Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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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 ...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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}\...
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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? $$df = f'(B_t) dB_t + \frac{1}{2} f''(B_t) dt$$
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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. ...
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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 $\int_0^T W_t \ln(W_t) dW_t$?
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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 ...
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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 ...
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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 $r_{t+1}$?
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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_{...
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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 + \...
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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^\...
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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-...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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 $f(t,x)$...
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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 ...
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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}{\...
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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 ...
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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 ...
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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 $T \rightarrow \infty$, ...
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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 ...
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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 ...
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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 + \...
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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 ...
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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 ...
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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 ...
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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 ...
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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_{...
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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 \...
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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 ...
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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, $\...
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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 ...
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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 $\...
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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 ...
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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 $\...
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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 ...
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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, ...
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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+\...
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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 ...
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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 ...
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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 ...
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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 $...

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