stochastic-calculus

In Mandelbrot(1968)'s paper, the fractional brownian motion, denoted by $B_{H}(t,\omega)$,(t>0) is defined by $$B_{H}(0,\omega)=b_{0}$$ $$B_{H}(t,\omega)-B_{H}(0,\omega)=\frac{1}{\Gamma(H+\frac{1}{2})}\{\int^{0}_{-\infty}[(t-s)^{H-1/2}-(-s)^{H-1/2}]dB(s,\omega)+\int^{t}_{0}(t-s)^{H-1/2}dB(s,\omega)\}$$ I have difficulty understanding fractional brownian motion by self study.Is there an intuitive…

I am trying to understand the interest that exists for SDEs without having done SDEs. It's very easy to get motivation for ODE and PDE. For ODE, just take Newton's laws, and basically most of what ...

I've bought Gatheral's book on Local Volatility and I have troubles with understanding a part where he shows that local variance is a conditional expectation of instantaneous variance. Why in the second equation from the bottom he just skips the term $\theta (S_T-K)dS_T$? He says that it's because $F_{t,T}$ is a martingale. I see that $F_{t,T}$ is a martingale, but don't know how this helps. Also…

A crash course on the theory underpinning Monte Carlo methods along with a brief survey of their practical applications in the realm of Quantitative Finance.

A Wiener process has infinitely many states of the world at any time step. Does that not mean that there are infinitely many EMM's for any model that uses the Wiener process? But then if there is only one EMM for this model, how is it possible that the Esscher transform can be exactly the right transform, out of all possible transforms, to get the exact Black-scholes pricing formula?

I am learning about stochastic processes. Let $X_i \sim \mathrm{Bin}(1,\frac12)$ and independent $M$ and $N$ be the stopping times for the first Tail (i.e. $0$) and first Head (i.e. $1$). My task is ...

A stochastic process describes a dynamical system evolving over a linearly ordered set (“time”), typically taken to be the (positive) integers or real numbers, whose dynamical laws of motion are morphisms in the Kleisli category of the Giry monad (or any other probability monad). By working in the larger category of algebras of that monad, a characterization of a stochastic processes can be model…

I am trying to prepare to exit academia and transition into quantitative finance from a statistical physics background (after spending 6 years as a postdoctoral researcher). My experience covers heavy computation (Matlab/C++), stochastic processes (Brownian motion, Langevin/Fokker-Planck equations), and statistical mechanics, but I have little formal exposure to asset pricing. Given this backgrou…

I have two coupled SDEs \begin{align*} dS_t=rS_tdt+V_tdW_t^{(1)},\\ dV_t=aV_tdt+b(V_t)dW_t^{(2)},\\ \end{align*} where $W_t^{(1)}$ and $W_t^{(2)}$ are independent Brownian motions, initial input data are $S_0$ and $V_0$ , $a(\cdot)$ and $b(\cdot)$ are sufficiently well-behaved, and I use an Euler-Maruyama discretisation with $N$ timesteps. How exactly should one calculate the derivative of a payo…

My question is simple, consider a European call with payoff max(S_T-K, 0), Let's suppose that the underlying stock follows a binomial tree with up and down factors I know as we take n goes to infinity that the stock is log-normally distributed at time t=T (I know how to derive it). The idea is to derive the B-S-M pricing formula as the expected value of the present value of max(S_t-K, 0) using t…

I am new to stochastic calculus. I would like to compute the closed-form solution for $$ \int_0^t \exp \left( \alpha s - \sigma W_s \right) \; {\rm d}s \tag{1}$$ $$ \int_0^t \exp \left( \alpha s - \sigma W_s \right) \; {\rm d} W_s \tag{2} $$ which I encountered when trying to solve the following stochastic differential equation (SDE) $$ dX_t = \theta(\mu - X_t)\; dt + \sigma X_t \; dW_t $$ How to…

I am a mathematician. What's the go-to reference for a proper math-based introduction to martingale theory and arbitrage pricing? The books I am being referred to deal mostly either with the discrete case, or, if its continuous, then it does not contain all the proofs and there's a lot of hand-waving (for example, Bjork's Arbitrage theory in Continuous time, which does not even contain proper pro…

I am analyzing the following function within a financial mathematics framework: $$ f(t) = \dfrac{B(S; S) \cdot m(t)}{B(t; S) \cdot m(S)} $$ where: $$ B(t; S) := \mathbb{E}_{t}^{\mathbb{P}} \left[\exp\left(-\int_{t}^{S}r_{f}(u)\, du\right)\right] $$ and $$ m(t) := \exp\left(\int_{0}^{t} r_{f}(u)\, du\right) $$ Definitions : $B(t; S)$ represents the price at time $t$ of a zero-coupon bond maturing …

To model a structured product, I thought of using a geometric Brownian motion model, where I choose a certain mean and variance for the normal distribution to make sure that a certain percentage of paths (as a result from Monte Carlo) cross a threshold value where different conditions apply. However my question is does this violate the risk-free and arbitrage free assumption? Meaning I can no lon…

Consider a HJM framework $$d f(t, T) = \sigma (t, T) d W_t^T$$ which is a SDE of instantaneous forward rates on $T$ -forward measure, and let $$P (t, T) = \exp (-\int_t^T f (t, u) d u)$$ $$B (t) = \exp (\int_0^t f(u, u) d u)$$ be a $T$ -discount bond price and a continuously compounded money market account. By definition, \begin{align} P(t, T) &= \exp (-\int_t^T f(0, u) d u - \int_t^T \int_0^t \s…

How to make decisions when your spreadsheet is lying about the future The post A Gentle Introduction to Stochastic Programming appeared first on Towards Data Science .

I'm looking to simulate the stochastic price and volatility process (Heston model) using some form of Euler method for Monte Carlo approximation of option prices. The results that I get are acceptable for deep in the money options and at the money options but not very satisfying at all for deep out of the money options. I want to reduce the variance for faster convergence and the importance sampl…

In binomial tree model, the stock price is modelled in the form of $S_{k\delta}=S_{(k-1)\delta}\exp(\mu\delta+\sigma\sqrt\delta Z_k)$ , where $\delta$ is time invertal between two observations $S_{k\delta},S_{(k-1)\delta}$ , $Z_k=1,-1$ for upward and downward scenarios of the stock price change. I noted some illustrations of variance and mean to explain why the model is set in the form, but I can…

Abstract This paper provides a rigorous deconstruction of the Narrquest framework (Chen, 2026) through the lens of Stochastic Control Theory and Information Physics. By defining "Invalidity Conditions" (IC-1 to IC-5) that decouple narrative structure from technical optimization and closed causal loops, Chen (2026) constructs an inherently unstable open-loop architecture. We demonstrate that such …

This video walks through the breakthrough research by Muhle-Karbe et al. linking order flow, market impact, and rough volatility through a single structural statistic — bridging microstructure and stochastic volatility theory. 🎥 Video Tutorial 🎥 Watch Video: https://youtu.be/wF1vaW8WwzU Topics: quantitative finance, investment analysis, financial education, financial education video, trading t…