Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
790
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Compare errors in estimating a probability
Let $X_t$ be a geometric Brownian motion: $dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t$ with $W_t$ a standard Brownian motion.
Given the intervals $[t_{j-1}, t_{j}]$ for $j\in {1,...,U,...,N}$, let $M_j$ ...
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2
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Derivation of static replication formula
I know that a way of computing the price of a derivative paying $S^2$ at time $T$ is by making use of the following strategy:
$V=\int_{0}^{\infty} s^2 \frac{\partial^2 C}{\partial K^2}(K=s)ds$
Where $\...
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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 ...
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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 ...
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HJM drift condition problem: Show that the HJM drift condition implies $b(t) \equiv b, \rho^{2}(t) \equiv a$
I need your help with understanding and solving the HJM framework. I am hoping I can get some help as I feel so lost with HJM and learning online because of the pandemic is adding more stress. Anyway ...
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How to compute $E[W(T)\exp(W(T)]$
I have got this interview question twice. Does anyone know from which interview question book or another source this question comes from? It may be some well known source as two different interviewers ...
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2
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Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?
Let $S_{t}$ denote the price of stock, $\beta_{t}$ denote the savings account. For each model below state with reason whether it admits arbitrage and whether it is complete.
(a) $\beta_{t}=e^{t}, S_{t}...
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2
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Heston: Variance of Integrated Variance
Consider the standard Heston model\begin{align*}
dX&=\left(r-\frac{1}{2}v\right)dt+\sqrt{v}dB,\\
dv&=\kappa(\theta-v)dt+\xi\sqrt{v}dW, \\
dBdW&=\rho dt.
\end{align*}
Computing $\mathbb{E}\...
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Derivation of Bergomi model
In Stochastic Volatility Modeling, L. Bergomi introduces in Chapter 7 the pricing equation (7.4) :
$$
\frac{dP}{dt}+(r-q)S\frac{dP}{dS}+\frac{\xi^t}{2}S^2\frac{d^2P}{dS^2}+\frac{1}{2}\int_t^Tdu\int_t^...
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Question on Ito's lemma involving $\mathrm{d}W(t)$
I am new to Ito-calculus, so please forgive me if the question is stupid.
Let $W(t)$ be a Brownian-Motion and $f(W(t))=W(t)^2$. If I want to calculate the differential $\mathrm{d}f(W(t))$, Ito's lemma ...
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Finding Option Probability Density Using Local Volatility from Dupire Model
This question is different than pricing using dupire local volatility model and Is Dupire's local volatility model path independent to recover historical option price?
I also asked this on Math ...
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Malliavin calculus on two different probability spaces
I'm studying Malliavin Calculus recently. I have two different text books, one is the lecture note written by Oksendal, and the other is a book (Introduction to Malliavin Calculus) by Nualart.
In this ...
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Equivalent local martingale measure vs. equvalent martingale measure in a Brownian setup
Assume you have the standard financial market built up of a Brownian motion. I have seen some books say that an equivalent local martingale measure imples no arbitrage, and some say that an equivalent ...
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In what cases characteristic function of (log-)price process is known?
Hey I know that we can use characteristic function of log-price process to price different options. But when we know the characteristic function? I know that we can take Levy processes and constant ...
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2
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Showing that the shortfall-to-quantile ratio of a normal distribution goes to one
I dont get why $$\lim_{x \to \infty}
\frac{\mu \{1 - \Phi(x)\} + \sigma \phi(x)}{(\mu + \sigma x) \{1 - \Phi(x)\} }
= \lim_{x \to \infty}
\frac{1}{1 - \sigma \frac{1 - \Phi(x)}{(\mu + \...
- 5,043
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0
answers
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Help in Bernoulli's differential equation
I want to solve the following Bernoulli differential equation:
$$A'(t)=A^2(t)[-2\sigma +1]-2aA(t)$$
where $\sigma$ and $a$ are real numbers.
Until now I have divided both sides of the equation with $A^...
