Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
2
votes
1answer
52 views
If S(t) is geometric Brownian motion, what is the distribution of S(t+h)-S(t)?
Suppose we have a geometric Brownian $S(t)$ which follows a lognormal process. Say
$$
\begin{equation}
dS_t = \mu S_t dt + \sigma S_tdW_t
\end{equation}
$$
My question is what is the distribution of $...
1
vote
2answers
86 views
How to numerically simulate exponential stochastic integral
For example given an integral
$$
\int^t_0 \exp(aW(t'))\,dt', t\in\mathbb R_+
$$
where $W(t')$ is a standard Wiener process.
I've been very confused about stochastic integrals like $\int^t_0 W(t')\,...
2
votes
0answers
53 views
SDE of futures price under non-constant interest rate and volatility process
I'm trying to figure out the form of the SDE of futures price under the risk neutral measure, when stock price follows GBM:
&...
1
vote
0answers
95 views
On quadratic covariation
I ran through an equality in a paper I was reading but couldn't check if it is correct.
Let $W^1_t$, $W^2_t$ and $W^3_t$ be three brownian motions, not necessarily independent, is it true that the ...
3
votes
1answer
84 views
HJM model Baxter Rennie: differentiating the discounted asset price using Ito
From Baxter and Rennie Page 145:
$Z(t,T) = exp(\int_{0}^{t}\Sigma(s,T)dW_s - \int_{0}^{T}f(o,u)du - \int_{0}^{t}\int_{s}^{T}\alpha(s,u)duds)$
where $\Sigma(t,T) = \int_{t}^{T}\sigma(t,u)du$
How ...
3
votes
1answer
120 views
Differential of integral of Wiener process over time
I am trying to compute this quantity:
$\frac{d}{dt}\int_{0}^{t} W_s ds $
Where $W_t$ is a Wiener process. Is there a theorem which tells how this can be computed?
I have tried https://en.wikipedia....
3
votes
1answer
83 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 ...
2
votes
1answer
128 views
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 ...
2
votes
4answers
151 views
Basic book on stochastic calculus, Itô and jump processes and Brownian Motion
I was looking for a good book that explains at beginner-level the basic of stochastic calculus, probability and random variables, Itô and jump processes as well as Brownian Motion.
At university we ...
6
votes
0answers
87 views
Random variable minus Integral of Ito Generator is a Martingale under what conditions?
I am reading about american option pricing and the variational inequality, and the book I am reading states, in the derivation of the variational inequality, the following is a martingale: $$M_s = U(s,...
0
votes
1answer
136 views
Why don't I get this right $\frac{d}{dt}\mathop{\mathbb{E}}\left[ e^{-\int_t^Tr(s)ds}|\mathscr{F}_t \right]$
Let $r$ a random process defined by :
$$dr_t=\theta(t)dt + \sigma dW_t$$
$\theta$ is deterministic in $t$ and $W$ a brownian motion.
I don't know where my calculation below is going wrong :
Let $...
1
vote
0answers
94 views
Ultra Powerfull Vibrato Montecarlo for delta sensitivities of a not regular payoff
Ciao,
I am working on a derivative with the following payoff at time $T$:
$$
\sqrt{(S_T - K)^+}
$$
where $S_T$ is the value of the stock at the expiring date.
As usual we will assume $S_t$ to be a ...
1
vote
0answers
54 views
kolmogorov backward equation intuition
The kolmogorov backward equation equation states that the probability density of a random variable $x$ which follows $dx= \mu dt + \sigma dw$
is given by
$-p_t = \mu p_x + 0.5\sigma^2 p_{xx} $
...
18
votes
1answer
411 views
What is the trickiest thing to get right in Rates Quant recently (2019)?
What are the biggest challenges for Rates Quants in 2019? Most quants have been through a lot over the past years-shifting their SABR models in JPY swaptions, fixing the FVA models for negative rates, ...
3
votes
2answers
72 views
Conditional Expectation with Indicator Functions for Poisson Process First Jump Time (Option Pricing PDE)
This is supposed to be for the derivation of a PDE for pricing a specific type of option, from the book 'Nonlinear Option Pricing' (Guyon).
The option delivers $g(\tau, X_{\tau})$ at time $\tau$ if $\...
0
votes
1answer
93 views
negative values in geometric brownian motion
A GBM
$ \frac{dx}{x} = \mu dx + \sigma dW $
solves to
$x_t = x_o e^{(\mu - \sigma^2)t + \sigma W_t}$
From the solution, it is clear that $x_t$ cannot become negative. However, it is not so clear ...
