stochastic-calculus

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 .

Friends, another possible route to solution of above problem when explaining variables are correlated is to somehow orthogonalize the explaining variables and then use the orthogonalized variables in our analysis. I had thought about this yesterday but problem is that every hermite polynomial component has its own correlation matrix and there is no single correlation matrix that would have to be …

Calculation of Conditional Density of Y Given K Explaining Random Variables  \(X_1, X_2, X_3, \ldots \,,X_K\) Are Correlated With Each Other. We re-write the equations of conditional mean of Y and other equations. Then conditional mean of Y given all of \(X_1, X_2, X_3, \ldots \,,X_K\) is given as \[\,E \left[Y\,| \left(Z_{x_1},\,Z_{x_2},\ldots,\,Z_{x_K}\right)\,\right]\,=\,\,ch_0\,+\,\sum\limits…

We copy the Bivariate Z-series of Y given X from previous post. Our task is to find correlations \(\rho_n\) in this given Z-series so that Hermite correlations of Y with X are perfectly retrieved. \begin{equation} Y(Z_x\,,\,Z_{\tilde{y}})=\,c_0\,+\sum\limits_{n=1}^{N} c_n\, \, {(\rho_n\, Z_x\,+ \,\zeta_n\,Z_{\tilde{y}})}^n \end{equation} When we expand all binomials in the above expansion, it res…

Friends, I will write about explaining the algorithms used in matlab program in the afternoon but summing up past few posts, I want to explain how all conditional central moments of a random variable dependent on specific realizations of correlated random variables can be found. Calculation of Conditional Central Moments of Y given Specific Realizations of K Correlated Random Variables \(X_1, X_2…

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…

Friends, I had promised to explain the details of the method used to write the matlab program for finding conditional density of a continuous random variable conditional upon the values taken by several other correlated continuous random variables. In the following few posts, I will write about preliminaries and context of the method and then move on to details of conditional density calculation …

Friends, I have been doing research to develop probability theory infrastructure around Z-series/Hermite-series for past four years and we have come a long way now and we have developed enough tools that we can successfully apply this infrastructure on practical applications that have still remained unsolved. This is a good development and I think all of us want to continue the momentum with furt…

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…

As many friends on wilmott already know from past, whenever I do good research, my persecution always increases as a rule and there are strong efforts by mind control agents to disable the very sequences of neurotransmitters on which I had been able to do good research. The present occasion is no exception to this rule. I have been able to do some good research in past few weeks and that has shar…

Friends, I went ahead and made some changes to the conditional density generation algorithm (I am posting the new matlab program at the end of this post). Now there are no complex coefficients in the analytical module. I turned out that problem was not with correlations, I rather realized that odd and even coefficients of hermite-series have to possibly not larger than ratios applied on hermite c…

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…

How to model two correlated stocks where the innovations are asymmetric and heavy-tailed? The model is estimated with MCMC on historical returns (not option data) and later used for VaR simulation. Consider the univariate SV model: $$ \begin{aligned} r_t &= \mu + e^{h_t}\epsilon_t \qquad \epsilon_t \sim N(0,1)\\ h_t &= \omega + \phi(h_{t-1}-\omega) + \bigl(J_t-\mathbb E[J_t]\bigr),\\ J_t &= \log …

Why does ordinary calculus fail when applied to the stock market? This video explores the "jagged" reality of financial assets and how Itô’s Lemma provides the mathematical bridge between deterministic physics and stochastic finance. 🎥 Video Tutorial 🎥 Watch Video: https://youtu.be/3-RdnIsr3f4 Topics: quantitative finance, investment analysis, financial education, financial education video, tr…

A comprehensive treatise on Itô's Lemma: the mathematical bridge between the smooth world of Newton and the jagged reality of financial markets. Master the fundamental theorem that transforms stochastic differential equations into the Black-Scholes framework. 📊 Deep Research Topics: quantitative finance, investment analysis, financial education, financial research, market analysis

Thomas Roos recently put a preprint on SSRN called Simple, Flexible, Analytic, Arbitrage Free Volatility Interpolation. Being interested in the subject, I had a detailed look at it. It turns out that Thomas stumbled upon spline stochastic collocation without realizing it. There are a few differences in his approach: The optimization is on the x’s instead of the y’s, meaning the strike…

Fabrice Rouah wrote two books on the Heston model: one with C# and Matlab code, and one with VBA code. The two books are very similar. They are good in that they tackle most of the important points with the Heston model, from calibration to simulation. The calibration part (chapter 6) is a bit too short, it would have been great if it presented the actual difficulties with calibration in practice…