Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
609
questions
0
votes
1answer
86 views
Pricing of $(S(T_0)-S(T))^+$
Problem: Consider a new derivative that at time $T$ pays $Y =(S(T_0) − S(T))^+$
where
$0 < T_0 < T$ is a fixed date.
(i) Show that the arbitrage-free of Y at time $t = T_0$ is given by $\pi_{...
-1
votes
1answer
180 views
How can I learn stochastic process & stochastic calculus in two weeks? [closed]
I am going for an interview for a quant job. The interview will focus on my mathematical knowledge about stochastic process & stochastic calculus, and I believe I will definitely be asked to solve ...
0
votes
0answers
22 views
Future forward convexity adjustment as the expected profit from reinvesting margin payments?
Having looked at the formula for the convexity adjustment as a function of the covariance between rates accruing till maturity and asset price, I have an intuition that the difference between fair ...
0
votes
0answers
62 views
SDF derivation by a stochastic process
I have a stochastic process to model the stochastic discount factor (SDF) with M:
\begin{equation}
dM_t = aM_tdt + bM_t d Z_t
\end{equation}
where, $Z_t$ is a standard brownian motion. How do I show ...
1
vote
1answer
109 views
What is the interpretation if the real world measure $\mathbb P$ is equal to the martingale measure $\mathbb Q$
Out of interest, is there anything noteworthy about a market when its real world measure $\mathbb P$ is actually also its martingale measure. In other words the real world measure $\mathbb P$ is equal ...
0
votes
1answer
75 views
Calibrating the Ornstein-Uhlenbeck process with an additional parameter
Firstly I find the spread between two cointegrated time-series $Y_t$ and $Z_t$ by finding the best slope parameter $\beta$ in the equation $spread_t = Y_t - \beta Z_t$ (via Cointegrated Dickey-Fuller ...
0
votes
0answers
56 views
What is the difference between the geometric brownian motion and cumulative product of percentage returns?
I wonder why the following code: one using GBM and the other using cumulative product of normally distributed percentage returns slightly different values.
...
5
votes
0answers
128 views
Summary of Stochastic Derivatives, Integrals, Expectations, and Variances
I wanted to make a summary table of stochastic functions to improve my understanding. Maybe the following should be a wiki page on this site so others can add functions and examples? Does the ...
1
vote
1answer
142 views
Trouble With Applying Ito's Lemma
I am having trouble applying Ito's Formula to the following:
Let $Z_t = W_{1t}^2 e^{W_{1t}+ \int_0^t W_{3s}dW_{2s}}$. Find $dZ_t$. $W_1,W_2,W_3$ are independent Brownian motions.
I know the formula ...
2
votes
0answers
122 views
Is it possibile to use Ito Formula here?
I have this process: $dY_s^y=\alpha(s,Y_s^y)ds + \frac{1}{2}\beta^2(Y_s^y)^2dW_s$ with inital value $Y_s^y=y$.
Moreover $\alpha(s,y)$ is a linear function in $y$ and bounded is $s$. I was wondering if ...
1
vote
1answer
157 views
the order of integral of Brownian motion
When we want to obtain the order of $\int_{0}^{T} B_{t} d t$, we can use the scale property of Brownian motion.
Let $B$ be a Brownian motion. Is the order of $\int_{0}^{T} B_{t} d t$ correctly ...
1
vote
0answers
185 views
Reduced form of credit model
The price for a simple credit bond, where a credit event is modeled as the first jump of a Poisson process $N$, with stochastic hazard rate $\lambda$, is given by
$$P_t = P(t, \lambda, N)$$
such that,
...
1
vote
0answers
115 views
Intuitive explanation for theta Hull-White
I am having a hard time coming up with an intuitive explanation for the long term mean $\theta$ in the Hull-White model:
$$\mathrm{d}r_t=[\theta(t)-\alpha r_t]\mathrm{d}t+ \sigma_t \mathrm{d}W_t$$
So ...
6
votes
3answers
629 views
Expectation of exponential of 3 correlated Brownian Motion
Consider,
are correlated Brownian motions with a given
I want to calculate the,
,
I can't think of a way to solve this although I have solved an expectation question with only a single exponential ...
0
votes
1answer
58 views
Pairs trading by transforming two cointegrated series into a mean-reverting process?
I am slightly confused about the following.
Let us assume I have two cointegrated time-series. I would like to model their 'cointegration' by a mean-reverting Ornstein-Uhlenbeck process since if they ...
4
votes
1answer
203 views
Expectation of $\int_0^t \frac{1}{1+W_s^2} \text dW_s$ [duplicate]
I am trying to calculate the expectation of
$$\int\limits_0^t \frac{1}{1+W_s^2} \text dW_s,$$
where $(W_t)$ is a Wiener process.
I was told that the value of this expectation is zero. Can someone ...
