11 votes

Risk neutral measure for jump processes

Assume a constant risk-free rate $r$ and no dividends. Generalisation is straightforward. To preclude arbitrage opportunities, under the risk-neutral measure $\Bbb{Q}$, the discounted asset price ...
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  • 14k
8 votes
Accepted

Merton's jump diffusion

Given for all $i$ the mean of $\epsilon_i$ is $k$ and that the $\{\epsilon_i\}_i$ are i.i.d., we have$^{\text{(1)}}$: $$\begin{align} E\left[\prod_{i=1}^{N_t}(1+\epsilon_i)\right] &=E\left[E\left[...
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7 votes
Accepted

Black-Scholes formula for Poisson jumps

We assume that the process $\{J_t, \, t\ge 0\}$ is defined at the jump times of the Poisson process $\{N_t, \, t \ge 0\}$, and all the jump sizes are independent and identically distributed. That is, \...
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  • 20.5k
6 votes
Accepted

Cadlag Property of Jump Proccesses

Intuitively, cadlag expresses the fact that we know a jump has occurred after the fact, but we never have advance knowledge that the jump is about to occur (i.e no knowledge of the starting point for ...
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  • 9,587
6 votes
Accepted

exercise on multivariate Ito's lemma + jumps (Poisson)

Answer Assuming the Poisson process $N_t$ is independent from the Brownian motions $(W_{1,t},W_{2,t})$, you'll have \begin{align} df(X_{1,t},X_{2,t}) &= \frac{\partial f}{\partial X_{1,t}} dX_{1,...
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  • 14k
5 votes
Accepted

Solution for a SDE for a Bond found in Bugard & Kjaer

I'll assume $$ J_t = \sum_{i=1}^{N_t} Z_i$$ be a compound Poisson process, with $(T_n)_{n\geq 1}$ being the jump times for Poisson process $(N_t)_{t\geq 0}$ and $(Z_i)_{i\geq 1}$ sequence of i.i.d. ...
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  • 5,038
4 votes

Levy process and random measure

There is a whole literature on risk-neutral modeling with Levy processes. Consider an arbitrage-free market where asset prices are modeled by a stochastic process $(S_t)_{t \in [0,T]}, \mathcal{F}_t$ ...
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  • 2,996
4 votes

Basic book on stochastic calculus, Itô and jump processes and Brownian Motion

The book Stochastic calculus for finance by Steven Shreve gives a good introduction to stochastic calculus applied to finance. A whole chapter is dedicated to the Itô Integral for example. It covers a ...
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4 votes

Basic book on stochastic calculus, Itô and jump processes and Brownian Motion

Elementary Stochastic Calculus by Thomas Mikosch is an excellent introduction to the topic in a very compact way. Alternatively, Stochastic Calculus for Finance II: Continuous-Time Models by Steven ...
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4 votes

SDE Jump-Diffusion

Let $$ J_t = \sum_{i=1}^{N_t} Z_i$$ be a compound Poisson process, with $(T_n)_{n\geq 1}$ being the jump times for Poisson process $(N_t)_{t\geq 0}$ and $(Z_i)_{i\geq 1}$ sequence of i.i.d. variables ...
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  • 5,038
3 votes

What is the intuition behind "jumps" causing volatility skew?

Jumps are an attempt to solve a math mistake in Modern Portfolio Theory. In the 19502-70s, economists were working on solving the variance-mean tradeoff. Furthermore, they needed to do so with ...
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  • 4,105
3 votes
Accepted

Binomial tree with jumps

Try this paper (although it's advanced): https://www.sciencedirect.com/science/article/pii/S0377042702009032 The topic you picked is not necessarily an easy one :)
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  • 5,191
3 votes

stochastic vol modelling not enough for smile

For diffusion models (i.e. no jumps): Local volatility models: match vanilla options market prices; give unrealistic volatility dynamic (smile flattens when we move forward in time); Stochastic ...
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  • 2,110
3 votes
Accepted

Euler Scheme for Jump-Diffusion models

Commonly, we employ the Euler scheme for $\Delta\ln(S_t)$, not for $\Delta S_t$. Let us specify the jump part as $$ S_{t+}=S_{t}J\Rightarrow dS_t=S_t(J-1) $$ where $J$ is a strictly positive random ...
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  • 5,783
3 votes
Accepted

