Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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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
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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 $Z(t)=exp\{-\theta B(t)-\dfrac{1}{2}\theta^2t\}$...
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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
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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
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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 ...
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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
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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
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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 ...
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_{...
4
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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
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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
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. ...
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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 ...
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+\...
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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+\...
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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 ...
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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 ...
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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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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
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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) = ...
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0answers
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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
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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 ...
2
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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(...
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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 ...
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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) = ...
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1answer
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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(...
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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,...
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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 \...
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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}+\...
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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 ...
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 ...
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 $$...
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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
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 ...
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1answer
86 views

Justify a backward differential equation

Regards of 4.5.1, how we get 4.5.5?
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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}...
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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 ...
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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 = ...
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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
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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
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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 ...
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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
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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
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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 $dX_t = t\,dS_t$.
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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.
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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
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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
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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(...