Questions tagged [itos-lemma]

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votes
0answers
22 views

Ito formula for $Y_t=tB_t$

someone can help me to solve this problem: $B_t$ is a Standard Brownian Motion. Let $Y_t=tB_t$. Using Ito formula, find drift and volatility of $Y_t$. The result I found is $dY_t=B_tdt+t\cdot dB_t$ ...
1
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1answer
82 views

Calculation of a process's drift

Let $X_t:=e^{W_t}$ where $W_t$ follows the Wiener process. Calculate the drift. The answer is given as $X_t/2$. My attempt at a solution (which I'm afraid is poor from a mathematical standpoint): I ...
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0answers
38 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 ...
0
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1answer
41 views

Compute dZ(t) : Ito's formula/lemma

We need to find dZ(t). I know I have to use Ito's formula. But I am confused because in the Ito's formula we have f(y,t) is a twice differentiable function with two variables But here Z(t) = 1/(2+x(t)...
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0answers
52 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
22 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 - ...
4
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1answer
251 views

Pricing call option using risk-neutral martingale approach with squared stock price boundary?

I have to use the risk-neutral martingale 5 step approach under BS pricing framework to price the following call option at time 0: $$X = \begin{cases}1, &{if} &S_T^2\geq K,\\0, & {...
3
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1answer
134 views

Solving Stochastic Differential Equation for Geometric Brownian Motion with time-dependent drift

Given the stochastic differential equation: $$dZ_t = -Z_t \theta_t dB_t, \quad Z_0 = 1.$$ for an adapted process $\theta_t$ and Brownian motion $B_t$, how exactly do I apply Itô's Lemma to obtain: ...
5
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0answers
63 views

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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2answers
137 views

Partial derivative of Ito integral without product rule

I'm thinking about the problem of deriving the stochastic differential of an integral with both time and state part of the integrand but not in a way that you can easily factor it out - for example I ...
2
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1answer
103 views

How To Understand the Drift of ln(S) if S Follows Geometric Brownian Motion

As we know, if an asset S follows geometric Brownian motion, under risk neutral measure, it can be expressed as $\frac{dS}{S}=rdt+\sigma dW$, by applying Ito's lemma, $d(lnS)=(r-0.5*σ^2)dt+σdW(t)$, ...
2
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0answers
112 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 ...
6
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2answers
306 views

Variance of a time integral with respect to a Brownian Motion function

Let process $$I_t = \int_0^t f(s) W_s \,\mathrm d s $$ where $W_s$ is standard Brownian motion. My question are the following: We know that $\mathbb{E} (I_{t})=0$ for all $t$ and $f$ a integrable ...
4
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2answers
92 views

Volatility of Exchange Option

I got a question and its partial solution, and have some doubts about the volatility of its geometric Brownian motion process: Question: How would you price an exchange call option that pays $max(S_{...
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0answers
44 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?
4
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1answer
111 views

How to determine components of Affine Term Structure for an Ohrnstein-Uhlenbeck process?

I wonder how I can determine the components $A(t,T)$ and $B(t,T)$ for the zero-coupon bond price process $p(t,T)=e^{A(t,T)-r(t)B(t,T)}$? The components are defined in the following link: https://en....
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0answers
30 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 ...
1
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0answers
90 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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1answer
48 views

How to determine the no arbitrage price of following claim? (change of numeraire)

How do I determine the no arbitrage price for claims such as $min(S_1(T),S_2(T))$ or $max(S_1(T),S_2(T))$? We can consider a standard Black Scholes model. Hence $S_i(T)=S_i(t)e^{(r-\sigma_i^2/2)(T-t)+\...
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1answer
74 views

Determining the No Arbitrage price of max[B(T), S(T)]

Following is given, $dB(t)=rB(t)dt$ $dS(t)= (r-\delta)S(t)dt+\sigma S(t)dW(t)$ where, $r$ is the risk-free interest rate, $\delta$ the continous dividend yield $\sigma$ is the stock asset ...
3
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2answers
172 views

Stochastic Calculus problem with three processes? (Itô calculus)

Can someone help me solve this following Itô Calculus problem? Let $Z(t):= [B(t)*X(t)]/S(t)$ We have the following dynamics of B(t), X(t) and S(t): $dS(t)=\alpha S(t)dt+\sigma S(t)dW(t)$ $dB(t)=rB(...
5
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1answer
65 views

Three proofs regarding brownian motions and martingales

1. Let $(B_t)_{t \geq 0}$ and $(W_t)_{t \geq 0}$ be two standard Brownian motions and let $X_t := B_t W_t$. Is $(X_t)_{t \geq 0}$ a martingale? The easiest way to proceed seems to be to apply Ito's ...
3
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1answer
114 views

Computing Itô differential of conditional expectation process (Heston SDE)

Going through this article on Heston's model, where the variance evolves following the SDE \begin{equation} \label{sd1} d\sigma^2_t = \kappa \bigg( m - \color{red}{\sigma^2_t} \bigg)dt + \nu \sqrt {\...
3
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1answer
182 views

How to calculate the mean and variance of this Ito integral?