1
vote
0answers
39 views
Moments of a SDE: a detail on the information set
Very basic questions. Let $(z_t)_{t \geq 0}$ be a standard Brownian motion and let
$$dS_t = \mu S_t dt + \sigma S_t dz_t.$$
When we write $E\left( S_t \right)$, do we mean $E\left( S_t \big| F_0 \...
1
vote
1answer
80 views
Illustrating the change of measure in Black-Scholes-Merton
Say that we have the following environment:
\begin{align}
dS_t &= \mu S_t dt + \sigma S_t dZ_t \\
dB_t &= r B_t dt
\end{align}
where $S_t$ is the price of a stock, $B_t$ is the price of ...
1
vote
1answer
80 views
What is the link between the SDF in the Black-Scholes-Merton model and the exponential process in Girsanov's theorem?
Question
I have been toying around to get some understanding of what the stochastic discount factor look likes in Black-Scholes-Merton and how it relates to the exponential process in Girsanov's ...
1
vote
0answers
39 views
Change of measure to get a determined drift
let's say I have a real stochastic process $dX_t=dt+\frac{1}{B_t}dB_t$ on $[0,T]$, with $B_t$ Brownian in $\mathbb{P}$ (not centered in 0) in $[0,\tau]$ with $\tau$ some adequate stopping time that ...
5
votes
2answers
193 views
Can a Process with a Stochastic Drift be a Martingale?
I have repeatedly come across the statement that "a process with a drift cannot be a martingale". Is this true also for stochastic drifts?
Suppose I have a process with a stochastic drift:
$$...
0
votes
0answers
50 views
Probability of Hitting time of Brownian motion
Let $B =\{ B(t); t \ge 0\}$ be Brownian motion. What is the probability that $B$
hits state one and then state minus one before time one?
My take: Let $T_x = \inf \{ t\ge 0 : B(t) = x\}$, the first ...
1
vote
2answers
174 views
Integral of the square of Brownian motion using definition of variance
Let $B = \{ B(t); t \ge 0\}$ and let $Z = \{ Z(t); t \ge 0 \}$ where $$Z(t) = \int_0^t B^2(s) ds.$$ How do we find $E[Z(t)]$ and $E[Z^2 (t)]$ in order to get the variance $Var [Z^2(t)] = E[Z^2 (t) ] -...
1
vote
1answer
140 views
Stochastic volatility Levy models
Hey I have some questions about stochastic volatility for Levy processes. If I understand correctly, if we change the time in Levy's process by CIR process, the newly received process is not Levy's ...
4
votes
0answers
73 views
Continuous option pricing: Brownian Bridge
I have a question on the proof of the formula of Sup(S) between 2 simulation points.
Do you know how the prove the following formula? Thanks
0
votes
0answers
40 views
American put option with $r=0$ [duplicate]
What the value of American put option in the case when $r=0$ with the payoff $\max(K-S(T),0)$, by using the Snell envelope Theorem?
Snell envelope theorem: the optimal value process $V$ is the Snell ...
0
votes
1answer
69 views
Let $W_t$ denote a standard Brownian motion. Evaluate this integral [closed]
$$
\int_{0}^{t}d(W_{u}^2)
$$
How can I deal with this kind of problem? If there is no function given to apply Itô's formula.
2
votes
1answer
125 views
Discounted price process - martingale
I have a process $S_{t}=S_{0}e^{\left(r-q\right)t+mt+X_{t}}$, where $X_t$ is a Levy process and I want to check for which $m$ the process $e^{-(r-q)t}S_t$ is a martingale. The third condition of a ...
1
vote
1answer
128 views
First Hitting Time and Monte Carlo simulation
I am interested in implementing a Monte Carlo simulation in Python of a first hitting time (first passage time) of an Ornstein-Uhlenbeck process (or similar). Specifically interested in fatter tails ...
4
votes
1answer
134 views
Correlated Stochastic Processes
Let say, I have 2 stochastic processes:
$$\begin{align}
dS_1 &= \left( r - q_1 \right)S_1 dt + \sigma_1 S_1 dW_1
\\
dS_2 &= \left( r - q_2 \right)S_2 dt + \sigma_2 S_2 dW_2
\end{align}$$
The ...
3
votes
0answers
77 views
Application of Ito's lemma relating to bond price
I'm interested in solving the following questions but I am confused on the second part because I do not know how to define/calculate the interest per "unit time", which I'm guessing is ...
3
votes
0answers
74 views
American Options in Merton's (1976) Jump Model
@LocalVolatility proves in this stellar answer that European call option prices in the Merton jump diffusion model are given by
$$ C_{Merton}(S_0,r,q,\sigma,K,T) = \sum_{n=0}^\infty e^{-\lambda T}\...
0
votes
0answers
39 views
Why can't we ignore the second term in Taylor Expansion in Ito's lemma? [duplicate]
Why can't we neglect the $dt$ there?
$$df = f'(B_t) dB_t + \frac{1}{2} f''(B_t) dt$$
0
votes
0answers
38 views
Time-changed Levy processes
in different articles the authors use the CIR process to change the time in different processes. They mostly use the CGMY, VG, NIG etc process, but I haven't noticed anybody using the Kou process. ...
