Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
683
questions
2
votes
0answers
40 views
Ansatz and HJB equation
Suppose we have an HJB equation of the form
$$
\frac{\partial v}{\partial t}+\frac{1}{2}\sigma^{2}\frac{\partial^{2}v}{\partial s^{2}}+max_{\delta^{a}}\left\{ \lambda^{a}(\delta^{a})\left[v(t,s,x+s+\...
-1
votes
0answers
38 views
Understanding of numeraire of the stock price [closed]
I have derived the stock price from the Stochastic Differential Equation of S_t (stock price under stochastic process)use using Ito's formula.
Given the SDE ===> πππ‘=πππ‘ππ‘+πΌππ‘πππ‘
we ...
-1
votes
0answers
38 views
Understanding the expectation of Stock price with martingale pricing theorem [closed]
how should understand the expectation of stock price under VT = (ST β K)+, trying to follow martingale pricing theorem, get V0/B0, but still have trouble to get connect with SDE of d(1/St)=-r(1/St)dt -...
3
votes
0answers
116 views
Deriving Bachelier Greeks
I am working on the Bachelier Model with r not equal to 0 as described in the first and most upvoted answer in following link:
Bachelier model call option pricing formula
This is fairly easy to code ...
4
votes
0answers
87 views
How to integrate Itô integral w.r.t time?
Let $W_t$ be a Brownian motion.
How to calculate the following integral
$$
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 ...
0
votes
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 ...
-1
votes
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 ...
1
vote
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 $M_j$ ...
3
votes
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 $\...
0
votes
0answers
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
votes
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
votes
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 ...
0
votes
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
votes
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
votes
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}...
7
votes
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}\...
2
votes
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^...
0
votes
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 ...
5
votes
0answers
78 views
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
votes
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
votes
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 ...
0
votes
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 ...
0
votes
0answers
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
votes
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 + \...
1
vote
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
votes
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:
$$\left(\frac{A_T - K}{A_T}\right)^{+} = \left(1 - \frac{K}{A_T}\right)^{+} = K \left(\frac{...
2
votes
2answers
126 views
Sampling change in the driving brownian motion of a CIR process
I have volatility driven by a CIR process:
$$\mathrm{d}v_t = \kappa (\bar{v}-v_t)\mathrm{d}t + \omega \sqrt{v_t}\mathrm{d}W_v\text{.}\tag{1}$$
I am working with several (complicated) approximations of ...
3
votes
1answer
178 views
Ito Lemma for Poisson Process
I'm new to stochastic calculus on jump processes and encountered a difficulty. I would appreciate some clarification from the community on the following question.
Let $g_t$ be a $\mathcal{F_t}$-...
0
votes
0answers
62 views
How to Discretize this SDE found in finance? (cross-posted)
Continuous-Time
In continuous-time form, the "Heston model" is written as
$$
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}
$$
...
1
vote
0answers
93 views
Equivalence of Call Option on $S_T$ and Put Option on $\frac{1}{S_T}$ in FX Markets
Part 1: I am trying to price an option in the FX world. It naturally pays in the domestic currency, but in this case the payout currency must be the foreign currency. For example, consider the payoff:
...
1
vote
0answers
75 views
Price difference digital option : constant vol vs local vol
I got the following interview question:
Consider a digital option, it will be priced by using two approaches: 1)constant volatility; 2)local volatility. At the strike, both volatilities are equal. (...
8
votes
0answers
127 views
Why were Laguerre polynomials a good choice of basis functions for American Monte Carlo?
I am implementing LSMC to price American options based on a custom model. I now need to make a choice of basis functions, so I am looking for the theoretical justification for using Laguerre ...
3
votes
1answer
157 views
Brownian Bridge general case
The SDE for the Brownian bridge is the following:
$dY_t=\frac{b-Y(t)}{1-t}dt+dW(t)$
with solution:
$Y(t)=Y(0)(1-t)+bt+(1-t)\int_0^t \dfrac{dW(s)}{1-s}$
Can someone help me on proving that $$\lim_{t\...
7
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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 ...
2
votes
0answers
77 views
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 ...
1
vote
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
votes
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
votes
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.
...
1
vote
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 ...
-2
votes
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
votes
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
votes
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 $\...
0
votes
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
votes
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
votes
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 $...
