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Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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37 views

Variance Equations is missing definition

here: https://www.nrc.gov/docs/ML1208/ML12088A329.pdf Campbell, Lo, Mackinlay: The Econometrics of Financial Markets on page 159 i am looking at equation 4.4.9 in the last line, = $I\sigma_{\...
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1answer
158 views

How to solve one-touch American call

I want to solve the one-touch American call at $t = 0$ with level $B,$ maturity $T$ under the following assumption: $$d S= rSd t + \sigma SdW,\quad S_0<B.$$ We have following formula: $$V(S_0,0) = \...
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1answer
56 views

Notation clarity on continous proesses [closed]

Can someone clarify differences between $dX_t,\frac{\partial X_t}{\partial t},\int_0^t X_{t'}dt',\int_0^tdX_{t'}$? Does $\int_0^t\frac{\partial X_{t'}}{\partial{t'}}d{t'}=X_t$?
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1answer
149 views

Notion of risk-less portfolio in derivation of Black-Scholes

EDIT: As pointed out by Gordon in the comments, the portfolio I considered in my original post is neither self-financing nor (locally) risk-free. Though the central question is still open. Suppose ...
2
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1answer
216 views

How to price a call option which depends on two Wiener processes?

Could someone explain to me why the regular call pricing formula works, just with $\sigma$ replaced by $\|\sigma\|$ in the case where the underlying asset depends on two Wiener processes? For example,...
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2answers
189 views

Stochastic process and brownian motion

I just read the following and i am having some difficulty to interpret it: We begin our analysis in the standard Black-Scholes world consisting of a bank account process of price denoted by $B_t$, ...
2
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1answer
82 views

Utility-optimal leverage with costs

Say I have a portfolio, $X_t$, using a leverage of $f$, such that the dynamics are given by \begin{equation} dX_t = \mu f X_t dt + \sigma f X_t dW_t \end{equation} I want to optimize the expected ...
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1answer
874 views

Geometric Brownian Motion: percentage returns vs log-returns

In classical calculus, we know that the limit of percentage return (ie $dS/S$) equals that of the log return (ie. $dln(S)$ ). With uncertainty, we rely on Ito Lemma to draw a relationship between the ...
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1k views

Jamshidian's trick for Swaptions

Following Brigo$^1$ p.77, we can decompose the price of a swaption as a sum of Zero-Coupon bond options (Jamshidian's Trick). To do so, the authors suggest to find $r^*$ the value of the spot rate at ...
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2answers
336 views

How to find the mean and variance of this stochastic process?

$ I_t = \int_0^t e^{i W_s} dWs $ where $W_s$ is the standard brownian motion and $i$ is the complex number. Any help will be appreciated!
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1answer
669 views

How to solve this PDE using Feynman-Kac?

I have the following problem right now: solve $$F_t(t,x) + rxF_x(t,x) + \frac{\sigma^2}{2}F_{xx}(t,x) = rF(t,x), \\ F(T,x) = (x - K)^2.$$ How do I solve this? There exists a theorem to solve this, ...
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1answer
54 views

Extreme cases of normal random numbers and NaN

While trying to implement my version of Euler's method for simulating a SDE in C++, I came up with a problem. It occurs in some cases that the path generated by my method ends up giving values which ...
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0answers
262 views

Different definitions of arbitrage

Consider the following setup: Let $S=\left(S_1,\ldots,S_n\right)$ be a $n$-dimensional price process and denote by $V$ its value process defined by $V_t=\phi_t\dot\ S_t$ for $t=0,\ldots,T$. In "...
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1answer
2k views

Given $S$ is a Geometric Brownian Motion, how to show that $S^n$ is also a Geometric Brownian Motion?

Suppose that a stock price $S$ follows Geometric Brownian Motion with expected return $\mu$ and volatility $\sigma:$ $$dS = \mu S dt +\sigma S dz$$ How to find out the process followed by variable $...
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2answers
215 views

Why is $Y(t)V^h(t)$ a martingale?

