Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
550
questions
5
votes
2answers
576 views
Ito vs. Stratonovich: Why is it the exact midpoint that renders Ito-correction zero?
Perhaps I am approaching this from the wrong direction but I was just thinking about the relationship between Ito and Stratonovich integrals:
It is a well known result that to convert one into the ...
5
votes
2answers
1k views
Uniqueness of equivalent martingale measure in Black Scholes-Model
Let's consider standard Black-Scholes model with price process $S_t$ satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where $B_t$ is standard Brownian Motion for probability $\mathbb{P}$. I ...
5
votes
1answer
181 views
Ito representation unique up to indistinguishability? Proof?
Given an Ito-process $X(t)$, $t\in[0,T]$
$$X(t)=X_{0}+\int_{0}^{t}F(s)ds + \int_{0}^{t}G(s)dW(s)$$
with $F\in \mathbb{L}^{1}(0,T)$ and $G\in\mathbb{L}^{2}(0,T)$.
It is now often claimed that this ...
5
votes
2answers
2k views
Geometric brownian motion vs. Ornstein Uhlenbeck
I'm looking at the SDE of Geometric brownian motion(*):
$$d X(t) = \sigma X(t) d B(t) + \mu X(t) d t$$
(with analytic solution $X(t) = X(0) e^{(\mu - \sigma^2 / 2) t + \sigma B(t)}$)
and the SDE of ...
5
votes
2answers
573 views
Why is the black-scholes model arbitrage free when σ>0?
I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
5
votes
1answer
214 views
Explicit solution SDE
I have the following SDE:
$$dY_{t}=A\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{1}+B\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{2}$$
where $W_{t}^{...
5
votes
1answer
228 views
What is the stochastic differential of a general semimartingale?
By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon's "Analysis of Fourier Transform Valuation Formulas and Applications", on page 3:
$$H = B + H^c + h(x) \...
5
votes
1answer
406 views
Bayes Theorem with change of measure
Tomas bjork- arbitrage theory in continuous time.
Appendix B, proposition B41 says:
The proof is not clear to me.
Thanks to Gordon's comment below of $E^Q (X/G)$ being $G$ measurable, I think the ...
5
votes
0answers
113 views
Complete Financial Market: Integrability condition for Contingent Claims
Consider an arbitrage-free and complete financial market with underlying filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\,\in\,[0,T]},\mathbb{Q})$, where $T\in(0,\infty)$ is ...
4
votes
3answers
1k 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(\...
4
votes
3answers
1k views
For $B_t$ a Brownian motion what is the probability that $B_1>0$ and $B_2<0$?
Let $B_t$ be a Brownian Motion. What's the probability that $B_1>0$ and $B_2<0$?
4
votes
3answers
453 views
Show that $E[B_t|\mathscr{F}_s] = B_s$ for $B_t = W_t^3 - 3 t W_t$
Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$
Let $(B_t)_{t \geq 0}$ where $B_t = W_t^3 - 3tW_t$. ...
4
votes
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$ ...
4
votes
3answers
370 views
Determine $E[W_p W_q W_r]$
Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$
Let 0 < p < q < r. Determine $E[W_p W_q W_r]$.
...
4
votes
2answers
2k views
Differential equation for log-returns
I have a question that might be trivial to most of you, but somehow I'm not able to solve it by myself. I have a disagreement with my colleague on the distributional properties of a Geometric Brownian ...
4
votes
1answer
1k 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$$
has normal distribution with mean zero. How can I find the variance of $X$?
4
votes
1answer
127 views
What the expectation of S^2 is from GBM? [closed]
I was at an interview and was asked to write down the SDE for GBM.
$$
dS = S\mu dt + S\sigma dX
$$
Then I was asked how I would compute the expectation of S^2. I didn't know where to start. Any ...
4
votes
1answer
265 views
How can I calculate $Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right)$
How can I calculate?
\begin{align}
Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right)
\end{align}
Thank you for your attention.
