Questions tagged [itos-lemma]

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2
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0answers
46 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+\...
2
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1answer
122 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,$...
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0answers
89 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, $ ...
0
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1answer
132 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 ...
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66 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 ...
3
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1answer
91 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. ...
6
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1answer
216 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 ...
6
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2answers
123 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 ...
0
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1answer
121 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 ...
2
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1answer
89 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 \...
3
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2answers
176 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 ...
6
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1answer
226 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 ...
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50 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
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1answer
189 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 $\mathcal{F_t}$-...
2
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1answer
112 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 ...
3
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0answers
98 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 $W_t$ ...
1
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2answers
112 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} + \...
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44 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 $...
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0answers
62 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. ...
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1answer
166 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 ...
3
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2answers
196 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] =...
3
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1answer
115 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)\...
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1answer
97 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 ...
4
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1answer
333 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 $$\mathbb{E} \exp \left(i u X_{t}\right)=\exp \left(-\frac{u^{2}}{2} \int_{0}^{t} h^{2}(s) d s\right),$$ ...
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1answer
93 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 $t,s \geq 0$. ...
1
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1answer
172 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 ...
2
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2answers
225 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
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1answer
67 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
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1answer
163 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 ...
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0answers
64 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
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2answers
266 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
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1answer
346 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
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2answers
144 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 ...
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2answers
359 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
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0answers
164 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 ...
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1answer
89 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
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1answer
184 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
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0answers
132 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
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1answer
54 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 $, ...
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0answers
58 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
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2answers
268 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) ] -...
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0answers
42 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
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1answer
485 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
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2answers
820 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
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1answer
339 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
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1answer
370 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
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2answers
303 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 ...
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0answers
88 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
267 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
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1answer
179 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 ...

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