Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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2
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0answers
40 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+\...
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Understanding of numeraire of the stock price [closed]

I have derived the stock price from the Stochastic Differential Equation of S_t (stock price under stochastic process)use using Ito's formula. Given the SDE ===> 𝑑𝑆𝑑=π‘Ÿπ‘†π‘‘π‘‘π‘‘+πžΌπ‘†π‘‘π‘‘π‘Šπ‘‘ we ...
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38 views

Understanding the expectation of Stock price with martingale pricing theorem [closed]

how should understand the expectation of stock price under VT = (ST βˆ’ K)+, trying to follow martingale pricing theorem, get V0/B0, but still have trouble to get connect with SDE of d(1/St)=-r(1/St)dt -...
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116 views

Deriving Bachelier Greeks

I am working on the Bachelier Model with r not equal to 0 as described in the first and most upvoted answer in following link: Bachelier model call option pricing formula This is fairly easy to code ...
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87 views

How to integrate Itô integral w.r.t time?

Let $W_t$ be a Brownian motion. How to calculate the following integral $$ I:=\int_0^t\left( \int_0^u(u-s)dW_s\right) du? $$ My attempt so far is: First note that $$ \int_0 ^u (u-s)dW_s = \int_0^u ...
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1answer
129 views

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 ...
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1answer
129 views

Is it fair in an introductory stochastic calculus/derivatives pricing class to ask for the price when absence of arbitrage is violated? [closed]

Re close votes: I believe this is a fair kind of opinion-based question because it's like those ethics questions in academia se or workplace se or because it's pedagogical. Context: I'm actually ...
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1answer
87 views

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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2answers
127 views

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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65 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 ...
6
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1answer
216 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 ...
2
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1answer
64 views

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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0answers
46 views

Vasicek Model Expected Short Rate

Please note I am new to quantitative finance and more so to stochastic calculus. I have what should be a relatively simple problem using the Vasicek model for estimating future parameters of the short ...
11
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5answers
485 views

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 ...
10
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2answers
672 views

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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2answers
336 views

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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0answers
131 views

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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1answer
121 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 ...
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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 ...
2
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1answer
64 views

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 ...
2
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1answer
88 views

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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1answer
66 views

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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86 views

Quantitative interview questions - fixed income [duplicate]

I will pass a quant interview (fixed income). The interviewer said the questions do not have a right answer, they are not math exercises. He said the questions are like their job,the objective is to ...
2
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1answer
43 views

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 + \...
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0answers
43 views

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^...
2
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1answer
80 views

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{...
2
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2answers
126 views

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 ...
3
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1answer
178 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 $\mathcal{F_t}$-...
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62 views

How to Discretize this SDE found in finance? (cross-posted)

Continuous-Time In continuous-time form, the "Heston model" is written as $$ dS_t = \mu S_t dt + \sqrt{\nu_t} S_t dW_t^S \\ d\nu_t = \kappa (\theta - \nu_t) dt + \xi \sqrt{v_t} dW_t^{\nu} $$ ...
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0answers
93 views

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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0answers
75 views

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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127 views

Why were Laguerre polynomials a good choice of basis functions for American Monte Carlo?

I am implementing LSMC to price American options based on a custom model. I now need to make a choice of basis functions, so I am looking for the theoretical justification for using Laguerre ...
3
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1answer
157 views

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\...
7
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0answers
212 views

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]},\...
3
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0answers
98 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 $W_t$ ...
5
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1answer
264 views

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 ...
3
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1answer
181 views

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 ...
1
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1answer
104 views

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 ...
4
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1answer
97 views

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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0answers
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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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0answers
72 views

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 ...
4
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1answer
199 views

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 ...
3
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0answers
62 views

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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1answer
163 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 ...
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2answers
234 views

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 ...
2
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1answer
112 views

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 ...
2
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1answer
102 views

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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0answers
44 views

Equivalence of two definitions of the stochastic integral

The Question I am reading Shreve's Stochastic Calculus for Finance, Volume II. On page 145, definition (4.4.20), he defines an integral with respect to an ItΓ΄ process. Definition 4.4.5. Let $X(t) = X(...
3
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1answer
115 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)\...
2
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1answer
94 views

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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