Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
1
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0answers
17 views
From one period to multi period risk neutral pricing
For a one period economy, we have the price of an asset as:
$ p_0 = E^Q [p_1 * \frac {B0}{B1}] $
where $B0 = e^{-r_0}$ = time 0 price of risk free bond maturing at time =1 and $r_0$ is known at t0. ...
0
votes
0answers
35 views
Correlated GBM and OU processes
I want to model two different stochastic processes, such that:
$X_t , V_t$ are correlated with coefficient $\rho$. Where:
$\frac{dX_t}{X_t}=\mu_1dt+\sigma_1 dW_{1,t}$ and $dV_t=\theta(\mu_2-V_t)dt+\...
1
vote
3answers
89 views
Need help to interpret the definition of a diffusion process
https://studentportalen.uu.se/uusp-filearea-tool/download.action?nodeId=1134155&toolAttachmentId=218130
In these lecture notes at page 15 and 16 I am looking at the definition of diffusion ...
3
votes
0answers
46 views
Price of a stochastic game between an agent and the market
In the article Pricing via utility maximization and entropy from Richard Rouge and Nicole El Karoui, they define the value function of the optimization problem as
\begin{align}
V(x,C) = \dfrac{1}{\...
0
votes
2answers
127 views
Integral of Brownian Motion w.r.t Time: what is wrong with this solution? [duplicate]
My question is about a stochastic integral of brownian motion w.r.t time.
Let $W(t)$ the Wiener process (or brownian motion). I want to calculate this:
\begin{eqnarray}
X(t)=\int_{0}^t dt' W(t').
\...
2
votes
1answer
70 views
The duality of the free energy and relative entropy used to deduce deduce the stochastic game between an agent and the market
I am reading the article Pricing via utility maximization and entropy by Richard Rouge and Nicole El Karoui. They talk about the relative entropy of a probability measure $Q$ with respect to the ...
2
votes
1answer
53 views
How to deduce the formula of the wealth process of a stochastic volatility model?
I am reading the paper Solution of the HJB Equations Involved
in Utility-Based Pricing from Daniel Hernandez and Shuenn Jyi Sheu.
The authors consider the utility function $U: \mathbb{R} \to \...
0
votes
1answer
41 views
standard brownian vs brownian motion
We say Xt with paramters (µ,σ) is brownian process if (Xt-s - X t) ~N (µs,σ2 s) AMONG other conditons .
Here we don't speak about any particular distribution for X t. We only say it is a brownian ...
1
vote
0answers
46 views
Pricing caplet with Bachelier (normal dynamic) using forward measure
I'm trying to price caplet with Bachelier under forward measure, but I can't find any solution.
Remind that Bachelier assumed rates follow a normal dynamic. So here what I was doing :
$C_t(T,T+d)$ ...
2
votes
2answers
94 views
Ho Lee model in Baxter&Rennie
I am currentyl reading Baxter&Rennie and I have a difficulty with understanding a derivation of formula for one function, $g(x,t,T)$ (this can be found on page 152 in the book). I know that there ...
1
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0answers
81 views
Compo Feature in Asian Option on Futures
I'm pricing an Asian option on futures using Turnbull–Wakeman (other suggestions welcome) where the average is defined as
$A _ { t _ { 1 } , t _ { n } } ^ { A , f } = \frac { 1 } { n } \sum _ { i = 1 }...
0
votes
0answers
31 views
If you have normally distributed returns, shouldn't you have the same adjustment factor as lognormally distributed?
We know that when using lognormal returns, the number you need to plug in is not the apparent return, but $\mu-\sigma^2/2$ because what you really have is, in essence, (1) a deterministic growth of $\...
2
votes
1answer
55 views
What does $\int dS \phi (S - K)$ mean in Gatheral's book?
In Gatheral's book on stochastic volatility, he writes the price of an option as $$\int_K^\infty dS \phi (S - K)$$
where $\phi$ is a density.
Where does this come from?
I have multiple questions:
...
