Questions tagged [itos-lemma]
The itos-lemma tag has no usage guidance.
214
questions
0
votes
1
answer
153
views
Deriving the stochastic process for a dividend-yielding stock (under Black-Scholes assumptions)
In order to derive the Black-Scholes equation for a stock $S(t)$ yielding dividends at the continuous rate $d$
$$
S(t) = S_0 e^{(\mu - d - \frac{\sigma^2}{2})t + \sigma \sqrt{t} N(0,1)} \text{,}
$$ M. ...
- 356
1
vote
1
answer
254
views
Integral of brownian motion wrt. time over [t;T]
From the post Integral of Brownian motion w.r.t. time we have an argument for
$$\int_0^t W_sds \sim N\left(0,\frac{1}{3}t^3\right).$$
However, how does this generalise for the interval $[t;T]$? I.e. ...
- 339
0
votes
1
answer
66
views
Integration of exponential raised with Brownian Motion wrt the Brownian Motion
I have to derive several things for my thesis, however, I have the following expression:
$$
\int^{t}_{0} \exp\{\sigma W_{t}\}.dW_{t}
$$
Does anyone know what the solution for this is?
Kind regards.
- 15
1
vote
0
answers
114
views
Value of trading strategy
A trading strategy is defined as follows: starting capital $v_0 = 5$ and 1 risky asset holdings $\varphi_t = 3W_t^2-3t$ where $W$ is a Wiener process.
The problem is to find the probability of the ...
- 11
1
vote
1
answer
105
views
Show that the solution to a SDE is strong
I have the following SDE
\begin{equation}
dX_t = - \frac{1}{1+t}X_t dt + \frac{1}{1+t}dB_t
\end{equation}
that has the solution:
\begin{equation}
\begin{aligned}
X_t = \frac{X_0 + B_t}{1+t} = \frac{...
2
votes
1
answer
181
views
Obtaining the dynamics of the Vasicek model using Itô
Consider the following expression for the short-term interest rate
$$r_t=r_0 e^{\beta t}+\frac{b}{\beta}\left(e^{\beta t}-1\right)+\sigma e^{\beta t}\int_0^te^{-\beta s}dW_s \tag{1},$$
which is ...
- 221
1
vote
1
answer
93
views
Ito's lemma for option pricing with Levy-alpha stable drift
Consider
$$dS=\omega\left(\Lambda-S\right)dt+\sigma_S S dW_t,$$
such that such that $W_t$ is a Wiener process, $\sigma_S$ is constant, $\omega: t\rightarrow\mathbb{R}$ represents anticipated drift and ...
- 110
0
votes
1
answer
92
views
Stochastic process as integral over window function
Consider the following stochastic integral of a deterministic function $f(t,s)$ with respect to the Wiener process $W_s$:
$$\int_0^\infty f(t,s) d W_s$$
My questions are:
Is such an integral ...
4
votes
1
answer
243
views
Difficulty with stochastic calculus problem
I'm currently working through Shreve's Volume II, and I'm having some difficulty on Exercise 5.4 of Chapter 5. The problem statement is:
Consider a stock whose price differential is
$$
dS(t) = r(t) S(...
- 163
2
votes
1
answer
133
views
Ansatz and HJB equation
Suppose we have an HJB equation of the form
$$
\frac{\partial v}{\partial t}+\frac{1}{2}\sigma^{2}\frac{\partial^{2}v}{\partial s^{2}}+max_{\delta^{a}}\left\{ \lambda^{a}(\delta^{a})\left[v(t,s,x+s+\...
- 81
2
votes
1
answer
182
views
Why this stochastic integral is calculated with Riemann integral
This picture is from Neftci's textbook, 'An Introduction to the Mathematics of Financial Derivatives, Third Edition'
What makes me uncomfortable is equation [10.61] In above picture.
In this equation,$...
- 303
0
votes
0
answers
93
views
Performance of dollar cost averaging
If we're investing money into a stock $S$ at a continuous rate, $C$, what is the probability distribution of the amount we have invested?
For example, modelling a stock as GBM without contributions, $ ...
- 101
0
votes
1
answer
156
views
Clarification on Paul Wilmott's derivation of Ito's Lemma
I'm currently self-studying to be quant and have been thoroughly enjoying PW's book. I have some questions regarding his derivation of Ito's lemma. Specifically, I can see that the first line in his ...
- 101
0
votes
0
answers
207
views
Ito's Lemma in option pricing for a stock satisfying $dS=\frac{P-S}{\omega}dt+SdW_t$
Suppose a stock follows the stochastic differential equation
$$dS=\frac{P-S}{\omega}dt+SdW_t,$$
such that $W_t$ is a wiener process, $\omega\in\mathbb{R}^+$, and $P_t,S_t\in\mathbb{R}$. If the value ...
- 110
3
votes
1
answer
126
views
Volatility and drift of the instantaneous forward rate under risk neutral measure using the zero coupon bond
I have question about this problem. I believe I have derived $f(t,T)$ correctly using the zero-coupon bond. But I am unsure about how to go forward with the question and how to use the second part.
