(\alpha-\beta\lambda-\frac{1}{2}\sigma^2)t+J_t \right]$$ where $W_t$ is a standard Brownian motion ...