Questions tagged [itos-lemma]

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138 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,...
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0answers
108 views
+50

Relating two equations in a jump-diffusion process

I am trying to understand an argument involving the pricing kernel $\xi_t$ in the context of a simple jump diffusion model for the price of an asset $S_t$: \begin{align} \xi_t = \exp \left[ -\theta ...
3
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0answers
227 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 ...
3
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0answers
66 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
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0answers
370 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 = ...
3
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0answers
440 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!
3
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0answers
112 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
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1answer
123 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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0answers
69 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 ...
2
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0answers
130 views

Itô’s formula and Wiener process

The Wikipedia page on the formula https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma and some textbooks I have looked at say we must assume that the relevant time-dependent function is over an Itô ...
2
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0answers
107 views

Ito's lemma for special case

Assume a HJM framework with the same Brownian motion driving the dynamics for every tenor. $$ df(t,T) = \alpha(t, T)dt + \sigma(t,T) dw_t \,, $$ with $\alpha(t, T) = \sigma(t,T)\int_t^T \sigma(t,s)ds$....
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0answers
211 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 $d\ln{X}$. ...
2
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0answers
241 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
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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 ...
2
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0answers
77 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^{-\...
2
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0answers
562 views

Stochastic Leibniz rule

We have the following single-factor HJM model $$d_tf(t,T)=\sigma(t,T)dW_t+\alpha(t,T)dt$$ $$f(t,T)=f(0,T)+\int_0^t\sigma(s,T)dW_s+\int_0^t\alpha(s,T)ds$$ The discounted T bond is then \begin{align} Z(...
2
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0answers
179 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+...
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32 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}...
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0answers
49 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
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0answers
64 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=...
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0answers
98 views

How to determine exchange rate dynamics in currency derivatives

I need some guidance regarding exchange rate dynamics in currency derivatives. Following three dynamics are defined below, $\frac{dS(t)}{S(t)}=\alpha dt+\sigma dW(t)$ ; the stock dynamics in the ...
1
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0answers
242 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 ...
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149 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?...
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82 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 ...
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0answers
31 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 ...
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0answers
44 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 ...
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0answers
29 views

Deriving coupling equation(s) for Heston Stochastic Volatility Model

In Bergomi Smile Dynamics (2003) Section 2.1 we are given the following coupled equations for the mean and for the variance of the hedger's portfolio: $ \begin{align*} \frac{dm}{dt} + \mathcal{L}m - ...
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45 views

How to proof the formula to be martingale under ITO process?

How can implies that is a martingale when using the defaultable bond price?
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33 views

Confirm If Risk-Neutral Measure is Unique in My Following Case

I'm reading a book that discusses about derivatives pricing and have some doubts about a particular problem and really appreciate your advice: Question: Assume a non-dividend paying stock follows a ...
0
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0answers
57 views

How does this follow from Ito's formula?

Let $S_t$ be a geometric brownian motion and let $M_t = \sup_{0 \le u \le t} S_t$ be the maximum process. Define $X_t = \frac{M_t}{S_t}$. A book I am reading states the following: How does this ...
-1
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1answer
82 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: $dX_t= \mu_t dt +\sigma_t dW_t$...