Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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Black and Normal Model for Caplet using Python

I am able to Price Caplet using Black 76 model in Python. However, I am unable to price the same with Normal Model. Can anyone suggest what is missing ? I am valuing caplet that caps interest rate on ...
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Code examples of solving Stochastic Optimal Control problems

I'm currently reading a book demonstrating how Stochastic Optimal Control can solve common optimization problems encountered within quantitative finance. I haven't covered much continuous mathematics ...
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markov property for stochastic differential equation

Suppose the stochastic equation： \begin{equation*} d X(u)=\beta(u,X(u))d u+\gamma(u,X(u))d W(u). \end{equation*} Suppose $X(T)$ is the solution of above stochastic differential equation with initial ...
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How to express the volatility of two correlated Ito processes $Wt_1, Wt_2$ expressed in terms of $W_t$?

Having two correlated Ito processes ($W_t^1$ and $W_t^2$ are correlated Brownian motions with correlation $\rho$) $dX_{t} =\mu_{1} dt + \sigma_1 dWt_1$ $dY_{t} = \mu_{2} dt + \sigma_2 dWt_2$ ...
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SVCJ (SVJJ) Duffie et. al Model implementation in Matlab

I'm attempting to implement aforementioned SVCJ model by Duffie et al in MATLAB. so far without success. It's supposed to price vanilla (european) calls . parameters provided, the expected price is: ~...
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I have some questions regarding a proof of Gÿongy's lemma given in 1 I would like to understand the following passage: $$\int_{s=t_0}^{s=t}\mathbb{E}\left[\delta(X_s-K)\langle dX_s\rangle^2 \right]= ... 1answer 186 views Limits of integration when applying stochastic Fubini theorem to Brownian motion I'm looking at the solution below from Quantuple, it's a nice, succinct solution but I'm confused about how the limits of the integrals in the second line come from. Could someone please elaborate on ... 3answers 172 views How to prove that X_s=\int^s_0 f(u)dW_u is independant from X_t-X_s I am asked to prove that X_s and X_t-X_s are independant for s<t then$$X_t=\int^t_0f(u)dW_u$$for a deterministic function f and brownian motion W_t. For the proof I am giving a hint to ... 2answers 231 views Moment Ito's Process Proof I have a following stochastic integral - related problem that I have difficulty to solve: Given $$dX_t = -\alpha X_tdt+\sigma\sqrt{X_t}dW_t$$ and the second moment of ... 1answer 390 views Dupire's formula proof I just have a question for the beginning of a proof: Suppose \frac{dS_{t}}{S_{t}}=(r_{t}-q_{t})dt+\sigma(t,S_{t})dW_{t} with r,q,S stochastic. In the book I read, it is written: We define the ... 2answers 756 views Question about the martingale property of stochastic integral Let W_{t} be a Wiener process, and let$$X_{t} = \int^{t}_{0}W_{\tau}d\tau$$Is X_{t} a martingale? We can rewrite in differential form as$$dX_{t} = W_{t}dt$$,which means X_{t} is a diffusion ... 1answer 891 views Does Ito/Malliavin calculus have any applications helpful for direction based trading? I'm an aspiring computer scientist who want to move into algorithmic trading at some point. At the moment I'm mostly focusing on courses in machine learning/data analysis etc. but I've noticed that ... 1answer 322 views What is augmented data when simulating stochastic differential equations using Gibbs Sampler? I am reading this paper on Bayesian Estimation of CIR Model. Basically, it is about estimating parameters using Bayesian inference. It estimates this stochastic differential equation:$$dy(t)=\{ \...
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Today seems to be question day for me, sorry. The complex process $$dX = i\sigma dW$$ where $i = \sqrt{-1}$ and $dW$ is a standard (real-valued) Brownian motion will have a negative variance ...