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10
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3answers
5k views

What is Ito's lemma used for in quantitative finance?

Further to my question asked here: prior post and which left some points unanswered, I have reformulated the question as follows: What is Ito's lemma used for in quantitative finance? and when is it ...
10
votes
5answers
1k views

Monte carlo methods for vanilla european options and Ito's lemma.

I understand that by applying Ito's lemma to the following SDE $$dX=\mu\,X\,dt+\sigma\,X\,dW$$ one obtains a solution to the above SDE which is as follows: $${X}\left( t\right) =\mathrm{X}\left( ...
9
votes
2answers
1k views

Why Ito calculus?

Coming from physics, I am used to the fact that the Ito interpretation of most natural stochastic equations is wrong, and one should be using Stratonovich calculus instead (of course they are ...
8
votes
2answers
4k views

Worked examples of applying Ito's lemma

In most textbooks Ito's lemma is derived (on different levels of technicality depending on the intended audience) and then only the classic examples of Geometric Brownian motion and the Black-Scholes ...
6
votes
2answers
820 views

Derivation of Ito's Lemma

My question is rather intuitive than formal and circles around the derivation of Ito's Lemma. I have seen in a variety of textbooks that by applying Ito's Lemma, one can derive the exact solution of a ...
6
votes
1answer
822 views

How to perform basic integrations with the Ito integral?

From the text book Quantitative Finance for Physicists: An Introduction (Academic Press Advanced Finance) I have this excercise: Prove that $$ ...
6
votes
2answers
481 views

Demonstration of Ito's correction term/lemma in binomial tree

I am preparing an undergraduate QuantFinance lecture. I want to demonstrate the ideas of Ito's correction term and Ito's lemma in the most accessible manner. My idea is to take the "working horse" of ...
5
votes
3answers
2k views

How to use Itô's formula to deduce that a stochastic process is a martingale?

I'm working through different books about financial mathematics and solving some problems I get stuck. Suppose you define an arbitrary stochastic process, for example $ X_t := W_t^8-8t $ where $ W_t ...
5
votes
1answer
97 views

Is this application of Ito's lemma correct?

Suppose that $S$ follows a geometric brownian motion $$dS=S(\mu dt+\sigma dB).$$ It is well understood that $$S_{T}=S_{0}exp((\mu-\dfrac{\sigma^{2}}{2})T+\sigma B_{T}).$$ Method 1 (I have no ...
4
votes
1answer
40 views

Multidimensional Ito's Lemma for Vector-Valued functions

Consider the vector of $n$ Ito processes $$ d \mathbf{X}_t = \mathbf{\mu}(\mathbf{X}_t,t)dt + \Sigma(\mathbf{X}_t,t)d\mathbf{W}_t $$ where $\mathbf{\mu} \in \mathbb{R}^n$ and $\Sigma \in ...
4
votes
1answer
264 views

How to measure a non-normal stochastic process?

If I understand right, Itô's lemma tells us that for any process $X$ that can be adapted to an underlying standard normal Wiener measure $\mathrm dB_t$, and any twice continuously differentiable ...
4
votes
1answer
211 views

Ito's Lemma - Integrand depends on upper limit of integration

A problem I came across while practicing using Ito's Lemma had a process with an integral whose integrand depends on the upper limit of integration (the goal is to find $dZ_{t}$): ...
3
votes
1answer
132 views

Integration of stochastic total derivative

Super basic question. I think I am doing this correctly, but just want a sanity check. Say I have a stochastic process $r(t)$. Say I have an equation $$d(e^{\beta (t-s)}r(s))=\dots$$ where the ...
3
votes
0answers
44 views

Dixit & Pindyck (1993) Chapter 4, equation 13

Starting with the Bellman equation for the optimal stopping problem: $$F(x,t)=max\{\Omega(x,t), \pi(x,t)+(1+\rho dt)^{-1} E[F(x+dx, t+dt)|x]\}$$ In the continuation region where the second term is the ...
2
votes
1answer
131 views

On an application of Ito's lemma

Assume that instantaneous returns are generated by the continuous time martingale: $$dp_t = \sigma_t dW_t$$ where $W_t$ denotes a standard Weiner process and One day returns are denoted by $r_{t+1} ...
2
votes
2answers
96 views

