Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
494
questions
2
votes
1answer
83 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
106 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 $Z(t)=exp\{-\theta B(t)-\dfrac{1}{2}\theta^2t\}$...
0
votes
1answer
79 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
0
votes
1answer
41 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
1
vote
1answer
59 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
238 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
84 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
228 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
89 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
91 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
140 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
61 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
85 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
161 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
77 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
55 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
70 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
168 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(...
12
votes
2answers
3k 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
57 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
26 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
60 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
171 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
81 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
143 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
$$
dY(t) = ...
5
votes
1answer
83 views
Why is the numeraire in the LGM model tradeable?
I'm trying to understand the LGM model, which Hagan defines as follows. The state variable $X$ evolves according to
$$dX(t) = \alpha(t) dW^N(t)$$
wrt the numeraire
$$N(t) = \frac{1}{P(0,t)} e^{H(t)X(...
1
vote
1answer
64 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
41 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
105 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
204 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
132 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
57 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 ...
2
votes
1answer
63 views
Independence of increments of the stochastic process $\frac{1}{t}\int_0^t u dW_u $
Let $X_t$ be a stochastic process such that
$$X_{t} =\frac{1}{t}\int_0^t u dW_u $$
I know that for
$$Y_{t} =\int_0^t u dW_u$$
$Y_t-Y_s$ is independent of $Y_s$ where $t>s$.
But is this also true ...
3
votes
2answers
132 views
Probability distribution of the stochastic process $\int_{0} ^{t}\frac{u}{t}dW_{u}$
I am wondering about the probability distribution of the stochastic process
$$X_t=\int_0^t \frac{u} {t} dW_{u}$$
I thought of using the Kolmogorov equation but after converting this into An SDE
$$...
0
votes
0answers
26 views
Call Option on the Square of a Log-Normal: Process of Underlying under Stock Measure and Risk Neutral Measure
I'm working on some quant interview questions from the book called Quant Job Interview Questions And Answers (by Mark Joshi and other authors).
Here are the questions from the bookd, and the answers ...
0
votes
1answer
39 views
Accumulation Rate of Variance in Random Walk
I am slightly confused with the terminology Shreve (2008), he states:
"The variance of the symmetric random walk accumulates at rate one per unit time, so that the variance of the increment over ...
1
vote
1answer
86 views
1
vote
1answer
63 views
Arithmetic Asian Option
Assume the risk-free bond Bt and the stock St follow the dynamics of the Black & Scholes model
without dividends (with interest rate r, stock drift $μ$ and volatility $σ$).
Let $A_T:=\frac{1}{T}...
1
vote
1answer
108 views
Asian Options-Change of Numeraire
Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model
without dividends (with interest rate r, stock drift $\mu$ and volatility $\sigma$).
Show that ...
7
votes
2answers
290 views
Stochastic Integral Graph
As we can represent the integration of $f(x)$ on $[a,b]$ with the graph below,
I was wondering how to represent the following integral with $X(t)$ a Brownian motion, $f(t)$ any function and $t_j = ...
4
votes
2answers
530 views
Finding price of the power option
Let's assume a market with $d=1$ and $X=X^1$ satisfying
$dX_t=\sigma X_t\,dW_t,\: \: X_0=1,$
where $(W_t)$ is a standard Brownian motion. Assume that $\mathbb{F}$ is the natural filtration of $X$ ...
4
votes
1answer
121 views
How do we calcualte $E[W_sW_t|W_s]$
$W_t$ is a Brownian motion. How do we calculate this expectation?
there are two cases:
$s < t$
$t < s$
Do we have to distinguish the two cases or there is a unified way of calculating it
4
votes
1answer
125 views
stochastic dominance displaced diffusions
Suppose I have two processes both satisfying a displace lognormal diffusion:
$$
dX(t) = \alpha(t)[X(t) - a] dW(t)
$$
$$
dY(t) = \beta(t)[Y(t) - b] dW(t)
$$
Note that the processes are perfectly ...
1
vote
2answers
91 views
Instantaneous change in value of portfolio
I am trying to figure out an intuitive explanation for the instantaneous change for the value of a portfolio (essentially I'm creating a self-financing portfolio to replicate a derivative payoff).
...
4
votes
1answer
102 views
What the expectation of S^2 is from GBM? [closed]
I was at an interview and was asked to write down the SDE for GBM.
$$
dS = S\mu dt + S\sigma dX
$$
Then I was asked how I would compute the expectation of S^2. I didn't know where to start. Any ...
4
votes
1answer
171 views
Evaluating the SDE $dX_t = t\,dS_t$
The process $S$ is a geometric Brownian motion with an SDE: $dS_t = S_t(\sigma\, dB_t + \mu\, dt)$. I'm stuck evaluating $E(X_t)$ and $V(X_t)$, where $dX_t = t\,dS_t$.
-1
votes
1answer
68 views
Stochastic Vol Mathematical derivation [closed]
I want to understand the mathematical steps done. Can someone please simplify the derivation of d(pi) from Pi? Thanks in advance.
6
votes
1answer
203 views
Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$
I want to solve the following SDE:
$$ dX_{t} = \mu X_{t} dt + \sigma dW_{t} \quad X_{0} = x_{0}$$
Integrating, I get:
$$ X_{t} - x_{0}= \mu \int_{0}^{t} X_{s} ds + \sigma \int_{0}^{T} dW_{t} $$
$$ ...
2
votes
0answers
58 views
Interchange Expectation and Supremum in Snell Envelope/American Options
I had a question about the properties of a snell envelope, $\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$, which came to me while studying American options.
I know that in general,...
2
votes
1answer
84 views
Are the Ito's Lemma given in Mark Joshi's Concept and Practice in Mathematical Finance same as what I learn?
In Joshi's Concepts and Practice in Mathematical Finance, page $110,$ he stated the Ito's Lemma:
Theorem $5.1$ (Ito's Lemma) Let $X_t$ be an Ito process satisfying
$$dX_t = \mu(X_t,t)dt + \sigma(...
