Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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1answer
176 views

Computing Correlation between Forward Rates

I have the feeling this question has an extremely simple answer but I'll put it out to the group anyway. Imagine I have data for 3M and 6M forward rates following a lognormal process, and that I ...
4
votes
1answer
370 views

discounted price is a martingale under any measure?

Assume I have some dynamics for the stock price under 2 different measures: risk-neutral and forward measures: $$dS_t=r S_tdt+\sigma S_td\tilde{W_t}$$ $$dS_t=\alpha S_tdt+\sigma S_td\hat{W_t}$$ now ...
5
votes
1answer
418 views

Martingale representation theorem

Let $r_t, \theta_t$ denote some stochastic processes driven by a $N$ dimensional Brownian motion $W_t$ (they are of course assumed adapted to the natural filtration $\mathcal{F}_t$ of that Brownian ...
1
vote
1answer
135 views

Piecewise Ito formula

Usually Ito's lemma is stated for $C^{1,2}(\mathbb{R}^{d+1},\mathbb{R})$ functions. My question is does Ito still hold if the domain is restricted. That is if the semi-martingale $Z_t$ is only ...
4
votes
1answer
813 views

What's the variance of this Ito integral?

I am reading stochastic calculus and I have understood that the process $$X=\int_{0}^{1}\sqrt{\frac{\tan^{-1}t}{t}}dW_t$$ has normal distribution with mean zero. How can I find the variance of $X$?
23
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3answers
16k views

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 ...
0
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1answer
2k views

Correlation coeffitiont between two stochastic processes

I want to find correlation coeffitiont between $W_t$ and $\int_{0}^{t}W_s ds$. I think that these are uncorrelated. But Why? So thanks
3
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1answer
323 views

Dupire's formula proof

I just have a question for the beginning of a proof: Suppose $\frac{dS_{t}}{S_{t}}=(r_{t}-q_{t})dt+\sigma(t,S_{t})dW_{t}$ with $r,q,S$ stochastic. In the book I read, it is written: We define the ...
4
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3answers
844 views

Variance of Brownian Motion

Can someone point me into the right direction to calculate this one: $E(B^4_t)=3t^2$ I had tried using the following property with no luck: $E(B^4_t)=E(B^2_tB^2_t)=E(\int B^2 dt )E(\int B^2 dt )=[E(\...
1
vote
1answer
350 views

Stratonovich Integral and Ito's lemma

Let $(\Omega, \mathcal{F},\mathbb{P},\{\mathcal{F}\}_t)$ be a filtered- probability space and $W_t$ be standard Wiener process. I want to show stratonovich integral of $W_t$, i.e $\int_{0}^{t} W_s ○ ...
-1
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2answers
341 views

Close form solution for Geometric Brownian Motion

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: $$\...
0
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1answer
104 views

Does the partition of time in a simple process depend on the omega in probability space?

In Steven Shreve's book "Stochastic Calculus for Finance 2", page 126, a simple process $\Delta(t)$ is a stochastic process such that there is a partition of time $0 < t_1 < ... < t_n \leq T$,...
2
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1answer
186 views

Mix of Arithmetic and Geometric Brownian Motion

Talking with some traders the other day, I found out that they were using a pricing model based on a mix between a geometric brownian motion and an arithmetic brownian motion to price certain ...
3
votes
1answer
551 views

What is the probability that a Brownian Bridge hits an upper barrier $U$ before a lower barrier $L$?

The probability that an arithmetic Brownian motion process $dt = \mu dt + \sigma dW$ hits an upper Barrier $U$ before it hits a lower barrier $L$ is given by $$ \mathbb{P}(\tau_U\leq \tau_L) = \frac{\...
5
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1answer
222 views

What is the probability that a OU process hits an upper barrier U before a lower barrier L?

What is the probability that the arithmetic OU process $dx_t= \theta(\mu-x_t)dt+\sigma dW_t$ hits barrier $U$ before hitting barrier $L$ when $L<x_0<U$ ?
1
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1answer
48 views

How to understand the following limits when kapa limits to Zero

The equation is quite simple, however it is not very obvious to me to have the following relationship: $$\begin{equation} \frac{1-exp(-\kappa(T-t))}{\kappa}\rightarrow(T-t) \quad \rm{when\space} \...
8
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1answer
630 views

Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \...
2
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1answer
136 views

stochastic interest rate $r_t=x_t+y_t$

Let $$dr_t=(\alpha(t)-\beta r_t)dt+\sigma dW_t$$ where $\alpha$ is non stochastic process and $\beta$ and $\sigma$ are constant. Can we write process $r_t$ in the form $$r_t=x_t+y_t$$ where the ...
1
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1answer
137 views

How to understand the following brownian integral using Fubini's method?

I am a little bit stucked with the following integral process, using Fubini's method, this is an intermediate step of short rate Merton Model. $\int_{t}^{T} W(s)ds=\int_{0}^{\hat {T}}ds\int_{0}^{s}...
7
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1answer
556 views

How to show that this process is “normally distributed”?