3
votes
1answer
81 views
Limits of integration when applying stochastic Fubini theorem to Brownian motion
I'm looking at the solution below from Quantuple, it's a nice, succinct solution but I'm confused about how the limits of the integrals in the second line come from. Could someone please elaborate on ...
2
votes
0answers
31 views
Stochastic Differential equation: CAPM
Let $R=(R_1, \dots ,R_M)$′ denote a vector of excess returns of M assets observed at $n$ time points, $0<t_1<t_2< \cdots <t_n<T$, within a time span $T>0$.
We wish to explain the ...
2
votes
0answers
76 views
Pre-requisites for Finance Mathematics
I would like to pursue research in the areas of Financial Mathematics. Hoping to look into Operations Research, Risk Management and Stochastic Modeling. Anyone got some suggestions on useful resources ...
3
votes
3answers
149 views
How to prove that $X_s=\int^s_0 f(u)dW_u$ is independant from $X_t-X_s$
I am asked to prove that $X_s$ and $X_t-X_s$ are independant for $s<t$ then $$X_t=\int^t_0f(u)dW_u$$ for a deterministic function $f$ and brownian motion $W_t$. For the proof I am giving a hint to ...
3
votes
0answers
30 views
Stochastic integral representation of $F(T-s,X_s)$-type equations
For $T\in R$ given and fixed consider:
$$
{\rm d}F(T-t,X_t)=g(T-t,X_t)\,{\rm d}W_t.
$$
where $g(t,x)$ is a given functions and $X_t$ is a given process driven by a brownian motion ($dX_t=(...)dt+(...)...
6
votes
1answer
127 views
Show that $(W_t, \int_0^t W_s ds)$ has a normal joint distribution
I have to show that, if $W_t$ is a 1-d Brownian motion then
$\biggl(W_t, \int_0^t W_s ds\biggr)$ has normal distribution.
Hint: apply Ito formula to this bivariate process.
Any idea or suggestion on ...
1
vote
1answer
56 views
How to check if $ E [\exp \{ \int_0^t \frac{Y_u^2}{1+Y_u^2}du \}]< \infty $
$dY_t=2Y_tdt+2\sqrt{1+Y_t^2}dW_t$ where $W_t$ is $P-$Brownian motion (Wiener process).
I have defined a new measure $Q$ where the Kernel density (In Girsanov theorem) is
$$ \phi_t = \frac{Y_t}{\sqrt{...
3
votes
2answers
185 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
82 views
How to express a process using Itos formula
Let $F(t,x)$ be the solution to the PDE
$$
F_t(t,x)=aF_x(t,x)+\frac{1}{2}F_{xx}(t,x),t>0
$$ $$F(0,x)=g(x)$$ for some function $g$.
Let $X_t$ be a process defined by
$$dx_t=aX(t)dt+dW(t)$$
Now ...
5
votes
1answer
92 views
Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?
Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$?
If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
4
votes
1answer
111 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(\...
0
votes
2answers
110 views
Integral of Wiener process over time
This should hopefully be an easy question to answer, but I am new to Stochastic Calculus and am gapping as to why the following is true, for a brownian motion $W_t$:
$$d(\int W_t dt ) = W_t dt$$
I ...
1
vote
1answer
100 views
Derive a mathematical equation for Eurodollar future rate
If we suppose that r(t) follows a Vasicek model, which is:
$$dr(t) = (\mu - \kappa r(t))dt + \sqrt\sigma dW(t)$$
How to derive an expression for Eurodollar future rate?
1
vote
1answer
44 views
Levy process and random measure
I am wondering if random measures are used under a Levy process and how this connects to finance (particularly pricing). Any paper or books for suggestions is welcomed.
3
votes
0answers
55 views
Solving BSDE in R
I was wondering how to implement a BSDE approximation in R. For example, if I have the toy BSDE
$$
dX_t = \mu dt + \sigma dW_t ; X_T\sim N(\mu_1,\sigma_1),
$$
for fixed real numbers $\mu,\mu_1,\sigma,...
1
vote
0answers
65 views
Question about Stochastic Calculus,(change of measure)?
Can any one give some hint for this question?
Let $\{S_t\}_{t=0}^\infty$ be an asset price process defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$. Assume that the log-return of $...
2
votes
1answer
54 views
Characteristic function and distribution of a random variable
This is exercise 4.3 in Bjork, Arbitrage Theory in Continous Time.
$$
X_t = \int^t_0 \sigma(s)dW_s
$$
$\sigma$ is a deterministic function and $W_t$ is brownian motion.
I am asked to find the ...
4
votes
1answer
220 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 ...
2
votes
2answers
91 views
Statistical estimation vs Stochastic calibration of models
I have never been able to deduce the precise differences between model building from the statistical perspective and the stochastic processes/calibration perspective. I can only infer that these are ...
