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0
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1answer
49 views

Is there a better, more rigorous explanation for why this partial derivative is 0 using Ito's Lemma?

I encountered the following slide in a lecture on Ito's Lemma. The lecturer explained that $$\frac{\partial V}{\partial t} = 0$$ because the first two derivatives on the slide already took into ...
1
vote
1answer
66 views

what is the meaning of the differential of an arbitrary adapted random process?

I was working on the definition of the self-financing portfolio. Say $V=\phi_tS_t+\psi_t A_t$ where $S_t$ and $A_t$ are the stock price and the money market price at time $t$, resp, and $\phi_t$ and $...
3
votes
1answer
83 views

Is this a poorly written example, or could volatility in fact be negative?

I'm self-studying and I encountered the following example. It seems to suggest that volatility is negative in this example. I was under the impression that volatility can never be negative, both from ...
4
votes
1answer
154 views

How to compute the expectation of integral of this random function?

Let $W_t$ be a standard wiener process and $$Y_t=\int_{0}^{t}\frac{W_s}{(1+W_s^2)^2}ds$$ If $W(t_0)=\sqrt{3}$, then how can we compute $\mathbb{E}[Y(t_0)]$? Is $\mathbb{E}[Y(t_0)]=0$?
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4answers
4k 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 ...
4
votes
1answer
66 views

clarification to log-stock price formula

Having financial market with safe rate r and risky asset S with dynamics under physical measure P $$\frac{dS_t}{S_t}=\mu dt +\sigma dW_t$$ what is the log-stock price? Using Ito formula it is ...
0
votes
1answer
42 views

stochastic discount factor transformation

I have $$\frac{dM_t}{M_t}=-\frac{\mu}{\sigma} dW_t + \gamma_t dB_t, \tag{1}$$ where $B_t$ and $W_t$ are two independent Brownian Motions, which was further presented as $$ M_t=\exp \left( -\frac{\mu}{...
4
votes
1answer
304 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 ...
2
votes
2answers
240 views

Ito calculus problem

given $S^1$ satifying the SDE $\quad dS_{t}^{1}=S_{t}^{1}((r+\mu)dt + \sigma dW_t), \quad S_{0}^{1}=1 $ and the safe asset $S_{t}^{0}$ $\quad S_{t}^{0}:=e^{rt} \quad for \quad r\geq 0$ Q1. how ...
0
votes
2answers
52 views

Multivariate Ito problem $M_t=\frac{X_t}{Y_t}$

I am analyzing a problem given in the lecture slides published here (Slide 7-8 Example of Multivariate Ito’s Lemma). Can anybody explain how the $M_t$ was calculated out of the Ito formula. I ...
1
vote
1answer
103 views

2 Ito processes - $d(X_{t} + X^{'}_{t})^2 = (Y_t + Y^{'}_{t})^2 dt$ why it is true?

Having two Ito processes $dX_{t} =z_{1} dt + Y_{t} dB_t $ $dX^{'}_{t} =z^{'}_{1} dt + Y^{'}_{t} dB_t $ I am analyzing a proof of the product rule $d(X_t X_t^{'})=X_t dX_t^{'}+ X_t^{'} dX_t + ...
4
votes
1answer
94 views

How to express the volatility of two correlated Ito processes $Wt_1, Wt_2$ expressed in terms of $W_t$?

Having two correlated Ito processes ($W_t^1$ and $W_t^2$ are correlated Brownian motions with correlation $\rho$) $dX_{t} =\mu_{1} dt + \sigma_1 dWt_1 $ $dY_{t} = \mu_{2} dt + \sigma_2 dWt_2 $ ...
9
votes
2answers
341 views

Solution of Merton's Jump-Diffusion SDE

In many textbooks and also in the original Merton's paper the solution of the SDE $$ dS_t = S_t\,\mu\,dt+S_t\,\sigma\,dW_t+S_{t^-}\,d\left(\sum_{j=1}^{N_t}V_j-1\right) $$ is written as $$ S_t = ...
5
votes
1answer
98 views