Say we have following SDE (Vasicek): $$dr(t) =(b-ar_t) dt + \sigma dW_t$$ I am able to reach an integral form of this SDE : $$r(t) = r(0) e^{-at} + \frac{b}{a}[1 - e^{-at}] + \sigma e^{-at}\int_0^t e^...
2
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1answer
83 views

Option price derivation with these dynamics

If my underlying follows a dynamics of the form \begin{align*} dF(t,T)/F(t,T)=\sigma_1(t,T)dW_1(t)+\sigma_2(t,T)dW_2(t), \end{align*} where $\sigma_1(t,T)=h_1e^{-\lambda(T-t)}+h_0$, and $\sigma_2(t,T)...
2
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1answer
164 views

How PCA is performed in the paper “Markov Models…”

can anyone explain in a bit detail on how PCA is performed in the paper "Markov Models for Commodity Futures: Theory and Practice" by Leif B. G. Andersen. I'm not clear on how the high dimension ...
3
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1answer
167 views

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?
7
votes
1answer
529 views

How were these SDE derived?

Can anyone give me a detailed explanation of how below equations (3) and (4) are derived from (1) and (2)? \begin{align*} \frac{dF_{t,T}}{F_{t,T}} &=\sigma e^{-\lambda(T-t)}dB_t, \tag{1}\\ \ln(F_{...
11
votes
2answers
2k views

FX forward with stochastic interest rates pricing

I would like to extend the following question about FX Forward rates in stochastic interest rate setup: "Expectation" of a FX Forward We consider a FX process $X_t = X_0 \exp( \int_0^t(r^...
4
votes
2answers
278 views

Are two stochastic processes independent if the Wiener processes inside are uncorrelated

Assume there are two stochastic processes: $dx_t = \alpha_1(x_t,t)dt + \beta_1(x_t,t)dW^1_t$ and $dy_t = \alpha_2(y_t,t)dt + \beta_2(y_t,t)dW^2_t$. Does $dW^1_t\times{dW^2_t} = 0$ imply that $\...
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2answers
433 views

Conditional probability of geometric brownian motion

I created paths using GBM to implement The stochastic mesh method. But the method requires the conditional distribution, given some S(t) the probability of S(t+1). I've searched and can't find this ...
4
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2answers
1k views

Square of Wiener process

In Ito's calculus one often comes $dW^2=dt$. How does this come about? What is it's relation to the Milstein method?
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1answer
113 views

Lebesgue-Stieltjes integration and related topics

The theory of stochastic integration relies on the concept of the Lebesgue-Stieltjes integral. However, it is hard to find a textbook that handles this concept in detail. Take, for instance, Chung ...
2
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1answer
148 views

Problem with derivating integral

I have a doubt : I know that if $x_{t}=\int_{0}^{t}\gamma(s)dW_{s}$ (with $W_{s}$ a brownian motion), we have : $dx_{t}=\gamma(t)dW_{t}$ What about if $x_{t}=\int_{0}^{t}\gamma(s,t)dW_{s}$. Do I have ...
4
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1answer
280 views

clarification to log-stock price formula

Having financial market with safe rate r and risky asset S with dynamics under physical measure P $$\frac{dS_t}{S_t}=\mu dt +\sigma dW_t$$ what is the log-stock price? Using Ito formula it is ...
2
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0answers
138 views

Quadratic variation

The following question is more math than quant, but since it arises from a mathematical finance textbook, I've figured the good people in this sub might be able to help me. So here goes. In the 3rd ...
1
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1answer
66 views

optimal strategy problem (using Jensen's inequality)

I have a strategy in Samuelson model with zero safe rate defined as $$Z_t^{\Pi}=\frac{X_t^{\Pi}}{X_t^{\rho}} \quad \quad (1)$$ where $$\frac{dX_t^{\Pi}}{X_t^{\Pi}} = \mu \pi dt + \sigma \pi \ dW_t \...
0
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1answer
104 views

stochastic discount factor transformation

I have $$\frac{dM_t}{M_t}=-\frac{\mu}{\sigma} dW_t + \gamma_t dB_t, \tag{1}$$ where $B_t$ and $W_t$ are two independent Brownian Motions, which was further presented as $$ M_t=\exp \left( -\frac{\mu}{...
2
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1answer
239 views

HJM framework problem - showing that HJM drift condition implies that $b(z)=b+βz$ and $(ρ)^2=α$

Hi I am looking for some general clarification to Heath–Jarrow–Morton framework. I am analyzing a problem where the forward rate is modeled as $$ f(t,T)=e^{\beta(T-t)} Z_t+h(T-t) \tag{1}$$ for some ...
3
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1answer
348 views

