Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
645
questions
1
vote
0answers
9 views
Integral of exponential brownian motion
I have the following process for dividends:
$\frac{D(s)}{D(t)} = \exp \left( \int_t^s \theta (u) du - \frac{1}{2} \sigma_D^2 (s-t) + \sigma_D (B_D (s) - B_D (t)) \right)$
And I need to calculate the ...
3
votes
0answers
50 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
133 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 ...
-3
votes
2answers
156 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
106 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
67 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
39 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 Itô process.
Definition 4.4.5. Let $X(t) = X(...
3
votes
1answer
90 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
$$\mathrm{d}X(t) = \sigma(t)\mathrm{d}W(t)+\alpha(t)\...
2
votes
1answer
83 views
Girsanov transform when drift coefficient is a function of the stock price
I'm working my way through an elementary stochastic calculus textbook. I'm having trouble with one of the questions:
Bachelier type stock price dynamics. Let the SDE for stock price $S$ be given by $...
1
vote
1answer
86 views
Hermite polynomials as martingales [closed]
Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite ...
4
votes
1answer
313 views
Ito calculus is Gaussian (using method of characteristic function)
Let $h$ be a deterministic function and define $X_{t}=\int_{0}^{t} h(s) d W_{s} .$ Show that
$$\mathbb{E} \exp \left(i u X_{t}\right)=\exp \left(-\frac{u^{2}}{2} \int_{0}^{t} h^{2}(s) d s\right),$$ ...
0
votes
1answer
85 views
Mutual variation of Brownian motions
Let $\{W^1\}_{t\geq0}$ and $\{W^2\}_{t\geq0}$ be two Brownian motions with correlation coefficient $\rho \in [0, 1]$, i.e., $\mathbb{E}[(W^1(t)-W^1(s))(W^2(t)-W^2(s))]=\rho(t-s)$ for all $t,s \geq 0$. ...
0
votes
0answers
56 views
What is the difference between “stochastic” heat equation and just heat equation?
I am trying to understand the difference between the "stochastic" heat equation and the heat equation. Will i be wrong to say the stochastic heat equation is just the heat equationg with the ...
0
votes
0answers
63 views
Stochastic optimization and mean field games : textbooks
Which textbooks and online courses would you recommend to learn :
stochastic optimization
mean field games
applied to quantitative finance.
My goal would be to read research articles like the ones ...
0
votes
0answers
37 views
Fisher information of an Ornstein-Uhlenbeck process
I would like to compute the Fisher information of an Ornstein-Uhlenbeck process $X_t = Y_t - \beta Z_t$ where $Y_t$ and $Z_t$ are two time-series.
My log-likelihood function in this case is:
$$\...
3
votes
1answer
155 views
Help on solving a stochastic differential equation
I am trying to solve the following SDE
$$dX(t)=rdt+aX(t)dW(t),\ t>0$$
$$X(0)=x$$
where W() is a Wiener process and r,a and x real numbers. I have proceeded by using the integrating factor
$$F(t)=...
2
votes
2answers
152 views
Proving that a stochastic process is a martingale using Ito's Lemma
Assume a Wiener process W and a bounded F-adjusted stochastic process a. Show that the following process is a martingale on F
$$X(t)=(\int_{0}^{t}a(s)dW(s))^{2}-\int_{0}^{t}a^{2}(s)ds,\ t\geq0$$
Can ...
0
votes
0answers
62 views
Query on Lebesgue Measure
I am reading Steven E. Shreve's book, titled "Stochastic Calculus for Finance II". I have a query w.r.t. an example given in the book which is as follows:-
5
votes
2answers
141 views
Expectation of integral where one of limits of integration is a random variable
Is it correct to write
\begin{equation}
E_t \int_0^{X_T} f(z) dz = \int_0^\infty \left(\int_0^x f(z) dz \right) p(x)dx \,\,?
\end{equation}
Here $X_T$ is a positive random variable with density $p(x)...
2
votes
1answer
54 views
Simplifying the expectation of the product of two stochastic integrals
Let $f(t, \omega), g(t, \omega)$ be functions that are independent of the increments of the Brownian motion $w(t, \omega)$ in the future. That is, $f(t, \omega), g(t, \omega)$ are independent of $w(t +...
3
votes
1answer
145 views
Derivative of Stochastic Integral
I am trying to take the derivative of the following stochastic integral,
$$d\left(\int g(S_t) dS_t \right),$$
where $dS(t) = \sigma S(t) dW_t$ and $g(.)$ is some (smooth) deterministic function. My ...