Square of arithmetic brownian motion process

We have an arithmetic Brownian motion process $X_t$ that follows $dX_t=\mu dt + \sigma dZ_t$ and we define the asset price $S_t=X_t^2$ and we are asked to find the stochastic differential equation ...
0
votes
0answers
63 views

Stochastic Integration

I have the following derivation question: A small company is investing resources in a risky project that it hopes will be profitable. The project could, for example, represent the manufacturing and ...
1
vote
2answers
106 views

Stochastic process theory question

*S follows a process $dS= mSdt + oSdz$ where m and o are constant. What is the probability followed by $ Y=(Se)^{(r-t)} $. If S follows a process $ dS= k (b-S) dt + oSdz $ where k, b, o are ...
0
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0answers
25 views

self financing property vs. unlimited borrowing

How the self financing property of a portfolio should be understood in the problems where the unlimited access to the borrowing is assumed?
2
votes
2answers
79 views

Problem with deriving the dynamics of a process

I'm trying to solve the following problem. Given a process $X_t$ and a process $Z_t$, with the dynamics of $X_t$ as $$ dX_t = (\alpha + \beta X_t)dt + (\gamma + \sigma X_t)dW_t $$ and $Z_t$ defined ...
2
votes
1answer
79 views

Brownian motion. Solve stoc. integral by using Ito's lemma

I want to show that following statement is true by using Ito's lemma to solve stochastic integrals: I define the functions in Ito's model: a()=0, b()= (2wt-2)^2. f(t)=Integrate[(2wt-2)^2] Then df=(b^...
1
vote
1answer
81 views

Link between two Itô's Lemma written in different ways

I have been told that these two expressions of Itô's Lemma are the same, but written in different ways : $$ f(t,X_t) = f(0, X_0) + \int_{0}^{t} \frac{\partial f}{\partial s} ds + \int_{0}^{t} \frac{\...
2
votes
2answers
1k 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 $dr=A\,dt+B\, dW$, ...
3
votes
1answer
161 views

Derivation using Ito's Lemma of price process

Define $q(t)$ as the log price minus a linear trend $$ q(t) = \ln P(t) - \mu t $$ Assume the log price process = Equation 1: $$ dq(t) = - \Theta q(t) dt + \sigma dW(t) $$ Can you show that the ...
1
vote
0answers
124 views

stochastic calculus and multidimentional itos lemma

I am considering a number of assets (N) in a portfolio. each asset follows a geometric Brownian motion process therefore the stochastic differential equation is dS(i) = S(i)μdt + S(i)σdX(i). The ...
5
votes
1answer
139 views

How to tackle this exercise about Ito's formula?

In the following exercise, I can't get started on question 2) as I am not sure what to do when there is an integral inside: Could you help me out?
1
vote
1answer
96 views

Stochastic calculus: what am I doing wrong?

it is just the computation of a second moment but however is creating debate !!... Can someone spot the error?
2
votes
0answers
134 views

Multivariate Itô's lemma

Hey guys I'm looking for worked examples who show how to apply Itô's lemma in several variables, starting from the very basics. Thank you in advance!
1
vote
3answers
1k 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 2....
2
votes
0answers
57 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 $T_0+...
5
votes
1answer
559 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 \mathbb{R}^{...
6
votes
2answers
601 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 ...
2
votes
2answers
136 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
125 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 $\gamma^{\...
3
votes
1answer
149 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
1answer
173 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} =...
1
vote
1answer
147 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 ...
0
votes
1answer
75 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 t}...
5
votes
1answer
141 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 ...
2
votes
2answers
133 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 ...
6
votes
2answers
1k 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 ...
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votes
2answers
9k 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 ...
2
votes
1answer
50 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 ...
3
votes
0answers
64 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 ...
0
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2answers
73 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 + (\...
1
vote
0answers
122 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 true?...
1
vote
1answer
599 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: $$S_t=S_0\,e^{\...
10
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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 ...
0
votes
1answer
147 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 $d(\ln \frac{f}{g})$ ...
3
votes
1answer
157 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 $...
4
votes
1answer
233 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 $dZ_{t}$): $Z_{t}=\int_{0}^{t}...
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( 0\...