# Questions tagged [itos-lemma]

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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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### 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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### 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, ...
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### 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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### 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 ...
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, ...
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. ...