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The continuous-time limit of asset price processes where there is more than one asset

I've read Merton's article "On the Mathematics and Economics Assumptions of Continuous-Time Models" (Reprinted in Continuous-time Finance, Chapter 3), where Merton proved that the price of ...
Steve Norkus's user avatar
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0 answers
29 views

Diffusive Limits for compound poisson process

I was reading about compounded Hawkes process and came across diffusive limit theorems. Where can I find diffusive limit theorems for Poisson processes. I am new to this area, is there a nice ...
user50123's user avatar
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1 answer
230 views

Kou model — solving PIDE for European and American options in Python

Toivanen proposed$^\color{magenta}{\star}$ a method to solve the partial integro-differential equation (PIDE) with a numerical scheme based on Crank-Nicolson. In particular, he proposed an algorithm ...
pierrot's user avatar
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2 votes
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Kou model - can't reproduce prices of European Option from Toivanen and Forsyth [duplicate]

I have implemented the Kou option model for pricing vanilla option. I have checked that my program can replicate the price of the option in the original paper of 2002. However, when I use it to price ...
pierrot's user avatar
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What would be the practitioner way of hedging jump risks?

I have developed a keen interest in volatility strategies and have implemented various approaches based on practitioner delta. This delta is meticulously calibrated using a no-arbitrage implied ...
Frank's user avatar
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1 answer
111 views

Pure jump process in Duffie, Pan and Singleton's paper

In page 1349 or Section 2.1 of "Duffie, D., Pan, J., & Singleton, K. (2000). Transform Analysis and Asset Pricing for Affine Jump-Diffusions. Econometrica, 68(6), 1343-1376" the pure ...
Roberto Palermo's user avatar
2 votes
0 answers
166 views

Is it possible to calibrate Mertons Jump Diffusion Model such that it matches mean and vola from a normal process without jumps? [closed]

I'm currently playing around with Mertons version of jump diffusion processes where i'm testing the predicitions of a trading model given a time series with and without jumps to isolate the effects of ...
T123's user avatar
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3 votes
2 answers
96 views

Does discretizing a diffusion model make it look like a jump diffusion model?

Can we distinguish a sample generated from a diffusion model with large time steps from a sample generated from a jump diffusion model. Not mathematically but numerically (if we ask a computer to tell ...
bigInner's user avatar
  • 171
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0 answers
86 views

Affine Jump Diffusion

I'm currently looking into affine jump-diffusions. I would like to get to know the literature better and I know the paper by Duffie, Pan, and Singleton (2000) is a very celebrated paper. Although I ...
Marc Allan's user avatar
1 vote
0 answers
129 views

Stochastic volatility with jumps [closed]

I'm reading the Duffie, Pan, and Singleton (2000) paper now and I've stumbled upon something that seems to me as an inconsistency. Whenever I look up the SVJJ model, I see that its log-transform is ...
CasMath's user avatar
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75 views

Mixing formula for SVJ models

I am trying to understand the mixing formula (Hull and White formula) for stochastic volatility models with jumps in the asset price. One article which discusses this is Lewis, The mixing approach to ...
p.sibuea's user avatar
5 votes
1 answer
590 views

Variance of the log returns in jump diffusion with time-varying jump sizes

I'm trying to calculate the variance $\mathrm{var}\left(\log\frac{S\left(t\right)}{S\left(0\right)}\right)$, where the dynamics of the stock $S$ follows a jump-diffusion process given by $$\frac{dS\...
Skumin's user avatar
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How do I estimate volatility for MPR historical data

How can I estimate volatility with historical data for Monetary Policy Rate (MPR) to use in a short rate model? I could use regular techniques like simple standard deviation or max likelihood, but the ...
Oliver Mohr Bonometti's user avatar
2 votes
1 answer
239 views

How to solve numerically the IDE of GUILBAUD & PHAM model?

By the Guilbaud & Pham model (Optimal high frequency trading with limit and market orders, 2011), the authors said that integro-differential-equation (IDE) can be easily solved by numerical method....
JMNQC's user avatar
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1 answer
214 views

Separating jumps and diffusion

I want to model energy prices. I have two markets, lets say market 1 and 2. Market 1 is continuously traded, and I will assume it follows brownian motion. So the value of the asset could be defined ...
charelf's user avatar
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1 answer
312 views

Jump Diffusion Process question

I have a European call option with time maturity $T=3$ years,$K=50$, and given that $S(t)$ refers to the derivative is being described by the geometric Brownian motion with $S_{0}=100$ and $r = 0.04$....
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1 vote
1 answer
128 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 ...
Giogre's user avatar
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3 votes
1 answer
896 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 ...
Math122's user avatar
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2 votes
0 answers
135 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 $...
Kapes Mate's user avatar
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483 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 ...
Paul's user avatar
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1 vote
2 answers
306 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 ...
Hans Larsen's user avatar
1 vote
0 answers
86 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,...
michael tancredi's user avatar
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0 answers
114 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 ...
HoldBreath's user avatar
2 votes
0 answers
180 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. ...
michael tancredi's user avatar
3 votes
0 answers
154 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}\...
Alex's user avatar
  • 688
2 votes
1 answer
85 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 ...
Physics Geek's user avatar
-1 votes
2 answers
93 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 ...
Oop's user avatar
  • 27
6 votes
2 answers
211 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 ...
CA-Quant's user avatar
1 vote
0 answers
167 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 ...
na1201's user avatar
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3 votes
2 answers
812 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 ...
RedZoro's user avatar
  • 33
3 votes
1 answer
200 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
Amir's user avatar
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5 votes
1 answer
323 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=-...
user258521's user avatar
1 vote
0 answers
162 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: $$dS = (r+\lambda*p)Sdt + \sigma*SdW+Sdq$$ $...
M00000001's user avatar
  • 647
1 vote
1 answer
198 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....
M00000001's user avatar
  • 647
4 votes
4 answers
476 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 ...
confused's user avatar
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2 votes
0 answers
53 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 ...
Alexa's user avatar
  • 21
1 vote
0 answers
34 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 ...
alexbougias's user avatar
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1 vote
0 answers
64 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 ...
confused's user avatar
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1 vote
1 answer
500 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 ...
Furqan Hashim's user avatar
2 votes
0 answers
169 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 ...
Furqan Hashim's user avatar
2 votes
0 answers
186 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] \...
godric97's user avatar
3 votes
4 answers
939 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 ...
KingOfOil's user avatar
6 votes
0 answers
83 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 ...
Lili M.'s user avatar
  • 61
8 votes
1 answer
5k 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 ...
PyRsquared's user avatar
2 votes
0 answers
775 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 ...
MrPefister's user avatar
1 vote
1 answer
179 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.
alexbougias's user avatar
  • 1,416
2 votes
0 answers
137 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 ...
user155214's user avatar
4 votes
1 answer
367 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 $N_t$ ...
Younes S's user avatar
3 votes
0 answers
111 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 ...
jhon's user avatar
  • 31
2 votes
1 answer
1k 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}...
locvol's user avatar
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