Let $\lambda$ be the market price of risk: $\frac{a - r}{\sigma}$, and define $Y(t) = e^{-\lambda W(t) - (r + \frac{\lambda^2}{2})t}$. Let $V^h(t)$ be the value process of any self-financing portfolio....
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1answer
374 views

Why do we have zero drift when switching to a martingale measure?

I am told that this is a consequence of the Girsanov theorem, yet I do not see how it it is. Consider some standard model with $dS_i = \mu S_i dt + \sigma S_i dW^P$. Let $Q$ be an equivalent ...
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1answer
1k views

CIR Process from Ornstein–Uhlenbeck process

The wikipedia entry on the CIR Model states that "this process can be defined as a sum of squared Ornstein–Uhlenbeck process" but provides no derivation or reference. Can any one do that? I could only ...
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1answer
2k views

Can I always use quadratic variation to calculate variance?

Suppose we have a Brownian Motion $BM(\mu,\sigma)$ defined as $X_t=X_0 + \mu ds + \sigma dW_t$ The quadratic variation of $X_t$ can be calculated as $dX_t dX_t = \sigma^2 dW_tdW_t = \sigma^2 dt$ ...
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2answers
254 views

Why won't Bjork ever show that the integrability condition is satisfied?

A major technique employed throughout Bjork's "Arbitrage theory in Continuous Time" is that when taking the expectation of a stochastic integral, the result is 0. This is based on a result presented ...
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0answers
173 views

How to understand the integral in the Girsanov theorem?

Let $W^P$ be a $d$-dimesional $P$-wiener procss. Define $L_t = > e^{\int_0^t \phi_s^T dW_s^P - \frac{1}{2} \int_0^t \| \phi_s\|^2 > ds}$.Assuming $E^PL_T = 1$, then the measure given by $dQ = ...
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1answer
201 views

How to calibrate an SDE's by finite difference equation?

I would like a general framework for the calibration of the unknown parameters in an arbitrary stochastic differential equation. I have a proposed method that seems reasonable in theory, but is ...
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0answers
79 views

Transformation of coupled forward-backward stochastic differential equations in 3 dimensions with Ito formula

Maybe this is the right place for my question: I have a system of coupled FBSDEs in 3 dimensions as follows (in cartesian coordinates): $$ \mathrm{d}\vec{r}(t) = \vec{u}(\vec{r}(t))\mathrm{d}t + \...
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1answer
241 views

How to show that $E\left[ \int_0^t \sigma(s) e^{iuX(s)} dW(s)\right] = 0$?

Let $\sigma(t)$ be a given deterministic function of time and define the process $X_t$ by $$X(t) = \int_0^t \sigma(s)dW(s)$$ I want to show $$E\left[ \int_0^t \sigma(s) e^{iuX(s)} dW(s)\right] = 0$$...
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2answers
163 views

Moment Ito's Process Proof

I have a following stochastic integral - related problem that I have difficulty to solve: Given \begin{equation} dX_t = -\alpha X_tdt+\sigma\sqrt{X_t}dW_t \end{equation} and the second moment of $...
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1answer
384 views

Step by Step Guide to Learn Quantitative Finance [closed]

Can some one help in creating step by step guide to learn Quantitative Finance? The suggestions should be in the lines of 1- Which Maths topics needs to be learn 1st 2- Which Maths Books or ...
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1answer
197 views

Why does the partial derivative, $X_t$, of an ABM $X(t)$ not involve standard Brownian motion $Z(t)$, even though $Z(t)$ varies with $t$?

Consider the arithmetic Brownian motion $X(t) = \alpha t + \sigma Z(t)$ and evaluating $dX(t)$ using Ito's lemma. We have $\frac{\partial X}{\partial t} = \alpha$, which does not involve $Z(t)$, even ...
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1answer
400 views

Mean Reverting to its own variance?