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FX Asian Option Moment-matching in Harmonic case
I need to price a "foreign-paying" fixed-strike Asian (i.e., average) option. Thus, the payoff is:
$$\left(\frac{A_T - K}{A_T}\right)^{+} = \left(1 - \frac{K}{A_T}\right)^{+} = K \left(\frac{...
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2
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Sampling change in the driving brownian motion of a CIR process
I have volatility driven by a CIR process:
$$\mathrm{d}v_t = \kappa (\bar{v}-v_t)\mathrm{d}t + \omega \sqrt{v_t}\mathrm{d}W_v\text{.}\tag{1}$$
I am working with several (complicated) approximations of ...
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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}$-...
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226
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Equivalence of Call Option on $S_T$ and Put Option on $\frac{1}{S_T}$ in FX Markets
Part 1: I am trying to price an option in the FX world. It naturally pays in the domestic currency, but in this case the payout currency must be the foreign currency. For example, consider the payoff:
...
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220
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Price difference digital option : constant vol vs local vol
I got the following interview question:
Consider a digital option, it will be priced by using two approaches: 1)constant volatility; 2)local volatility. At the strike, both volatilities are equal. (...
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387
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On a time integral of Brownian motion up to the hitting time
Just come up with a 'simple' and interesting problem that I've been struggling to deal with for some time. Consider a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,T]},\...
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1
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209
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Stochastic optimization and mean field games : textbooks
Which textbooks and online courses would you recommend to learn :
stochastic optimization
mean field games
applied to quantitative finance.
My goal would be to read research articles like the ones ...
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341
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Brownian Bridge general case
The SDE for the Brownian bridge is the following:
$dY_t=\frac{b-Y(t)}{1-t}dt+dW(t)$
with solution:
$Y(t)=Y(0)(1-t)+bt+(1-t)\int_0^t \dfrac{dW(s)}{1-s}$
Can someone help me on proving that $$\lim_{t\...
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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 $W_t$ ...
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Is first order stochastic dominance conserved under change of measure?
As the title states, my question is whether first order stochastic dominance is conserved under change of measure, for instance from the $\mathbb{P}$ measure to $\mathbb{Q}$ measure and change of ...
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611
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Bergomi Volatility Model
I was studying on the Bergomi volatility model(using forward variance represented as $\xi_{t}^{T}$).However I don't understand how the author passes from the sde to the first step by only integrating ...
- 5,672
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193
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Deriving Law of Motion by Ito's Lemma
I've been trying to derive the law of motion for the stochastic process above using Ito's Lemma, given Geometric Brownian Motion with it's law of motion shown below:
I've managed to take the partial ...
- 466
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429
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Weak solution of a SDE
$\text { Consider the } \operatorname{SDE} d X_{t}=\operatorname{sign}\left(X_{t}\right) d t+d B_{t} \text { on } 0 \leq t \leq T, \text { where } \operatorname{sign}(x)=1\\
\text { for } x>0 \text ...
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Solving SDE Dubins-Schwarz Theorem
$\text{ Let } X_{t}=1+t+B_{t}, \text { and } T=\inf \left\{t: X_{t}=0\right\} . \text { Define } G(t)=\int_{0}^{t \wedge T} \frac{d s}{X_{s}}. $
$\text { Let }\
\tau_{t}=G^{-1}(t) \text { be the ...
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Conditional expectation of integral of brownian motion
I am trying to calculate
$$\mathbb{E}\biggl[\biggl(\int_s^t W_u du\biggl)^2 \biggl|W_s=x, W_t=y\biggl] $$
where $W$ is a Standard Brownian Motion and $s\leq u \leq t$.
Any help or tips would be ...
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Changing order of integration on stochastic term in Vasicek [closed]
This question is in relation to the vasicek model, where i am trying to find the solution.
I have this term: $-\int_{t}^{T} \sigma \int_{t}^{s} e^{-\kappa(s-u)} d W(u) d s$
I need to change the ...
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Is it a problem that there are so few stocks in the generalized Black Scholes market? [duplicate]
In the standard Black Scholes market there is only one stock. In the generealized market there can be a finite amount, but my impression is that there are few stocks in the market. The real world ...
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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 ...
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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.
...