...
- 373
6
votes
1
answer
232
views
Parametric Stochastic Integral
I need help.
Defining the parametric stochastic integral
$$
F_t = \int_t^T\xi(t,s)g(s)ds
$$
$\\\\$
with $\xi$ a generic stochastic process such that $d\xi(t,s) = \mu(t,s)dt + \sigma(t,s)dW_t$, I'm ...
- 61
6
votes
2
answers
135
views
Covariance of the product of log normal process and normal procces
I tried to compute the following covariance :
$$Cov(e^{\int_{t}^{T}W^1_sds},\int_{t}^{t+1}W^2_sds)$$
where $W^1_t$ and $W^2_t$ are Brownian motions such that $dW_t^1dW_t^2=\rho dt $
My idea was to ...
- 338
0
votes
1
answer
133
views
Question on Ito's lemma involving $\mathrm{d}W(t)$
I am new to Ito-calculus, so please forgive me if the question is stupid.
Let $W(t)$ be a Brownian-Motion and $f(W(t))=W(t)^2$. If I want to calculate the differential $\mathrm{d}f(W(t))$, Ito's lemma ...
- 436
2
votes
1
answer
96
views
Confused by derivation of variance swap payoff
I'm trying to follow
https://en.wikipedia.org/wiki/Variance_swap#Pricing_and_valuation
where it seems to me that they're just subtracting a simple return:
$$ R_t = \frac{\mathrm{d}S_t}{S_t} = \mu \...
- 71
3
votes
2
answers
240
views
Ito's lemma $f(t,W_t^2)$
Let $f$ be a function of $t$ and $W_t^2$.
a)Find a function $f$ such that $f(t,W_t^2)$ is a $F_{t^-}$ martingale, with $F$ the Brownian filtration.
b)Use Ito's lemma to show that $f(t,W_t^2)$ is a ...
- 133
6
votes
1
answer
277
views
Ito multiplication
Let $\{N_t|0<t\leqslant T \}$ and $\{M_t|0<t\leqslant T \}$ be two Poisson processes with intensities $\lambda_n, \lambda_m>0$, respectively.
Based on the implicit results of Corollaries 1 ...
user57062
0
votes
0
answers
62
views
Ito's lemma results in negative volatility processes
I struggle with the interpretation of a process I derive from Ito's Lemma. Let's say I have function f(S,t) which is twice differentiable wrt S.
I thus can apply Ito's Lemma to get $df(S,t)$. So far ...
3
votes
1
answer
526
views
Ito Lemma for Poisson Process
I'm new to stochastic calculus on jump processes and encountered a difficulty. I would appreciate some clarification from the community on the following question.
Let $g_t$ be a $\mathcal{F_t}$-...
- 184
2
votes
1
answer
159
views
Solving an SDE using Ito's Lemma
Suppose that
$Z(t)=e^{-\int_0^t \theta'(s)dW(s)-\frac{1}{2}\int_0^t ||\theta(s)||^2ds}$
with $\theta()=\sigma^{-1}()[b()-r()]$, $\sigma()>0$ and invertable and $W()$ a Wiener process
There is also ...
- 129
3
votes
0
answers
112
views
MGF of Generalised Itô Integral
The following derivation produces a moment closure problem - I would appreciate any insight. It may seem trivial at first glance, but the key aspect is the integrand dependence on $t$.
Consider $W_t$ ...
- 31
1
vote
2
answers
299
views
Drift Term in Black-Scholes Model Martingale
How would I prove that a Black-Scholes Model is not a Martingale if it has drift. In many cases it is just stated as a fact (without proof).
For instance if Im looking at:
$$dS_{t} = \mu S_{t} + \...
- 21
0
votes
0
answers
51
views
Multidimentional Black Scholes Formula
I need to write the Black-Scholes formula for option $V = (S_1, S_2, t)$, where:
$$d S_1 = \mu_1 S_1 dt + \sigma_1 S_1 d W_1,$$
$$ d S_2 = \mu_2 S_2 dt + \sigma_2 S_2 d W_2.$$
We know that $W_1$ and $...
- 1
3
votes
0
answers
69
views
Derivation of option pricing PIDE: Why does the drift need to be zero?
I started studying PIDE methods for option pricing and am struggling to understand or find the necessary theory that shows why the PIDE is obtained by the condition that the drift term has to be zero.
...
- 161
1
vote
1
answer
212
views
How is the formula of Quadratic Variation of Brownian Motion derived? [closed]
This is a follow up on this question on quant SE:
The question mentions for a Brownian motion :
$X_t = X_0 + \int_0^t\mu ds + \int_0^t\sigma dW_t $
, the quadratic variation is calculated as
$dX_t ...
- 13
3
votes
2
answers
424
views
Covariance between integral of brownian motion and brownian motion
Let
$$
I = \int_0^1W_tdt,
$$
where $W_t$ is a Brownian motion.
From Integral of Brownian motion w.r.t. time we have that
$$
\mathbb{E}[I]=0,
$$
by Fubini's theorem. And that
$$
\mathbb{V}\text{ar}[I] =...
