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8
votes
2answers
174 views

Stochastic Calculus Rescale Exercise

I have the following system of SDE's $ dA_t = \kappa_A(\bar{A}-A_t)dt + \sigma_A \sqrt{B_t}dW^A_t \\ dB_t = \kappa_B(\bar{B} - B_t)dt + \sigma_B \sqrt{B_t}dW^B_t $ If $\sigma_B > \sigma_A$ I ...
0
votes
2answers
44 views

Conditional probability of geometric brownian motion

I created paths using GBM to implement The stochastic mesh method. But the method requires the conditional distribution, given some S(t) the probability of S(t+1). I've searched and can't find this ...
2
votes
2answers
183 views

Square of Wiener process

In Ito's calculus one often comes $dW^2=dt$. How does this come about? What is it's relation to the Milstein method?
-1
votes
0answers
41 views

Stochatic Ito formula [closed]

I have $$dX_t= a dt + H dB_t $$ With$ B_t$ brownien motion and H is a function t such as $$E(H^2) \leq \infty $$ And$$ c< X_t < b$$ and $$t->X_t$$ is an increasing dunction I have to show ...
0
votes
1answer
46 views

Lebesgue-Stieltjes integration and related topics

The theory of stochastic integration relies on the concept of the Lebesgue-Stieltjes integral. However, it is hard to find a textbook that handles this concept in detail. Take, for instance, Chung ...
1
vote
1answer
123 views

Problem with derivating integral

I have a doubt : I know that if $x_{t}=\int_{0}^{t}\gamma(s)dW_{s}$ (with $W_{s}$ a brownian motion), we have : $dx_{t}=\gamma(t)dW_{t}$ What about if $x_{t}=\int_{0}^{t}\gamma(s,t)dW_{s}$. Do I have ...
3
votes
1answer
54 views

clarification to log-stock price formula

Having financial market with safe rate r and risky asset S with dynamics under physical measure P $$\frac{dS_t}{S_t}=\mu dt +\sigma dW_t$$ what is the log-stock price? Using Ito formula it is ...
8
votes
1answer
278 views

Distribution of hitting time of the integrated CIR process

If an increasing process $X_t$ has a known Laplace transform $\mathbb{E} e^{-s X_t} = m_t(s)$, define its hitting time $\tau$ to some level $B$ to be $$ \tau = \inf\{ u > 0 : X_u \geq B \}. $$ Can ...
2
votes
3answers
95 views

How to understand nonrandom/random process in Shreve book?

I have been reading Chapter 4 of Shreve's Stochastic Calculus for Finance II. It is easy to understand the simple process, $\Delta(t)$, defined on Page 126, which is just a constant inside a given ...
1
vote
0answers
59 views

Quadratic variation

The following question is more math than quant, but since it arises from a mathematical finance textbook, I've figured the good people in this sub might be able to help me. So here goes. In the 3rd ...
1
vote
1answer
39 views

optimal strategy problem (using Jensen's inequality)

I have a strategy in Samuelson model with zero safe rate defined as $$Z_t^{\Pi}=\frac{X_t^{\Pi}}{X_t^{\rho}} \quad \quad (1)$$ where $$\frac{dX_t^{\Pi}}{X_t^{\Pi}} = \mu \pi dt + \sigma \pi \ dW_t ...
1
vote
0answers
48 views

stochastic log utility maximization problem, portfolio optimal strategy

Looking for a help with explaining some steps of the logarithmic utility maximization problem where given market with a zero safe rate and risky asset with dynamics $$ \frac{dS_{t}}{S_{t}}=\mu B_t ...
0
votes
1answer
34 views

stochastic discount factor transformation

I have $$\frac{dM_t}{M_t}=-\frac{\mu}{\sigma} dW_t + \gamma_t dB_t, \tag{1}$$ where $B_t$ and $W_t$ are two independent Brownian Motions, which was further presented as $$ M_t=\exp \left( ...
0
votes
1answer
179 views

close form for stochastic integral

I am new to stochastic calculus. Can I know how to compute the close-form solution for $$\int_0^t \exp(\alpha s - \sigma W_s) \; ds$$ and $$\int_0^t \exp(\alpha s - \sigma W_s) \; dW_s.$$ I encounter ...
4
votes
1answer
298 views

How to measure a non-normal stochastic process?