1
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0answers
40 views
Extension of HJM to multiple factors
The HJM model calibrates the entire forward curve using the existing yield curve data and this results in the following expression for its instantaneous forward rate-
$$df(t,T)=\sigma(t,T)\int_0^T\...
3
votes
1answer
73 views
Application of Vibrato Montecarlo methods
Ciao,
I was studying Vibrato Montecarlo methods and I came up with a very simple question: what is an real application of this method? Let me explain.
In short the main idea of the method is the ...
0
votes
0answers
56 views
Discretizing a Continuous Time Stochastic Volatility Model
Forgive me for cross-posting.
I have the following continuous time SDE for a stochastic volatility model. $S_t$ is the price, and $v_t$ is a variance process.
$$
dS_t = \mu S_tdt + \sqrt{v_t}S_t dB_{...
0
votes
0answers
26 views
Impact analysis of parameters in the 2 Factor Hull White Model
Through the 2-Factor-Hull White Model you can model the yield curve if you have the parameters $a, b, \sigma, \eta$ given.
Is there any way to measure the impact of these parameters on the yield ...
0
votes
0answers
79 views
Dynamics of an option on a future
I have trouble understanding why $$V_s=exp(\int_s^t r_u du) \int_s^t exp(−\int_t^v r_u du)\theta_v dW_v$$ is the solution to the following SDE
$dV_s=\theta_s dW_s+r_s V_s ds$.
I tried of course with ...
2
votes
0answers
32 views
Prove the given stochastic integral are equally distributed
Let $W^i_t$ and $W_t$ be pairwise independent Brownian motions for $i \in \{1, \dots , d\}$.
Let $X_t^i$ be $d$ independent Ornstein–Uhlenbeck processes for $i \in \{1, \dots , d\}$, i.e. each $X_t^i$...
1
vote
1answer
118 views
Two papers - two different solutions of the Ornstein-Uhlenbeck process
Bernal 2016 says that the solution of
$$ dr_{t}=\lambda*(\mu-r_{t})*dt+\sigma dW_{t} \qquad (eq.1) $$
equals
$$ r_{t}=r_0*exp(-\lambda t)+\mu(1-exp(-\lambda t))+\sigma \int_{0}^{t} exp(-\lambda t)...
1
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0answers
46 views
Bond prices at future times under Vasick one-factor model
In Vasicek one-factor model (and in other affine models), the price of a zero-coupon bond at time $t$ conditional on the information at this time is
$$P(t,T) = E[e^{-\int^T_tr(u)du}|F_t] = A(t,T)e^{-...
2
votes
1answer
93 views
Calculating the stochastic integral of $\exp(-rt)S_t$
I am currently reading lecture notes which aim to show that if
$$
S_t = S_0 \exp (\mu t + \sigma W_t)
$$
then, under the probability measure $\tilde{\mathbb{P}}$ with density
$$
\gamma_T = \exp (c W_T ...
1
vote
0answers
84 views
Applying Ito's formula to complex functions
Within my lecture notes, the following definition is given:
We say that the stochastic process $X_t$ has stochastic differential
$$
dX_t = b_t dt + \sigma_t dW_t
$$
if and only if
$$
X_t = ...
1
vote
0answers
35 views
autocorrelation function of the trending OU process
What is the autocorrelation function of the trending Ornstein-Uhlenbeck (OU) process?
First, the OU process
$dX_t = -\frac{1}{\mu} X_t + \sqrt{\frac{2\sigma^2}{\mu}} dW_t $
generates coloured noise ...
1
vote
1answer
60 views
Girsanov's Theorem for Multiple Risky Assets
Girsanov's theorem provides the measure transformation from probability measure P to Q such that-
$dW_t^Q=dW_t^P+\lambda dt\implies \xi_tW_t^Q$ is a martingale under the P measure where $\xi_t=e^{-\...
2
votes
1answer
75 views
Black Scholes in the case of dividends
Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a ...
