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Questions tagged [jump]

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Bates Model Jump Percentage Parameters

I am trying to calculate the jump parameters for the Bates volatility jumps, specifically, the mean of the jump percentages, $\mu_j$. For the value of $J$, I am using jumps $|\frac{s_{i}-s_{i-1}}{s_{i-...
4
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1answer
114 views

Geometric Brownian Motion unable to model / predict jumps

In my finance course, we were talking about the flaws of modelling Stock Prices with the geometric Brownian Motion. According to my professor: "Since the geometric Brownian Motion has continous time ...
5
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1answer
93 views

What is the purest way to get exposure to Jump risk premia, is there a jump swap

So to get exposure to Variance risk premia one could use variance swaps, is there a equivalent security for jumps. Hedging against jump but not diffusion risk could allow one to take targeted exposure ...
2
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1answer
139 views

Characteristic function of CGMY model

I have a basic question about the CGMY model which has characteristic function $$ \Gamma(-Y_p)\left((M-iu)^{Y_p}-M^{Y_p}\right)+\frac{C_n}{C_p}\Gamma(-Y_n)\left((G+iu)^{Y_n}-G^{Y_n}\right) $$ whith $...
2
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1answer
240 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
50 views

Computing squared returns given differential equation for prices

I am looking for general advice on how to start tackling the problem below. My background in math is fairly bad when it comes to stochastic differential equations, but if you have any recommendations ...
4
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1answer
170 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
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1answer
558 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
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0answers
134 views

Jim Gatheral's claim on the decay of the effect of jumps on the final return distribution

I got a full answer for my question on Jim Gatheral's book The Volatility Surface. I am going to try my luck again on another question on the same book. In Section The Decay of Skew Due to Jumps on ...
3
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0answers
83 views

Marked poisson process vs compounded

I am a bit fuzzy about difference between compounded poisson process defined as $$\sum_{i=1}^{N_t} D_i $$ where $N_t$ is poisson process and $ D_i $ are iid random variables and marked poisson ...
2
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2answers
426 views

Hawkes process intensity solution

Hail to all, I am struggling to solve the following SDE for intensity: $d\lambda_t = \kappa(\rho(t) - \lambda_t)dt + \delta dN_t $ I know to expect the solution in the form of $\lambda_t = c(0)e^{-...
0
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1answer
304 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} ...
0
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2answers
216 views

How to compute the conditional variance of this jump process?

Let $N_t$ be a Poisson process with intensity $\lambda>0$ and $S_t$ follows a pure jump process $$dS_t=S_t(J_t-1)dN_t$$ where $J_t$ is the jump size variable if $N_t$ jumps at time $t$. Also, ...
1
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0answers
464 views

Could someone please share the Matlab code for the stochastic volatility jump diffusion option pricing model? (Bates model) [closed]

I have not been able to write a Matlab code for the Bates model without errors. Could someone share theirs please?
4
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0answers
1k views

Calibration of Merton's jump diffusion model

Setting In my financial engineering project I'm working on a new calibration formalism for jump-diffusion models and in particular Merton's jump diffusion model. A jump diffusion process $\{X(t), t \...
1
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1answer
276 views

Numerical Methods for Merton Model

The stochastic differential equation for an underlying with jumps in Merton model is: $$d{{S}_{t}}=\mu \,{{S}_{t}}dt+\sigma \,{{S}_{t}}\,d{{W}_{t}}^{P}+(J-1){{S}_{t}}d{{q}_{t}}$$ where $t \quad\,\,\, ...
3
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1answer
352 views

Simple question on jump-diffusion

In the textbook by Shreve in sec. 11.7.2 a jump-diffusion process is introduced. More precisely $$ dS_t = \alpha\,S_t\,dt+\sigma\,S_t\,dW_t+S_{t-}\,d\left(Q_t-\beta\,\lambda\,t\right)\quad (1) $$ ...
12
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2answers
3k views

Solution of Merton's Jump-Diffusion SDE

In many textbooks and also in the original Merton's paper the solution of the SDE $$ dS_t = S_t\,\mu\,dt+S_t\,\sigma\,dW_t+S_{t^-}\,d\left(\sum_{j=1}^{N_t}V_j-1\right) $$ is written as $$ S_t = ...
1
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2answers
105 views

How to price jumps in payoffs

I specifically want to know how to model a jump condition while valuing a derivative.Example :- the jumps which are observed in digital product payoffs, or barriers and knockouts. Although a ...
12
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2answers
410 views

Realized variance in SVJJ (Heston with jumps) model

I am working with the stochastic volatility model with jumps in both the price and volatility dynamics, ie. the risk neutral dynamics are of the form: $$\mathrm{d}V_t = \kappa(\theta - V_t)\mathrm{d}t ...
1
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3answers
299 views

Why Jumps in Option Pricing models?

The Bates model adds a Jump process to the Underlying. I understand this may represent observed time series more realistically, but why would one care about this in option pricing? The option price ...
4
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1answer
2k views

Option prices in Bates SVJ model?

In this [post] discussed the European put and call price formulas under the Heston Stochastic Volatility model. There exists an important extension of Heston model to include diffusion jumps, known ...
8
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2answers
603 views

how we can derive $PIDE$ of double exponential Jump-diffusion model (we know as kou model)?

I'm working in double exponential Jump-diffusion model (we know as kou model) with following form , under the physical probability measure $P$: \begin{equation} ‎\frac{dS(t)}{S(t-)}=\mu‎‏ ‎dt+\sigma ‎...
2
votes
2answers
158 views

Where to find pricing formulas for affine stochastic volatility jump-diffusion models?

Does anyone know a reference where I can find the pricing formulas for vanilla calls in the affine stochastic volatility jump diffusion class of models such as SVJ and SVJJ? I am looking for ...
1
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1answer
398 views

Validation of Bates SVJ model

I have just finished implementing the Bates model for pricing European call options. To check results, I have been looking for a validation set where I could see the Bates parameter values and ...
3
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1answer
471 views

Simulating Brownian motion with jumps

I am trying to improve my understanding of jump processes. As a first step, I want to simulate sample paths for the process $$dX(t) = dw(t) + dJ(t)$$ where $dw(t)$ is a Brownian motion and $dJ(t)$ ...
2
votes
1answer
484 views

Do we need Feller condition if volatility process jumps?

It is fairly known that in affine processes, as Heston model \begin{equation} \begin{aligned} dS_t &= \mu S_t dt + \sqrt{v_t} S_t dW^{S}_{t} \\ dv_t &= k(\theta - v_t) dt + \xi \sqrt{v_t} dW^{...
3
votes
2answers
817 views

Valuation of barrier options in Jump diffusion model

I am trying to evaluate the value of a Barrier option using Monte carlo method. The stock follows a jump diffusion model. I am using the method described in Metwally and Atiya. The authors describe ...
4
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0answers
159 views

Simple question concerning Jump process (Lévy process) model for a risky actif price process [closed]

Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is $$ \nu \left( dx\right) = A \sum_{n=...
4
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0answers
337 views

Discrete-time Jump-Diffusion Model

I am wondering if anybody could point me to any literature that talks about a discrete time version of the jump-diffusion model, I am aware that there is a paper by Amin (1993) that shows a discrete ...