Questions tagged [jump-diffusion]

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Mixing formula for SVJ models

I am trying to understand the mixing formula (Hull and White formula) for stochastic volatility models with jumps in the asset price. One article which discusses this is Lewis, The mixing approach to ...
5 votes
1 answer
252 views

Variance of the log returns in jump diffusion with time-varying jump sizes

I'm trying to calculate the variance $\mathrm{var}\left(\log\frac{S\left(t\right)}{S\left(0\right)}\right)$, where the dynamics of the stock $S$ follows a jump-diffusion process given by $$\frac{dS\...
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How do I estimate volatility for MPR historical data

How can I estimate volatility with historical data for Monetary Policy Rate (MPR) to use in a short rate model? I could use regular techniques like simple standard deviation or max likelihood, but the ...
2 votes
1 answer
111 views

How to solve numerically the IDE of GUILBAUD & PHAM model?

By the Guilbaud & Pham model (Optimal high frequency trading with limit and market orders, 2011), the authors said that integro-differential-equation (IDE) can be easily solved by numerical method....
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1 answer
110 views

Separating jumps and diffusion

I want to model energy prices. I have two markets, lets say market 1 and 2. Market 1 is continuously traded, and I will assume it follows brownian motion. So the value of the asset could be defined ...
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SDE of a Geometric Levy process with compound Poisson process

Suppose that a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ is given. A geometric Levy process is defined in the form of $S_t=S_0 exp(X_t)$ where $S_0$, let's say, is the initial price and $...
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1 answer
249 views

Jump Diffusion Process question

I have a European call option with time maturity $T=3$ years,$K=50$, and given that $S(t)$ refers to the derivative is being described by the geometric Brownian motion with $S_{0}=100$ and $r = 0.04$....
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1 vote
1 answer
113 views

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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2 votes
1 answer
414 views

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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2 votes
0 answers
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The distribution of the jump diffusion process

In the Merton jump diffusion model the process of the share price can be expressed as $$S_{t}=S_{0}\cdot\exp\left\{ X_{t}\right\} ,$$ where $$X_{t}=\mu t+\sigma W_{t}+\sum_{i=1}^{N_{t}}Y_{i}.$$ Here $...
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256 views

Efficient way to perform MLE on Merton Jump Diffusion model parameters?

I understand that under Merton Jump Diffusion Model, if we are going to estimate the parameters $ \alpha, \sigma,\mu_J, \delta, \lambda $, we can use maximum likelihood estimation on the probability ...
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2 answers
222 views

Jump diffusion simulation

I want to simulate a geometric Brownian motion and we assume that the volatility of the stock can take just two values $\sigma_1=0.2$ and $\sigma_2=0.8$. We also assume that the jumps up from lower ...
1 vote
0 answers
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"Pricing European Options in a Stochastic-Volatility-Jump Diffusion Model" ,does anyone have this article?

I can't find the article "Pricing European Options in a Stochastic-Volatility-Jump Diffusion Model" of Thomas Knudsen and Laurent Nguyen-Ngoc, Journal of Financial and Quantitative Analysis,...
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0 answers
47 views

How option value default adjusted in jump diffusion model

According to the doc here: http://faculty.baruch.cuny.edu/jgatheral/JumpDiffusionModels.pdf. Formula 7 specifies that the option value under jump diffusion model becomes: So when the default ...
2 votes
0 answers
120 views

Characteristic function for heston model with jumps in price and variance

I need the characteristic function of the Heston model with jumps in price and variance, or in other words, the characteristic function of the Bates model (1996) adding jumps in the variance dynamics. ...
3 votes
0 answers
105 views

American Options in Merton's (1976) Jump Model

@LocalVolatility proves in this stellar answer that European call option prices in the Merton jump diffusion model are given by $$ C_{Merton}(S_0,r,q,\sigma,K,T) = \sum_{n=0}^\infty e^{-\lambda T}\...
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2 votes
1 answer
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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 ...
-1 votes
2 answers
80 views

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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6 votes
2 answers
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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 ...
1 vote
0 answers
93 views

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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3 votes
2 answers
488 views

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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3 votes
1 answer
134 views

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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4 votes
1 answer
191 views

Expected Value of Mean-Reverting Jump Process

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=-...
1 vote
0 answers
112 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: $$dS = (r+\lambda*p)Sdt + \sigma*SdW+Sdq$$ $...
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1 vote
1 answer
160 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....
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4 votes
4 answers
346 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 ...
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2 votes
0 answers
49 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 ...
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1 vote
0 answers
30 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 ...
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1 vote
0 answers
59 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 ...
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1 vote
1 answer
375 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 ...
2 votes
0 answers
132 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 ...
2 votes
0 answers
170 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] \...
3 votes
4 answers
620 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 ...
6 votes
0 answers
79 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 ...
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8 votes
1 answer
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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 ...
2 votes
0 answers
606 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 ...
1 vote
1 answer
135 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.
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2 votes
0 answers
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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 ...
4 votes
1 answer
311 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 $N_t$ ...
3 votes
0 answers
85 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 ...
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2 votes
1 answer
973 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}...
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3 votes
1 answer
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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 ...
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1 vote
1 answer
132 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?
5 votes
1 answer
227 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 ...
1 vote
1 answer
1k 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 ...
2 votes
1 answer
816 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') &= \...
5 votes
1 answer
914 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 ...
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3 votes
1 answer
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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 ...
0 votes
1 answer
786 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} ...
7 votes
1 answer
1k views

Black-Scholes formula for Poisson jumps

For underlying asset $$d S = r S dt + \sigma S d W + (J-1)Sd N$$ here $W$ is a Brownian motion, $N(t)$ is Poisson process with intensity $\lambda.$ Suppose $J$ is log-normal with standard deviation $\...
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