Questions tagged [itos-lemma]

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Volatility of a stochastic Process given by an SDE

I am currently working on this thesis: http://arks.princeton.edu/ark:/88435/dsp01vd66w212h and i am stuck on page 199. There we have a portfolio $P=\alpha F+\beta G $ with $\alpha +\beta =1$ and ...
Valentin's user avatar
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Delta Gamma Hedging Portfolio of Multiple Options Derivation

I am trying to make the correct derivation of the Delta Gamma Hedge of a portfolio composed of a multi-option strategy, like a Straddle with the following parameters Long 1 Call K = 100, Long 1 Put K =...
Coco Garazzo's user avatar
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Solving the SDE for GBM [closed]

Let's assume that we have the following stochastic differential equation: $dX_t = \mu X_t dt + \sigma X_tdW_t$ and that we have to prove that this is its solution: $X_t = X_0 \exp\left(\left(\mu -{\...
Alessandro's user avatar
2 votes
0 answers
86 views

Ito formula and confusion with the differential operator $d$

Thanks for visiting my question. Im am currently working on this paper (https://arxiv.org/abs/2305.02523) and I am stuck at page 21 (Theorem 14 proof). First these SDE's were defined: \begin{align*} ...
Valentin's user avatar
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What’s the Ito’s lemma of compound Poisson process with two-sided jump and mean-reverting jump size?

In the book Cartea and Jaimungal (Algorithmic and High Frequency Trading, page 249.), the midprice dynamics follows $ dS_t=\sigma dW_t+\epsilon^+ dM^+_t-\epsilon^- dM^-_t$ (10.22) where $M^+_t$ and $M^...
EdisonKIng's user avatar
2 votes
0 answers
137 views

Proof: Deterministic Ito Integral (Thomas Mikosh Chapter 2)

I'm referencing Elementary Stochastic Calculus with Finance in View by Thomas Mikosch between chapters of Shreve's Volume II text. In one section Mikoshch text makes the following claim without proof: ...
James Bender's user avatar
2 votes
1 answer
360 views

Integrated Brownian motion

I occasionally see a post here: Integral of brownian motion wrt. time over [t;T]. This post has the conclusion that $\int_t^T W_s ds = \int_t^T (T-s)dB_s$. However, here is my derivation which is ...
Wang Jing's user avatar
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How to express the process of number of stock (nt) in a portfolio using ito's lemma

We have a regular self-financing portfolio $W_t$: $$dW_t = n_t dS_t + (W_t − n_t S_t) r dt$$ Where $W_t$ is total wealth, $n_t$ is amount of stock, $S_t$ is stock price, $r$ is the risk-free rate. And ...
nearhome's user avatar
4 votes
1 answer
163 views

Deriving an Analytical Expression for Standard Deviation of Log Returns

I am looking to find an expression for the standard deviation log returns of a stock price process. I have a stock price which follows the following dynamics: $dY(t) = Y(t)(r(t)dt + η(t)dW(t))$ Here,...
user67245's user avatar
1 vote
1 answer
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Pricing various classes of derivatives and replicating them

Consider the following three derivative styles and assume zero dividends for simplicity. The "american style", "european style", and "infinite" style: $$L_{A}(S,K,t,T)=f(...
Cris's user avatar
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Volatility of the product of two correlated asset following a log normal distribution [duplicate]

I am trying to solve the problem: Given two assets X and Y that follow a log normal distribution with volatility $\sigma_1$ and $\sigma_2$ respectively and with correlation $\rho$, what is the ...
kakarito's user avatar
1 vote
1 answer
230 views

How calculate expectation and variation of stochastic integral Based on Heston model?

I was calculated Heston volatility model. But I think it is wrong. $dS_t = \mu dt + \sqrt V_t dW_t^s$ $dV_t = k(\theta - V_t)dt + \sigma \sqrt V_t dW_t^v$. $dW^s_t dW^v_t = \rho dt$ take integral to ...
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Is the time derivative of asset returns expressible as an SDE?