I tried to calculate this integral use Ito's lemma, $W_{t}$ is the Wiener Process. $$I_{T}=\int_{0}^{T}\sqrt{|W_{t}|}dW_{t}$$ We have $d f\left(W_{t}\right)=f^{\prime}\left(W_{t}\right) d W_{t}+\...
2
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1answer
68 views

Independence of increments of the stochastic process $\frac{1}{t}\int_0^t u dW_u $

Let $X_t$ be a stochastic process such that $$X_{t} =\frac{1}{t}\int_0^t u dW_u $$ I know that for $$Y_{t} =\int_0^t u dW_u$$ $Y_t-Y_s$ is independent of $Y_s$ where $t>s$. But is this also true ...
3
votes
2answers
148 views

Probability distribution of the stochastic process $\int_{0} ^{t}\frac{u}{t}dW_{u}$

I am wondering about the probability distribution of the stochastic process $$X_t=\int_0^t \frac{u} {t} dW_{u}$$ I thought of using the Kolmogorov equation but after converting this into An SDE $$...
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0answers
48 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 ...
5
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2answers
341 views

More questions about integral of Brownian Motion w.r.t time

A similar question have been posted earlier but one part has remained unanswered. Let us define: $$X_t = \int_0^t W_s ds,$$ where $W_t$ is a standard Brownian Motion. Is $X_t$ an Itô process or a ...
2
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1answer
101 views

Are the Ito's Lemma given in Mark Joshi's Concept and Practice in Mathematical Finance same as what I learn?

In Joshi's Concepts and Practice in Mathematical Finance, page $110,$ he stated the Ito's Lemma: Theorem $5.1$ (Ito's Lemma) Let $X_t$ be an Ito process satisfying $$dX_t = \mu(X_t,t)dt + \sigma(...
2
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1answer
103 views

Stochastic Processes (Applying Ito's Lemma on Ho-Lee Model )

I seek a basic form (SDE) to understand the Ho-Lee model. I already understand the models from Vasicek, Merton and Cox-Ingereoll-Ross, etc.. For example, \begin{align*} dX_t &= -1/2 \alpha X_t ...
1
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1answer
64 views

How to derive the expression for the forward rate?

The following RN dynamics of a ZCB maturing at time is given: $$\frac{dZ(t,T)}{Z(t,T)} = r_tdt + \sigma_Z(t,T)dX_t$$ and the forward rate is given: $$f(t,T,T+\delta) = \frac{ln(Z(t,T)) - ln(Z(t,T,...
1
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1answer
113 views

Integration and expectation of geometric Brownian motion

Let the stock price S follows the geometric brownian motion: $$dS=\mu Sdt+\sigma Sdz$$ $$\frac{dS}S=\mu dt+\sigma dz$$ where $dz$ is a wiener process. Naively integrating the second equation above ...
2
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1answer
92 views

Zero-coupon bond pricing equation derivation

I'm trying to understand how in Chawla's paper that I've linked below, how he obtains equation (2.5) for the zero coupon bond pricing equation? The equation is: $\frac{\partial B}{\partial t} + \...
3
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1answer
154 views

List: Behavioural characteristics of key Ito processes used in finance

My hope from this question is to become a repository of the behavioural characteristics, use-cases and interesting features of the key Ito processes used in quantitative finance - examples being GBM, ...
2
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0answers
98 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 $\alpha(t, T) = \sigma(t,T)\int_t^T \sigma(t,s)ds$....
3
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1answer
160 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
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1answer
115 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, ...
2
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1answer
192 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. ...
1
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1answer
86 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 ...
-1
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1answer
77 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 $S_T$ ...
4
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1answer
489 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}
3
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4answers
725 views

Log of square of Geometric Brownian Motion

Which of the two calculations below, is wrong? Why? $dF = \sigma F dW$ First: $dF^2 = (F^2)' dF + \frac{1}{2}(F^2)''dF.dF$ $dF^2 = 2F dF + dF.dF$ $dF^2 = 2 \sigma F^2 dW + \sigma^2 F^2 dt$ $\...
3
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0answers
59 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
142 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
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0answers
160 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 $d\ln{X}$. ...
5
votes
1answer
212 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
120 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
0answers
124 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
117 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(...
4
votes
2answers
417 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 $Z_s=f(t-s,X_s)$ ...