2
votes
1answer
135 views
Show that the Ito integral is Gaussian
Let $f(t), 0 \leq t \leq T$ be a deterministic function with $f(t) = \sum_{i=1}^na_{i-1}1_[t_{i=1}, t_i)(t)$ with $0 \leq t_0<t_1<...<t_{n-1} = T$. Show that the stochastic integral $I_t(f) ...
-2
votes
1answer
100 views
squaring stochastic calculus and other solutions [closed]
It is well-known that the solution to the stochastic SDE
$$ dS = S_0(\mu dt + \sigma dWt) $$
is
$$ S_t=S_0 e^{(\mu-\frac{\sigma^2}{2})t+W_t} $$
Were $\sigma=0$, this is simply the formula for ...
2
votes
1answer
84 views
Distribution in Heston
$$dV_t=-k(V_t-1)dt+ \epsilon\sqrt{V_t}dW_t$$
$W_t$ is wiener process and the rest is just some parameters.
For $T_{i+1}>T_{i}$ how do I find the expectation and variance of $V_{T_{i+1}}$ ...
1
vote
1answer
98 views
Finding the process of $X/Y$
This comes from Mark Joshi's concepts of mathematical finance exercise 4 chapter 11.
If
$$dX_t = \alpha X_t dt + \beta X_t dW_t$$
$$dY_t = \alpha Y_t dt + \gamma Y_t d\tilde{W}_t$$
with $W$ ...
3
votes
2answers
146 views
Isn't GBM the equivalent of adding infinitessimally small normally distributed returns?
The classic treatment of GBM for asset pricing leads to a point where eventually one gets a solution that is the same as assuming an underlying arithmetic Brownian motion, $X_t$, which has (over unit ...
2
votes
1answer
78 views
How to derive the dynamic of the log forward price?
I have the following Schwartz model:
$$dS_t=a(\mu-\ln S_t)S_tdt+\sigma S_tdW_t$$
$$X_t=\ln S_t$$
$$dX_t=a(\hat{\mu}-X_t)dt+\sigma dW_t$$
with $\hat{\mu}=\mu-\frac{\sigma^2}{2a}\sigma$
$$F_t(T)= \exp\...
1
vote
1answer
91 views
Properties of Stochastic Exponential
Let $\{X_t\}_{t \ge 0},\{Y_t\}_{t \ge 0}$ be a continuous semi-martingale with $X_0 = Y_0 = 0$, let ${\cal E}(X)$ to be the unique solution of:
$dZ_t = Z_t dX_t$ with $Z_0=1$.
We can show that ${\cal ...
4
votes
1answer
400 views
Integral of Wiener process w.r.t. time
I have a doubt with regards to the calculation of the below integral-
$\int_0^t W_sds$
where $W_s$ is the Wiener Process.
This has been solved very ably in the following page. It turns out to be a ...
3
votes
0answers
48 views
Computing Malliavin Derivative for European Call Payoff
Let $X_t$ be a continuous local-martingale modeling the stock price given by
$$
X_t = \int_0^t \sigma_t(T,K)dW_t
,
$$
and $\sigma_t(T,K)$ is an $L^2$-measurable process not adapted to $W_t$'s ...
0
votes
0answers
89 views
Ito's & correlated Brownian proof
$W_{1}(t)$ , $W_{2}(t)$ are correlated Brownian motion with quadratic-var $\langle W_{1}, W_{2}\rangle_{t}=\rho.$
$Z_{t}=e^{\int_{0}^{t}W_{1}(s)ds}+e^{W_{1}(t)}$ and $X_{t}=\cos(W_{2}(t))$
Find:
$$...
3
votes
2answers
124 views
Show that the two solutions of the SDE are equivalent
I have a process:
$$dr_t = (W_t^1 - ar_t)dt +\sigma dW_t^2$$
where $W_t^1$ and $W_t^2$ are brownian motions with instantaneous correlation coefficient $\rho$.
I want to show that the solution of this ...
1
vote
1answer
127 views
Have I used correct state space formulation of Bivariate Trending OU process for Kalman Filter estimation?
