Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
551
questions
4
votes
0answers
48 views
Confused about discretization
I am reading a paper here: https://pdfs.semanticscholar.org/5f91/2d46b02b03230a4ffaaa42d655b2b6147d56.pdf
The following is my confusion.
The paper has the following continuous time model for the price ...
2
votes
0answers
19 views
Expression for the expectation of Integrated variance in case of GARCH(1,1) process
I have the following SDE (GARCH(1,1)) for the instantaneous variance:
$$ d\sigma_t^2 = \kappa (\theta - \sigma_t^2) dt + \psi \sigma_t^2 dW_t $$
I would like to find an expression for $IV_t = E[\int_{...
1
vote
1answer
43 views
Covariation of Ito semimartingales
If we have two Ito semimartingales over $[0,T]$:
$$d X_t^i=a^i_tdt+\sigma_t^idW_t^i,\quad i=1,2$$
What is the relationship between
$$\langle X^1,X^2 \rangle_t \quad \text{and} \quad \langle W^1,W^2 \...
2
votes
2answers
115 views
Itos Lemma Derivation notation
So in Hull (2012) the main point is that $\Delta x^2 = b^2 \epsilon ^2 \Delta t + $higher order terms$ $ has a term of order $\Delta t$ and can not be ignored as the Brownian motion exhibits the ...
1
vote
1answer
64 views
Taking Expectation of Stopping Time and Integral Manipulation
Consider a stopping time $\tau$ that represents the point in time when the first credit event (e.g. default) occurs on a compact interval $[0,T]$.
Consider the expectation of the indicator function, $\...
2
votes
0answers
33 views
mixing fractional Brownian motions
Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by
$$dW_t^1 = \rho d W_t^2+\sqrt{1-\rho^2}dZ_t$$
where $W_t^{2}$ and $Z_t$ are independent of each other.
My question then: is ...
3
votes
1answer
156 views
Application of Ito's Lemma in expected utility theory
An investor with utility curve $U(.)$ has wealth $X_t$ at time t. He invests
A proportion $p$ of his wealth in a risky asset that follows a geometric Brownian motion, with parameters $\mu$ and $\...
1
vote
1answer
48 views
The most general conditions under which Ito lemma holds
Prompted by a question that came up in the comments here, namely why we can apply the Ito lemma to a function of the form $f(x)=(x-K)^{+}$, I would be interested in knowing what are the least ...
0
votes
0answers
22 views
Ito's formula with a random jump measure
Suppose all processes and functions defined are nice enough such that all the following definitions make sense.
On a probability space $(\Omega,\mathcal{F},\mathbb{P})$ equipped with a filtration $\...
3
votes
0answers
105 views
Rigorous proof of Dupire formula (e.g. using Gyöngy's theorem)
Where can I find a rigorous proof of the Dupire formula (for example, using using Gyƶngy's theorem)? I imagine this would be covered by a paper or by a standard financial math text, but I could not ...
0
votes
1answer
40 views
Serial correlation, quadratic variation and variance of returns
On p. 3 of Lorenzo Bergomi's book on Stochastic Volatility Modeling, there is the following assertion:
Indeed, to a good approximation, the variance of returns scales linearly with their time scale, ...
3
votes
1answer
101 views
Under which conditions the given random process is martingale and under which submartingale?
Let $a_t $ be adapted to the filtration random process $a_t: P\{\int _0^T|a_t|dt < \infty \} = 1 $ and $ b_t \in M_T^2. \quad$ Under which conditions the random process $$X_t = exp\{\int _0^ta_sds+\...
0
votes
0answers
14 views
Statistical test for comparing two different speed of mean reversion parameters for CIR model
I am trying to compare two different values of speed of mean reversion parameter for CIR model.
I would like to know if there exists a statistical test for comparing these two parameters.
the estimate ...
1
vote
0answers
107 views
Dynamic programming and Bellman equation to obtain the maximum
This is the problem of Marhsall (1992) "Inflation and Asset Returns in a Monetary Economy" and Balvers and Huang (2009) "Money and the C-CAPM"
Suppose an endowment economy where the representative ...
0
votes
1answer
100 views
How to replicate the future instantaneous short rate?
Suppose we have an interest rate model $R(t)=\alpha(t)d(t)+\sigma d\tilde{W}(t)$, where the brownian motion is under the risk neutral measure. Suppose $S(t)$ is the price at time $t$ for a contract ...
2
votes
0answers
25 views
Differentiation of value function in perpetual american option
I am trying to solve the perpetual American option problem. Currently I'm following this (slide 9). The stock price is modelled as Ito's process.
$dS_t = (\mu-D_0)S_tdt\ +\ \sigma S_tdW_t $
where $...
3
votes
0answers
92 views
Boundary condition in perpetual american option problem
I am trying to solve the perpetual American option problem. Currently I'm following this (slide 9). The stock price is modelled as Ito's process.
$dS_t = (\mu-D_0)S_tdt\ +\ \sigma S_tdW_t $
where $...
0
votes
0answers
35 views
Stochastic differential equation of Avellaneda model
I was reading this paper and page 14 the model is given.
