Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
584
questions
0
votes
0answers
35 views
American put option with $r=0$
What the value of American put option in the case when $r=0$ with the payoff $\max(K-S(T),0)$, by using the Snell envelope Theorem?
Snell envelope theorem: the optimal value process $V$ is the Snell ...
0
votes
1answer
54 views
Let $W_t$ denote a standard Brownian motion. Evaluate this integral [closed]
$$
\int_{0}^{t}d(W_{u}^2)
$$
How can I deal with this kind of problem? If there is no function given to apply Itô's formula.
2
votes
1answer
113 views
Discounted price process - martingale
I have a process $S_{t}=S_{0}e^{\left(r-q\right)t+mt+X_{t}}$, where $X_t$ is a Levy process and I want to check for which $m$ the process $e^{-(r-q)t}S_t$ is a martingale. The third condition of a ...
0
votes
1answer
64 views
First Hitting Time and Monte Carlo simulation
I am interested in implementing a Monte Carlo simulation in Python of a first hitting time (first passage time) of an Ornstein-Uhlenbeck process (or similar). Specifically interested in fatter tails ...
3
votes
1answer
111 views
Correlated Stochastic Processes
Let say, I have 2 stochastic processes:
$$\begin{align}
dS_1 &= \left( r - q_1 \right)S_1 dt + \sigma_1 S_1 dW_1
\\
dS_2 &= \left( r - q_2 \right)S_2 dt + \sigma_2 S_2 dW_2
\end{align}$$
The ...
3
votes
0answers
73 views
Application of Ito's lemma relating to bond price
I'm interested in solving the following questions but I am confused on the second part because I do not know how to define/calculate the interest per "unit time", which I'm guessing is ...
3
votes
0answers
64 views
American Options in Merton's (1976) Jump Model
@LocalVolatility proves in this stellar answer that European call option prices in the Merton jump diffusion model are given by
$$ C_{Merton}(S_0,r,q,\sigma,K,T) = \sum_{n=0}^\infty e^{-\lambda T}\...
0
votes
0answers
38 views
Why can't we ignore the second term in Taylor Expansion in Ito's lemma? [duplicate]
Why can't we neglect the $dt$ there?
$$df = f'(B_t) dB_t + \frac{1}{2} f''(B_t) dt$$
0
votes
0answers
33 views
Time-changed Levy processes
in different articles the authors use the CIR process to change the time in different processes. They mostly use the CGMY, VG, NIG etc process, but I haven't noticed anybody using the Kou process. ...
1
vote
1answer
283 views
If $W_t$ is standard Brownian motion, what is $\int_0^T W_t \ln(W_t) dW_t$?
If $W_t$ is standard Brownian motion, what is meant by $\int_0^T W_t dW_t$ in finance?
Furthermore, what then is the meaning of $\int_0^T W_t \ln(W_t) dW_t$?
8
votes
2answers
406 views
Anticipating stochastic integral $\int_0^T W_T dW_t$
Using basic techniques from Malliavin calculus it can be shown that
$$
\int_0^T W_T dW_t = W_T^2 - T
$$
As can be seen the above integral is a non-adapted stochastic integral.
We also know using Ito ...
2
votes
0answers
50 views
Explicit expression for option prices in SABR?
I am trying to get a grip of the current state of research regarding option pricing in the SABR model.
Am I correct in that, so far, there is no known general formula for the option price in the SABR ...
1
vote
1answer
72 views
How can the increments of a CIR process be derived?
For a CIR process, which has SDE
$$
dr_t = \alpha (\mu - r_t) dt + \sigma \sqrt{r_t} dW_t
$$
how can I derive the increments over the discrete time-interval from $r_t$ to $r_{t+1}$?
3
votes
0answers
36 views
Characteristic function of time-changed Levy processes
Let $X_t$ be a Levy process, and $Y_t$ be a subordinator i.e. process with nondecreasing trajectories. I have to find characteristic function of $X_{Y_t}$. I know that I have to calculate:
$$E[e^{iuX_{...
2
votes
1answer
109 views
Stochastic growth model
In this problem we consider a model of stochastic growth. In particular, consider the
following system of SDEs:
\begin{align}
dX_t &= Y_t dt + \sigma_XdZ_{1t}\\
dY_t &= -\lambda Y_t dt + \...
0
votes
0answers
63 views
Price of a Forward Contract
I have the following,
Let ${F_t,t\geq0}$ be the price process of the forward contract on the
risky asset with maturity $T' > 0$. Since interest rates are
deterministic, we have
$$F_t=S_t\ e^{r(T^\...
2
votes
0answers
31 views
solution of Jump Diffusion SDE (Kou, Merton)
Hey in Kou 2002 paper he write SDE as:
$$\frac{dS(t)}{S(t-)}=\mu dt+\sigma dW(t)+d\left( \sum_{i=1}^{N(t)}\left( V_{i}-1\right) \right)$$
Is it equivalent with:
$$dS(t)=S(t)\mu dt+S(t)\sigma dW(t)+S(t-...
1
vote
1answer
53 views
Characterizing distribution of a stochastic intergal
characterize the distribution of $\int_0^T f(t)Z_tdt$. In
particular, verify that it is a Gaussian distribution and compute its moments.
3
votes
1answer
146 views
Mean Reverting Heston Model?
Is there a name for a variation on the Heston Stochastic Process Model where not only the underlying volatility but the asset price itself is mean-reverting? I'm looking to model long term equity ...
1
vote
1answer
112 views
Covariance of mean-reverting Vasicek process?