CIR model - nth moment generation $E^*[r_T^n]$

I am analyzing the nth moment generation process for $r_t$ with dynamics defined by CIR model $r_t$ has following dynamics $$dr_t=a(b-r_t)dt+\sigma \sqrt{r_t} dW_t^* \quad \quad (1)$$ for some ...
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2answers
701 views

CIR model problem - deriving PDE, Feynman-Kac

I am reviewing a CIR model problem, where $r_t$ has following dynamics $$dr_t=a(b-r_t)dt+\sigma \sqrt{r_t} dW_t^* \quad \quad (1)$$ for some constants $ab>\frac{\sigma^2}{2} \quad$ Letting T ...
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1answer
75 views

Expected Value of Products of Processes

Suppose I have two processes. $A_t = A_0 \exp((a-\frac{1}{2}\sigma_A^2)t+\sigma_A W_t^A$ $B_t = B_0 \exp((b-\frac{1}{2}\sigma_B^2)t+\sigma_B W_t^B$ I would like to calculate $E[A_s B_t]$ where s &...
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1answer
368 views

Ho-Lee model - A and B derivation for $P(t,T)=e^{-A(t,T)-B(t,T)r_t}$

I am analyzing the transition of the bond prices in the affine models in the form of $P(t,T)=e^{-A(t,T)-B(t,T)r_t}$ using the property that the diffusion and the drift of an affine model can be ...
2
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1answer
274 views

“Expectation” of a FX Forward

I have an FX process $X_t = X_0 \exp((r_d-r_f)t+ \sigma W_t)$. Now clearly $E[X_t] = F_{0,t}^X$. i.e. a forward contract of the process $X$ starting at time 0 and maturing at time $t$. What if I ...
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1answer
464 views

Vasicek model problem

I am analyzing a problem where the below is given Vasicek model with risk-neutral dynamics $$dr_t = \kappa (\theta - r_t)dt + \sqrt{r_t} dW_t \quad \quad (1) $$ bond prices $$P(t,T)=e^{A(t,T)-B(t,T)...
4
votes
1answer
3k views

Girsanov Theorem for Quanto/Compo adjustment

Assume that I have a foreign asset $$Y_t = Y_0 \exp \left((r_f-\frac{1}{2}\sigma^2_Y)t+\sigma_Y W_t^1\right)$$ and an exchange rate $$X_t = X_0 \exp\left((r_d-r_f-\frac{1}{2}\sigma^2_X)t+\sigma_X ...
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1answer
577 views

Quanto/Compo adjustments - Product of two geometric brownian motion

Let's say I have two processes $X_t =X_0 \exp((a-\frac{1}{2}\sigma_X^2)t +\sigma_X dW_t^1)$ and $Y_t=Y_0 \exp((b-\frac{1}{2}\sigma_Y^2)t +\sigma_Y dW_t^2)$ and I then multiply them together (like ...
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1answer
129 views

approximating fBm stochastic integral

Suppose I have the following stochastic integral: $$\int_a^b f(t)dB_H(t)$$ with the term $dB_H(t)$ a fractional brownian motion with associated $H$ parameter. Is it true that for $H \in (1/2,1)$, ...
4
votes
1answer
183 views

On the reflection of a stochastic integral

Let ${(I_t)}_{t\geq 0}$ be a stochastic integral defined by $$ I_t=\int_{0}^{t}\theta_sdW_t, $$ where $W$ is a standard Brownian motion defined on $(\Omega,\mathcal{F},{(\mathcal{F}_t)}_{t\geq 0},\...
0
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1answer
55 views

trading strategy problem - initial capital x buys S over time [0,T] at the constant rate of x/T euros per unit of time

I am looking for clarification to the trading strategy problem where the number of stocks is depending on time. In the Market with zero safe rate and stock dynamics defined as $$\frac{dS_t}{S_t}=\...
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2answers
141 views

Multivariate Ito problem $M_t=\frac{X_t}{Y_t}$

I am analyzing a problem given in the lecture slides published here (Slide 7-8 Example of Multivariate Ito’s Lemma). Can anybody explain how the $M_t$ was calculated out of the Ito formula. I ...
2
votes
3answers
182 views

Perpetual American Put Supermartingale property

Discounted price process of an american put (perpetual) has a $dt$ part in it, which is negative if the price at time $t$ is less than the optimal exercise price. This is the only thing that drags the ...
1
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1answer
587 views

Integration in the Hull-White SDE

I'm stuck in solving the SDE in Hull-White interest rate model. I do not have a thorough background in math (only Real Analysis during my blissful undergrad years), so I am having trouble ...
2
votes
1answer
88 views

investor terminal value of portfolio with two risky assets 1) correlated 2)not correlated $\phi_t^1=S^{2}_{t}, \ \phi_t^2=S^{1}_{t}$

I am analyzing a problem where I have two stocks described by the equations $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t}$$ $$ \frac{dS^{2}_{t}}{S^{2}_{t}}=\mu_{2} dt + \sigma_{2}...