Good morning all, When trying to decipher some documentation I have come across this stochastic process which seems to me much like a Ornstein-Uhlenbeck (or Vasicek) process. $$dX_t=-\kappa(X_t-\...
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1answer
256 views

How to define the $f$ function to apply Ito's lemma?

\begin{equation} Z(t) = \exp (a W(t)) \end{equation} I am asked to find $dZ$. I am pretty sure it can be done using Ito's lemma. But in all my textbook (Bjork) examples Ito's lemma is giving from a $...
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1answer
330 views

Is this process of Brownian motion?

Background Information: The process $W = (W_t:t\geq 0)$ is a $\mathbb{P}$-Brownian motion if and only if i) $W_t$ is continuous, and $W_0 = 0$ ii) the value of $W_t$ is distributed, under $\mathbb{...
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2answers
213 views

How to compute the conditional variance of this jump process?

Let $N_t$ be a Poisson process with intensity $\lambda>0$ and $S_t$ follows a pure jump process $$dS_t=S_t(J_t-1)dN_t$$ where $J_t$ is the jump size variable if $N_t$ jumps at time $t$. Also, ...
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1answer
859 views

How is the Wiener integral $\int{WdW}$ calculated?

I want to calculate $\int ^t _0 W_tdW_t$ I know that the reasoning is the following: Let $x(t)=W(t)$ with $a=0$ and $b=1$ in the definition of an Ito Process, and $f(t,x)=x^2$. Then, applying Ito'...
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1answer
228 views

Closed- solution for Convertible bond price two factor model

I am trying to find the closed- solution of convertible bond $V(s,r,t)$ under Vasicek model of two factor model of PDE shown in below link Ito lemma of Convertible Bond under Two-factor Model ...
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2answers
256 views

Ito lemma of Convertible Bond under Two-factor Model Interest Rate

@Behrouz Maleki has provided the PDE of two factor model in other post so could anyone please provide Ito lemma of this equation and how this PDE was derived from Vasicek model. as far as I know it ...
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1answer
495 views

Application of Ito's Lemma, finding the condition for the martingale

The Vasicek short rate model is $$dr_t=\kappa(\theta-r_t)dt+\sigma dW_t$$ Define the processes $x_t$ and $f(x,t)$ $$x_t=\frac{r_t}{\kappa}(1-e^{-\kappa(T-t)})+\int_0^tr_sds$$ $$f(x,t)=e^{a(T-t)-x_t}$$ ...
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0answers
77 views

Approximating an SDE for Volatility Estimation

Consider the SDE $$ dT(t) = ds(t) + a(s(t) - T(t))dt + \sigma dW(t) $$ where $s(t)$ is a deterministic function that turns out to be the long-term mean (this SDE is used to model daily temperature, so ...
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2answers
300 views

What is the strong solution for this SDE

I want to calculate $E_t[(X_T-K)^+]$ where $$dX_t=\frac{3}{X_t}dt+2X_t dW_t$$ and $X_0=x$. I don't know how extact the strong solution of this SDE. Indeed I used Ito's lemma but it was not usefule. ...
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1answer
388 views

Chain rule for Ito's Lemma

The CIR short rate model is $$dr_t=k(\theta-r_t)dt+\sigma\sqrt{r_t}dW_t$$ under the risk-neutral measure. The bond price is of the form $$P(t,T)=A(t,T)e^{-B(t,T)r_t}$$ where the continuously ...
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1answer
180 views

Analytic ZCB call option under Vasicek

The call option with strike $X$ and maturity $T$ on a ZCB maturing at time $S$, where $T\le S$, is $$ZBO(t,T,S,X)=E_t[e^{-\int_t^Tr_sds}(P(T,S)-X)^+]$$ The ZCB price is denoted by $$P(t,T)=E_t[e^{-\...
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3answers
3k views

Variance of time integral of squared Brownian motion

I want to calculate the variance of $$I = \int_0^t W_s^2 ds$$ I was thinking I could define the function $f(t,W_t) = tW_t^2$ and then apply Ito's lemma so I get $$f(t,W_t)-f(0,0) = \int_0^t \frac{\...
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0answers
341 views