If I understand right, Itô's lemma tells us that for any process $X$ that can be adapted to an underlying standard normal Wiener measure $\mathrm dB_t$, and any twice continuously differentiable ...
2
votes
2answers
235 views

Ito calculus problem

given $S^1$ satifying the SDE $\quad dS_{t}^{1}=S_{t}^{1}((r+\mu)dt + \sigma dW_t), \quad S_{0}^{1}=1 $ and the safe asset $S_{t}^{0}$ $\quad S_{t}^{0}:=e^{rt} \quad for \quad r\geq 0$ Q1. how ...
1
vote
1answer
63 views

HJM framework problem - showing that HJM drift condition implies that $b(z)=b+βz$ and $(ρ)^2=α$

Hi I am looking for some general clarification to Heath–Jarrow–Morton framework. I am analyzing a problem where the forward rate is modeled as $$ f(t,T)=e^{\beta(T-t)} Z_t+h(T-t) \tag{1}$$ for some ...
3
votes
1answer
54 views

CIR model - nth moment generation $E^*[r_T^n]$

I am analyzing the nth moment generation process for $r_t$ with dynamics defined by CIR model $r_t$ has following dynamics $$dr_t=a(b-r_t)dt+\sigma \sqrt{r_t} dW_t^* \quad \quad (1)$$ for some ...
3
votes
2answers
94 views

CIR model problem - deriving PDE, Feynman-Kac

I am reviewing a CIR model problem, where $r_t$ has following dynamics $$dr_t=a(b-r_t)dt+\sigma \sqrt{r_t} dW_t^* \quad \quad (1)$$ for some constants $ab>\frac{\sigma^2}{2} \quad$ Letting T ...
1
vote
1answer
54 views

Expected Value of Products of Processes

Suppose I have two processes. $A_t = A_0 \exp((a-\frac{1}{2}\sigma_A^2)t+\sigma_A W_t^A$ $B_t = B_0 \exp((b-\frac{1}{2}\sigma_B^2)t+\sigma_B W_t^B$ I would like to calculate $E[A_s B_t]$ where s ...
0
votes
0answers
45 views

approximating fBm stochastic integral

Suppose I have the following stochastic integral: $$\int_a^b f(t)dB_H(t)$$ with the term $dB_H(t)$ a fractional brownian motion with associated $H$ parameter. Is it true that for $H \in (1/2,1)$, ...
0
votes
1answer
54 views

Ho-Lee model - A and B derivation for $P(t,T)=e^{-A(t,T)-B(t,T)r_t}$

I am analyzing the transition of the bond prices in the affine models in the form of $P(t,T)=e^{-A(t,T)-B(t,T)r_t}$ using the property that the diffusion and the drift of an affine model can be ...
1
vote
1answer
71 views

Quanto/Compo adjustments - Product of two geometric brownian motion

Let's say I have two processes $X_t =X_0 \exp((a-\frac{1}{2}\sigma_X^2)t +\sigma_X dW_t^1)$ and $Y_t=Y_0 \exp((b-\frac{1}{2}\sigma_Y^2)t +\sigma_Y dW_t^2)$ and I then multiply them together (like ...
1
vote
1answer
40 views

“Expectation” of a FX Forward

I have an FX process $X_t = X_0 \exp((r_d-r_f)t+ \sigma W_t)$. Now clearly $E[X_t] = F_{0,t}^X$. i.e. a forward contract of the process $X$ starting at time 0 and maturing at time $t$. What if I ...
6
votes
1answer
108 views

Stochastic control (HJB) for wealth process involving stopping times

Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ where $\theta_t$ is the worth of holding at time $t$ and $c_t$ is the consumption stream. ...
3
votes
1answer
89 views

Girsanov Theorem for Quanto/Compo adjustment

Assume that I have a foreign asset $$Y_t = Y_0 \exp \left((r_f-\frac{1}{2}\sigma^2_Y)t+\sigma_Y W_t^1\right)$$ and an exchange rate $$X_t = X_0 \exp\left((r_d-r_f-\frac{1}{2}\sigma^2_X)t+\sigma_X ...
1
vote
1answer
53 views

Vasicek model problem

I am analyzing a problem where the below is given Vasicek model with risk-neutral dynamics $$dr_t = \kappa (\theta - r_t)dt + \sqrt{r_t} dW_t \quad \quad (1) $$ bond prices ...
3
votes
0answers
49 views

On the reflection of a stochastic integral

Let ${(I_t)}_{t\geq 0}$ be a stochastic integral defined by $$ I_t=\int_{0}^{t}\theta_sdW_t, $$ where $W$ is a standard Brownian motion defined on $(\Omega,\mathcal{F},{(\mathcal{F}_t)}_{t\geq ...
0
votes
1answer
37 views

trading strategy problem - initial capital x buys S over time [0,T] at the constant rate of x/T euros per unit of time

I am looking for clarification to the trading strategy problem where the number of stocks is depending on time. In the Market with zero safe rate and stock dynamics defined as ...
3
votes
1answer
86 views

Why is the value of an adaptive stochastic process known at time t?