1
vote
1answer
317 views
If $W_t$ is standard Brownian motion, what is $\int_0^T W_t \ln(W_t) dW_t$?
If $W_t$ is standard Brownian motion, what is meant by $\int_0^T W_t dW_t$ in finance?
Furthermore, what then is the meaning of $\int_0^T W_t \ln(W_t) dW_t$?
9
votes
2answers
450 views
Anticipating stochastic integral $\int_0^T W_T dW_t$
Using basic techniques from Malliavin calculus it can be shown that
$$
\int_0^T W_T dW_t = W_T^2 - T
$$
As can be seen the above integral is a non-adapted stochastic integral.
We also know using Ito ...
4
votes
0answers
75 views
Explicit expression for option prices in SABR?
I am trying to get a grip of the current state of research regarding option pricing in the SABR model.
Am I correct in that, so far, there is no known general formula for the option price in the SABR ...
1
vote
1answer
81 views
How can the increments of a CIR process be derived?
For a CIR process, which has SDE
$$
dr_t = \alpha (\mu - r_t) dt + \sigma \sqrt{r_t} dW_t
$$
how can I derive the increments over the discrete time-interval from $r_t$ to $r_{t+1}$?
4
votes
0answers
41 views
Characteristic function of time-changed Levy processes
Let $X_t$ be a Levy process, and $Y_t$ be a subordinator i.e. process with nondecreasing trajectories. I have to find characteristic function of $X_{Y_t}$. I know that I have to calculate:
$$E[e^{iuX_{...
5
votes
2answers
201 views
Stochastic growth model
In this problem we consider a model of stochastic growth. In particular, consider the
following system of SDEs:
\begin{align}
dX_t &= Y_t dt + \sigma_XdZ_{1t}\\
dY_t &= -\lambda Y_t dt + \...
1
vote
0answers
70 views
Price of a Forward Contract
I have the following,
Let ${F_t,t\geq0}$ be the price process of the forward contract on the
risky asset with maturity $T' > 0$. Since interest rates are
deterministic, we have
$$F_t=S_t\ e^{r(T^\...
2
votes
0answers
49 views
solution of Jump Diffusion SDE (Kou, Merton)
Hey in Kou 2002 paper he write SDE as:
$$\frac{dS(t)}{S(t-)}=\mu dt+\sigma dW(t)+d\left( \sum_{i=1}^{N(t)}\left( V_{i}-1\right) \right)$$
Is it equivalent with:
$$dS(t)=S(t)\mu dt+S(t)\sigma dW(t)+S(t-...
1
vote
1answer
61 views
Characterizing distribution of a stochastic intergal
characterize the distribution of $\int_0^T f(t)Z_tdt$. In
particular, verify that it is a Gaussian distribution and compute its moments.
3
votes
1answer
205 views
Mean Reverting Heston Model?
Is there a name for a variation on the Heston Stochastic Process Model where not only the underlying volatility but the asset price itself is mean-reverting? I'm looking to model long term equity ...
1
vote
1answer
119 views
Covariance of mean-reverting Vasicek process?
I am dealing with a mean-reverting Vasicek process defined as:
\begin{equation}
S_t = S_0 e^{-at} + b(1-e^{(-at)}) + \sigma e^{(-at)} \int_{0}^{t} e^{(-as)} \ W_t
\end{equation}
I want to ...
1
vote
0answers
27 views
Change of numeraire between t1-forward mesure and t2-forward mesure
Let denote $\mathbb{Q}_{t_1}$ the $t_1$-forward mesure associated to zero coupon bond $B(.,t_1)$.
Let denote $\mathbb{Q}_{t_2}$ the $t_2$-forward mesure associated to zero coupon bond $B(.,t_2)$.
I am ...
1
vote
1answer
96 views
Transition density of geometric Brownian motion with time-dependent drift and volatility
Can you provide a reference to the transition density of the scalar geometric Brownian Motion with time-dependent drift and volatility, i.e. the scalar process $X = (X_t)_{t\geq 0}$ defined by the SDE
...
2
votes
0answers
69 views
Correct application of Feynman Kac formula
I have a question on Feynman-Kac formula but can I ask the community if I have done it correctly? If no, may you point out to where I went wrong? Thanks!
The original FK formula states: Assume $f(t,x)$...
6
votes
2answers
505 views
Clarification on Deriving Ito's Lemma
The classical approach to deriving Ito's Lemma is to assume we have some smooth function $f(x,t)$ which is at least twice differentiable in the first argument and continuously differentiable in the ...
1
vote
0answers
68 views
How to solve this particular PDE using Feynman-Kac formula?
I have to solve the PDE
$$
\begin{align}
\frac{\partial F}{\partial t} + \frac{1}{2}\frac{\partial^2 F}{\partial x^2} + \frac{1}{2}\frac{\partial^2 F}{\partial y^2} + \frac{1}{2}\frac{\partial^2 F}{\...