4
votes
1answer
136 views
How do we calcualte $E[W_sW_t|W_s]$
$W_t$ is a Brownian motion. How do we calculate this expectation?
there are two cases:
$s < t$
$t < s$
Do we have to distinguish the two cases or there is a unified way of calculating it
4
votes
2answers
459 views
Application of Ito's lemma
Let $X_t$ be some stochastic process driven by wiener process ($W_t)$ so it can be expressed as:
$$dX_t=(...)dt+(...)dW_t$$
Let $f(t,x)$ be some $C^2$ function. Define the process $Z_s=f(t-s,X_s)$ ...
4
votes
1answer
302 views
For the Brownian motion integrate
I want to calculate
$$\operatorname{E} \left[ \int_0^1{W(t)dt \cdot \int_0^1{t^2W(t)dt}} \right].$$
I discovered that the first integral is $\operatorname{N}(0, \frac{1}{3})$ but I don't know how to ...
4
votes
1answer
612 views
Merton's jump diffusion
Can someone help me finding the expected value of the solution to Merton's jump diffusion model:
\begin{align}
S_t &= S_0 \exp \left( \left(r - \frac{\sigma^2}{2} - \lambda k \right) t + \sigma ...
4
votes
1answer
4k views
Girsanov Theorem for Quanto/Compo adjustment
Assume that I have a foreign asset
$$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 ...
4
votes
1answer
167 views
Stochastic Differential
Let $W_t$ be a Wiener process. It is clear to me that $dW_t$ is of size $\sqrt{dt}$. This can be seen because
$$
\mathrm{Var}(W_{t+\Delta} - W_{t})=\Delta.
$$
But am I allowed to actually write $(...
4
votes
2answers
1k views
Financial Mathematics - Martingales example
Was hoping somebody could help me with the following question.
Prove that under the risk-neutral probability $\tilde{\mathsf P}$ the stock and the bank account have the same average rate of growth. ...
4
votes
1answer
502 views
Ito`s Lemma problem
Can someone help me with calculus for this problem.
I have these 3 equations and with Ito`s Lemma I have to find $dXt$.
\begin{cases} dY= μYdt+σYdB
\\ X=\frac{1}{2}cY\\ dc =-aαcdt\end{cases}
4
votes
2answers
157 views
Show that the two solutions of the SDE are equivalent
I have a process:
$$dr_t = (W_t^1 - ar_t)dt +\sigma dW_t^2$$
where $W_t^1$ and $W_t^2$ are brownian motions with instantaneous correlation coefficient $\rho$.
I want to show that the solution of this ...
4
votes
2answers
2k 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?
4
votes
2answers
701 views
Asymptotic behavior property of geometric Brownian Motion proof
Online I found the asymptotic behavior property of geometric Brownian Motion $X_t$as:
If $\mu$ (drift parameter) is $\ge$ $\sigma^2/2$ where $\sigma$ is the volatility parameter, then $X_t \...
4
votes
2answers
472 views
question on Leif Andersen's “Interest Rate Modeling, vol 2 Term Structure Models”
I'm reading Leif Andersen's "Interest Rate Modeling, vol 2 Term Structure Models" and met a problem on Chapter 14 LM Dynamics and Measures, $\S$ 14.2.5 Stochastic Volatility, Lemma 14.2.6, on page 602....
4
votes
1answer
185 views
Are all change of measure operations between equivalent probability measures Doléans-Dade exponentials?
Let $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space, where $\mathbb{F}=\left(\mathcal{F}\right)_{t\in[0;T]}$ and $\mathcal{F}=\mathcal{F}_T$. Let $(W_t)_{t\in[0;T]}$ be ...
4
votes
1answer
134 views
Invariance Scaling of Brownian Motion
Prove $\frac{1}{\sqrt{t}}\log\left(\int_0^t \exp(B_s)\mathrm{d}s\right)$ converges to $\sup\limits_{t\in [0,1]}B_t$ in distribution as $t\to\infty$. I have a sense to use scaling invariance, but no ...
4
votes
1answer
125 views
Compute distribution of a stochastic variable
$sign(x)=1$ if $x\geq0$
$sign(x)=-1$ if $x< 0$
Consider
$$
X_t = \int^t_0 sign(W_u)dW_u
$$
where $W_t$ is a wiener proces.
How can I determine the distribution of $X_t$ and compute $E[\exp(\...
