All Questions
Tagged with stochastic-processes stochastic-calculus
310
questions
0
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answers
41
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how to calculate pdf and cdf for an Ornstein-Uhlenbeck process
I have the
Task. For Ornstein-Uhlenbeck process generate a path and plot a)
cumulative distribution (cdf), b) density function (pdf), c) calculate the 95%-quantile.
My solution.
From the literature we ...
- 257
1
vote
0
answers
125
views
Calibrating Hull-White model using historical data
I'm in search of a way to calibrate a very simple Hull-White model with a constant volatility and a constant mean-reversion speed, purely based on historical zero rates.
$$dr(t) = (\theta(t) - \alpha ...
- 101
0
votes
0
answers
42
views
Reference request: Approximate mapping of a multi-factor stochastic volatility model to single-factor stochastic volatility model
I am looking for approaches to transform a more complicated stochastic volatility model such as the one shown in Section 2.2 of Smile Dynamics II to a single-factor model such as the one shown in ...
- 552
3
votes
0
answers
56
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Feymann Kac pde with correlated process
I have to solve the following PDE:
\begin{equation}
\begin{cases}
\dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{1}{...
- 163
1
vote
1
answer
54
views
Dynamics of discounted prices (multi-dimensional)
My objective is to find the dynamics of the discounted prices, given by $\mathbf{y}_{t} = \mathbf{P}_{t}\mathrm{e}^{-\int^{t}_{0} r_{s} ds}$. I know the dynamics should be $d\mathbf{y}_{t} = \mathrm{...
- 61
2
votes
0
answers
57
views
Munk (2011) exercise 3.6
I'm trying to solve the exercise in Munk (2011). The exercise reads:
"Find the dynamics of the process: $\xi^{\lambda}_{t} = \exp\left\{-\int^{t}_{0} \lambda_{s} dz_{s} - \frac{1}{2}\int^{t}_{0} \...
- 61
1
vote
0
answers
91
views
Analytical expression for SDE
I'm trying to find an analytical expression for the following. Suppose $X$ is a geometric Brownian motion, such that: $dX_{t} = \mu X_{t} dt + \sigma X_{t} dW_{t}$. Suppose furthermore, that the ...
- 61
3
votes
1
answer
136
views
Integral of Function of Brownian Motion w.r.t Time (Context: Computing Quadratic Variation)
I am looking to compute the quadratic variation of $$S_t = S_0e^{\sigma B_t}$$ where $B_t$ is Brownian Motion. Applying Itô's lemma, I having the following
$$(dS_t)^2 = S_0^2\sigma^2e^{2\sigma B_t}dt$$...
2
votes
0
answers
153
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If $\Delta \log(V_{t})$ behaves like the increments of fractional Brownian motion, why do we model the rough volatility as follows
From Gatheral's paper, Volatility is rough and empirical evidence, it is clear that $\big\{\log(V_{t+1})-\log(V_{t})\big\}_{t}$ behaves like the increments of fractional Brownian motion $B^{H}$ with ...
- 73
4
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0
answers
137
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optimal stopping time problem
I'm currently reading a paper (The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing, American Journal of Operations Research, March ...
2
votes
1
answer
117
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Deriving the variance of G2++ Model
I'm studying G2++ Model in Brigo(2007)'s book.
The model constructed as follows,
$$
r(t) = x(t) + y(t) + φ(t), \quad r(0) = r_0\\
$$
with the dynamics of $dx(t)$ and $dy(t)$ described by:
\begin{align}...
- 303
1
vote
1
answer
291
views
Euler Discretization python code
Write the Euler discretization of the 1-dimensional stochastic equation
$dXt = b (t, X_t) \space dt + \sigma (t, X_t) \space dW_t$
For this part I would say all right because it is a purely ...
- 35
4
votes
2
answers
409
views
Transformation of local volatility model
Assume we have an SDE
$$dX_t=\mu(X_t)dt + \sigma(X_t)dW_t$$
where $\sigma>0$ and $W_t$ is a Wiener process. Is there a transformation $y(X_t)$ that will make the dynamics of the transformed process ...
- 43
-1
votes
1
answer
118
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Integration of exponential raised with Brownian Motion wrt the Brownian Motion
I have to derive several things for my thesis, however, I have the following expression:
$$
\int^{t}_{0} \exp\{\sigma W_{t}\}.dW_{t}
$$
Does anyone know what the solution for this is?
