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Questions tagged [itos-lemma]

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3
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1answer
90 views

What is the easiest way to learn Option pricing with PDE?

I was reading about Ito's formula and Girsanov theorem, but I am still struggling to grasp how in reality these are combined to compute the price of an option. What are the main source to understand ...
1
vote
1answer
91 views

Brownian Motions theorems

I know that if $W$ and $W′$ are two independent brownian motions, then $dWt \ dWt′$ = 0. How can I prove/demonstrate this theorem? Additionaly, how can we prove that if $W$ and $W′$ are dependent, ...
4
votes
1answer
407 views

Ito`s Lemma problem

Can someone help me with calculus for this problem. I have these 3 equations and with Ito`s Lemma I have to find $dXt$. \begin{cases} dY= μYdt+σYdB \\ X=\frac{1}{2}cY\\ dc =-aαcdt\end{cases}
0
votes
1answer
96 views

Ito formula (lemma) problem

I am trying to solve this problem Consider the following one-dim. stochastic process $$dX_t = b_t dt + \sigma_t dW_t$$ where $W$ is a one-dim. Brownian motion. The above SDE is well-defined. ...
0
votes
1answer
34 views

Stochastic solution (mean, variance) to lognormal drift and normal volatility

I have trouble deriving the state equations for a mixture of normal/lognormal stochastic differential, namely for its a) expected mean, (b) variance, and (c) drift adjustment for LMM - libor model I ...
7
votes
1answer
166 views

Basic question on Ito integrals

$Let \space X(t) =\begin{cases} 2, \qquad\text{if} \space 0\le t \le 1 \\ 3, \qquad\text{if} \space 1 < t \le 3 \\ -5, \qquad\text{if}\space 3 < t \le 4 \end{cases} $ or in one forumala $...
-1
votes
1answer
65 views

Different Forms of Geometric Brownian Motion [closed]

If the stock price S follows the geometric brownian motion: $$dS=\mu Sdt+\sigma Sdz$$ $$\frac{dS}S=\mu dt+\sigma dz$$ Where $dz=\epsilon\sqrt{dt}$ is a wiener process. Integrating this to get $S_T$ ...
3
votes
4answers
389 views

Log of square of Geometric Brownian Motion

Which of the two calculations below, is wrong? Why? $dF = \sigma F dW$ First: $dF^2 = (F^2)' dF + \frac{1}{2}(F^2)''dF.dF$ $dF^2 = 2F dF + dF.dF$ $dF^2 = 2 \sigma F^2 dW + \sigma^2 F^2 dt$ $\...
2
votes
0answers
44 views

Ito Diffusion with Change of Measure

Let $(X_t)$ be an Ito diffusion with speed $(V_t)$, under a probability measure P. Could there exist a change of measure to a probability measure Q, with Q ~ P, under which $(X_t)$ is an Ito diffusion ...
3
votes
1answer
106 views

Expectation in a stochastic differential equation

I'm new to stochastic calculus, I want to find the mean of $X_2$ with $X_t = \exp(W_t)$, with $W_t$ a Wiener process. I used Ito's Lemma is arrive at the SDE: \begin{align} d(X_t) = \frac{1}{2}X_t dt ...
2
votes
0answers
64 views

For an Ito Process, $d\ln{X} \neq \frac{dX}{X}$ and $(d\ln{X})^2 = (\frac{dX}{X})^2$, but $d\ln{X} \neq \pm \frac{dX}{X}$

In normal calculus we can write $d\ln{x} = \frac{dx}{x}$ since there is no quadratic variation to deal with. This isn't true for stochastic processes, and Ito's Lemma is used to calculate $d\ln{X}$. ...
4
votes
1answer
157 views

Ito's Lemma for this problem

I'm attempting to prove a lemma from a paper, in the context of optimal contracts. $r,\rho,\gamma,\alpha,\sigma$ are all known constants. $dR_t = (\alpha + r)dt + \sigma dZ_t$ where $Z_t$ is a ...
3
votes
1answer
107 views

HJM model Baxter Rennie: differentiating the discounted asset price using Ito

From Baxter and Rennie Page 145: $Z(t,T) = exp(\int_{0}^{t}\Sigma(s,T)dW_s - \int_{0}^{T}f(o,u)du - \int_{0}^{t}\int_{s}^{T}\alpha(s,u)duds)$ where $\Sigma(t,T) = \int_{t}^{T}\sigma(t,u)du$ How ...
7
votes
1answer
470 views

Baxter & Rennie HJM: differentiating Ito integral

From Baxter and Rennie, page 138: $$f(t,T)=\sigma W_t+f(0,T)+\int_0^t\alpha(s,T)ds$$ $$Z_t=\exp-\bigg(\sigma(T-t)W_t+\sigma\int_0^tW_sds+\int_0^Tf(0,u)du+\int_0^t\int_s^T\alpha(s,u)ds\bigg)$$ $$dZ_t=...
6
votes
0answers
105 views

Random variable minus Integral of Ito Generator is a Martingale under what conditions?

