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# Questions tagged [itos-lemma]

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### Worked examples of applying Ito's lemma

In most textbooks Ito's lemma is derived (on different levels of technicality depending on the intended audience) and then only the classic examples of Geometric Brownian motion and the Black-Scholes ...
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### What is Ito's lemma used for in quantitative finance?

Further to my question asked here: prior post and which left some points unanswered, I have reformulated the question as follows: What is Ito's lemma used for in quantitative finance? and when is it ...
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### Demonstration of Ito's correction term/lemma in binomial tree

I am preparing an undergraduate QuantFinance lecture. I want to demonstrate the ideas of Ito's correction term and Ito's lemma in the most accessible manner. My idea is to take the "working horse" of ...
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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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### 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 ...
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### Random Walk with normal increments and n time periods why is the increment $\sqrt{(t/n)}$?

Question is basically in the title. I have found several sources stating that $R_i = \sqrt{\frac{t}{n}}$, but I couldn't find the intuition behind taking the square root. And it seems to be crucial ...
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### How to express the volatility of two correlated Ito processes $Wt_1, Wt_2$ expressed in terms of $W_t$?

Having two correlated Ito processes ($W_t^1$ and $W_t^2$ are correlated Brownian motions with correlation $\rho$) $dX_{t} =\mu_{1} dt + \sigma_1 dWt_1$ $dY_{t} = \mu_{2} dt + \sigma_2 dWt_2$ ...
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Super basic question. I think I am doing this correctly, but just want a sanity check. Say I have a stochastic process $r(t)$. Say I have an equation $$d(e^{\beta (t-s)}r(s))=\dots$$ where the $... 1answer 128 views ### Expectation in a stochastic differential equation I'm new to stochastic calculus, I want to find the mean of$X_2$with$X_t = \exp(W_t)$, with$W_ta Wiener process. I used Ito's Lemma is arrive at the SDE: \begin{align} d(X_t) = \frac{1}{2}X_t dt ... 1answer 114 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$How ... 1answer 353 views ### Stochastic Differentials - Ito's formula for a self-financing portfolio Suppose I have a portfolio of stocks$(S)$and savings account ($\beta_t$) then, the value is $$V = a_t S_t + b_t \beta_t$$ and for this portfolio to be self replicating, we need by Ito's lemma $$dV ... 1answer 167 views ### Why is Ito applied this way? Given the price of a call option :$$C = \mathbb{E}\left[ D_{0,T} (s-K)1_{s>K} |\mathcal{F_0}\right] $$with D_{0,T}=e^{-\int_0^Tr(u)du} I read somewhere that applying Itô gives :$$dC = \... 1answer 90 views ### How to derive the dynamic of the log forward price? I have the following Schwartz model: $$dS_t=a(\mu-\ln S_t)S_tdt+\sigma S_tdW_t$$ $$X_t=\ln S_t$$ $$dX_t=a(\hat{\mu}-X_t)dt+\sigma dW_t$$ with$\hat{\mu}=\mu-\frac{\sigma^2}{2a}\sigma\$ $$F_t(T)= \exp\... 1answer 218 views ### Derivation using Ito's Lemma of price process Define q(t) as the log price minus a linear trend$$ q(t) = \ln P(t) - \mu t $$Assume the log price process = Equation 1:$$ dq(t) = - \Theta q(t) dt + \sigma dW(t) $$Can you show that the ... 1answer 66 views ### Show that Z(t)/Z(0) is a positive mean-1 martingale We look at a standard no dividends Black-Scholes model and here we have a process Z, which is defined by: Z(t)=(S(t)/H)^p , where H is a positive constant and p=1-2r/sigma^2 I am now asked to show ... 0answers 54 views ### Ito Diffusion with Change of Measure Let (X_t) be an Ito diffusion with speed (V_t), under a probability measure P. Could there exist a change of measure to a probability measure Q, with Q ~ P, under which (X_t) is an Ito diffusion ... 0answers 277 views ### Applying Ito's formula to complex functions Within my lecture notes, the following definition is given: We say that the stochastic process X_t has stochastic differential$$ dX_t = b_t dt + \sigma_t dW_t $$if and only if$$ X_t = ...
Starting with the Bellman equation for the optimal stopping problem: $$F(x,t)=max\{\Omega(x,t), \pi(x,t)+(1+\rho dt)^{-1} E[F(x+dx, t+dt)|x]\}$$ In the continuation region where the second term is the ...