Questions tagged [itos-lemma]
The itos-lemma tag has no usage guidance.
160
questions
1
vote
2answers
90 views
Compute the price of a derivative which pays $\log(S_T)S_T$ in the Black Scholes world
Compute the price of a derivative which has pays $\log(S_T)S_T$, you can assume that the Black Scholes model is valid.
Using the stock measure we can write the expectation as
$$D(0) = S_0 \mathbb{E}...
3
votes
1answer
143 views
Gamma PnL from Itô's Lemma derivation
The change in a call portfolio ($f$), derived from Itô's Lemma, is:
\begin{align*}
\left( \frac{\partial f}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2}\right)\mathrm{d}t &=...
2
votes
2answers
116 views
Itos Lemma Derivation notation
So in Hull (2012) the main point is that $\Delta x^2 = b^2 \epsilon ^2 \Delta t + $higher order terms$ $ has a term of order $\Delta t$ and can not be ignored as the Brownian motion exhibits the ...
3
votes
1answer
157 views
Application of Ito's Lemma in expected utility theory
An investor with utility curve $U(.)$ has wealth $X_t$ at time t. He invests
A proportion $p$ of his wealth in a risky asset that follows a geometric Brownian motion, with parameters $\mu$ and $\...
1
vote
1answer
48 views
The most general conditions under which Ito lemma holds
Prompted by a question that came up in the comments here, namely why we can apply the Ito lemma to a function of the form $f(x)=(x-K)^{+}$, I would be interested in knowing what are the least ...
0
votes
1answer
40 views
Serial correlation, quadratic variation and variance of returns
On p. 3 of Lorenzo Bergomi's book on Stochastic Volatility Modeling, there is the following assertion:
Indeed, to a good approximation, the variance of returns scales linearly with their time scale, ...
3
votes
1answer
101 views
Under which conditions the given random process is martingale and under which submartingale?
Let $a_t $ be adapted to the filtration random process $a_t: P\{\int _0^T|a_t|dt < \infty \} = 1 $ and $ b_t \in M_T^2. \quad$ Under which conditions the random process $$X_t = exp\{\int _0^ta_sds+\...
0
votes
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 ...
4
votes
1answer
77 views
Differential of time over Browninan motion
I know that $\frac{dW_t}{dt}$, with $W_t$ a brownian motion, does not exist. However, does $\frac{dt}{dW_t}$ exists? Or does it even make sense? I am trying to calculate the quotient of two ...
2
votes
0answers
62 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 ...
1
vote
0answers
30 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 $[0,T]\times \mathbb{R}$:
$$
\begin{cases}
\dfrac{\partial F}{\partial t}+\mu(t,x) \dfrac{\partial F}{\partial x}+ \frac12 \sigma^2(t,x)\dfrac{\partial^2 F}{\partial x^2}...
3
votes
0answers
224 views
Black and Scholes equation for portfolio **with** arbitrage
I am well aware of how the ordinary Black and Scholes equation is derived, under the assumption of an arbitrage free portfolio, $V=G-hS$. Here $S$ is the price of the underlying and $G$ is the option ...
1
vote
1answer
60 views
Ito's lemma for a Forward
I'm trying to understand the derivation of Ito's process with respect to a Forward $F$ on a stock $S$ that pays a constant dividend yield, say $y$. Stock follows brownian motion $\\$
$dS_{t} = S_{t}(\...
0
votes
0answers
40 views
Compo/Quanto Adjustment & Multivariate Ito
Related to the issue that I have raised here, I am facing another question. As the rule here is 1 question / 1 post, I take the opportunity to ask it below:
By exploring StackExchange, I noticed the ...
3
votes
2answers
180 views
Why is $S(t) = e^{\alpha + \beta t + \sigma W(t)}$ used as a model for prices?
Why is the Geometric Brownian Motion defined as $S(t) = e^{\alpha + \beta t + \sigma W(t)}$ used as a model for stock prices? $S(t)$ has a lognormal distribution which is right skewed. Another problem ...
2
votes
1answer
54 views
Generalization of Ito's Lemma to composite function
Ito's Lemma gives that for a function $F$ of a stochastic variable $X$, $dF = \frac{dF}{dX}dX + \frac{1}{2}\frac{d^2F}{dX^2}dt$
Given a stochastic differential equation $dS = a(S) dt + b(S) dX$ and a ...
1
vote
1answer
62 views
Derivation of stock price formula John C. Hull 9th Ed p309
It says assuming a no-uncertainty Weiner process that models stock price:
$$
\Delta S = \mu S\Delta t
$$
Can be rearranged to (after taking the limit of $\Delta t \to 0$...
$$
\frac{dS}{S}=\mu dt
$$
...
