# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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### How do one solve $\int_t^T \exp[\int_0^u-( r-\delta_s)ds] dW_u$? Double integral with general deterministic function $\delta(t)$

How do one solve $\int_t^T \exp[\int_0^u-\left( r-\delta_s\right)ds] dW_u$ ? $\delta(t)$ is a general deterministic function. $r$ is constant.
1answer
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### Derivation using Ito's Lemma of price process

Define $q(t)$ as the log price minus a linear trend $$q(t) = \ln P(t) - \mu t$$ Assume the log price process = Equation 1: $$dq(t) = - \Theta q(t) dt + \sigma dW(t)$$ Can you show that the ...
1answer
276 views

### About the boundary conditions of the Black-Scholes-Merton PDE

I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve. Let $c(t,x)$ be the value of the ...
3answers
677 views

### Why is the CAPM securities market line straight?

Let $\gamma$ be the expected return, in terms of its exponential growth rate, of the market asset. If we set $\gamma=\mu-\sigma^2/2$ as explained by the Doléans-Dade exponential, then the expected ...
0answers
115 views

### Rigorous proof of Dupire formula (e.g. using Gyöngy's theorem)

Where can I find a rigorous proof of the Dupire formula (for example, using using Gyöngy's theorem)? I imagine this would be covered by a paper or by a standard financial math text, but I could not ...
0answers
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### Jamshidian's trick for Swaptions

Following Brigo$^1$ p.77, we can decompose the price of a swaption as a sum of Zero-Coupon bond options (Jamshidian's Trick). To do so, the authors suggest to find $r^*$ the value of the spot rate at ...
0answers
44 views

### Regularity requirement for convergence of Euler scheme for stochastic integral?

Let $S_t$ be follow Black Scholes, then I am interesting in simulating the process $\int ^t _0 e^{-rt}1_{\{S_t\leq K\}}dS_t$ which is like a naive hedge of a European put, which does not work in ...
0answers
437 views

### Multivariate Itô's lemma

Hey guys I'm looking for worked examples who show how to apply Itô's lemma in several variables, starting from the very basics. Thank you in advance!
0answers
283 views

### PDE and Black Scholes problem

Consider Black Scholes problem $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$ with boundary condition $V(S,T)=f(S)$, ...
0answers
671 views

### Test for stationarity and make use of non-stationary points in financial market?

I have two questions to ask: What are the best methods to determine stationarity in a financial market (such as stocks) using MATLAB? What methods would you recommend to use in order to change from ...
0answers
239 views

### Measure change in a bond option problem

This is not a homework or assignment exercise. I'm trying to evaluate $\displaystyle \ \ I := E_\beta \big[\frac{1}{\beta(T_0)} K \mathbf{1}_{\{B(T_0,T_1) > K\}}\big]$, where $\beta$ is the ...
0answers
231 views

### Stochastic discount factor (aka deflator or pricing kernel) and class D processes

When (under what assumptions on the model) does a Stochastic Discount Factor need to be of Class D? What would be the implications if it was not? Is it connected to one of the no-arbitrage notions?
2answers
739 views

### Is the Brownian motion multiplication rule a definition or is it a theorem?

Is the Brownian motion multiplication rule a definition or is it a theorem? Refer to the highlight part of http://i.stack.imgur.com/doQuT.png where $dw_1(t)dw_1(t)=dt$
1answer
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### Finding the process of $X/Y$

This comes from Mark Joshi's concepts of mathematical finance exercise 4 chapter 11. If $$dX_t = \alpha X_t dt + \beta X_t dW_t$$ $$dY_t = \alpha Y_t dt + \gamma Y_t d\tilde{W}_t$$ with $W$ ...
2answers
321 views

### What is the strong solution for this SDE

I want to calculate $E_t[(X_T-K)^+]$ where $$dX_t=\frac{3}{X_t}dt+2X_t dW_t$$ and $X_0=x$. I don't know how extact the strong solution of this SDE. Indeed I used Ito's lemma but it was not usefule. ...
1answer
427 views

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### How to compute the dynamic of stock using Geometric Brownian Motion?

I have been given the following question: Given that $S_t$ follows Geometric Brownian Motion, write down the dynamic of $S_t$ and then compute the dynamic of $f(t,S_t) = e^{tS^{2}}$ For the first ...