I am reading about american option pricing and the variational inequality, and the book I am reading states, in the derivation of the variational inequality, the following is a martingale: $$M_s = U(s,...
4
votes
1answer
114 views

Expected payoff at future time

Let $a$, $b$, $c$, and $e$ be constants, $W_1$ and $W_2$ be Brownian motions with correlation $\rho$, and $f(t)$ and $g(t)$ be deterministic functions of time. Let $X$ satisfy $$d(X(t))=(aX(t)+ef(t)g(...
3
votes
2answers
261 views

Application of Ito's lemma

Let $X_t$ be some stochastic process driven by wiener process ($W_t)$ so it can be expressed as: $$dX_t=(...)dt+(...)dW_t$$ Let $f(t,x)$ be some $C^2$ function. Define the process $Z_s=f(t-s,X_s)$ ...
4
votes
1answer
85 views

How to express a process using Itos formula

Let $F(t,x)$ be the solution to the PDE $$ F_t(t,x)=aF_x(t,x)+\frac{1}{2}F_{xx}(t,x),t>0 $$ $$F(0,x)=g(x)$$ for some function $g$. Let $X_t$ be a process defined by $$dx_t=aX(t)dt+dW(t)$$ Now ...
5
votes
1answer
201 views

Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?

Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$? If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
3
votes
4answers
11k 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....
3
votes
2answers
100 views

Random Walk with normal increments and n time periods why is the increment $\sqrt{(t/n)}$?

Question is basically in the title. I have found several sources stating that $R_i = \sqrt{\frac{t}{n}}$, but I couldn't find the intuition behind taking the square root. And it seems to be crucial ...
-1
votes
1answer
67 views

Differential product Correlated processes

I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now: We have the following two stochastic processes: $dX_t= \mu_t dt +\sigma_t dW_t$...
2
votes
1answer
158 views

Differential of integrating factor $d(e^{at}r_t)$ in Vasicek model

I am attempting to solve the Vasicek model SDE (using Wikipedia parametrisation): $$ dr_t = a(b-r_t)dt + \sigma dW_t $$ Every solution is proceeding to multiply both sides of the equation by the ...
3
votes
1answer
162 views

Why is Ito applied this way?

Given the price of a call option : $$C = \mathbb{E}\left[ D_{0,T} (s-K)1_{s>K} |\mathcal{F_0}\right] $$ with $D_{0,T}=e^{-\int_0^Tr(u)du}$ I read somewhere that applying Itô gives : $$dC = \...
-1
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2answers
73 views
2
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0answers
98 views

Pricing caplet with Bachelier (normal dynamic) using forward measure

I'm trying to price caplet with Bachelier under forward measure, but I can't find any solution. Remind that Bachelier assumed rates follow a normal dynamic. So here what I was doing : $C_t(T,T+d)$ ...
2
votes
1answer
85 views

How to derive the dynamic of the log forward price?

I have the following Schwartz model: $$dS_t=a(\mu-\ln S_t)S_tdt+\sigma S_tdW_t$$ $$X_t=\ln S_t$$ $$dX_t=a(\hat{\mu}-X_t)dt+\sigma dW_t$$ with $\hat{\mu}=\mu-\frac{\sigma^2}{2a}\sigma$ $$F_t(T)= \exp\...
2
votes
0answers
237 views

Applying Ito's formula to complex functions

Within my lecture notes, the following definition is given: We say that the stochastic process $X_t$ has stochastic differential $$ dX_t = b_t dt + \sigma_t dW_t $$ if and only if $$ X_t = ...
2
votes
1answer
194 views

Black Scholes in the case of dividends

Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a ...
2
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0answers
64 views

$\int_{0}^1W_x(t)dW_y(t)/(\int_{0}^1W_x^2(t)dt)^{1/2}$ normally-distributed?

I have came across the following stochastic integrals: $$\frac{\int_{0}^1W_x(t)dW_y(t)}{(\int_{0}^1W_x^2(t)dt)^{1/2}}$$ which was claimed to be standard normally distributed ($W_x$ and $W_y$ are ...
4
votes
1answer
165 views

Ito representation unique up to indistinguishability? Proof?

Given an Ito-process $X(t)$, $t\in[0,T]$ $$X(t)=X_{0}+\int_{0}^{t}F(s)ds + \int_{0}^{t}G(s)dW(s)$$ with $F\in \mathbb{L}^{1}(0,T)$ and $G\in\mathbb{L}^{2}(0,T)$. It is now often claimed that this ...
0
votes
1answer
121 views

Self finance conditions - proof check

Find expressions for the process $\psi=(\psi(t),\ 0\leq t\leq T)$ , so the portfolio $(\phi,\ \psi)$ is self-financing when: (1) $\phi(t)= \int_{0}^{t}S_{s}ds $ (2) $\phi(t)=S_{t}$ where $\phi(t)$ ...
3
votes
2answers
137 views