1
vote
1answer
270 views
Reason for 0 in discounted stock price process
Let's assume $dD_t = rD_tdt$ ($D_t$ is Bond Price)
and $dS_t = rS_tdt + σS_tdW_t$
The reference said $dD_tdS_t = 0$
But I don't understand the reason why it is zero.
It said, the Bond Price is ...
1
vote
0answers
53 views
Replicating portfolio in the Heston model
Given the Heston model $$dS_t=\mu S_tdt+\sqrt{\nu_t}S_tdB_{1,t}\\ d\nu_t=k(\theta-\nu_t)dt+\eta\sqrt\nu_tB_{2,t}$$
how should the replicating portfolio $V_t$ for the derivative $F_t$ be composed?
I ...
1
vote
0answers
50 views
Milstein Scheme for Jump-Diffusion models
Hey in this report (Approximation of Jump Diffusions in Finance and Economics by Bruti-Liberati and Platen) is described the Milstein formula (3.5) for simulation SDE with jump component. How it is ...
1
vote
0answers
58 views
How to solve/evaluate an Ito Integral?
I'm given the following Ito integral which I need to evaluate. $Z_t$ is the Brownian motion. My problem is that online resources aren't making much sense because of the notation, so it ends up leaving ...
1
vote
1answer
76 views
Simulation of Gamma process (distribution of increments)
The gamma process is a Levy process $X$, where $X_t$ has gamma distribution with parameters $at,b>0$ and density
$$f\left(x\right)=\frac{b^{at}}{\Gamma\left(at\right)}x^{at-1}e^{-bx}$$
I want to ...
0
votes
1answer
74 views
Solution to geometric Brownian motion with time dependent volatility and drift?
I am able to compute the general solution of a standard geometric Brownian motion, but I'm struggling to find the general solution for a GBM where volatility and mean depend on time, $$\text{d}S_t = \...
4
votes
2answers
277 views
Heston stochastic volatility, Girsanov theorem
How can we apply Girsanov's theorem to a stochastic volatility model?
In Heston's model the dynamics are given by
\begin{align*}
dS_t &= \mu S_t dt + \sqrt{v_t}S_t d\widehat{W}^\mathbb{P}_{1,t}, ...
1
vote
0answers
92 views
A Cauchy Problem: How can I find the following solution?
Suppose that we have the following time-dependent partial differential equation:
\begin{equation}
\frac{\partial V(t, x)}{\partial t} = \frac{1}{2}\frac{\partial^2 V(t, x)}{\partial x^2} - wxV(t, x), \...
0
votes
0answers
49 views
Is there any method/module/library to directly solve an SDE in python? Especially if it's just geometric brownian motion
Now, I'm given an SDE $$dS_t = 2S_t\,dt + 4 S_t\,dW_t$$ which I need to find the solution of. I have the solution on paper, but I want to know if there's any way I can solve this directly in python. ...
3
votes
2answers
242 views
Calculate Ito integral $\int_0^t W_s^2\text dW_s$ from first principles
I am stuck on the 1st equation of the solution where the Wiener process $W_{t_i}^2$ is expanded so that the Itô integral (in terms of infinite sums) looks like the RHS of the first equation of the ...
1
vote
0answers
38 views
Do we model stock prices using non-Markovian processes in continuous setting?
In a continuous setting, is it common to model stock prices using non-Markovian processes ? If so, do you have some examples of models ? Or is Markovianity something "embedded" in the ...
2
votes
0answers
52 views
Testing the fit of an Ornstein-Uhlenbeck process
I would like to check if a time-series follows an Ornstein-Uhlenbeck process defined by an SDE:
$$dX_t - \lambda (\mu - X_t) dt = \sigma dW_t$$
where
$\lambda > 0$ is the mean-reversion ...
3
votes
0answers
42 views
Why does it hold true that $\theta_{t} d\overline{X}_{t}$ is a local $Q$ martingale if $\overline{X}$ is a local $Q$ martingale
I am learning from Bernt Oksendal's Stochastic Differential Equations and on page 276 Lemma 12.1.6, it is stated that:
The existence of an equivalent martingale measure $Q$ on the discounted price ...
1
vote
1answer
81 views
How to take the expectation of an exponential martingale? And an exponential with a random value?
I am reading Shreve's Stochastic Calculus for Finance II. He states on pages 110 and 111 that,
$$E[exp(\sigma m-\frac{1}{2}\sigma^2 \tau_m)] = 1$$
$$E[exp(-\frac{1}{2}\sigma^2 \tau_m)] = e^{-\sigma m}$...
