Questions tagged [jump-diffusion]
The jump-diffusion tag has no usage guidance.
0
votes
0answers
20 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 ...
1
vote
0answers
29 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]
\...
2
votes
4answers
164 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
56 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 ...
4
votes
0answers
114 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
75 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
48 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
55 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 ...
3
votes
1answer
141 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
50 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
137 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 $\frac{S_n}{S_{n-1}...
1
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0answers
86 views
Barrier Option under Jump Diffusion
I am trying to price a Barrier Option under a model with jumps. I am using a brownian bridge approach but struggle with the jumps around these bridges and don't know how to handle this.
My main ...
0
votes
0answers
47 views
Jump diffusion model and Firm probability of default
I want to examine whether corporate events affect firm's probability of default. My initial thought was a jump diffusion model, although in the literature, the only work I found, involved CDS market ...
1
vote
1answer
102 views
stochastic vol modelling not enough for smile
It seems in practice models that include Stochastic Volatility alone do not have enough power to produce actual observed implied vol surfaces. Is there recent empirical literature documenting this?
4
votes
1answer
144 views
Simulate double exponential process with correlated jumps?
So, I'm trying to simulate a correlated double exponential jump process for two assets, and I understand the pure exponential jump process ($\eta_1$ and $\eta_2$, the probability of an upward jump ...
1
vote
1answer
363 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 ...
0
votes
1answer
262 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') &= \...
4
votes
1answer
360 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 ...
2
votes
1answer
102 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
258 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}
...
5
votes
1answer
598 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
55 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: ...
0
votes
0answers
69 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
372 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
$$dr_t=\kappa(\theta-r_t)dt+\sigma\sqrt{r_t}\,dW_t+d\left(\sum\limits_{i=1}^{N_t}\,...
4
votes
1answer
456 views
Risk neutral measure for jump processes
Assume we model the dynamics of a tradable asset as follows
$$ 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 ...
