Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
512
questions
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votes
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8 views
Proving an Identity between a pair of correlated Wiener processes
Suppose we have the following subordinated stochastic differential equations:
$dR(t)=\mu dt+\sigma (Y(t))dW_{1}(t)$
$dY(t)=f(Y)dt+g(Y)dW_{2}(t)$,
where $W_i$'s are standard Wiener process such that ...
0
votes
1answer
51 views
Trouble understanding Notation in Stochastic Calculus (wedge symbol ∧)
I am a beginner in Stochastic Calculus. I am having trouble understanding the meaning behind a specific notation which appears in the topic of Ito process which in differential notation can be written ...
0
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1answer
38 views
Compute dZ(t) : Ito's formula/lemma
We need to find dZ(t). I know I have to use Ito's formula. But I am confused because in the Ito's formula we have f(y,t) is a twice differentiable function with two variables
But here Z(t) = 1/(2+x(t)...
0
votes
0answers
49 views
On Geometric Brownian motion and Itô's formula
Let $S_t$ be a geometric brownian motion such as
$$d S(t) = rS(t)dt +\sigma S(t)dW(t),$$
where $W$ is a standard Brownian motion.
With Itô's lemma and formulas $(dt)^2=dtdW_t=dW_tdt=0$ and $(dW_t)^2=...
3
votes
1answer
73 views
question about spot/vol correlation
In this paper The Interplay between Stochastic Volatility and Correlations in Equity Autocallables by Alvise De Col, Patrick Kuppinger (2017) https://papers.ssrn.com/sol3/papers.cfm?abstract_id=...
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19 views
Deriving coupling equation(s) for Heston Stochastic Volatility Model
In Bergomi Smile Dynamics (2003) Section 2.1 we are given the following coupled equations for the mean and for the variance of the hedger's portfolio:
$
\begin{align*}
\frac{dm}{dt} + \mathcal{L}m - ...
4
votes
1answer
79 views
Advantages of pathwise calculus over stochastic calculus in continuous self-financing trading models
I am new to stochastic calculus but the statement below confuses me:
Beside the issue of the impossible consensus on a probability measure,
the representation of the gain from trading lacks a ...
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0answers
22 views
Order of expectation versus expectation of order (error terms in Taylor expansion)
Given a payoff function $F(X)$ of a random variable $X$, and a Taylor expansion of $F(X)$ around $X=a$, then the expecation of $F(X)$ can be written as
$$
E[F(X)] = F(a) + E[ O((X-a))]
$$
Under what ...
1
vote
1answer
60 views
Why are quadratic variation and rough paths so important in quantitative finance?
I am new to quant finance - come from a mathematics background. I am starting stochastic calculus and have been particularly interested in some papers pathwise integration and rough calculus in ...
1
vote
1answer
61 views
Expectations in Infinite Probability Spaces with Sub Sigma-Algebras [closed]
Let $X$ be an (integrable) random variable on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Suppose $\mathcal{G}$ is a sub-$\sigma$-algebra of $\mathcal{F}$ and let $Z=\mathbb{E}(X|\mathcal{...
1
vote
1answer
51 views
Finding Differential and Quadratic Variation Squared Process
A question based from Springer's Stochastic Calculus for Finance II book - I've tried working this out, but keep ending up in circles.
Let $S(t)$ be given by the usual formula for an asset price ...
0
votes
0answers
46 views
Convert drift and diffusion term in terms of time in the Geometric Brownian Motion framework
Assume that we have daily prices covering the period of 10 years. For calibrating the drift and diffusion parameters of the GBM model
$$S_{t+1} = S_{t}e^{[(\mu-\sigma^2/2)]\Delta t + \sigma \sqrt{\...
2
votes
0answers
55 views
Is it meaningful to look at $\int f(W_t, t) \,dt$?
CONTEXT (can skip):
My textbook looks at two things -
1) Ito integrals for deterministic functions—i.e. $\int f(t) \,dW_t$. We are able to say that they are normally distributed, with a mean of 0 ...
3
votes
1answer
134 views
Solving Stochastic Differential Equation for Geometric Brownian Motion with time-dependent drift
Given the stochastic differential equation:
$$dZ_t = -Z_t \theta_t dB_t, \quad Z_0 = 1.$$
for an adapted process $\theta_t$ and Brownian motion $B_t$, how exactly do I apply Itô's Lemma to obtain:
...
1
vote
1answer
103 views
Going from $\mathcal{P}$ to $\mathcal{Q}$
Under $\mathcal{P}$, we have the Heston Model given by:
$$
d S_{t}=\mu S_{t} d t+\sqrt{\nu_{t}} S_{t} d W_{t}^{S},\\
d \nu_{t}=\kappa\left(\theta-\nu_{t}\right) d t+\xi \sqrt{\nu_{t}} d W_{t}^{\nu}.
$...
3
votes
2answers
135 views
Partial derivative of Ito integral without product rule
I'm thinking about the problem of deriving the stochastic differential of an integral with both time and state part of the integrand but not in a way that you can easily factor it out - for example I ...