3
votes
1answer
112 views
Pricing Call Option on Coupon Bond under Vasicek
Consider a the Vascicek model, and let A and B denote the functions such that $P(t,T)=\exp(A(t,T)-B(t,T)r(t))$. We now look at a coupon bond that makes deterministic payments $\alpha_1,...,\alpha_N$ ...
3
votes
0answers
62 views
Is $E_t^{Q}(g(Y))=E_t^{Q^Z}(g(Y))$?
Consider $$Z(t)=\left(\frac{S(t)}{H}\right)^p$$where $S$ has a standard Black-scholes Dynamics for a stock, $H$ is a postive constant and $p =1 - \frac{2r}{\sigma^2}$
and a simple claim with a pay-off ...
2
votes
1answer
70 views
COS Method and existence of density
Hey in the COS method we use characteristic function of $\ln{S_T}$ to price european options (by recovering density from characteristic function). But how do we know that density exists? For example I ...
1
vote
0answers
100 views
Quanto put hedge\ replication with a brownian motion
Consider $d B_{us}(t)=r_{us} B_{us}(t) dt\\dX(t)=X(t)(r_{us}-r_J)dt+X(t)\sigma^T_J dW(t)\\d B_J(t)=r_{J} B_{J}(t) dt\\dS_J(t)=S_J(t)(r_J-\sigma^T_X\sigma_J)dt+S_J(t)\sigma^T_J dW(t)$
where the $\sigma$...
3
votes
0answers
84 views
Explicit form for forwards Feynman-Kac formula
This might be a simple question, but I'm having trouble with it.
Consider the Cauchy problem with final condition.
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) ...
5
votes
2answers
768 views
A stochastic differential equation
Consider the following stochastic differential equation (SDE)
$$d X_s= \mu (X_s + b)ds + \sigma X_s d w_s $$
where constants $\mu, \sigma, b > 0$ and initial position $X_0$ are given.
If $b=0$, ...
0
votes
1answer
115 views
Pricing of $(S(T_0)-S(T))^+$
Problem: Consider a new derivative that at time $T$ pays $Y =(S(T_0) − S(T))^+$
where
$0 < T_0 < T$ is a fixed date.
(i) Show that the arbitrage-free of Y at time $t = T_0$ is given by $\pi_{...
-1
votes
1answer
222 views
How can I learn stochastic process & stochastic calculus in two weeks? [closed]
I am going for an interview for a quant job. The interview will focus on my mathematical knowledge about stochastic process & stochastic calculus, and I believe I will definitely be asked to solve ...
0
votes
0answers
27 views
Future forward convexity adjustment as the expected profit from reinvesting margin payments?
Having looked at the formula for the convexity adjustment as a function of the covariance between rates accruing till maturity and asset price, I have an intuition that the difference between fair ...
0
votes
0answers
70 views
SDF derivation by a stochastic process
I have a stochastic process to model the stochastic discount factor (SDF) with M:
\begin{equation}
dM_t = aM_tdt + bM_t d Z_t
\end{equation}
where, $Z_t$ is a standard brownian motion. How do I show ...
1
vote
1answer
120 views
What is the interpretation if the real world measure $\mathbb P$ is equal to the martingale measure $\mathbb Q$
Out of interest, is there anything noteworthy about a market when its real world measure $\mathbb P$ is actually also its martingale measure. In other words the real world measure $\mathbb P$ is equal ...
0
votes
1answer
98 views
Calibrating the Ornstein-Uhlenbeck process with an additional parameter
Firstly I find the spread between two cointegrated time-series $Y_t$ and $Z_t$ by finding the best slope parameter $\beta$ in the equation $spread_t = Y_t - \beta Z_t$ (via Cointegrated Dickey-Fuller ...
0
votes
0answers
61 views
What is the difference between the geometric brownian motion and cumulative product of percentage returns?
I wonder why the following code: one using GBM and the other using cumulative product of normally distributed percentage returns slightly different values.
...
4
votes
0answers
158 views
Summary of Stochastic Derivatives, Integrals, Expectations, and Variances
I wanted to make a summary table of stochastic functions to improve my understanding. Maybe the following should be a wiki page on this site so others can add functions and examples? Does the ...
1
vote
1answer
147 views
Trouble With Applying Ito's Lemma
I am having trouble applying Ito's Formula to the following:
Let $Z_t = W_{1t}^2 e^{W_{1t}+ \int_0^t W_{3s}dW_{2s}}$. Find $dZ_t$. $W_1,W_2,W_3$ are independent Brownian motions.
I know the formula ...