Stochastic Leibniz rule

We have the following single-factor HJM model $$d_tf(t,T)=\sigma(t,T)dW_t+\alpha(t,T)dt$$ $$f(t,T)=f(0,T)+\int_0^t\sigma(s,T)dW_s+\int_0^t\alpha(s,T)ds$$ The discounted T bond is then \begin{align} Z(...
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1answer
128 views

Simple HJM model, differentiating the bond price

We have the following simple HJM model $$f(t,T)=f(0,T)+\int_0^t\alpha(s,T)ds+\sigma W_t$$ $$r_t=f(0,t)+\int_0^t\alpha(s,t)ds+\sigma W_t$$ $$P(t,T)=\exp-\bigg(\int_t^Tf(0,u)du+\int_0^t\int_t^T\alpha(s,...
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1answer
160 views

Variance of the Cox-Ingersoll-Ross short rate

Shreve II page 151, the Cox-Ingersoll-Ross model is defined as $$dr_t=(\alpha-\beta r_t)dt+\sigma\sqrt{r_t}dW_t$$ By applying Ito's Lemma, we obtain \begin{align} r_t&=r_0e^{-\beta t}+\frac{\alpha}...
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1answer
471 views

Baxter & Rennie HJM: differentiating Ito integral

From Baxter and Rennie, page 138: $$f(t,T)=\sigma W_t+f(0,T)+\int_0^t\alpha(s,T)ds$$ $$Z_t=\exp-\bigg(\sigma(T-t)W_t+\sigma\int_0^tW_sds+\int_0^Tf(0,u)du+\int_0^t\int_s^T\alpha(s,u)ds\bigg)$$ $$dZ_t=...
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1answer
597 views

Ito's Lemma, differentiating an integral with Brownian motion

In How were these SDE derived? I don't understand one part of Gordon's answer, specifically: $$\ln S_t=\ln F_{0,t}-\frac{\sigma^2}{4\lambda}(1-e^{-2\lambda t})+\sigma e^{-\lambda t}\int_0^t e^{\...
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1answer
279 views

Differential of integral of a stochastic process

Let $Y_{t}$ be \begin{equation} Y_{t}=\int_{\Omega} g(X_{u}) du \end{equation} where $g(.)$ is a deterministic function and $\Omega=[t_{0},t]$ continuos partition of $\mathbb{R}$. Furthermore let $...
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0answers
53 views

Using malliavin derivative to find the worst Delta-positive hedge?

Background: I've heard that Malliavin Calculus can be used to show the explicit form of a delta-neutral hedge (given an SDE driven market model). For example, here is a sketch here on page 21 on how ...
1
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1answer
170 views

How do I find this Expectation?

I have an expectation given as: $\mathbb{E}\left(S_{T}\mathbb{1}_{S_{T}\geq K} \right)$ where $K$ is just an arbitrary number (i.e. the strike price, but that's unimportant) and $S$ can be modelled ...
0
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1answer
142 views

Integral with respect of $(dW_s)^n$

I know $$\int _0^t dW_s=W_t-W_0=W_t$$ Since $ dW_s dW_s=ds$ , so $$\int _0^t( dW_s)^2=\int_0^t ds=t-0=t$$ I Want to know why for $n\ge 3$ we have $$\int _0^t (dW_s)^n=0$$ My try $$(dW_s)^2 dW_s (...
9
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4answers
851 views

Free or Relatively Less Pricey Quant Finance courses online

I am trying to figure out what all online Quant Finance courses are out there which are free or relatively less pricey? CQF is not less pricey Financial Engineering course on Coursera - Not so great ...
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1answer
159 views

Discrete Time to Continuous Time and Summation of Two Geometric Brownian Motions

Could someone please suggest with detailed steps and/or a reference, 1) How to convert the below discrete time summation to continuous time form and write it as an integral? 2) Any methods to ...