# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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### How to determine components of Affine Term Structure for an Ohrnstein-Uhlenbeck process?

I wonder how I can determine the components $A(t,T)$ and $B(t,T)$ for the zero-coupon bond price process $p(t,T)=e^{A(t,T)-r(t)B(t,T)}$? The components are defined in the following link: https://en....
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### Mark Joshi uses forward price to price an option that pays $S_t^2-K$ if $S_t^2>K$ and zero otherwise? Why can we do that?

The following question is taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, Exercise $6.6$ Suppose a stock follows geometric Brownian motion in a Black-Scholes ...
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### Show that $Ae^{rt}$ is a solution of the Black-Scholes equation. Why should this be so?

The following is taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, exercise $5.6$. Question: Show that $Ae^{rt}$ is a solution of the Black-Scholes equation. ...
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### How to determine exchange rate dynamics in currency derivatives

I need some guidance regarding exchange rate dynamics in currency derivatives. Following three dynamics are defined below, $\frac{dS(t)}{S(t)}=\alpha dt+\sigma dW(t)$ ; the stock dynamics in the ...
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### Heston Model and antithetic variables

I was implementing some variance reduction techniques for the heston model and came up with a question when implementing the antithetic variable technique. Namely, I was not sure if I had to implement ...
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### Determining the No Arbitrage price of max[B(T), S(T)]

Following is given, $dB(t)=rB(t)dt$ $dS(t)= (r-\delta)S(t)dt+\sigma S(t)dW(t)$ where, $r$ is the risk-free interest rate, $\delta$ the continous dividend yield $\sigma$ is the stock asset ...
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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 ...
As we can represent the integration of $f(x)$ on $[a,b]$ with the graph below, I was wondering how to represent the following integral with $X(t)$ a Brownian motion, $f(t)$ any function and $t_j = ... 2answers 571 views ### Finding price of the power option Let's assume a market with$d=1$and$X=X^1$satisfying$dX_t=\sigma X_t\,dW_t,\: \: X_0=1,$where$(W_t)$is a standard Brownian motion. Assume that$\mathbb{F}$is the natural filtration of$X$... 1answer 136 views ### How do we calcualte$E[W_sW_t|W_s]W_t$is a Brownian motion. How do we calculate this expectation? there are two cases:$s < tt < s\$ Do we have to distinguish the two cases or there is a unified way of calculating it
Suppose I have two processes both satisfying a displace lognormal diffusion: $$dX(t) = \alpha(t)[X(t) - a] dW(t)$$ $$dY(t) = \beta(t)[Y(t) - b] dW(t)$$ Note that the processes are perfectly ...