Questions tagged [jump-diffusion]
The jump-diffusion tag has no usage guidance.
48
questions
1
vote
1answer
97 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 ...
2
votes
1answer
201 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
81 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
votes
0answers
118 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
...
0
votes
0answers
115 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
176 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
0answers
65 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
votes
0answers
32 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
0answers
83 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
0answers
92 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
$$ C_{Merton}(S_0,r,q,\sigma,K,T) = \sum_{n=0}^\infty e^{-\lambda T}\...
2
votes
1answer
75 views
What can the area under a GBM jump curve tell you
So I used matlab and simulated stock prices with the Merton diffusion model. Now I want to take the integral of the area. Now would there be any financial insight by taking the integral of a stock ...
0
votes
2answers
70 views
I just got Matlab, what are some options that I should model in a jump diffusion
Don't worry I understand mathematics: ito's calc, martingales, etc. I am just curious what options I should test, and from what indices. Is there stuff I can test from the 2008 crash to measure their ...
6
votes
2answers
124 views
Solution for a SDE for a Bond found in Bugard & Kjaer
I'm going over the paper -Partial Differential Equation Representation of Derivatives with Bilateral Counterparty Risk and Funding Costs- from Burgard and Kjaer. There the following SDE is given for ...
1
vote
0answers
67 views
Unique risk neutral measure for jumps or incomplete markets for jumps
I wanted to understand why the market is incomplete in jump-diffusion models. whereas if we have a model following geometric Brownian motion then we can get a risk-neutral measure and hence a complete ...
3
votes
2answers
313 views
SDE Jump-Diffusion
If you combine the compound Poisson process with the Brownian motion you obtain the simplest case
of a Jump-diffusion. Let’s define
$$X_t = \mu t + \sigma W_t + J_t$$
where $W_t$ is a Wiener process ...
3
votes
1answer
104 views
Binomial tree with jumps
I am struggling with developing a binomial tree with jumps. although there are models such as CRR, could you suggest a book or have any idea to proceed?
Thanks,
Amir
4
votes
1answer
136 views
Expected Value of Mean-Reverting Jump Process
I cant see the link between my method of calculation and the method done in the book Cartea and Jaimungal (Algorithmic and High Frequency Trading, page 220.) We have a mean-reverting process
$$d\mu_t=-...
1
vote
0answers
95 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
vote
1answer
144 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
votes
4answers
274 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
votes
0answers
48 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
vote
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
vote
0answers
54 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
310 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
124 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
0answers
150 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
519 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
0answers
72 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 ...
8
votes
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
445 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
105 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
votes
0answers
77 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
votes
1answer
286 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
747 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 $\frac{S_n}{S_{n-1}...
3
votes
1answer
228 views
Barrier Option under Jump Diffusion
I am trying to price a Barrier Option under a model with jumps. I am using a brownian bridge approach but struggle with the jumps around these bridges and don't know how to handle this.
My main ...
1
vote
1answer
119 views
stochastic vol modelling not enough for smile
It seems in practice models that include Stochastic Volatility alone do not have enough power to produce actual observed implied vol surfaces. Is there recent empirical literature documenting this?
5
votes
1answer
209 views
Simulate double exponential process with correlated jumps?
So, I'm trying to simulate a correlated double exponential jump process for two assets, and I understand the pure exponential jump process ($\eta_1$ and $\eta_2$, the probability of an upward jump ...
1
vote
1answer
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
1answer
681 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
808 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
167 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
votes
1answer
634 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
63 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
vote
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
484 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
$$dr_t=\kappa(\theta-r_t)dt+\sigma\sqrt{r_t}\,dW_t+d\left(\sum\limits_{i=1}^{N_t}\,...
5
votes
1answer
878 views
Risk neutral measure for jump processes
Assume we model the dynamics of a tradable asset as follows
$$ 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 ...
