Questions tagged [itos-lemma]
The itos-lemma tag has no usage guidance.
183
questions
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vote
1answer
20 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 ...
2
votes
1answer
51 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 +...
3
votes
1answer
131 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 ...
1
vote
0answers
51 views
How to solve/evaluate an Ito Integral?
I'm given the following Ito integral which I need to evaluate. $Z_t$ is the Brownian motion. My problem is that online resources aren't making much sense because of the notation, so it ends up leaving ...
3
votes
2answers
232 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 ...
-1
votes
1answer
218 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 ...
1
vote
2answers
131 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 ...
9
votes
2answers
313 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 $...
4
votes
0answers
153 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 ...
0
votes
1answer
62 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 ...
1
vote
1answer
143 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
128 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 ...
0
votes
1answer
52 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 $, ...
0
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0answers
53 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
203 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) ] -...
0
votes
0answers
40 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?
$$df = f'(B_t) dB_t + \frac{1}{2} f''(B_t) dt$$
4
votes
1answer
298 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 ...
5
votes
2answers
595 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 ...
2
votes
1answer
277 views
Pricing Swaption Analytically using Libor Market Model
I was asked the following question in a recent interview: "(i) Express a forward swap rate in terms of forward Libor rates. (ii) Apply Ito's lemma to this expression to derive the process for the ...
1
vote
1answer
350 views
Can I write Ito's Lemma as a taylor expension?
instead of using Wikipedia's definition:
$$
{d}(f(X_t,t)) = \frac{\partial f}{\partial t}(X_t,t)\,\mathrm{d}t + \frac{\partial f}{\partial x}(X_t,t) \, \mathrm{d}X_t + \frac{1}{2} \frac{\partial^2 f}{\...
1
vote
2answers
183 views
Ito's lemma and Lognormal Property
What would be the difference between:
\begin{align}
dS = udt + \sigma dz
\end{align}
and
\begin{align}
dS=u*S*dt + \sigma*S*dzdS
\end{align}
Is that the former is in absolute terms and the latter is ...
0
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0answers
87 views
Ito's differential in portfolio dynamics
I try to be as concise as possible. Basically I'm following the text "Arbitrage Theory in Continuous Time", by Tomas Bjork.
I put here the point where I'm stuck: Chapter 6 - Portfolio ...
2
votes
1answer
173 views
How can Ito's Lemma be used to show that a delta-neutral portfolio is instantaneously risk-free?
The lecture notes I am currently reading give the following example of a delta-neutral portfolio:
minus one derivative (whose value at time $t$, when the value of the underlying is $S_t$, is denoted $...
2
votes
1answer
149 views
Clarification of Ito's lemma
I was looking at the various examples provided in the discussion Worked examples of applying Ito's lemma
One such example is 9.1 (c). This states that -
if $S_t =\! S_0 + \int\limits_{0}^{t} \mu_u ...
0
votes
1answer
44 views
Applicability of the Ito's lemma [duplicate]
Ito's lemma is used to find the stochastic process of the function of a ...
1
vote
1answer
148 views
Question about using Ito's lemma in Gamma PnL
While deriving the delta hedge error if we hedge with implied vol, and the true vol is different, we say that the PnL of the call option is:
$$dC=C_tdt+C_SdS+0.5C_{ss}<QV>dt - (1)$$
Where $<...
2
votes
0answers
43 views
Solution to Stock Price SDE with mean reversion [duplicate]
Suppose $S_t$ follows the process (notice the $S_t$ term in the diffusion part):
$$ S_t := S_0 + \int_{h=t_0}^{h=t}\alpha(\mu -S_h)dh + \int_{h=t_0}^{h=t}\sigma S_h dW(h) $$.
I actually don't know how ...
1
vote
2answers
149 views
Compute the price of a derivative which pays $\log(S_T)S_T$ in the Black Scholes world
Compute the price of a derivative which has pays $\log(S_T)S_T$, you can assume that the Black Scholes model is valid.
Using the stock measure we can write the expectation as
$$D(0) = S_0 \mathbb{E}...
3
votes
1answer
249 views
Gamma PnL from Itô's Lemma derivation
The change in a call portfolio ($f$), derived from Itô's Lemma, is:
\begin{align*}
\left( \frac{\partial f}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2}\right)\mathrm{d}t &=...
2
votes
2answers
161 views
Itos Lemma Derivation notation
So in Hull (2012) the main point is that $\Delta x^2 = b^2 \epsilon ^2 \Delta t + $higher order terms$ $ has a term of order $\Delta t$ and can not be ignored as the Brownian motion exhibits the ...
3
votes
1answer
195 views
Application of Ito's Lemma in expected utility theory
An investor with utility curve $U(.)$ has wealth $X_t$ at time t. He invests
A proportion $p$ of his wealth in a risky asset that follows a geometric Brownian motion, with parameters $\mu$ and $\...
3
votes
1answer
152 views
The most general conditions under which Ito lemma holds
Prompted by a question that came up in the comments here, namely why we can apply the Ito lemma to a function of the form $f(x)=(x-K)^{+}$, I would be interested in knowing what are the least ...
1
vote
1answer
99 views
Serial correlation, quadratic variation and variance of returns
On p. 3 of Lorenzo Bergomi's book on Stochastic Volatility Modeling, there is the following assertion:
Indeed, to a good approximation, the variance of returns scales linearly with their time scale, ...