I am dealing with a mean-reverting Vasicek process defined as:
\begin{equation}
S_t = S_0 e^{-at} + b(1-e^{(-at)}) + \sigma e^{(-at)} \int_{0}^{t} e^{(-as)} \ W_t
\end{equation}
I want to ...
1
vote
0answers
25 views
Change of numeraire between t1-forward mesure and t2-forward mesure
Let denote $\mathbb{Q}_{t_1}$ the $t_1$-forward mesure associated to zero coupon bond $B(.,t_1)$.
Let denote $\mathbb{Q}_{t_2}$ the $t_2$-forward mesure associated to zero coupon bond $B(.,t_2)$.
I am ...
1
vote
1answer
77 views
Transition density of geometric Brownian motion with time-dependent drift and volatility
Can you provide a reference to the transition density of the scalar geometric Brownian Motion with time-dependent drift and volatility, i.e. the scalar process $X = (X_t)_{t\geq 0}$ defined by the SDE
...
2
votes
0answers
49 views
Correct application of Feynman Kac formula
I have a question on Feynman-Kac formula but can I ask the community if I have done it correctly? If no, may you point out to where I went wrong? Thanks!
The original FK formula states: Assume $f(t,x)$...
5
votes
2answers
354 views
Clarification on Deriving Ito's Lemma
The classical approach to deriving Ito's Lemma is to assume we have some smooth function $f(x,t)$ which is at least twice differentiable in the first argument and continuously differentiable in the ...
1
vote
0answers
62 views
How to solve this particular PDE using Feynman-Kac formula?
I have to solve the PDE
$$
\begin{align}
\frac{\partial F}{\partial t} + \frac{1}{2}\frac{\partial^2 F}{\partial x^2} + \frac{1}{2}\frac{\partial^2 F}{\partial y^2} + \frac{1}{2}\frac{\partial^2 F}{\...
1
vote
1answer
145 views
Black Scholes to Heat Equation
Equation (2) was derived by setting r=0 in the Black-Scholes equation for the Bachelier model (1).
Can someone please help me understand all the steps for how we get from the heat equation under time ...
0
votes
1answer
74 views
Infinitesimal generator - Is it obtained from a stochastic process or It can construct the process
We can see here that the generator is an operator which can be determined for a stochastic process. But, in the answers and comments here we can see that the brownian motion on sphere can be ...
1
vote
0answers
74 views
Option that never expires
I have been struggling with the problem below for quite some time now. I really don't know how to approach it. All I could think of is to use the Black-Scholes formula with $T \rightarrow \infty$, ...
2
votes
4answers
513 views
Ito Integral of functions of Brownian motion
How does one show that:
$$ \mathbb{E}\left[ \int f(W_s)dWs \right] = 0 $$
For all $f()$ that are powers of $W(s)$?? I assume that one would have to go via the definition of Ito integral and express ...
4
votes
0answers
70 views
Why is the Schöbel-Zhu model affine?
In the Schöbel-Zhu model, the stochastic volatility process is $dv_t=\kappa(\theta-v_t)dt+\sigma dW_t$.
The characteristic function of the stock process can be found by arguing that the model is ...
0
votes
0answers
59 views
Arbitrage free pricing of option to trade stocks
Consider Black-Scholes model with constant interest rate r and stocks with prices $S_t^A$ and $S_t^B$ that satisfy the SDE's $dS_t^A = S_t^A(\mu^A dt + \sigma^A dB_t)$ and $dS_t^B = S_t^B(\mu^B dt + \...
2
votes
0answers
43 views
Solution to Stock Price SDE with mean reversion [duplicate]
Suppose $S_t$ follows the process (notice the $S_t$ term in the diffusion part):
$$ S_t := S_0 + \int_{h=t_0}^{h=t}\alpha(\mu -S_h)dh + \int_{h=t_0}^{h=t}\sigma S_h dW(h) $$.
I actually don't know how ...
3
votes
2answers
605 views
Intuition for Martingale Representation Theorem
Can you please explain Martingale Representation Theorem in a non-technical way that what is it and why is it required?
Most of the stuffs I studied so far are ...
6
votes
2answers
101 views
Solution for a SDE for a Bond found in Bugard & Kjaer
I'm going over the paper -Partial Differential Equation Representation of Derivatives with Bilateral Counterparty Risk and Funding Costs- from Burgard and Kjaer. There the following SDE is given for ...
4
votes
0answers
61 views
Confused about discretization
I am reading a paper here: https://pdfs.semanticscholar.org/5f91/2d46b02b03230a4ffaaa42d655b2b6147d56.pdf
The following is my confusion.
The paper has the following continuous time model for the price ...
2
votes
0answers
25 views
Expression for the expectation of Integrated variance in case of GARCH(1,1) process
I have the following SDE (GARCH(1,1)) for the instantaneous variance:
$$ d\sigma_t^2 = \kappa (\theta - \sigma_t^2) dt + \psi \sigma_t^2 dW_t $$
I would like to find an expression for $IV_t = E[\int_{...
1
vote
1answer
63 views
Covariation of Ito semimartingales
If we have two Ito semimartingales over $[0,T]$:
$$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
147 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
68 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, $\...
4
votes
0answers
77 views
mixing fractional Brownian motions
Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by
$$W_t^1 = \rho W_t^2+\sqrt{1-\rho^2}Z_t$$
where $W_t^{2}$ and $Z_t$ are independent of each other.
My question then: is there ...
3
votes
1answer
183 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 $\...
2
votes
1answer
117 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
25 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
125 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 Gyöngy's theorem)? I imagine this would be covered by a paper or by a standard financial math text, but I could not ...
1
vote
1answer
70 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
110 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
21 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
115 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
118 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
27 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 $...