Consider the following SDE for $(S_t)_{t\geq 0}$ under $\mathbb{Q}$, \begin{equation} \mathop{dS_t}=S_t\left(r\mathop{dt}+\sigma(t,S_t)\mathop{dW_t}\right), \end{equation} which (in Langevin form) may ...
UNOwen's user avatar
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The solution of SDE after Itô lemma for diffusion process [closed]

Consider the one-dimensional diffusion process $dX_t = \mu_tdt + \sigma_tdB_t$ and function $f : \mathbb{R} \to \mathbb{R}$, which is twice differentiable. Here we have another SDE by using Itô lemma ...
user64779's user avatar
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Conditional Expectation of Integral of Squared Brownian Motion - PDE Approach

I am looking to compute the following using Ito's formula. $$u(t,\beta_t) = \mathbb{E}(\int_t^T\beta_s^2ds|\beta_t)$$ Knowing the properties of brownian motion, it is rather easy to show that the ...
ilikemath3.14's user avatar
1 vote
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99 views

Analytical expression for SDE

I'm trying to find an analytical expression for the following. Suppose $X$ is a geometric Brownian motion, such that: $dX_{t} = \mu X_{t} dt + \sigma X_{t} dW_{t}$. Suppose furthermore, that the ...
John Stevens's user avatar
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489 views

Deriving the stochastic process for a dividend-yielding stock (under Black-Scholes assumptions)

In order to derive the Black-Scholes equation for a stock $S(t)$ yielding dividends at the continuous rate $d$ $$ S(t) = S_0 e^{(\mu - d - \frac{\sigma^2}{2})t + \sigma \sqrt{t} N(0,1)} \text{,} $$ M. ...
Giogre's user avatar
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Integral of brownian motion wrt. time over [t;T]

From the post Integral of Brownian motion w.r.t. time we have an argument for $$\int_0^t W_sds \sim N\left(0,\frac{1}{3}t^3\right).$$ However, how does this generalise for the interval $[t;T]$? I.e. ...
Landscape's user avatar
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Integration of exponential raised with Brownian Motion wrt the Brownian Motion

I have to derive several things for my thesis, however, I have the following expression: $$ \int^{t}_{0} \exp\{\sigma W_{t}\}.dW_{t} $$ Does anyone know what the solution for this is? Kind regards.
cem's user avatar
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Value of trading strategy

A trading strategy is defined as follows: starting capital $v_0 = 5$ and 1 risky asset holdings $\varphi_t = 3W_t^2-3t$ where $W$ is a Wiener process. The problem is to find the probability of the ...
Simplexity's user avatar
1 vote
1 answer
119 views

Show that the solution to a SDE is strong

I have the following SDE \begin{equation} dX_t = - \frac{1}{1+t}X_t dt + \frac{1}{1+t}dB_t \end{equation} that has the solution: \begin{equation} \begin{aligned} X_t = \frac{X_0 + B_t}{1+t} = \frac{...
Alejandro Andrade's user avatar
2 votes
1 answer
459 views

Obtaining the dynamics of the Vasicek model using Itô

Consider the following expression for the short-term interest rate $$r_t=r_0 e^{\beta t}+\frac{b}{\beta}\left(e^{\beta t}-1\right)+\sigma e^{\beta t}\int_0^te^{-\beta s}dW_s \tag{1},$$ which is ...
Mr Frog's user avatar
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1 answer
168 views

Ito's lemma for option pricing with Levy-alpha stable drift

Consider $$dS=\omega\left(\Lambda-S\right)dt+\sigma_S S dW_t,$$ such that such that $W_t$ is a Wiener process, $\sigma_S$ is constant, $\omega: t\rightarrow\mathbb{R}$ represents anticipated drift and ...
UNOwen's user avatar
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1 answer
112 views

Stochastic process as integral over window function

Consider the following stochastic integral of a deterministic function $f(t,s)$ with respect to the Wiener process $W_s$: $$\int_0^\infty f(t,s) d W_s$$ My questions are: Is such an integral ...
broken_urn's user avatar
4 votes
1 answer
348 views

Difficulty with stochastic calculus problem

I'm currently working through Shreve's Volume II, and I'm having some difficulty on Exercise 5.4 of Chapter 5. The problem statement is: Consider a stock whose price differential is $$ dS(t) = r(t) S(...
Alex Lapanowski's user avatar
2 votes
1 answer
188 views