1
vote
0answers
25 views
Variance of integrated dynamical system
Define time increment $\mu:=t_{k+1}-t_{k}$. Consider the signal $x(\mu)-\mathbb{E}[x(\mu)]$ defined as
$x(\mu)-\mathbb{E}[x(\mu)]=\frac{1}{\mu}\int_{t_{k}}^{t_{k+1}}\int_{0}^{\tau}e^{A(\tau-\delta)}...
2
votes
1answer
153 views
Black and Normal Model for Caplet using Python
I am able to Price Caplet using Black 76 model in Python. However, I am unable to price the same with Normal Model. Can anyone suggest what is missing ?
I am valuing caplet that caps interest rate on ...
2
votes
1answer
129 views
How to calculate the covariance between two stochastic integrals?
How to calculate the covariance between the integral of a Brownian motion at different times:
$$\text{Cov}\left(\int^{t_1}_0\sigma(t)dW_t,\int^{t_2}_0\sigma(t)dW_t\right)\ ?$$
I know the answer is:
$$\...
1
vote
0answers
37 views
Model of asset substitution/risk shifting in continuous time
Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion:
$$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$
where $r>0$ is the risk-free ...
3
votes
0answers
47 views
Euler discretization with jumps
There is a process
$B_t = B_0\prod_{i=1}^{N_t}(1-Z_n)$,
where $Z_n=e^{-ξ_n}$ for i.i.d exponentially distributed random variables $(ξn)_{n≥1}$ with rate $ρ=20$.
${N_t}$ is a counting process ...
1
vote
0answers
21 views
From one period to multi period risk neutral pricing
For a one period economy, we have the price of an asset as:
$ p_0 = E^Q [p_1 * \frac {B0}{B1}] $
where $B0 = e^{-r_0}$ = time 0 price of risk free bond maturing at time =1 and $r_0$ is known at t0. ...
1
vote
0answers
63 views
Correlated GBM and OU processes
I want to model two different stochastic processes, such that:
$X_t , V_t$ are correlated with coefficient $\rho$. Where:
$\frac{dX_t}{X_t}=\mu_1dt+\sigma_1 dW_{1,t}$ and $dV_t=\theta(\mu_2-V_t)dt+\...
1
vote
3answers
99 views
Need help to interpret the definition of a diffusion process
https://studentportalen.uu.se/uusp-filearea-tool/download.action?nodeId=1134155&toolAttachmentId=218130
In these lecture notes at page 15 and 16 I am looking at the definition of diffusion ...
3
votes
0answers
52 views
Price of a stochastic game between an agent and the market
In the article Pricing via utility maximization and entropy from Richard Rouge and Nicole El Karoui, they define the value function of the optimization problem as
\begin{align}
V(x,C) = \dfrac{1}{\...
0
votes
2answers
354 views
Integral of Brownian Motion w.r.t Time: what is wrong with this solution? [duplicate]
My question is about a stochastic integral of brownian motion w.r.t time.
Let $W(t)$ the Wiener process (or brownian motion). I want to calculate this:
\begin{eqnarray}
X(t)=\int_{0}^t dt' W(t').
\...
2
votes
1answer
101 views
The duality of the free energy and relative entropy used to deduce deduce the stochastic game between an agent and the market
I am reading the article Pricing via utility maximization and entropy by Richard Rouge and Nicole El Karoui. They talk about the relative entropy of a probability measure $Q$ with respect to the ...
2
votes
1answer
65 views
How to deduce the formula of the wealth process of a stochastic volatility model?
I am reading the paper Solution of the HJB Equations Involved
in Utility-Based Pricing from Daniel Hernandez and Shuenn Jyi Sheu.
The authors consider the utility function $U: \mathbb{R} \to \...
0
votes
1answer
43 views
standard brownian vs brownian motion
We say Xt with paramters (µ,σ) is brownian process if (Xt-s - X t) ~N (µs,σ2 s) AMONG other conditons .
Here we don't speak about any particular distribution for X t. We only say it is a brownian ...
2
votes
0answers
81 views
Pricing caplet with Bachelier (normal dynamic) using forward measure
I'm trying to price caplet with Bachelier under forward measure, but I can't find any solution.
Remind that Bachelier assumed rates follow a normal dynamic. So here what I was doing :
$C_t(T,T+d)$ ...
2
votes
2answers
205 views
Ho Lee model in Baxter&Rennie
I am currentyl reading Baxter&Rennie and I have a difficulty with understanding a derivation of formula for one function, $g(x,t,T)$ (this can be found on page 152 in the book). I know that there ...
