Questions tagged [jump-diffusion]
The jump-diffusion tag has no usage guidance.
50
questions
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1answer
106 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 ...
2
votes
0answers
66 views
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 $...
0
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0answers
81 views
Vasicek Model With Jumps
I'm trying to calibrate a mean-reverting, jump diffusion model using the outline provided on page 11 here:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.40.3489&rep=rep1&type=pdf
...
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0answers
43 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 ...
1
vote
2answers
154 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
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0answers
56 views
“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,...
0
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0answers
31 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
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0answers
55 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
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0answers
81 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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0answers
43 views
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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52 views
Merton's Jump-Diffusion Model (learning materials)
what materials / books do you recommend for learning the Merton model? So that it was reasonably well explained and that the formula for the price of the european call option was somehow reasonably ...
2
votes
1answer
72 views
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 ...
0
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2answers
69 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 ...
6
votes
2answers
110 views
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
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0answers
52 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 ...
3
votes
2answers
192 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$$
where $W_t$ is a Wiener process ...
3
votes
1answer
79 views
Binomial tree with jumps
I am struggling with developing a binomial tree with jumps. although there are models such as CRR, could you suggest a book or have any idea to proceed?
Thanks,
Amir
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0answers
22 views
Exact Simulation algorithm SVCJ (Broadie Kaja)
I'm trying to write the code for Exact simulation algorithm SVCJ http://www.columbia.edu/~mnb2/broadie/Assets/broadie_kaya_WSC2004.pdf The code seems to be working but fluctuates a lot. Could anyone ...
4
votes
1answer
77 views
Expected Value of Mean-Reverting Jump Process
I cant see the link between my method of calculation and the method done in the book Cartea and Jaimungal (Algorithmic and High Frequency Trading, page 220.) We have a mean-reverting process
$$d\mu_t=-...
1
vote
0answers
80 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:
$$dS = (r+\lambda*p)Sdt + \sigma*SdW+Sdq$$
$...
1
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1answer
127 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....
4
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4answers
226 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 ...
2
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0answers
45 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 ...
1
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0answers
29 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 ...
1
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0answers
49 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 ...
1
vote
1answer
269 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
0answers
115 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
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0answers
136 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
4answers
443 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
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0answers
68 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 ...
7
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1answer
2k 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 ...
2
votes
0answers
362 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
1answer
93 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.
2
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0answers
70 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 ...
4
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1answer
270 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
0answers
69 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 ...
2
votes
1answer
614 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 $\frac{S_n}{S_{n-1}...
3
votes
1answer
219 views
Barrier Option under Jump Diffusion
I am trying to price a Barrier Option under a model with jumps. I am using a brownian bridge approach but struggle with the jumps around these bridges and don't know how to handle this.
My main ...
1
vote
1answer
118 views
stochastic vol modelling not enough for smile
It seems in practice models that include Stochastic Volatility alone do not have enough power to produce actual observed implied vol surfaces. Is there recent empirical literature documenting this?
5
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1answer
195 views
Simulate double exponential process with correlated jumps?
So, I'm trying to simulate a correlated double exponential jump process for two assets, and I understand the pure exponential jump process ($\eta_1$ and $\eta_2$, the probability of an upward jump ...
1
vote
1answer
924 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
1answer
590 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
1answer
727 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 ...
3
votes
1answer
157 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 ...
0
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1answer
539 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
1answer
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 $\...
0
votes
1answer
60 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:
...
1
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0answers
83 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_{...
4
votes
1answer
454 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
$$dr_t=\kappa(\theta-r_t)dt+\sigma\sqrt{r_t}\,dW_t+d\left(\sum\limits_{i=1}^{N_t}\,...
5
votes
1answer
788 views
Risk neutral measure for jump processes
Assume we model the dynamics of a tradable asset as follows
$$ 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 ...