Stochastic Differentials - Ito's formula for a self-financing portfolio

Suppose I have a portfolio of stocks $(S)$ and savings account ($\beta_t$) then, the value is $$V = a_t S_t + b_t \beta_t$$ and for this portfolio to be self replicating, we need by Ito's lemma $$dV ...
2
votes
2answers
85 views

Markov Pricing kernel

I'm reading about Markov pricing kernels in the lecture notes of a course I'm following, but I have a big doubt on an application of Ito's lemma. The setting is the following: We define the pricing ...
2
votes
1answer
42 views

Show that Z(t)/Z(0) is a positive mean-1 martingale

We look at a standard no dividends Black-Scholes model and here we have a process Z, which is defined by: Z(t)=(S(t)/H)^p , where H is a positive constant and p=1-2r/sigma^2 I am now asked to show ...
2
votes
0answers
21 views

Bond yield: is it martingale with respect to risk-neutral probability measure of some numeraire?

Let $t$ mean current time, let $T_0, T_n$ mean two times such that $T_0\le T_n$, and let $y_t[T_0, T_n]$ mean the forward swap rate of a swap starting at $T_0$ and ending at $T_n$. (I am ignoring ...
1
vote
1answer
395 views

How to get Geometric Brownian Motion's closed-form solution in Black-Scholes model?

The Black Scholes model assumes the following dynamics for the underlying, well known as the Geometric Brownian Motion: $$dS_t=S_t(\mu dt+\sigma dW_t)$$ Then the solution is given: ...
1
vote
1answer
95 views

stochastic calculus - Itô formula?

I encounter a problem in the proof below: I don't know how to proove the first line in yellow (cf below): it makes me think about the Itô formula a lot I don't undertand the deduction (ok ...
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vote
2answers
96 views

How can I make this portfolio self-financing?

$a_t S_t$ = number of shares ($S_t$ is stock price at $t$), $S_0 = 1$ $b_t \beta _t$ = saving account value , $d \beta_t = r \beta_t dt$, $r=$ interest rate So the value of the portfolio: $$V_t = ...
1
vote
1answer
124 views

What are the dynamics of the reverse of this FX process?

Assuming the dynamics of the exchange rate between two currencies at time $t$ is given by: $$ dX_t=\Delta r X_t dt+ σ X_t dW_t$$ Is the FX Reverse process $\frac{1}{X_t}$ a brownian motion? How can ...
1
vote
3answers
36 views

Difference between ito process, brownian motion and random walk

Can someone explain to a non-math person (myself) what is the difference between these three? If they are so different that a comparison does not even make sense, please point it out. 1.Ito process ...
1
vote
2answers
408 views

How to express the Black Derman & Toy Model in a $dr=A\,dt+B\, dW$ form?

The Black Derman & Toy (BDT) model is given by $$d(\ln\,r)=\left(\theta(t)-\frac {d(\ln\sigma(t))}{dt}\ln r\right)\,dt+\sigma(t) \, dW.$$ How can one rewrite the BDT model as $dr=A\,dt+B\, dW$, ...
1
vote
0answers
106 views

In what kind of stochastic process Ito's lemma is adopted?

I have been told that Ito's lemma serves as the stochastic calculus counterpart of the chain rule. And yet again my tutor mentioned it is not used for all stochastic processes. Is this statement ...
0
votes
2answers
67 views

Shreve book II Question 4.6 Error?

I'm working through Shreve II, and on question 4.6, you are asked to compute $d(S_t^p)$ where $S_t$ = $S_0e^{\sigma W_t + (\alpha - \frac{1}{2}\sigma^2)t}$ I get the answer $pS_t^p[\sigma dW_t + ...
0
votes
1answer
100 views

How to derive equivalent martingale measure using Ito's Lemma

Can someone explain how to get equation 27.14 below? I understand the first usage of Ito's Lemma to get $d(\ln f-\ln g)$ but I do not understand how to use Ito's Lemma to go from $d(\ln \frac{f}{g})$ ...
0
votes
1answer
31 views

Simulating a GBM with martingale condition - Ito process moving downwards

I want to correctly simulate a $\mathcal{Q}$ - martingale $S$, which is a geometric Brownian motion and an exponential of a process $X$, \begin{equation} X_t = X_0 + \mu t + \sigma B_t = X_{t-\Delta ...