I am having a hard time to understand the concept of an adapted stochastic process. Using an analogy to finance, I have been told we can think of adaptiveness of a stock price process as having an ...
0
votes
2answers
51 views

Multivariate Ito problem $M_t=\frac{X_t}{Y_t}$

I am analyzing a problem given in the lecture slides published here (Slide 7-8 Example of Multivariate Ito’s Lemma). Can anybody explain how the $M_t$ was calculated out of the Ito formula. I ...
1
vote
1answer
84 views

Second Moment of Stock Process

I have a stock process which I have decided to model as $$S_T=S_t\exp((r-q-\frac{1}{2}\sigma^2)(T-t)+\sigma(W_T-Wt))-D_T$$ where $D_T$ is a cash dividend at time $T$. This dividend is known. I then ...
11
votes
2answers
270 views

Deriving the definition of stochastic integrals with respect to Ito processes from first principles

When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition ...
3
votes
3answers
45 views

Perpetual American Put Supermartingale property

Discounted price process of an american put (perpetual) has a $dt$ part in it, which is negative if the price at time $t$ is less than the optimal exercise price. This is the only thing that drags the ...
1
vote
1answer
47 views

investor terminal value of portfolio with two risky assets 1) correlated 2)not correlated $\phi_t^1=S^{2}_{t}, \ \phi_t^2=S^{1}_{t}$

I am analyzing a problem where I have two stocks described by the equations $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t}$$ $$ \frac{dS^{2}_{t}}{S^{2}_{t}}=\mu_{2} dt + ...
1
vote
1answer
53 views

Integration in the Hull-White SDE

I'm stuck in solving the SDE in Hull-White interest rate model. I do not have a thorough background in math (only Real Analysis during my blissful undergrad years), so I am having trouble ...
4
votes
1answer
90 views

How to express the volatility of two correlated Ito processes $Wt_1, Wt_2$ expressed in terms of $W_t$?

Having two correlated Ito processes ($W_t^1$ and $W_t^2$ are correlated Brownian motions with correlation $\rho$) $dX_{t} =\mu_{1} dt + \sigma_1 dWt_1 $ $dY_{t} = \mu_{2} dt + \sigma_2 dWt_2 $ ...
6
votes
6answers
864 views

Why the expected return rate of a stock has nothing to do with its option price?

OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ...
1
vote
0answers
66 views

SDE for a portfolio of two correlated assets $ Y_{t} = 2 S^{1}_{t} S^{2}_{t}$

I am analysing a problem where I have two correlated stocks described by Brownian motions $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t} \quad \quad (1)$$ $$ ...
6
votes
1answer
123 views

pdf 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 ...
5
votes
1answer
94 views

Square of arithmetic brownian motion process

We have an arithmetic Brownian motion process $X_t$ that follows $dX_t=\mu dt + \sigma dZ_t$ and we define the asset price $S_t=X_t^2$ and we are asked to find the stochastic differential equation ...
7
votes
5answers
811 views

Geometric Brownian motion - Volatility Interpretation (in the drift term)

A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ...
8
votes
0answers
164 views

Real world application of stochastic portfolio theory

There is a branche of stochastic portfolio theory (see also this question). Fernholz and Karatzas have published research in this field (e.g. "Diversity and relative arbitrage in equity markets") and ...
3
votes
2answers
122 views

Intergral of Brownian motion w.r.t. Brownian motion

I don't understand why $S$ (highlight on picture), I learned $$\int_0^t W(s) dW(s) = \left. \frac{1}{2} (W^2(s)-s) \right \vert_0^t $$ everyone please explain for me. Thank you
3
votes
2answers
78 views

ARMA-GARCH model, bset model selection and confidence levels calculations

I'm a newbie in GARCH models. I tried to realize ARMA(p, q)-GARCH(u, v) model via fGarch. So, 2 main questions. 1) Can I use BIC/AIC for selection best model for all (p, q)-(u, v) models? So, is it ...
4
votes
1answer
82 views

Lookback option to find stock price

Consider the payoff equation for the lookback option $\psi(T)= max(S_t-S_T)$, where $t\in[0,T]$ and $S_t$ is modeled by the geometric Brownian motion with constant parameters. Find the price of stock ...
1
vote
2answers
49 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 ...
0
votes
0answers
57 views

Stochastic Integration

I have the following derivation question: A small company is investing resources in a risky project that it hopes will be profitable. The project could, for example, represent the manufacturing and ...
5
votes
1answer
202 views

Implications of shifting the lognormal model for forward rates from a probability perspective

I have a question regarding the application of a shift to the Black-Scholes formula for negative forward rates. I am reading in the Brigo book that "increasing the shift $\alpha$ shifts the ...
3
votes
1answer
31 views

prove the normality, with given moments, of this process:

I have this process: $dx_t = -\frac{k}{2}x_tdt + \frac{\beta}{2}dz_t$ and must prove it's normally distributed with first two moments: $\mu = e^{-\frac{1}{2}kt}x_0$ $\sigma^2 = ...