Kind regards.
- 5
1
vote
1
answer
134
views
Calculating Expectation of Stochastic Volatility
I have a question while reading THE NELSON–SIEGEL MODEL OF THE TERM
STRUCTURE OF OPTION IMPLIED VOLATILITY
AND VOLATILITY COMPONENTS by Guo, Han, and Zhao.
I don't understand why the above equations ...
- 11
1
vote
0
answers
107
views
Differential vs. derivative in the Vasicek model [closed]
Can anyone help me in understanding how we get the line I have marked with a red arrow?
I guess I have trouble in understanding the difference between differentials and derivatives, i.e. what is the ...
- 11
7
votes
3
answers
703
views
Why does the diffusion term remain the same when we change pricing measure?
Consider some Itô process $dS(t)=\mu(t)dt+\sigma(t)dW^{\mathbb P}_{t}$ under the measure $\mathbb P$, where $W^{\mathbb P}$ is a $\mathbb P$-Brownian motion
In plenty of interest rate examples, I have ...
- 73
2
votes
1
answer
222
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Obtaining the dynamics of the Vasicek model using Itô
Consider the following expression for the short-term interest rate
$$r_t=r_0 e^{\beta t}+\frac{b}{\beta}\left(e^{\beta t}-1\right)+\sigma e^{\beta t}\int_0^te^{-\beta s}dW_s \tag{1},$$
which is ...
- 221
1
vote
1
answer
264
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Jump Diffusion Process question
I have a European call option with time maturity $T=3$ years,$K=50$, and given that $S(t)$ refers to the derivative is being described by the geometric Brownian motion with $S_{0}=100$ and $r = 0.04$....
user57440
3
votes
1
answer
240
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Pricing of European options on two underlying assets
Is anybody able to give the solution to the following problem?
Suppose we have two assets, each of which follows a GBM process, and where $dW_S$ and $dW_X$ are correlated $(dW_SdW_X=\rho)$.
$dS=\mu_s ...
- 33
2
votes
2
answers
515
views
Solving SDE using integration factor and Ito's lemma [closed]
I don't understand how to define such integration factor in order to solve SDE, for example, as was shown in Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$ and Solving Stochastic Differential ...
- 21
4
votes
0
answers
128
views
Where is the Quadratic Variation Coming from in this One-Factor Cheyette Model?
I am having difficulty switching from a general interest rate model (the quasi-gaussian or cheyette model) and a specific version of this model. In particular, I assume the following instantaneous ...
- 41
0
votes
0
answers
150
views
Black Scholes derivation: Why treat Delta as a constant?
In the derivation of the Black-Scholes equation, it is argued (e.g. in the original paper and in Hull) that
$$dV(S_t, t)=(…)dt + \frac{\partial V}{\partial S} dS_t,$$
where $V(S_t, t)$ is the value at ...
- 183
1
vote
0
answers
168
views
Change of Numeraire technique (Cross-currency models)
Hey I have problem with understanding change of numeraire technique. For example we have
$dr^d(t)=\kappa_1(\theta_1(t)-r^d(t))dt+\sigma_1 dW_1$ (under measure $Q^1$ associated with domestic bank ...
- 423
0
votes
0
answers
298
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Ito's Lemma in option pricing for a stock satisfying $dS=\frac{P-S}{\omega}dt+SdW_t$
Suppose a stock follows the stochastic differential equation
$$dS=\frac{P-S}{\omega}dt+SdW_t,$$
such that $W_t$ is a wiener process, $\omega\in\mathbb{R}^+$, and $P_t,S_t\in\mathbb{R}$. If the value ...
- 100
6
votes
1
answer
236
views
Parametric Stochastic Integral
I need help.
Defining the parametric stochastic integral
$$
F_t = \int_t^T\xi(t,s)g(s)ds
$$
$\\\\$
with $\xi$ a generic stochastic process such that $d\xi(t,s) = \mu(t,s)dt + \sigma(t,s)dW_t$, I'm ...
- 61
2
votes
1
answer
146
views
HJM drift condition problem: Show that the HJM drift condition implies $b(t) \equiv b, \rho^{2}(t) \equiv a$
I need your help with understanding and solving the HJM framework. I am hoping I can get some help as I feel so lost with HJM and learning online because of the pandemic is adding more stress. Anyway ...
- 373
10
votes
2
answers
1k
views
Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?