I'm trying to find the steps to get to the SDE given.
OI is the open interest, E is the elasticity of demand.
$$
\frac{\Delta S}{S} ā¼ EĀ·Q^p \...
3
votes
0answers
224 views
Black and Scholes equation for portfolio **with** arbitrage
I am well aware of how the ordinary Black and Scholes equation is derived, under the assumption of an arbitrage free portfolio, $V=G-hS$. Here $S$ is the price of the underlying and $G$ is the option ...
1
vote
1answer
60 views
Ito's lemma for a Forward
I'm trying to understand the derivation of Ito's process with respect to a Forward $F$ on a stock $S$ that pays a constant dividend yield, say $y$. Stock follows brownian motion $\\$
$dS_{t} = S_{t}(\...
1
vote
1answer
65 views
Properties of integrated GBM
(I asked this question on MSE but I think it might have more success here)
Good day,
I was going over some exercises and I stumbled upon a question that, for its solution, requires me to find/...
0
votes
1answer
72 views
Stochastic Interest Rates in Option pricing
My lecturer has written the slide below. The function B^T(t) is a zero coupon bond. I don't understand how V(t) can be a negative integral from 0 to ...
2
votes
1answer
67 views
Brownian motion and Stochastic Integration
I have two questions relating stochastic integration which perhaps could be answered together.
First question:
First of all, I don't really understand why we can't use Riemann-Stieltjes integration ...
0
votes
1answer
78 views
Normal vs. Lognormal Greeks for Negative Rates Options
My understanding is that for some of the G10 currencies with negative rates (CHF, EUR), Swaption and Cap / Floor prices are quoted in terms of BOTH, normal and log-normal Vols. That in itself is not ...
2
votes
0answers
65 views
Option on $ \left( \int_0^T dW_t \right)^2$
Silly question, but how would you actually price
$$
E_0 \left( \left( \int_0^T dW_t \right)^2 - K \right)_+
$$
where $dW_t$ are standard Brownian motions. Is there a closed form analytical solution?...
0
votes
0answers
40 views
Compo/Quanto Adjustment & Multivariate Ito
Related to the issue that I have raised here, I am facing another question. As the rule here is 1 question / 1 post, I take the opportunity to ask it below:
By exploring StackExchange, I noticed the ...
1
vote
1answer
36 views
Differential bond price stochastic rates
Suppose that the short rate follows the process
$$dr(t) = a(t, r(t))dt + \sigma(t, r(t))dW(t)$$
If $B(t) = exp(-\int_0^t r(u) d u)$, can one still write the differential $dB(t)$ a-la-Ito? Thanks.
1
vote
2answers
64 views
Proof that $f$ is continuous if and only if it has 0 quadratic variation?
I understand that $f$ continuous $\Rightarrow Q(f) = 0$ where this is defined over a bounded interval [0,T] as then we may use uniform continuity and the mean value theorem. But I am not sure how the ...
0
votes
0answers
54 views
Hedging a short position in the Lookback Option
SOLUTION
I got the correct answer using this formula
$X_2(HH)=(1+r)*[X_1(H)-\Delta_1(H)*S_1(H)]+\Delta_1(H)*S_2(HH)$
$(1+0.25)[2.24-(.06667*8)]+0.06667*16=3.20$
6
votes
2answers
120 views
Radon Nikodym derivative when changing numeraires
I note from Wikipedia that if $Q$ and $Q^N$ are two measures corresponding to numeraires $M$ and $N$, then the Radon Nikodym derivative is given by: $$\frac{dQ^N}{dQ} = \frac{M(0)}{M(T)}\frac{N(T)}{N(...
1
vote
1answer
66 views
Idea of using logarithm for solving SDE in Black-Scholes model
In the Black-Scholes model they consider that the stock follows this stochastic differential equation: $$ dS = \mu S dt + \sigma S\ dW $$
I was wondering, was it common at the time they work on this ...
2
votes
1answer
47 views
Exact solution stock price with Vasicek interest rate model
Define two correlated stock price- and interest rate (Vasicek) processes, governed by the Wiener processes $W^{S}(t)$ and $W^{r}(t)$
$$dS(t)=r(t)S(t)dt+\sigma S(t)dW^{S}(t)$$
$$dr(t)=\kappa(\theta-r(...
1
vote
0answers
37 views
Simulate correlated Brownian motions conditioned on future state(s)
Consider a model defined by 2 geometric Brownian motions
$$dY_{1}(t) = \sigma_{2} Y_{1}(t)dW_{1}(t)$$
$$dY_{2}(t) = \sigma_{2} Y_{2}(t)dW_{2}(t)$$
with $Y_{1}(0) = y_{1}$, $Y_{2}=y_{2}$ and $dW_{1}(...
2
votes
1answer
54 views
Generalization of Ito's Lemma to composite function
Ito's Lemma gives that for a function $F$ of a stochastic variable $X$, $dF = \frac{dF}{dX}dX + \frac{1}{2}\frac{d^2F}{dX^2}dt$
Given a stochastic differential equation $dS = a(S) dt + b(S) dX$ and a ...