Ansatz and HJB equation

Suppose we have an HJB equation of the form $$ \frac{\partial v}{\partial t}+\frac{1}{2}\sigma^{2}\frac{\partial^{2}v}{\partial s^{2}}+max_{\delta^{a}}\left\{ \lambda^{a}(\delta^{a})\left[v(t,s,x+s+\...
sle's user avatar
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1 answer
417 views

Why this stochastic integral is calculated with Riemann integral

This picture is from Neftci's textbook, 'An Introduction to the Mathematics of Financial Derivatives, Third Edition' What makes me uncomfortable is equation [10.61] In above picture. In this equation,$...
user13232877's user avatar
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107 views

Performance of dollar cost averaging

If we're investing money into a stock $S$ at a continuous rate, $C$, what is the probability distribution of the amount we have invested? For example, modelling a stock as GBM without contributions, $ ...
Zaz's user avatar
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Clarification on Paul Wilmott's derivation of Ito's Lemma

I'm currently self-studying to be quant and have been thoroughly enjoying PW's book. I have some questions regarding his derivation of Ito's lemma. Specifically, I can see that the first line in his ...
user3613025's user avatar
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401 views

Ito's Lemma in option pricing for a stock satisfying $dS=\frac{P-S}{\omega}dt+SdW_t$

Suppose a stock follows the stochastic differential equation $$dS=\frac{P-S}{\omega}dt+SdW_t,$$ such that $W_t$ is a wiener process, $\omega\in\mathbb{R}^+$, and $P_t,S_t\in\mathbb{R}$. If the value ...
UNOwen's user avatar
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3 votes
1 answer
298 views

Volatility and drift of the instantaneous forward rate under risk neutral measure using the zero coupon bond

I have question about this problem. I believe I have derived $f(t,T)$ correctly using the zero-coupon bond. But I am unsure about how to go forward with the question and how to use the second part. ...
codelearner's user avatar
6 votes
1 answer
257 views

Parametric Stochastic Integral

I need help. Defining the parametric stochastic integral $$ F_t = \int_t^T\xi(t,s)g(s)ds $$ $\\\\$ with $\xi$ a generic stochastic process such that $d\xi(t,s) = \mu(t,s)dt + \sigma(t,s)dW_t$, I'm ...
Deros's user avatar
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6 votes
2 answers
161 views

Covariance of the product of log normal process and normal procces

I tried to compute the following covariance : $$Cov(e^{\int_{t}^{T}W^1_sds},\int_{t}^{t+1}W^2_sds)$$ where $W^1_t$ and $W^2_t$ are Brownian motions such that $dW_t^1dW_t^2=\rho dt $ My idea was to ...
DeepInTheQF's user avatar
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1 answer
155 views

Question on Ito's lemma involving $\mathrm{d}W(t)$

I am new to Ito-calculus, so please forgive me if the question is stupid. Let $W(t)$ be a Brownian-Motion and $f(W(t))=W(t)^2$. If I want to calculate the differential $\mathrm{d}f(W(t))$, Ito's lemma ...
Count's user avatar
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2 votes
1 answer
115 views

Confused by derivation of variance swap payoff

I'm trying to follow https://en.wikipedia.org/wiki/Variance_swap#Pricing_and_valuation where it seems to me that they're just subtracting a simple return: $$ R_t = \frac{\mathrm{d}S_t}{S_t} = \mu \...
robsmith11's user avatar
3 votes
2 answers
741 views

Ito's lemma $f(t,W_t^2)$

Let $f$ be a function of $t$ and $W_t^2$. a)Find a function $f$ such that $f(t,W_t^2)$ is a $F_{t^-}$ martingale, with $F$ the Brownian filtration. b)Use Ito's lemma to show that $f(t,W_t^2)$ is a ...
Mathxx's user avatar
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1 answer
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Ito multiplication

Let $\{N_t|0<t\leqslant T \}$ and $\{M_t|0<t\leqslant T \}$ be two Poisson processes with intensities $\lambda_n, \lambda_m>0$, respectively. Based on the implicit results of Corollaries 1 ...
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Ito's lemma results in negative volatility processes