4
votes
1answer
770 views
Ito's Lemma: Multiplication Rule
I have a conceptual question about Ito's lemma, in particular, the multiplication.
Ito's multiplication rule states, that multiplying dt by itself or by dx (the stochastic differential) equals zero. ...
4
votes
1answer
528 views
Step by Step Guide to Learn Quantitative Finance [closed]
Can some one help in creating step by step guide to learn Quantitative Finance?
The suggestions should be in the lines of
1- Which Maths topics needs to be learn 1st
2- Which Maths Books or ...
4
votes
1answer
541 views
Application of Ito's Lemma, finding the condition for the martingale
The Vasicek short rate model is
$$dr_t=\kappa(\theta-r_t)dt+\sigma dW_t$$
Define the processes $x_t$ and $f(x,t)$
$$x_t=\frac{r_t}{\kappa}(1-e^{-\kappa(T-t)})+\int_0^tr_sds$$
$$f(x,t)=e^{a(T-t)-x_t}$$
...
4
votes
1answer
385 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 ...
4
votes
1answer
336 views
Closed form solution of PDE of Option Price
Let $V=V(S_t,t)$ be the option price and
\begin{align}
V_t+\mu\,S\,V_S+\frac{1}{2}\sigma^2\,S^2\,V_{SS}=0\\
V(S_T,T)=\ln (S_T)^{2}.
\end{align}
My question: How can I obtain a closed form solution of ...
4
votes
1answer
138 views
Discounted risky asset stochastic process problem
$S_t$ is the random variable representing the risky asset price at time $t$.
M_t is the riskless asset. They are governed by the equations
$\frac{dS_t}{dt}=\mu dt + \sigma dZ_t$ and
$dM_t = rM_t ...
4
votes
3answers
117 views
Volatility of Exchange Option
I got a question and its partial solution, and have some doubts about the volatility of its geometric Brownian motion process:
Question:
How would you price an exchange call option that pays $max(S_{...
4
votes
1answer
182 views
Evaluating the SDE $dX_t = t\,dS_t$
The process $S$ is a geometric Brownian motion with an SDE: $dS_t = S_t(\sigma\, dB_t + \mu\, dt)$. I'm stuck evaluating $E(X_t)$ and $V(X_t)$, where $dX_t = t\,dS_t$.
4
votes
1answer
391 views
Pricing the Passport option
Suppose underlying asset $S$
$$dS = \mu Sdt + \sigma Sd W$$
our portfolio $\pi$ consist with $q(t)$ stock $S$ and cash $\pi - qS$...
4
votes
1answer
173 views
Notion of risk-less portfolio in derivation of Black-Scholes
EDIT: As pointed out by Gordon in the comments, the portfolio I considered in my original post is neither self-financing nor (locally) risk-free.
Though the central question is still open. Suppose ...
4
votes
1answer
932 views
How to solve this PDE using Feynman-Kac?
I have the following problem right now: solve
$$F_t(t,x) + rxF_x(t,x) + \frac{\sigma^2}{2}F_{xx}(t,x) = rF(t,x), \\ F(T,x) = (x - K)^2.$$
How do I solve this?
There exists a theorem to solve this, ...
4
votes
2answers
327 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 $\...
4
votes
2answers
1k views
Libor Market Model: numeraire change
I am currently studying the Libor forward market model, and although I get the mechanics behind the main arguments, I still do not have an intuitive idea of what's exactly the objective behind ...
4
votes
1answer
498 views
unique equivalent martingale measure in incomplete markets
Do you have any idea about how we can prove, and under which conditions, that an equivalent martingale measure (EMM) in an incomplete market is unique? The assumptions we have made are:
1) that the ...
4
votes
1answer
123 views
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....
4
votes
1answer
270 views
Bond-price dynamics in the Vasicek model
Hello I am studying about interest rate modeling
There is one good source about Vasicek (link: https://web.mst.edu/~bohner/fim-10/fim-chap4.pdf). However there is one equation that I try but unable ...
4
votes
1answer
253 views
Feynman-Kac converse
If the pricing function $F$ satisfies the black scholes PDE, then I can obtain risk-neutral evaluation formula from Feynman-Kac.
If I already have the risk-neutral evaluation formula, can I still use ...
