Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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Jamshidian's trick for Swaptions

Following Brigo$^1$ p.77, we can decompose the price of a swaption as a sum of Zero-Coupon bond options (Jamshidian's Trick). To do so, the authors suggest to find $r^*$ the value of the spot rate at ...
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Regularity requirement for convergence of Euler scheme for stochastic integral?

Let $S_t$ be follow Black Scholes, then I am interesting in simulating the process $\int ^t _0 e^{-rt}1_{\{S_t\leq K\}}dS_t$ which is like a naive hedge of a European put, which does not work in ...
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PDE and Black Scholes problem

Consider Black Scholes problem $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$ with boundary condition $V(S,T)=f(S)$, ...
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Test for stationarity and make use of non-stationary points in financial market?

I have two questions to ask: What are the best methods to determine stationarity in a financial market (such as stocks) using MATLAB? What methods would you recommend to use in order to change from ...
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Measure change in a bond option problem

This is not a homework or assignment exercise. I'm trying to evaluate $\displaystyle \ \ I := E_\beta \big[\frac{1}{\beta(T_0)} K \mathbf{1}_{\{B(T_0,T_1) > K\}}\big]$, where $\beta$ is the ...
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Stochastic discount factor (aka deflator or pricing kernel) and class D processes

When (under what assumptions on the model) does a Stochastic Discount Factor need to be of Class D? What would be the implications if it was not? Is it connected to one of the no-arbitrage notions?
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How does this transformation for Euler Scheme in mean reverting SDEs alleviate instability?

I saw this text in the book - Interest Rate Modelling by Andersen volume 1 on Page 112: I am unable to understand: How does instability arise when we use the Euler scheme on X(t)? What change does ...
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Novikov condition for Vasicek process

Suppose that we have a money account $S^{(0)}$ with dynamics \begin{align} dS^{(0)}_{t} = r_{t} S^{(0)}_{t}\, dt, \end{align} where \begin{align} dr_t = a(b-r_t)\, dt + \sigma_{r} \, dW_t^{(0)}. \...
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Ito Diffusion with Change of Measure

Let $(X_t)$ be an Ito diffusion with speed $(V_t)$, under a probability measure P. Could there exist a change of measure to a probability measure Q, with Q ~ P, under which $(X_t)$ is an Ito diffusion ...
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For an Ito Process, $d\ln{X} \neq \frac{dX}{X}$ and $(d\ln{X})^2 = (\frac{dX}{X})^2$, but $d\ln{X} \neq \pm \frac{dX}{X}$

In normal calculus we can write $d\ln{x} = \frac{dx}{x}$ since there is no quadratic variation to deal with. This isn't true for stochastic processes, and Ito's Lemma is used to calculate $d\ln{X}$. ...
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SDE of futures price under non-constant interest rate and volatility process

I'm trying to figure out the form of the SDE of futures price under the risk neutral measure, when stock price follows GBM:             &...
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Pre-requisites for Finance Mathematics

I would like to pursue research in the areas of Financial Mathematics. Hoping to look into Operations Research, Risk Management and Stochastic Modeling. Anyone got some suggestions on useful resources ...
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Pricing caplet with Bachelier (normal dynamic) using forward measure

I'm trying to price caplet with Bachelier under forward measure, but I can't find any solution. Remind that Bachelier assumed rates follow a normal dynamic. So here what I was doing : $C_t(T,T+d)$ ...
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Prove the given stochastic integral are equally distributed

Let $W^i_t$ and $W_t$ be pairwise independent Brownian motions for $i \in \{1, \dots , d\}$. Let $X_t^i$ be $d$ independent Ornstein–Uhlenbeck processes for $i \in \{1, \dots , d\}$, i.e. each $X_t^i$...
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Computing Malliavin Derivative for European Call Payoff

Let $X_t$ be a continuous local-martingale modeling the stock price given by $$X_t = \int_0^t \sigma_t(T,K)dW_t ,$$ and $\sigma_t(T,K)$ is an $L^2$-measurable process not adapted to $W_t$'s ...
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Normalized Gains Process is a Q-Martingale - Proof and Intuition

I'm trying to work the proof that the normalized gains process, $G^z_t = \frac{S_t}{B_t}+\int^t_0\frac{1}{B_s}dD_s$ is a Q-martingale under Q (the risk-neutral measure). I'll show what I've worked ...
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Laplace Exponent of a Jump-Diffusion Process

I'm currently reading a paper (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2543702) which uses the following process to describe the dynamics of a firm's asset value: \begin{equation} V_t = ...
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Change of measure from physical to risk-neutral under Radon-Nikodym and Girsanov Theorem

Given a stochastic process, how do we prove and generate the change-of-measure? I have been trying to prove the change-of-measure as under the Radon-Nikodym theorem and Girsanov Theorem, but ...
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Proving Flow Property of Stochastic Differential Equation

I am trying to show that $X_t^{s,x} = X_t^{r, X_r^{s,x}}$ for $0 \leq s \leq r \leq t$, $x \in \mathbb{R}^n$ is a given initial condition for time $s$, for some SDE: \begin{equation*} d X(u)=b(X(u))d ...
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I ran through an equality in a paper I was reading but couldn't check if it is correct. Let $W^1_t$, $W^2_t$ and $W^3_t$ be three brownian motions, not necessarily independent, is it true that the ...
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Ultra Powerfull Vibrato Montecarlo for delta sensitivities of a not regular payoff

Ciao, I am working on a derivative with the following payoff at time $T$: $$\sqrt{(S_T - K)^+}$$ where $S_T$ is the value of the stock at the expiring date. As usual we will assume $S_t$ to be a ...
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kolmogorov backward equation intuition

The kolmogorov backward equation equation states that the probability density of a random variable $x$ which follows $dx= \mu dt + \sigma dw$ is given by $-p_t = \mu p_x + 0.5\sigma^2 p_{xx}$ ...
Can any one give some hint for this question? Let $\{S_t\}_{t=0}^\infty$ be an asset price process defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$. Assume that the log-return of $... 0answers 26 views Variance of integrated dynamical system Define time increment$\mu:=t_{k+1}-t_{k}$. Consider the signal$x(\mu)-\mathbb{E}[x(\mu)]$defined as$x(\mu)-\mathbb{E}[x(\mu)]=\frac{1}{\mu}\int_{t_{k}}^{t_{k+1}}\int_{0}^{\tau}e^{A(\tau-\delta)}...
Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion: $$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$ where $r>0$ is the risk-free ...