# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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### Resources to learn stochastic calculus

I'm looking for a good resource to learn stochastic calculus but I'm not very good at calculus . So could anyone please tell whether the book Stochastic Calculus for Finance by Steven Shreve is a ...
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### Stochastic exponential and Girsanov

Let's say I have an instrument $V$ with payoff at maturity: $P(S_T)$ (for example $1_{S > K}$) that I want to price under stochastic rates: $dr_t = \mu dt + \theta_t dW_t$, with $W_t$ a Brownian ...
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### ABM Crossing Times

Suppose I have a process that follows an arithmetic brownian motion $dX_t = \sigma dW_t$ How do I calculate, within a certain interval $\Delta t$ , the expected number of times that the process will &...
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### Kolmogorov's backward equation with initial value

I am refreshing basic financial mathematics concepts and self-learning from the text, A first course in Stochastic Calculus, by Louis Pierre Arguin. I understand that, the transition probability ...
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### How calculate expectation and variation of stochastic integral Based on Heston model?

I was calculated Heston volatility model. But I think it is wrong. $dS_t = \mu dt + \sqrt V_t dW_t^s$ $dV_t = k(\theta - V_t)dt + \sigma \sqrt V_t dW_t^v$. $dW^s_t dW^v_t = \rho dt$ take integral to ...
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### Weak stationarity of continuous ARMA process from Brockwell

I am currently working on Brockwell "Levy-driven CARMA processes" (2001) and I am stuck in the introduction. So we have a continuous AR process (CAR(p)) \begin{align*} X_t=e^{At}X_0+\...
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### Moments of the integral of the exponential of Brownian motion/Normal random variable

I'm studying arithmetic Asian options and there is integral of the following form: $$X_T=\int_0^T e^{\sigma W_t+\left(r-\frac{\sigma^2}{2}\right)t}dt,$$ where $W_t$ is a Brownian motion/Wiener process....
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### Approximation of an Itô integral with python

Exercise 3.11 (Approximation of an Itô Integral). In this example, the stochastic integral $\int^t_0tW(t)dW(t)$ is considered. The expected value of the integral and the expected value of the square ...
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### Distribution of Geometric Brownian with time-dependant volatility

The process $S(t) =\exp\left(\mu.t + \int_0^t\sigma(s) \text{d}W(s) - \int_0^t \frac{1}{2}\sigma^2(s)\text{d}s\right)$ where $\sigma(s) = 0.03s$ is log-normally distributed, but i'm not sure about the ...
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### Price a contingent claim with payoff $(S_{1T}-S_{2T})^+$ at time $T$

Two stocks are modelled as follows: $$dS_{1t}=S_{1t}(\mu_1dt+\sigma_{11}dW_{1t}+\sigma_{12}dW_{2t})$$ $$dS_{2t}=S_{2t}(\mu_2dt+\sigma_{21}dW_{1t}+\sigma_{22}dW_{2t})$$ with $dW_{1t}dW_{2t}=\rho dt$....
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### Why is Feynman-Kac formula applicable in Burgard-Kjaers PDE paper?

In the paper Partial Differential Equation Representation of Derivatives with Bilateral Counterparty Risk and Funding Costs by Burgard and Kjaer, they say we may formally apply the Feynman-Kac theorem ...
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### Balland - SABR goes normal

To summarise this very long post : please help me understand the undetailed proof of the quoted paper. I am not comfortable using a result I do not fully understand. I am reading Balland & Tran ...
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### State space equation of CARMA(p,q) processes

Thanks for visting my question:) I am currently working on Carma(p,q) processes and do not understand how to derive the state equation. So the CARMA(p,q) process is defined by: for $p>q$ the ...
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### How to understand Short Gamma and Long Volatility for Leveraged ETFs?

In the book Leveraged Exchange-Traded Funds: Price Dynamics and Options Valuation, it describes a static delta-hedged long volatility position by simultaneously shorting regular/inverse leveraged ETFs ...
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### What is the PDE for this interest rate derivative?

We have the following model for the short rate $r_t$under $\mathbb{Q}$: $$dr_t=(2\%-r_t)dt+\sqrt{r_t+\sigma_t}dW^1_t\\d\sigma_t=(5\%-\sigma_t)dt+\sqrt{\sigma_t}dW^2_t$$ What is the PDE of which the ...
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### Ito formula and confusion with the differential operator $d$

Thanks for visiting my question. Im am currently working on this paper (https://arxiv.org/abs/2305.02523) and I am stuck at page 21 (Theorem 14 proof). First these SDE's were defined: \begin{align*} ...
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### Can the PDE of Black and Scholes really be derived from the CAPM?

Black and Scholes (1973) argue that their option pricing formula can directly be derived from the CAPM. Apparently, this was the original approach through which Fischer Black derived the PDE, although ...
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### Kalman Filtering to estimate parameters of G2++ Model

I'm trying to use Kalman Filtering to estimate the parameters of the G2++ short rate model. For this, I've been using Implementing Short Rate Models: A Practical Guide by F.C. Park. For reference, he ...
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### Integral of Brownian motion w.r.t. time

Let $$X_t = \int_0^t W_s \,\mathrm d s$$ where $W_s$ is our usual Brownian motion. My questions are the following: Expectation? Variance? Is it a martingale? Is it an Ito process or a Riemann ...