# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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### How to derive an option price for an asset with these dynamics?

Assuming my underline asset price follows the process: $$d\ln (F_{t,T})=-(1/2)\sigma ^2e^{-2\lambda(T-t)}dt+\sigma e^{-\lambda(T-t)}dB_t$$ How should I derive an option price formula?
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### Ito lemma of Convertible Bond under Two-factor Model Interest Rate

@Behrouz Maleki has provided the PDE of two factor model in other post so could anyone please provide Ito lemma of this equation and how this PDE was derived from Vasicek model. as far as I know it ...
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### What are the units of the variables appearing in a standard stochastic differential equation for a Wiener process?

The Black Scholes model assumes the following form for the Wiener process describing the evolution of the stock price S: $dS=\mu S dt + \sigma S dX$ Clearly $S$ ...
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### Asian Options-Change of Numeraire

Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends (with interest rate r, stock drift $\mu$ and volatility $\sigma$). Show that ...
659 views

### FX Rate dynamics

Let's suppose USD/EUR price in USD follows a GBM with $$dS_t = rS_tdt + \sigma S_tdW_t$$ What process does EUR/USD follow in EUR?
Suppose we are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $... 1answer 385 views ### Vector of differences of Brownian motion integrals is multivariate normal Given a 2-dimensional Wiener process$(W_{1},W_{2})$with correlation$\rho$. Let \begin{equation*} X(t):= F(t) + \int_{0}^{t} f(s) dW_{1}(s) + \int_{0}^{t} g(s) dW_{2}(s)\end{equation*} for some ... 1answer 2k views ### Given$S$is a Geometric Brownian Motion, how to show that$S^n$is also a Geometric Brownian Motion? Suppose that a stock price$S$follows Geometric Brownian Motion with expected return$\mu$and volatility$\sigma:$$$dS = \mu S dt +\sigma S dz$$ How to find out the process followed by variable$...
I have a very fundamental problem, please help me out. I am little confused with the derivation for the close form solution for the Geometric Brownian Motion, from the very fundamental stock model: \...