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Calculating futures price
Consider a world as follows:
$$\frac{dB}{B} = r_tdt$$
$$\frac{dS}{S} = r_tdt - 0.05dW_1 + 0.5dW_2$$
$$dr_t = 0.2 dW_1$$
where $r_0=0$. The Wiener processes $W_1$ and $W_2$ are independent. The price ...
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Integral of Brownian motion w.r.t. time and integral not starting at zero
I'm new to stochastic calculus and try to calculate (1) mean and (2) variance of
$$\int_s^t W_u du$$
where $W_u$ is a Brownian motion. I already found this helpful answer, where it was shown that $\...
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1
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Girsanov transform when drift coefficient is a function of the stock price
I'm working my way through an elementary stochastic calculus textbook. I'm having trouble with one of the questions:
Bachelier type stock price dynamics. Let the SDE for stock price $S$ be given by $...
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128
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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
$$\mathrm{d}X(t) = \sigma(t)\mathrm{d}W(t)+\alpha(t)\...
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Hermite polynomials as martingales [closed]
Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite ...
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Mutual variation of Brownian motions
Let $\{W^1\}_{t\geq0}$ and $\{W^2\}_{t\geq0}$ be two Brownian motions with correlation coefficient $\rho \in [0, 1]$, i.e., $\mathbb{E}[(W^1(t)-W^1(s))(W^2(t)-W^2(s))]=\rho(t-s)$ for all $t,s \geq 0$. ...
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Ito calculus is Gaussian (using method of characteristic function)
Let $h$ be a deterministic function and define $X_{t}=\int_{0}^{t} h(s) d W_{s} .$ Show that
$$\mathbb{E} \exp \left(i u X_{t}\right)=\exp \left(-\frac{u^{2}}{2} \int_{0}^{t} h^{2}(s) d s\right),$$ ...
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What is the difference between "stochastic" heat equation and just heat equation?
I am trying to understand the difference between the "stochastic" heat equation and the heat equation. Will i be wrong to say the stochastic heat equation is just the heat equationg with the ...
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3
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1
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288
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Help on solving a stochastic differential equation
I am trying to solve the following SDE
$$dX(t)=rdt+aX(t)dW(t),\ t>0$$
$$X(0)=x$$
where W() is a Wiener process and r,a and x real numbers. I have proceeded by using the integrating factor
$$F(t)=...
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2
votes
2
answers
703
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Proving that a stochastic process is a martingale using Ito's Lemma
Assume a Wiener process W and a bounded F-adjusted stochastic process a. Show that the following process is a martingale on F
$$X(t)=(\int_{0}^{t}a(s)dW(s))^{2}-\int_{0}^{t}a^{2}(s)ds,\ t\geq0$$
Can ...
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0
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Query on Lebesgue Measure
I am reading Steven E. Shreve's book, titled "Stochastic Calculus for Finance II". I have a query w.r.t. an example given in the book which is as follows:-
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2
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4k
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Heston stochastic volatility, Girsanov theorem
How can we apply Girsanov's theorem to a stochastic volatility model?
In Heston's model the dynamics are given by
\begin{align*}
dS_t &= \mu S_t dt + \sqrt{v_t}S_t d\widehat{W}^\mathbb{P}_{1,t}, ...
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votes
1
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421
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Simplifying the expectation of the product of two stochastic integrals
Let $f(t, \omega), g(t, \omega)$ be functions that are independent of the increments of the Brownian motion $w(t, \omega)$ in the future. That is, $f(t, \omega), g(t, \omega)$ are independent of $w(t +...
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Asymptotic behavior property of geometric Brownian Motion proof
Online I found the asymptotic behavior property of geometric Brownian Motion $X_t$as:
If $\mu$ (drift parameter) is $\ge$ $\sigma^2/2$ where $\sigma$ is the volatility parameter, then $X_t \rightarrow ...
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1
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Derivative of Stochastic Integral
I am trying to take the derivative of the following stochastic integral,
$$d\left(\int g(S_t) dS_t \right),$$
where $dS(t) = \sigma S(t) dW_t$ and $g(.)$ is some (smooth) deterministic function. My ...
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