Second variation of a Brownian motion under jump-diffusion process

$$ X_t = B_t 1_{t<0.5} + (x+ B_t) 1_{t\geq 0.5} = B_t + x1_{t\geq 0.5}$$ $$ [X, X]_t = [B, B]_t + x^2 1_{t\geq 0.5} = t+ x^2 1_{t\geq 0.5}$$ (the author probably intended to use $0.5$ as jump size ...
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  • 5,038
2 votes

Basic book on stochastic calculus, Itô and jump processes and Brownian Motion

Consider “Paul Wilmott Introduces Quantitative Finance” if you look for an enjoyable read, good intuition and a not too academic approach
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  • 1,581
2 votes

Understanding and simulating the jumps in Merton's Jump-Diffusion SDE?

For anyone else searching for good Merton Jump Diffusion examples, found a much better notated reference here: https://www.codearmo.com/python-tutorial/merton-jump-diffusion-model-python
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2 votes
Accepted

Bond PDE under an Affine Jump Diffusion model

Let $P(t, r_t, T)$ be the bond price at time $t$, where $0 \leq t \leq T$. Then, by Ito's formula, \begin{align*} &\ P(t, r_t, T) \\ =& P(0, r_0, T) + \int_0^t\partial_s P(s, r_s, T) ds + \...
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  • 20.5k
2 votes
Accepted

SDE Jump-Diffusion

$dJ_{t}$ can be understood as a Steljes measure , when you want to define jumps using bounded variation function , but you can simply understand it as $J_{t}-J_{t-}$ Those processes belong to a more ...
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  • 118
2 votes
Accepted

Expected Value of Mean-Reverting Jump Process

First, we need to be careful about putting the condition at the right place: \begin{align} e^{kt}\mathbb{E}[\mu_t] -\mu_0 &= \mathbb{E}\bigg[\sum_{m=1}^{N_t} e^{k\tau_m}\eta_m\bigg]\\ &= \...
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  • 136
2 votes
Accepted

Vanilla Call Option Priced Using Jump Diffusion Model

No arbitrage means that you can't have a portfolio with a positive expectation without risk. let's suppose that the value of option with jumps is lower than $C_{BS}(0,S_0)$ Please consider the ...
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2 votes

Solution for a SDE for a Bond found in Bugard & Kjaer

As a complement to @ir7’s comprehensive derivation, in the case of Burgard and Kjaer’s the jump process $J_t$ models the default of the issuer. You specialize the process by setting $Z_1=-1$, while ...
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2 votes

Basic book on stochastic calculus, Itô and jump processes and Brownian Motion

If you're interested in learning about stochastic calculus outside of the context of quant finance (which I think is a better approach than learning about it solely in the context of finance), check ...
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  • 1,132
2 votes

What can the area under a GBM jump curve tell you

I suppose the expectation could be used to get at some time-weighted average price (TWAP) where we assume each instant of observation has infinitesimal and equal weight $\frac{dt}{T}$: $\bar{S}_T := \...
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  • 2,750
2 votes

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

Don't bother. I implemented the strategy and took it live and only lost 4000 dollars after turning over 10 million dollars worth of stock. The Cox process assumption is the flaw in this paper. It ...
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1 vote

I just got Matlab, what are some options that I should model in a jump diffusion

The bellwether Indices for testing, are NASDAQ, Technology sector, S & P 500 Big 500 capital weighted Stocks, Russell 2000, MID sector stocks and some small stocks. It is better to use the data ...
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1 vote

What is the intuition behind "jumps" causing volatility skew?

Jumps do not imply fat tails. See the simulation in R. Note that the excess kurtosis of [normal variable + jump] is negative. ...
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  • 242
1 vote

What is the intuition behind "jumps" causing volatility skew?

Actually, I do not think it's true. Jumps, when added to the Black-Scholes (BS) dynamics, do modify the volatility surface. However, the volatility skew may get inverted: the implied BS volatility may ...
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  • 242
1 vote

Predicting time series using Jump Diffusion model and Neural Networks

Most of the work you will find on jump diffusion models will be in derivative pricing or related work on insurance. In essence, they tend to be interesting ways to think about future distributions. ...
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  • 2,366
1 vote

Crash cliquet price

Defining $\tilde{S}_n = S_n/S_{n-1}$ (which is well defined, assuming $S_n > 0$ for all $n$), the problem becomes that of barrier option pricing. In particular, you're looking to price a down-and-...
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  • 171

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