# Questions tagged [itos-lemma]

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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $max(S_{... 0answers 45 views ### How to proof the formula to be martingale under ITO process? How can implies that is a martingale when using the defaultable bond price? 1answer 130 views ### How to determine components of Affine Term Structure for an Ohrnstein-Uhlenbeck process? I wonder how I can determine the components$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)}$? The components are defined in the following link: https://en.... 0answers 33 views ### Confirm If Risk-Neutral Measure is Unique in My Following Case I'm reading a book that discusses about derivatives pricing and have some doubts about a particular problem and really appreciate your advice: Question: Assume a non-dividend paying stock follows a ... 0answers 98 views ### How to determine exchange rate dynamics in currency derivatives I need some guidance regarding exchange rate dynamics in currency derivatives. Following three dynamics are defined below,$\frac{dS(t)}{S(t)}=\alpha dt+\sigma dW(t)$; the stock dynamics in the ... 1answer 68 views ### How to determine the no arbitrage price of following claim? (change of numeraire) How do I determine the no arbitrage price for claims such as$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)+\...
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 ...