Let $S_{t}$ denote the price of stock, $\beta_{t}$ denote the savings account. For each model below state with reason whether it admits arbitrage and whether it is complete.
(a) $\beta_{t}=e^{t}, S_{t}...
- 103
1
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0
answers
223
views
Derivation of Bergomi model
In Stochastic Volatility Modeling, L. Bergomi introduces in Chapter 7 the pricing equation (7.4) :
$$
\frac{dP}{dt}+(r-q)S\frac{dP}{dS}+\frac{\xi^t}{2}S^2\frac{d^2P}{dS^2}+\frac{1}{2}\int_t^Tdu\int_t^...
- 552
0
votes
1
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138
views
Question on Ito's lemma involving $\mathrm{d}W(t)$
I am new to Ito-calculus, so please forgive me if the question is stupid.
Let $W(t)$ be a Brownian-Motion and $f(W(t))=W(t)^2$. If I want to calculate the differential $\mathrm{d}f(W(t))$, Ito's lemma ...
- 471
0
votes
1
answer
238
views
In what cases characteristic function of (log-)price process is known?
Hey I know that we can use characteristic function of log-price process to price different options. But when we know the characteristic function? I know that we can take Levy processes and constant ...
- 43
1
vote
0
answers
56
views
Help in Bernoulli's differential equation
I want to solve the following Bernoulli differential equation:
$$A'(t)=A^2(t)[-2\sigma +1]-2aA(t)$$
where $\sigma$ and $a$ are real numbers.
Until now I have divided both sides of the equation with $A^...
- 129
3
votes
1
answer
736
views
Ito Lemma for Poisson Process
I'm new to stochastic calculus on jump processes and encountered a difficulty. I would appreciate some clarification from the community on the following question.
Let $g_t$ be a $\mathcal{F_t}$-...
- 184
3
votes
1
answer
243
views
Brownian Bridge general case
The SDE for the Brownian bridge is the following:
$dY_t=\frac{b-Y(t)}{1-t}dt+dW(t)$
with solution:
$Y(t)=Y(0)(1-t)+bt+(1-t)\int_0^t \dfrac{dW(s)}{1-s}$
Can someone help me on proving that $$\lim_{t\...
- 129
8
votes
0
answers
282
views
On a time integral of Brownian motion up to the hitting time
Just come up with a 'simple' and interesting problem that I've been struggling to deal with for some time. Consider a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,T]},\...
- 81
3
votes
0
answers
117
views
MGF of Generalised Itô Integral
The following derivation produces a moment closure problem - I would appreciate any insight. It may seem trivial at first glance, but the key aspect is the integrand dependence on $t$.
Consider $W_t$ ...
- 31
3
votes
1
answer
302
views
Bergomi Volatility Model
I was studying on the Bergomi volatility model(using forward variance represented as $\xi_{t}^{T}$).However I don't understand how the author passes from the sde to the first step by only integrating ...
- 436
4
votes
1
answer
300
views
Weak solution of a SDE
$\text { Consider the } \operatorname{SDE} d X_{t}=\operatorname{sign}\left(X_{t}\right) d t+d B_{t} \text { on } 0 \leq t \leq T, \text { where } \operatorname{sign}(x)=1\\
\text { for } x>0 \text ...
2
votes
0
answers
88
views
Solving SDE Dubins-Schwarz Theorem
$\text{ Let } X_{t}=1+t+B_{t}, \text { and } T=\inf \left\{t: X_{t}=0\right\} . \text { Define } G(t)=\int_{0}^{t \wedge T} \frac{d s}{X_{s}}. $
$\text { Let }\
\tau_{t}=G^{-1}(t) \text { be the ...
- 373
3
votes
0
answers
76
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.
...
- 161
1
vote
1
answer
262
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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 ...
- 13
2
votes
1
answer
135
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$$
where $r_0=0$. The Wiener processes $W_1$ and $W_2$ are independent. The price ...
- 287
3
votes
1
answer
226
views
Help on solving a stochastic differential equation
I am trying to solve the following SDE
$$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)=...
- 129
2
votes
2
answers
409
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 ...
- 129
2
votes
1
answer
242
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 +...
- 121
1
vote
0
answers
100
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 ...
- 423
1
vote
1
answer
145
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 ...
- 423
9
votes
2
answers
2k
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}, ...
- 141
0
votes
0
answers
75
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. ...
- 101
1
vote
0
answers
92
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 ...
- 183