Introduction
I'm trying to estimate the parameters of an Ornstein Uhlenbeck process for a risky asset using the Kalman Filter but have doubts about the state space formulation that I am using. Also, ...
6
votes
1answer
129 views
Integral of the OU (Ornstein Uhlenbeck) process conditioned on hitting a threshold value for the first time
Let say I have a zero-mean OU process as follows:
$dX_t = -\alpha X_t + dW_t$
The process starts at $x_0 = 0%$ and I'm interested in the event in which the process hits the value $x_{\tau} = a$ for ...
2
votes
1answer
256 views
How to take the differential of a stochastic integral?
Denote
$$X_t = \int^t_0\sigma e^{-k(t-s)}dW_s$$
here $W_s$ is the Brownian motion, $k,\sigma$ are constants.
I want to calculate $d X_t$ and the variance $Var[X_t].$ I know how to take the ...
0
votes
0answers
36 views
Relationship between two Brownian motions generated by equivalent martingale measures
If $Q^1$ and $Q^2$ is equivalent martingale measures ($Q^1\sim Q^2$) and the following condition holds: $$\frac{dQ^2}{dQ^1}=(\frac{X_T}{X_0})^q$$
for some positive constant $q$ and the $Q^1$ dynamics ...
0
votes
2answers
94 views
Show a process is Martingale
$$Z(t)=(\frac{S(t)}{H})^p$$where $S$ has a standard Black-scholes Dynamics for a stock, $H$ is a postive constant and $p =1 - \frac{2r}{\sigma^2}$. How can I show that $Z(t)/Z(0)$ is a postive Q-...
0
votes
0answers
81 views
What is the distribution of 1/t*[integral from 0 to t of (Wt*dt)]? [duplicate]
$$\frac{1}{t}\int_0^t W_t dt $$
I know that Brownian motion is usually distributed normally with mean 0 and variance t but I'm not sure how to find the distribution within this integral? The answer ...
0
votes
0answers
49 views
Computing the Expectation to a Max function
if $X_T$ is log-normally distributed and $k$ is a constant, how do I compute:
$$E[\max(X_T-k,0)]$$
I can compute $E[X_T-k]$ and $P(X_T-k>0)$. I was thinking that an approach will be compute to $$E[...
3
votes
1answer
350 views
Equivalent Martingale Measure(EMM) of Inverse of Stock Price
I met this question says how to price a vanilla call option $C(St,t,T,K) = \frac{1}{S_T}$which pays the inverse of a stock $V_{t} = \frac{1}{S_{t}}$ at maturity if the stock price follows a geometric ...
1
vote
0answers
68 views
Pricing a structured note instrument
I am trying to work out the following fixed income problem, where I am asked to price a structured note in Excel, which seems to me to be a reverse collar.
My purpose was replicating this structured ...
5
votes
0answers
81 views
Complete Financial Market: Integrability condition for Contingent Claims
Consider an arbitrage-free and complete financial market with underlying filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\,\in\,[0,T]},\mathbb{Q})$, where $T\in(0,\infty)$ is ...
1
vote
1answer
64 views
Stochastic Calculus: How to test for dependency of random variables
If I let $g(x)$ be a deterministic function of a real variable $x$ and define $X(t)$ as:
$$X_T=\int_{0}^{T}f(u)dW_u$$ with $W_t$ being a wiener process.
For $s<t$, Will $X_s$ and $X_s-X_t$ then be ...
1
vote
0answers
65 views
Geometric Brownian Motion: Drawdown as a function of time
Suppose I have a strategy (model it as the usual geometric Brownian motion with a drift). Question is, how does max drawdown grow as a function of duration?
1
vote
1answer
115 views
Test if a process (with no drift) is a martingale
Consider the process
$$Z(t)=\int_{0}^{t} \frac{u^a}{t^a}dW_u$$ for some real constant $a$ and $W_t$ is a wiener process. I want to check whether this process is a $F_t^W$-martingale.
I noticed Lemma 4....
