Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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2
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1answer
69 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
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1answer
99 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
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1answer
78 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
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 ...
0
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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
4
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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_{...
11
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2answers
7k views

Why is this stochastic integral a martingale?

Suppose that: $W^*_t$ is a Wiener process under probability measure $\mathbb{P}^*$ and; $\tilde{S}_t=S_0+\sigma\int_{0}^{t}S(u)dW^*_s$. In my lecture notes, it says that $\tilde{S}_t$ is a ...
10
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2answers
222 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
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1answer
79 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
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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 ...
3
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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 ...
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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}...
4
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1answer
89 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....
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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 ...
1
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1answer
60 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. ...
6
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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
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2answers
76 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+\...
3
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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(...
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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 ...
10
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1answer
247 views

Probability density function of simple equation, compound Poisson noise

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where: \begin{equation*} dX_t = -aX_t dt + d N_t, \end{equation*} $a$ is a constant and $N_t$ is a ...
0
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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 ...
2
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2answers
159 views

How to numerically simulate exponential stochastic integral

For example given an integral $$ \int^t_0 \exp(aW(t'))\,dt', t\in\mathbb R_+ $$ where $W(t')$ is a standard Wiener process. I've been very confused about stochastic integrals like $\int^t_0 W(t')\,...
12
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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, ...
12
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1answer
502 views

Transformation of Volatility - BS

I have recently seen a paper about the Boeing approach that replaces the "normal" Stdev in the BS formula with the Stdev \begin{equation} \sigma'=\sqrt{\frac{ln(1+\frac{\sigma}{\mu})^{2}}{t}} \end{...
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5answers
21k views

Integral of Brownian motion w.r.t. time

Let $$X_t = \int_0^t W_s \,\mathrm d s$$ where $W_s$ is our usual Brownian motion. My questions are the following: Expectation? Variance? Is it a martingale? Is it an Ito process or a Riemann ...
2
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1answer
55 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) = ...
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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 ...
7
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1answer
141 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) = ...
3
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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 ...
4
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2answers
624 views

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 \...
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1answer
63 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,...
2
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2answers
168 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
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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 ...
5
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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(...
0
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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
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2answers
946 views

Why does the short rate in the Hull White model follow a normal distribution?

Consider Hull White model $dr(t)=[\theta(t)-\alpha(t)r(t)]dt+\sigma(t)dW(t)$ when we solve the SDE above we have $r(t)=e^{-\alpha t}r(0)+\frac{\theta}{\alpha}(1-e^{-\alpha t})+\sigma e^{-\alpha t}\...
3
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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 $$...
3
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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
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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
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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}+\...
13
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6answers
4k views

Self-financing and Black-Scholes-Merton formula

Self-financing is an important concept in financial product replicating, normally used in pricing. I read about several ways to derive Black-Scholes-Merton (BSM) formula. Seems some approaches ...
1
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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 ...
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1answer
86 views
2
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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 ...
4
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1answer
271 views

Bayes Theorem with change of measure

Tomas bjork- arbitrage theory in continuous time. Appendix B, proposition B41 says: The proof is not clear to me. Thanks to Gordon's comment below of $E^Q (X/G)$ being $G$ measurable, I think the ...
0
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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
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1answer
79 views

Event Occurs Almost Surely

Consider an uncountably infinite space, an infinite coin-tossing. Let $(\Omega,\mathcal{F},\mathbb{P})$ be the probability space. If a set $A\in\mathcal{F}$ satisfies $\mathbb{P(A)=1},$ then we say ...
0
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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
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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
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2answers
288 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 = ...