Questions tagged [jump-diffusion]

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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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40 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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68 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 ...
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52 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 ...
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0answers
46 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] \...
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4answers
183 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 ...
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58 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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306 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 ...
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0answers
109 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 ...
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1answer
54 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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1answer
158 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 ...
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0answers
60 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 ...
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1answer
154 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 $N_t$ ...
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1answer
379 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}\,...
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56 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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1answer
188 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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0answers
101 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 ...
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1answer
104 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?
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1answer
461 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 ...
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1answer
310 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') &= \...
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1answer
439 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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1answer
113 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 ...
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1answer
289 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} ...
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1answer
688 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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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: ...
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1answer
521 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 ...
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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_{...