4
votes
1answer
188 views
Are all change of measure operations between equivalent probability measures Doléans-Dade exponentials?
Let $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space, where $\mathbb{F}=\left(\mathcal{F}\right)_{t\in[0;T]}$ and $\mathcal{F}=\mathcal{F}_T$. Let $(W_t)_{t\in[0;T]}$ be ...
1
vote
1answer
62 views
Derivation of stock price formula John C. Hull 9th Ed p309
It says assuming a no-uncertainty Weiner process that models stock price:
$$
\Delta S = \mu S\Delta t
$$
Can be rearranged to (after taking the limit of $\Delta t \to 0$...
$$
\frac{dS}{S}=\mu dt
$$
...
1
vote
1answer
70 views
Integration of a deterministic function w.r.t. a Brownian motion
Help me solve this problem:
Let $W_t$ be a Brownian motion and suppose
$X_t = \int_{0}^{t}\delta _{s}dW_{s}$
where $\delta _{s}$ is a deterministic function. Then show that $X_t$ is a Gaussian ...
4
votes
1answer
53 views
Expected Value of Mean-Reverting Jump Process
I cant see the link between my method of calculation and the method done in the book Cartea and Jaimungal (Algorithmic and High Frequency Trading, page 220.) We have a mean-reverting process
$$d\mu_t=-...
2
votes
3answers
232 views
How can I use the Radon-Nikodym theorem to show that forward measure is indeed measure?
The following statements are taken from the Wikipedia page for forward measure.
Let
$$B(T)=\exp \left(\int _{0}^{T}r(u)\,du\right)$$
be the bank account or money market account numeraire and
$...
2
votes
0answers
32 views
Building up an Economic Scenario Generator [closed]
I am trying to build an Economic Scenario Generator in VBA or Python. Can anyone please help me with some good resources which I can follow or some basic procedures which explains how to go about in ...
1
vote
1answer
41 views
Proving an Identity between a pair of correlated Wiener processes
Suppose we have the following subordinated stochastic differential equations:
$dR(t)=\mu dt+\sigma (Y(t))dW_{1}(t)$
$dY(t)=f(Y)dt+g(Y)dW_{2}(t)$,
where $W_i$'s are standard Wiener process such that ...
0
votes
1answer
65 views
Trouble understanding Notation in Stochastic Calculus (wedge symbol ā§)
I am a beginner in Stochastic Calculus. I am having trouble understanding the meaning behind a specific notation which appears in the topic of Ito process which in differential notation can be written ...
1
vote
1answer
72 views
Compute dZ(t) : Ito's formula/lemma
We need to find dZ(t). I know I have to use Ito's formula. But I am confused because in the Ito's formula we have f(y,t) is a twice differentiable function with two variables
But here Z(t) = 1/(2+x(t)...
1
vote
0answers
63 views
On Geometric Brownian motion and Itô's formula
Let $S_t$ be a geometric brownian motion such as
$$d S(t) = rS(t)dt +\sigma S(t)dW(t),$$
where $W$ is a standard Brownian motion.
With ItƓ's lemma and formulas $(dt)^2=dtdW_t=dW_tdt=0$ and $(dW_t)^2=...
3
votes
1answer
119 views
question about spot/vol correlation
In this paper The Interplay between Stochastic Volatility and Correlations in Equity Autocallables by Alvise De Col, Patrick Kuppinger (2017) https://papers.ssrn.com/sol3/papers.cfm?abstract_id=...
0
votes
0answers
25 views
Deriving coupling equation(s) for Heston Stochastic Volatility Model
In Bergomi Smile Dynamics (2003) Section 2.1 we are given the following coupled equations for the mean and for the variance of the hedger's portfolio:
$
\begin{align*}
\frac{dm}{dt} + \mathcal{L}m - ...
4
votes
1answer
92 views
Advantages of pathwise calculus over stochastic calculus in continuous self-financing trading models
I am new to stochastic calculus but the statement below confuses me:
Beside the issue of the impossible consensus on a probability measure,
the representation of the gain from trading lacks a ...
0
votes
0answers
23 views
Order of expectation versus expectation of order (error terms in Taylor expansion)
Given a payoff function $F(X)$ of a random variable $X$, and a Taylor expansion of $F(X)$ around $X=a$, then the expecation of $F(X)$ can be written as
$$
E[F(X)] = F(a) + E[ O((X-a))]
$$
Under what ...
2
votes
1answer
76 views
Why are quadratic variation and rough paths so important in quantitative finance?
I am new to quant finance - come from a mathematics background. I am starting stochastic calculus and have been particularly interested in some papers pathwise integration and rough calculus in ...
1
vote
1answer
84 views
Expectations in Infinite Probability Spaces with Sub Sigma-Algebras [closed]
Let $X$ be an (integrable) random variable on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Suppose $\mathcal{G}$ is a sub-$\sigma$-algebra of $\mathcal{F}$ and let $Z=\mathbb{E}(X|\mathcal{...