1
vote
1answer
71 views
Integration of a deterministic function w.r.t. a Brownian motion
Help me solve this problem:
Let $W_t$ be a Brownian motion and suppose
$X_t = \int_{0}^{t}\delta _{s}dW_{s}$
where $\delta _{s}$ is a deterministic function. Then show that $X_t$ is a Gaussian ...
4
votes
1answer
91 views
Ito formula for $Y_t=tB_t$
someone can help me to solve this problem:
$B_t$ is a Standard Brownian Motion.
Let $Y_t=tB_t$.
Using Ito formula, find drift and volatility of $Y_t$.
The result I found is $dY_t=B_tdt+t\cdot dB_t$ ...
1
vote
1answer
99 views
Calculation of a process's drift
Let $X_t:=e^{W_t}$ where $W_t$ follows the Wiener process. Calculate the drift.
The answer is given as $X_t/2$. My attempt at a solution (which I'm afraid is poor from a mathematical standpoint):
I ...
1
vote
0answers
43 views
Application Itô's Lemma: Forward to Spot process
I am working on the following equation (I want to apply Ito's lemma on it):
and I know that:
and also
and
My problem is that I want the dynamic of F(S,T) without S because I need first to ...
1
vote
1answer
72 views
Compute dZ(t) : Ito's formula/lemma
We need to find dZ(t). I know I have to use Ito's formula. But I am confused because in the Ito's formula we have f(y,t) is a twice differentiable function with two variables
But here Z(t) = 1/(2+x(t)...
1
vote
0answers
63 views
On Geometric Brownian motion and Itô's formula
Let $S_t$ be a geometric brownian motion such as
$$d S(t) = rS(t)dt +\sigma S(t)dW(t),$$
where $W$ is a standard Brownian motion.
With Itô's lemma and formulas $(dt)^2=dtdW_t=dW_tdt=0$ and $(dW_t)^2=...
0
votes
0answers
25 views
Deriving coupling equation(s) for Heston Stochastic Volatility Model
In Bergomi Smile Dynamics (2003) Section 2.1 we are given the following coupled equations for the mean and for the variance of the hedger's portfolio:
$
\begin{align*}
\frac{dm}{dt} + \mathcal{L}m - ...
4
votes
1answer
294 views
Pricing call option using risk-neutral martingale approach with squared stock price boundary?
I have to use the risk-neutral martingale 5 step approach under BS pricing framework to price the following call option at time 0:
$$X = \begin{cases}1, &{if} &S_T^2\geq K,\\0, & {...
3
votes
1answer
156 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
votes
0answers
66 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
votes
2answers
150 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
votes
1answer
108 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
votes
0answers
124 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
votes
2answers
331 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 ...
5
votes
3answers
119 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_{...
0
votes
0answers
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?
4
votes
1answer
123 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....
0
votes
0answers
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 ...
1
vote
0answers
94 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
vote
1answer
61 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)+\...
1
vote
1answer
76 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
votes
2answers
173 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
votes
1answer
76 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
votes
1answer
124 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
votes
1answer
259 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
votes
1answer
71 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
169 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
$$...
0
votes
0answers
53 views
How does this follow from Ito's formula?
Let $S_t$ be a geometric brownian motion and let $M_t = \sup_{0 \le u \le t} S_t$ be the maximum process.
Define $X_t = \frac{M_t}{S_t}$.
A book I am reading states the following:
How does this ...
6
votes
2answers
525 views
More questions about integral of Brownian Motion w.r.t time
A similar question have been posted earlier but one part has remained unanswered. Let us define:
$$X_t = \int_0^t W_s ds,$$
where $W_t$ is a standard Brownian Motion. Is $X_t$ an Itô process or a ...
2
votes
1answer
103 views
Are the Ito's Lemma given in Mark Joshi's Concept and Practice in Mathematical Finance same as what I learn?
In Joshi's Concepts and Practice in Mathematical Finance, page $110,$ he stated the Ito's Lemma:
Theorem $5.1$ (Ito's Lemma) Let $X_t$ be an Ito process satisfying
$$dX_t = \mu(X_t,t)dt + \sigma(...
2
votes
1answer
134 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
vote
1answer
65 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:
$$f(t,T,T+\delta) = \frac{ln(Z(t,T)) - ln(Z(t,T,...
1
vote
1answer
129 views
Integration and expectation of geometric Brownian motion
Let the stock price S follows the geometric brownian motion:
$$dS=\mu Sdt+\sigma Sdz$$
$$\frac{dS}S=\mu dt+\sigma dz$$
where $dz$ is a wiener process.
Naively integrating the second equation above ...