3
votes
1answer
105 views
Why sub-replication is not studied in literature
There are numerous paper about super-hedging and super-replication in an incomplete market where the risk neutral measures are not unique. The most fundamental result is that the super-replication ...
1
vote
0answers
33 views
Sub replication of contingent claims
I am trying to prove subreplication cost is the infimum of risk neutral expectation of the contingent claim. I wonder if the following equality holds, which is the key to the proof.
$$
\lim_n \inf_{Q\...
3
votes
1answer
113 views
Process with negative quadratic variation
Today seems to be question day for me, sorry.
The complex process
$$
dX = i\sigma dW
$$
where $i = \sqrt{-1}$ and $dW$ is a standard (real-valued) Brownian motion will have a negative variance ...
2
votes
1answer
156 views
Steven Shreve: Stochastic Calculus and Finance
The lecture notes have the following theorem:
Let $\theta\in \mathbb{R}$ be given and $B(t)$ stands for the Brownian motion which is a martingale, then $Z(t)=exp\{-\theta B(t)-\dfrac{1}{2}\theta^2t\}$...
1
vote
1answer
92 views
What is the SDE of this equation? [closed]
I am new and struggling to understand how to solve this using Ito lemma.
Can someone please explain it to me:
$$dS_t=-\frac{1}{2}\sigma^2 S_t dW_t$$
what is the solution with explanation please
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1answer
47 views
What is Variance of delta of brownian motion [closed]
I am new to this.
If variance of Brownian motion b is t, what is the variance of db?
db is delta of b
2
votes
1answer
80 views
Covariance of logarithms of geometric Brownian motion
Suppose I have a Geometric Brownian Motion process,
$$dX_t=\mu X_t dt + \sigma X_t dW_t$$
I'd like to find the covariance of $\log(X_t)$ and $\log(X_s)$ where $s<t$. We can write $\log(X_t)$ in ...
11
votes
2answers
280 views
Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$
Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation where $a$, $b$, and $c$ are positive constants, so I tried to solve it but I got stuck in ...
1
vote
1answer
89 views
Construction of Butterfly Spread as sum of Call Options
I have rigorously stated my problem here.
The task at hand is to express a butterfly spread [no transaction fees] as a sum of long and short call options.
I have found the solution on Wikipedia:
$$\...
5
votes
2answers
339 views
Can strike prices of options be negative?
I am trying to understand the stochastic model of a financial market in one period by [Föllmer, Schied]. They introduce call and put options for the primary assets, which are non-negative. They do not ...
4
votes
2answers
92 views
Volatility of Exchange Option
I got a question and its partial solution, and have some doubts about the volatility of its geometric Brownian motion process:
Question:
How would you price an exchange call option that pays $max(S_{...
4
votes
1answer
111 views
How to determine components of Affine Term Structure for an Ohrnstein-Uhlenbeck process?
I wonder how I can determine the components $A(t,T)$ and $B(t,T)$ for the zero-coupon bond price process $p(t,T)=e^{A(t,T)-r(t)B(t,T)}$? The components are defined in the following link: https://en....
3
votes
1answer
148 views
Mark Joshi uses forward price to price an option that pays $S_t^2-K$ if $S_t^2>K $ and zero otherwise? Why can we do that?
The following question is taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, Exercise $6.6$
Suppose a stock follows geometric Brownian motion in a Black-Scholes ...
1
vote
1answer
76 views
Show that $Ae^{rt}$ is a solution of the Black-Scholes equation. Why should this be so?
The following is taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, exercise $5.6$.
Question: Show that $Ae^{rt}$ is a solution of the Black-Scholes equation. ...
1
vote
0answers
89 views
How to determine exchange rate dynamics in currency derivatives
I need some guidance regarding exchange rate dynamics in currency derivatives.
Following three dynamics are defined below,
$\frac{dS(t)}{S(t)}=\alpha dt+\sigma dW(t)$ ; the stock dynamics in the ...
6
votes
1answer
166 views
Periodic functions when determining No Arbitrage price
Is it possible to value a T-claim which has a periodic component? For example a claim such as $X = cos(S(T))$.
We assume here that $S(T)$ is the stock price derived from the dynamics $dS(t)=rS(t)dt+\...
2
votes
2answers
79 views
Question regarding No Arbitrage price of a call option
I have a question regarding how to solve the NA price for a slightly modified call option.
Say that I have a money account $B(T)=e^{r(T-t)}$ and a stock dynamic $\frac{dS(t)}{S(t)}=(r-\delta)dt+\...
0
votes
2answers
63 views
Heston Model and antithetic variables
I was implementing some variance reduction techniques for the heston model and came up with a question when implementing the antithetic variable technique. Namely, I was not sure if I had to implement ...
