Questions tagged [itos-lemma]

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215 views

Clarification on Deriving Ito's Lemma

The classical approach to deriving Ito's Lemma is to assume we have some smooth function $f(x,t)$ which is at least twice differentiable in the first argument and continuously differentiable in the ...
2
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1answer
186 views

Pricing Swaption Analytically using Libor Market Model

I was asked the following question in a recent interview: "(i) Express a forward swap rate in terms of forward Libor rates. (ii) Apply Ito's lemma to this expression to derive the process for the ...
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1answer
328 views

Can I write Ito's Lemma as a taylor expension?

instead of using Wikipedia's definition: $$ {d}(f(X_t,t)) = \frac{\partial f}{\partial t}(X_t,t)\,\mathrm{d}t + \frac{\partial f}{\partial x}(X_t,t) \, \mathrm{d}X_t + \frac{1}{2} \frac{\partial^2 f}{\...
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104 views

Ito's lemma and Lognormal Property

What would be the difference between: \begin{align} dS = udt + \sigma dz \end{align} and \begin{align} dS=u*S*dt + \sigma*S*dzdS \end{align} Is that the former is in absolute terms and the latter is ...
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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 ...
2
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1answer
119 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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1answer
93 views

Clarification of Ito's lemma

I was looking at the various examples provided in the discussion Worked examples of applying Ito's lemma One such example is 9.1 (c). This states that - if $S_t =\! S_0 + \int\limits_{0}^{t} \mu_u ...
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1answer
40 views

Applicability of the Ito's lemma [duplicate]

Ito's lemma is used to find the stochastic process of the function of a ...
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1answer
119 views

Question about using Ito's lemma in Gamma PnL

While deriving the delta hedge error if we hedge with implied vol, and the true vol is different, we say that the PnL of the call option is: $$dC=C_tdt+C_SdS+0.5C_{ss}<QV>dt - (1)$$ Where $<...
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0answers
39 views

Solution to Stock Price SDE with mean reversion [duplicate]

Suppose $S_t$ follows the process (notice the $S_t$ term in the diffusion part): $$ S_t := S_0 + \int_{h=t_0}^{h=t}\alpha(\mu -S_h)dh + \int_{h=t_0}^{h=t}\sigma S_h dW(h) $$. I actually don't know how ...
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2answers
111 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}...
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1answer
170 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 &=...
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2answers
121 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
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1answer
174 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 $\...
2
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1answer
97 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 ...
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1answer
61 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
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1answer
103 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+\...
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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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1answer
82 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 ...
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68 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 ...
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31 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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226 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 ...
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1answer
63 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}(\...
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43 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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2answers
186 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 ...
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1answer
55 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 ...
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1answer
64 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 $\Delta t \to 0$... $$ \frac{dS}{S}=\mu dt $$ ...
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1answer
107 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 ...
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1answer
94 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 $dY_t=B_tdt+t\cdot dB_t$ ...
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1answer
107 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
45 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 ...
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1answer
75 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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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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28 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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1answer
316 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, & {...
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1answer
163 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: ...
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67 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 ...
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2answers
157 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 ...
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1answer
112 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, $d(lnS)=(r-0.5*σ^2)dt+σdW(t)$, ...
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0answers
127 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 ...
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2answers
351 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 ...
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3answers
124 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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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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1answer
124 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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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 ...
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0answers
96 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 ...
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1answer
65 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
77 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 ...
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2answers
177 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(...
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1answer
80 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 ...