3
votes
1answer
115 views
Under which conditions the given random process is martingale and under which submartingale?
Let $a_t $ be adapted to the filtration random process $a_t: P\{\int _0^T|a_t|dt < \infty \} = 1 $ and $ b_t \in M_T^2. \quad$ Under which conditions the random process $$X_t = exp\{\int _0^ta_sds+\...
0
votes
0answers
32 views
Discreet-time stochastic difference equation and Ito thorem
In continuous time, when we want to find the dynamics of a function of a stochastic process, we need to use Ito's lemma which gives an "extra"" term for the drift. What if we are in discreet time and ...
4
votes
1answer
91 views
Differential of time over Browninan motion
I know that $\frac{dW_t}{dt}$, with $W_t$ a brownian motion, does not exist. However, does $\frac{dt}{dW_t}$ exists? Or does it even make sense? I am trying to calculate the quotient of two ...
2
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0answers
137 views
understanding of Ito's lemma applied to stock price?
I am currently reading John Hull's book and am a bit confused about the Ito's lemma when it is applied to the stock price. Given $dS=\mu Sdt+\sigma Sdz$, by applying Ito's lemma to $G=\ln S$, we have ...
1
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0answers
39 views
Feynman-Kac formula for $\mu(t,x)=-\frac{1}{1-t}, \sigma(t,x)=1$ and $g(t,x)=x^2$
Consider the following PDE on $[0,T]\times \mathbb{R}$:
$$
\begin{cases}
\dfrac{\partial F}{\partial t}+\mu(t,x) \dfrac{\partial F}{\partial x}+ \frac12 \sigma^2(t,x)\dfrac{\partial^2 F}{\partial x^2}...
3
votes
0answers
233 views
Black and Scholes equation for portfolio **with** arbitrage
I am well aware of how the ordinary Black and Scholes equation is derived, under the assumption of an arbitrage free portfolio, $V=G-hS$. Here $S$ is the price of the underlying and $G$ is the option ...
1
vote
1answer
92 views
Ito's lemma for a Forward
I'm trying to understand the derivation of Ito's process with respect to a Forward $F$ on a stock $S$ that pays a constant dividend yield, say $y$. Stock follows brownian motion $\\$
$dS_{t} = S_{t}(\...
0
votes
0answers
60 views
Compo/Quanto Adjustment & Multivariate Ito
Related to the issue that I have raised here, I am facing another question. As the rule here is 1 question / 1 post, I take the opportunity to ask it below:
By exploring StackExchange, I noticed the ...
3
votes
2answers
213 views
Why is $S(t) = e^{\alpha + \beta t + \sigma W(t)}$ used as a model for prices?
Why is the Geometric Brownian Motion defined as $S(t) = e^{\alpha + \beta t + \sigma W(t)}$ used as a model for stock prices? $S(t)$ has a lognormal distribution which is right skewed. Another problem ...
2
votes
1answer
85 views
Generalization of Ito's Lemma to composite function
Ito's Lemma gives that for a function $F$ of a stochastic variable $X$, $dF = \frac{dF}{dX}dX + \frac{1}{2}\frac{d^2F}{dX^2}dt$
Given a stochastic differential equation $dS = a(S) dt + b(S) dX$ and a ...
1
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1answer
70 views
Derivation of stock price formula John C. Hull 9th Ed p309
It says assuming a no-uncertainty Weiner process that models stock price:
$$
\Delta S = \mu S\Delta t
$$
Can be rearranged to (after taking the limit of $\Delta t \to 0$...
$$
\frac{dS}{S}=\mu dt
$$
...
1
vote
1answer
777 views
Integration of a deterministic function w.r.t. a Brownian motion
Help me solve this problem:
Let $W_t$ be a Brownian motion and suppose
$X_t = \int_{0}^{t}\delta _{s}dW_{s}$
where $\delta _{s}$ is a deterministic function. Then show that $X_t$ is a Gaussian ...
4
votes
1answer
105 views
Ito formula for $Y_t=tB_t$
someone can help me to solve this problem:
$B_t$ is a Standard Brownian Motion.
Let $Y_t=tB_t$.
Using Ito formula, find drift and volatility of $Y_t$.
The result I found is $dY_t=B_tdt+t\cdot dB_t$ ...
1
vote
1answer
126 views
Calculation of a process's drift
Let $X_t:=e^{W_t}$ where $W_t$ follows the Wiener process. Calculate the drift.
The answer is given as $X_t/2$. My attempt at a solution (which I'm afraid is poor from a mathematical standpoint):
I ...
1
vote
0answers
66 views
Application Itô's Lemma: Forward to Spot process
I am working on the following equation (I want to apply Ito's lemma on it):
and I know that:
and also
and
My problem is that I want the dynamic of F(S,T) without S because I need first to ...
1
vote
1answer
106 views
Compute dZ(t) : Ito's formula/lemma
We need to find dZ(t). I know I have to use Ito's formula. But I am confused because in the Ito's formula we have f(y,t) is a twice differentiable function with two variables
But here Z(t) = 1/(2+x(t)...
1
vote
0answers
73 views
On Geometric Brownian motion and Itô's formula
Let $S_t$ be a geometric brownian motion such as
$$d S(t) = rS(t)dt +\sigma S(t)dW(t),$$
where $W$ is a standard Brownian motion.
With Itô's lemma and formulas $(dt)^2=dtdW_t=dW_tdt=0$ and $(dW_t)^2=...