I struggle with the interpretation of a process I derive from Ito's Lemma. Let's say I have function f(S,t) which is twice differentiable wrt S. I thus can apply Ito's Lemma to get $df(S,t)$. So far ...
ValuePartner's user avatar
3 votes
1 answer
1k views

Ito Lemma for Poisson Process

I'm new to stochastic calculus on jump processes and encountered a difficulty. I would appreciate some clarification from the community on the following question. Let $g_t$ be a $\mathcal{F_t}$-...
finmathstudent's user avatar
2 votes
1 answer
249 views

Solving an SDE using Ito's Lemma

Suppose that $Z(t)=e^{-\int_0^t \theta'(s)dW(s)-\frac{1}{2}\int_0^t ||\theta(s)||^2ds}$ with $\theta()=\sigma^{-1}()[b()-r()]$, $\sigma()>0$ and invertable and $W()$ a Wiener process There is also ...
Martin_Gale's user avatar
3 votes
0 answers
130 views

MGF of Generalised Itô Integral

The following derivation produces a moment closure problem - I would appreciate any insight. It may seem trivial at first glance, but the key aspect is the integrand dependence on $t$. Consider $W_t$ ...
DavidJ's user avatar
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1 vote
2 answers
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Drift Term in Black-Scholes Model Martingale

How would I prove that a Black-Scholes Model is not a Martingale if it has drift. In many cases it is just stated as a fact (without proof). For instance if Im looking at: $$dS_{t} = \mu S_{t} + \...
Sam Loi's user avatar
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Multidimentional Black Scholes Formula

I need to write the Black-Scholes formula for option $V = (S_1, S_2, t)$, where: $$d S_1 = \mu_1 S_1 dt + \sigma_1 S_1 d W_1,$$ $$ d S_2 = \mu_2 S_2 dt + \sigma_2 S_2 d W_2.$$ We know that $W_1$ and $...
Dracks's user avatar
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3 votes
0 answers
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Derivation of option pricing PIDE: Why does the drift need to be zero?

I started studying PIDE methods for option pricing and am struggling to understand or find the necessary theory that shows why the PIDE is obtained by the condition that the drift term has to be zero. ...
Leguan3000's user avatar
1 vote
1 answer
678 views

How is the formula of Quadratic Variation of Brownian Motion derived? [closed]

This is a follow up on this question on quant SE: The question mentions for a Brownian motion : $X_t = X_0 + \int_0^t\mu ds + \int_0^t\sigma dW_t $ , the quadratic variation is calculated as $dX_t ...
Jay Mangal's user avatar
3 votes
2 answers
942 views

Covariance between integral of brownian motion and brownian motion

Let $$ I = \int_0^1W_tdt, $$ where $W_t$ is a Brownian motion. From Integral of Brownian motion w.r.t. time we have that $$ \mathbb{E}[I]=0, $$ by Fubini's theorem. And that $$ \mathbb{V}\text{ar}[I] =...
Oliver's user avatar
  • 33
3 votes
1 answer
129 views

Justification for substituting "Itô differentials"

I'm reading Shreve's Stochastic Calculus for Finance, Volume II. In it, he uses the stochastic differential notation. For example, he may write $$\mathrm{d}X(t) = \sigma(t)\mathrm{d}W(t)+\alpha(t)\...
user54908's user avatar
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1 vote
1 answer
132 views

Hermite polynomials as martingales [closed]

Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite ...
qszbwldxz's user avatar
4 votes
1 answer
353 views

Ito calculus is Gaussian (using method of characteristic function)

Let $h$ be a deterministic function and define $X_{t}=\int_{0}^{t} h(s) d W_{s} .$ Show that $$\mathbb{E} \exp \left(i u X_{t}\right)=\exp \left(-\frac{u^{2}}{2} \int_{0}^{t} h^{2}(s) d s\right),$$ ...
qszbwldxz's user avatar
0 votes
1 answer
138 views

Mutual variation of Brownian motions

Let $\{W^1\}_{t\geq0}$ and $\{W^2\}_{t\geq0}$ be two Brownian motions with correlation coefficient $\rho \in [0, 1]$, i.e., $\mathbb{E}[(W^1(t)-W^1(s))(W^2(t)-W^2(s))]=\rho(t-s)$ for all $t,s \geq 0$. ...
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