0
votes
1answer
74 views
Determining the No Arbitrage price of max[B(T), S(T)]
Following is given,
$dB(t)=rB(t)dt$
$dS(t)= (r-\delta)S(t)dt+\sigma S(t)dW(t)$
where, $r$ is the risk-free interest rate, $\delta$ the continous dividend yield $\sigma$ is the stock asset ...
3
votes
2answers
172 views
Stochastic Calculus problem with three processes? (Itô calculus)
Can someone help me solve this following Itô Calculus problem?
Let $Z(t):= [B(t)*X(t)]/S(t)$
We have the following dynamics of B(t), X(t) and S(t):
$dS(t)=\alpha S(t)dt+\sigma S(t)dW(t)$
$dB(t)=rB(...
13
votes
2answers
4k views
Why is Brownian motion useful in finance?
The following is an interview question from Mark Joshi et al. Quant Job Interview.
Question: Why is Brownian motion useful in finance?
I am from a Pure Maths PhD background (functional analysis, ...
2
votes
1answer
77 views
Expectation and variance of $\int_0^t (W_s)^n ds$ for any positive integer $n$?
It is well known that the integral
$$\int_0^t W_s ds,$$
where $(W_s)_s$ is a Brownian motion, can be derived using Ito's Lemma. More precisely, Ito's lemma on $d(tW_t)$ implies that
$$d(tW_t) = ...
1
vote
0answers
29 views
Realized Variance as an approximation of the Integrated Variance
Realized Variance is written as $RV_{[0,T]}^{n} = \sum_{j = 1}^{n} r_{j,n}^2$, where $r_{j,n}$ is the log return for the $j$th increment, and $n$ is the total number of sample points in the time ...
3
votes
1answer
62 views
Boundaries for Call Spread
I'm reading an interview book called A Practical Guide to Quantitative Finance Interview and I have some doubts regarding part of its solution and highlighted them in bold:
Question:
What are the ...
2
votes
2answers
213 views
Assumptions in using risk-neutral pricing formula
The well-known risk-neutral pricing formula goes as follows (extracted from Shreve's Volume 2, section $5.2.4$ (Pricing Under the Risk-Neutral Measure)):
Given any $T>0$ and any $t\in[0,T],$ if $V(...
2
votes
0answers
82 views
Martingale representation of European option
Let stock price $S$ satisfy
$$S(t)=S(0)e^{(\int_0^t\sigma(s)dB_s-\frac{1}{2}\int_0^t\sigma(s)^2ds)}$$
I want to calculate the Martingale representation $V(t)=E(F|F_t)$ of European option with strike ...
7
votes
1answer
151 views
Option pricing with Brownian Bridge
Say I have an asset following arithmetic Brownian motion
$$
dX(t) = \sigma dW^\bot (t)
$$
with $\sigma$ constant, and I have prices of vanilla options on $X$.
I introduce a Brownian bridge
$$
dY(t) = ...
5
votes
1answer
90 views
Why is the numeraire in the LGM model tradeable?
I'm trying to understand the LGM model, which Hagan defines as follows. The state variable $X$ evolves according to
$$dX(t) = \alpha(t) dW^N(t)$$
wrt the numeraire
$$N(t) = \frac{1}{P(0,t)} e^{H(t)X(...
1
vote
1answer
122 views
How does longer time to maturity affect standard European call and put option values?
Denote American call and put option values as $C$ and $P$ respectively.
Similarly, denote European call and put options values as $c$ and $p$.
It is well known that time to maturity affects all $C,P,...
0
votes
1answer
43 views
Compute value of $\mathbb{E}(B_3)$
I wonder would anybody tell me how to calculate $\mathbb{E}(B_3)$
Assuming that $\int_0^{t}r_s\,ds\sim N(0.03t,0.25t)$, then is
=====
I have similar problem solved:
Assuming that $\int_0^t r_s ds \...
3
votes
1answer
112 views
Computing Itô differential of conditional expectation process (Heston SDE)
Going through this article
on Heston's model, where the variance evolves following the SDE
\begin{equation}
\label{sd1}
d\sigma^2_t = \kappa \bigg( m - \color{red}{\sigma^2_t} \bigg)dt + \nu \sqrt {\...
3
votes
1answer
218 views
Discretization of Wiener process
The Wiener process $(W_t)$ is a continuous stochastic process that satisfies the following there conditions:
$W_0 = 0$,
the increments $\mathrm{d}W_t = W_{t + \mathrm{d}t} - W_t$ are normally ...
3
votes
1answer
169 views
How to calculate the mean and variance of this Ito integral?
I tried to calculate this integral use Ito's lemma, $W_{t}$ is the Wiener Process.
$$I_{T}=\int_{0}^{T}\sqrt{|W_{t}|}dW_{t}$$
We have
$d f\left(W_{t}\right)=f^{\prime}\left(W_{t}\right) d W_{t}+\...
1
vote
1answer
63 views
Intuition behind Scaling Symmetric Random Walk
I am reading a section in Shreve (2008) where we are scaling down the step size but speeding up the time a symmetric random walk, so that in the limit, we produce a Brownian motion.
I understand the ...