Show that the two solutions of the SDE are equivalent

I have a process: $$dr_t = (W_t^1 - ar_t)dt +\sigma dW_t^2$$ where $W_t^1$ and $W_t^2$ are brownian motions with instantaneous correlation coefficient $\rho$. I want to show that the solution of this ...
1
vote
1answer
302 views

exercise on multivariate Ito's lemma + jumps (Poisson)

Given the two jump-diffusions: \begin{equation} \begin{aligned} dX_{1,t} &= a_1 dt + b_1 dW_t + c_1 dN_t(\lambda) \\ dX_{2,t} &= a_2 dt + b_2 dW'_t + c_2 dN_t(\lambda) \\ corr(dW,dW') &= \...
1
vote
1answer
130 views

Mark Joshi, Chapter 5 Problem 2 of The concepts and practice of mathematical finance

If $$dX_t = \mu(t,X_t)dt + \sigma(X_t)dW_t$$ with $\sigma$ positive, show there exists a function $f$ such that $$d\left(f(X_t)\right) = v(t,X_t)dt + V dW_t$$ where $V$ is constant. How unique is $f$...
2
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0answers
64 views

How to calculate the product of forward rates with different reset times using Ito's lemma?

I am curious about a calculation I saw in this question. Specifically in this equation: \begin{align*} &\ L(T_s, T_p, T_e) L(T_s, T_s, T_e) \\ =&\ L(t_0, T_p, T_e) L(t_0, T_s, T_e) e^{-\...
3
votes
1answer
577 views

Ito's Lemma: Multiplication Rule

I have a conceptual question about Ito's lemma, in particular, the multiplication. Ito's multiplication rule states, that multiplying dt by itself or by dx (the stochastic differential) equals zero. ...
2
votes
1answer
323 views

Intuition behind Ln transformation of stock price when applying Ito lemma [closed]

I am able to replicate steps and arrive to the option price using Black Scholes framework. Here however I am more interested to understand, at least intuitively, why the ln transformation of price ...
2
votes
1answer
260 views

Quantile normal and lognormal

Let's assume we have a normal distribution $X\sim \mathcal{N}(\mu,\sigma^2)$. In a normal distribution the quantile can be calculated as follows: \begin{equation} \Phi_X ^{-1}(p)=\mu +\sigma {\sqrt {...
8
votes
3answers
1k 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
153 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 $...
-2
votes
1answer
819 views

On the application of Itos lemma to Geometric Brownian motion [closed]

I recently read this from a book: The canonical SDE in financial math, the geometric Brownian motion, ${{d{S_t}} \over {{S_t}}} = \mu dt + \sigma d{W_t}$ has solution $${S_t} = {S_0}{e^{(\mu -...
0
votes
1answer
206 views

Self-Financing Portfolio

Why when we are using self-financing portfolios to replicate some external payoff we do not consider the quadratic variation of the portfolio weights? Say, in Black-Scholes world, when we are using $...
1
vote
0answers
79 views

Transformation of coupled forward-backward stochastic differential equations in 3 dimensions with Ito formula

Maybe this is the right place for my question: I have a system of coupled FBSDEs in 3 dimensions as follows (in cartesian coordinates): $$ \mathrm{d}\vec{r}(t) = \vec{u}(\vec{r}(t))\mathrm{d}t + \...
1
vote
1answer
76 views

Is it possible to approach finding the risk premium of this derivative using Ito's Lemma?

I understand the author's intended solution to the below problem, but I thought I would see if I could solve this using first principles and Ito's Lemma instead for practice. Let $V(S(t), t) = e^{rt}\...
19
votes
2answers
26k 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 ...
-1
votes
1answer
2k views

Given $S$ is a Geometric Brownian Motion, how to show that $S^n$ is also a Geometric Brownian Motion?

Suppose that a stock price $S$ follows Geometric Brownian Motion with expected return $\mu$ and volatility $\sigma:$ $$dS = \mu S dt +\sigma S dz$$ How to find out the process followed by variable $...
2
votes
1answer
241 views

How to show that $E\left[ \int_0^t \sigma(s) e^{iuX(s)} dW(s)\right] = 0$?

Let $\sigma(t)$ be a given deterministic function of time and define the process $X_t$ by $$X(t) = \int_0^t \sigma(s)dW(s)$$ I want to show $$E\left[ \int_0^t \sigma(s) e^{iuX(s)} dW(s)\right] = 0$$...
-1
votes
1answer
525 views

Bond price and its process

Suppose that x is the yield to maturity with continuous compounding on a discount bond that pays off $1 at time T. Assume that the x follows the process $dx=a(x_0-x)dt + sxdz$ where $a, x_0$ and $s$ ...
3
votes
2answers
163 views

Moment Ito's Process Proof

I have a following stochastic integral - related problem that I have difficulty to solve: Given \begin{equation} dX_t = -\alpha X_tdt+\sigma\sqrt{X_t}dW_t \end{equation} and the second moment of $...