- 33
3
votes
1
answer
121
views
Justification for substituting "Itô differentials"
I'm reading Shreve's Stochastic Calculus for Finance, Volume II. In it, he uses the stochastic differential notation. For example, he may write
$$\mathrm{d}X(t) = \sigma(t)\mathrm{d}W(t)+\alpha(t)\...
- 437
1
vote
1
answer
102
views
Hermite polynomials as martingales [closed]
Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite ...
- 69
4
votes
1
answer
339
views
Ito calculus is Gaussian (using method of characteristic function)
Let $h$ be a deterministic function and define $X_{t}=\int_{0}^{t} h(s) d W_{s} .$ Show that
$$\mathbb{E} \exp \left(i u X_{t}\right)=\exp \left(-\frac{u^{2}}{2} \int_{0}^{t} h^{2}(s) d s\right),$$ ...
- 69
0
votes
1
answer
98
views
Mutual variation of Brownian motions
Let $\{W^1\}_{t\geq0}$ and $\{W^2\}_{t\geq0}$ be two Brownian motions with correlation coefficient $\rho \in [0, 1]$, i.e., $\mathbb{E}[(W^1(t)-W^1(s))(W^2(t)-W^2(s))]=\rho(t-s)$ for all $t,s \geq 0$. ...
- 69
1
vote
1
answer
197
views
What does it mean to "compute" an Itô integral?
I'm reading Shreve's Stochastic Calculus for Finance II. On page 191, Exercise 4.6, we are given the problem
Exercise 4.6. Let $S(t)=S(0)\exp\Big \{\sigma W(t)+(\alpha-\frac{1}{2}\sigma^2)t\Big\}$ be ...
- 437
2
votes
2
answers
331
views
Proving that a stochastic process is a martingale using Ito's Lemma
Assume a Wiener process W and a bounded F-adjusted stochastic process a. Show that the following process is a martingale on F
$$X(t)=(\int_{0}^{t}a(s)dW(s))^{2}-\int_{0}^{t}a^{2}(s)ds,\ t\geq0$$
Can ...
- 129
2
votes
1
answer
178
views
Simplifying the expectation of the product of two stochastic integrals
Let $f(t, \omega), g(t, \omega)$ be functions that are independent of the increments of the Brownian motion $w(t, \omega)$ in the future. That is, $f(t, \omega), g(t, \omega)$ are independent of $w(t +...
- 121
3
votes
1
answer
182
views
Derivative of Stochastic Integral
I am trying to take the derivative of the following stochastic integral,
$$d\left(\int g(S_t) dS_t \right),$$
where $dS(t) = \sigma S(t) dW_t$ and $g(.)$ is some (smooth) deterministic function. My ...
- 31
3
votes
2
answers
373
views
Calculate Ito integral $\int_0^t W_s^2\text dW_s$ from first principles
I am stuck on the 1st equation of the solution where the Wiener process $W_{t_i}^2$ is expanded so that the Itô integral (in terms of infinite sums) looks like the RHS of the first equation of the ...
- 133
-1
votes
1
answer
506
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 ...
- 11
1
vote
2
answers
181
views
Show that $\int_0^T(T-t)dW_t=\int_0^TW_t \ dt$
I want to show that $$\int_0^T(T-t)dW_t=\int_0^TW_t \ dt \tag1$$
By Ito's lemma we can write $$d((T-t)W_t)=(T-t)dW_t-W_t \ dt.\tag{2}$$
Now If I integrate this expression and use that $W_0=0$ I should ...
- 221
9
votes
2
answers
446
views
conditional expectation of stochastic integral
let $M_t$ be the following stochastic integral
$$
M_t = \int_0^t \sigma_s dW_s
$$
where $\sigma_t$ is a sufficiently regular deterministic function and $W_t$ is a standard Wiener process (that is $...
- 680
4
votes
0
answers
172
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 ...
- 113
0
votes
1
answer
131
views
Differentiability of solutions of a stochastic differential equation
I would like to clarify a confusion I have.
It is well known that a Wiener process (Brownian motion) is nowhere differentiable. I have no difficulty in understanding that. But I am wondering about the ...
- 101
2
votes
1
answer
237
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 ...
- 41
2
votes
0
answers
139
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 ...
- 33
0
votes
1
answer
60
views
Volatility of a function of an asset
Suppose that $ G $ is a function of the underlying asset $ S $, which follows a geometric Brownian motion. Suppose that $ \sigma_{S} $ and $ \sigma_{G} $ are the volatilities of $ S $ and $ G $, ...
1
vote
2
answers
608
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) ] -...
- 121
0
votes
0
answers
44
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?
$$df = f'(B_t) dB_t + \frac{1}{2} f''(B_t) dt$$
- 1
4
votes
1
answer
715
views
Deriving the solution for European call option in the Heston Model
I'm deriving the solution for European call option in the Heston Model. I follow the original paper by
Heston and Fabrice Douglas Rouah's derivations in his book The Heston Model and Its Extensions in ...
- 127