Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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44 views

What exactly does a “closed-form solution” mean

The question is pretty trivial but still I am not 100% sure about it. For example, geometric Brownian motion follows the below SDE: $$dS_t=S_t(\mu dt+\sigma dW_t)$$ and after some manipulation it is ...
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2answers
643 views

A stochastic differential equation

Consider the following stochastic differential equation (SDE) $$d X_s= \mu (X_s + b)ds + \sigma X_s d w_s $$ where constants $\mu, \sigma, b > 0$ and initial position $X_0$ are given. If $b=0$, ...
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95 views

Pricing of $(S(T_0)-S(T))^+$

Problem: Consider a new derivative that at time $T$ pays $Y =(S(T_0) − S(T))^+$ where $0 < T_0 < T$ is a fixed date. (i) Show that the arbitrage-free of Y at time $t = T_0$ is given by $\pi_{...
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188 views

How can I learn stochastic process & stochastic calculus in two weeks? [closed]

I am going for an interview for a quant job. The interview will focus on my mathematical knowledge about stochastic process & stochastic calculus, and I believe I will definitely be asked to solve ...
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22 views

Future forward convexity adjustment as the expected profit from reinvesting margin payments?

Having looked at the formula for the convexity adjustment as a function of the covariance between rates accruing till maturity and asset price, I have an intuition that the difference between fair ...
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62 views

SDF derivation by a stochastic process

I have a stochastic process to model the stochastic discount factor (SDF) with M: \begin{equation} dM_t = aM_tdt + bM_t d Z_t \end{equation} where, $Z_t$ is a standard brownian motion. How do I show ...
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1answer
111 views

What is the interpretation if the real world measure $\mathbb P$ is equal to the martingale measure $\mathbb Q$

Out of interest, is there anything noteworthy about a market when its real world measure $\mathbb P$ is actually also its martingale measure. In other words the real world measure $\mathbb P$ is equal ...
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1answer
78 views

Calibrating the Ornstein-Uhlenbeck process with an additional parameter

Firstly I find the spread between two cointegrated time-series $Y_t$ and $Z_t$ by finding the best slope parameter $\beta$ in the equation $spread_t = Y_t - \beta Z_t$ (via Cointegrated Dickey-Fuller ...
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56 views

What is the difference between the geometric brownian motion and cumulative product of percentage returns?

I wonder why the following code: one using GBM and the other using cumulative product of normally distributed percentage returns slightly different values. ...
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128 views

Summary of Stochastic Derivatives, Integrals, Expectations, and Variances

I wanted to make a summary table of stochastic functions to improve my understanding. Maybe the following should be a wiki page on this site so others can add functions and examples? Does the ...
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1answer
142 views

Trouble With Applying Ito's Lemma

I am having trouble applying Ito's Formula to the following: Let $Z_t = W_{1t}^2 e^{W_{1t}+ \int_0^t W_{3s}dW_{2s}}$. Find $dZ_t$. $W_1,W_2,W_3$ are independent Brownian motions. I know the formula ...
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122 views

Is it possibile to use Ito Formula here?

I have this process: $dY_s^y=\alpha(s,Y_s^y)ds + \frac{1}{2}\beta^2(Y_s^y)^2dW_s$ with inital value $Y_s^y=y$. Moreover $\alpha(s,y)$ is a linear function in $y$ and bounded is $s$. I was wondering if ...
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1answer
158 views

the order of integral of Brownian motion

When we want to obtain the order of $\int_{0}^{T} B_{t} d t$, we can use the scale property of Brownian motion. Let $B$ be a Brownian motion. Is the order of $\int_{0}^{T} B_{t} d t$ correctly ...
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189 views

Reduced form of credit model

The price for a simple credit bond, where a credit event is modeled as the first jump of a Poisson process $N$, with stochastic hazard rate $\lambda$, is given by $$P_t = P(t, \lambda, N)$$ such that, ...
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116 views

Intuitive explanation for theta Hull-White

I am having a hard time coming up with an intuitive explanation for the long term mean $\theta$ in the Hull-White model: $$\mathrm{d}r_t=[\theta(t)-\alpha r_t]\mathrm{d}t+ \sigma_t \mathrm{d}W_t$$ So ...
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631 views

Expectation of exponential of 3 correlated Brownian Motion

Consider, are correlated Brownian motions with a given I want to calculate the, , I can't think of a way to solve this although I have solved an expectation question with only a single exponential ...
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1answer
58 views

Pairs trading by transforming two cointegrated series into a mean-reverting process?

I am slightly confused about the following. Let us assume I have two cointegrated time-series. I would like to model their 'cointegration' by a mean-reverting Ornstein-Uhlenbeck process since if they ...
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1answer
206 views

Expectation of $\int_0^t \frac{1}{1+W_s^2} \text dW_s$ [duplicate]

I am trying to calculate the expectation of $$\int\limits_0^t \frac{1}{1+W_s^2} \text dW_s,$$ where $(W_t)$ is a Wiener process. I was told that the value of this expectation is zero. Can someone ...
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Moments of a SDE: a detail on the information set

Very basic questions. Let $(z_t)_{t \geq 0}$ be a standard Brownian motion and let $$dS_t = \mu S_t dt + \sigma S_t dz_t.$$ When we write $E\left( S_t \right)$, do we mean $E\left( S_t \big| F_0 \...
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1answer
81 views

Illustrating the change of measure in Black-Scholes-Merton

Say that we have the following environment: \begin{align} dS_t &= \mu S_t dt + \sigma S_t dZ_t \\ dB_t &= r B_t dt \end{align} where $S_t$ is the price of a stock, $B_t$ is the price of ...
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1answer
80 views

What is the link between the SDF in the Black-Scholes-Merton model and the exponential process in Girsanov's theorem?

Question I have been toying around to get some understanding of what the stochastic discount factor look likes in Black-Scholes-Merton and how it relates to the exponential process in Girsanov's ...
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39 views

Change of measure to get a determined drift

let's say I have a real stochastic process $dX_t=dt+\frac{1}{B_t}dB_t$ on $[0,T]$, with $B_t$ Brownian in $\mathbb{P}$ (not centered in 0) in $[0,\tau]$ with $\tau$ some adequate stopping time that ...
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2answers
197 views

Can a Process with a Stochastic Drift be a Martingale?

I have repeatedly come across the statement that "a process with a drift cannot be a martingale". Is this true also for stochastic drifts? Suppose I have a process with a stochastic drift: $$...
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51 views

Probability of Hitting time of Brownian motion

Let $B =\{ B(t); t \ge 0\}$ be Brownian motion. What is the probability that $B$ hits state one and then state minus one before time one? My take: Let $T_x = \inf \{ t\ge 0 : B(t) = x\}$, the first ...
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2answers
179 views

Integral of the square of Brownian motion using definition of variance

Let $B = \{ B(t); t \ge 0\}$ and let $Z = \{ Z(t); t \ge 0 \}$ where $$Z(t) = \int_0^t B^2(s) ds.$$ How do we find $E[Z(t)]$ and $E[Z^2 (t)]$ in order to get the variance $Var [Z^2(t)] = E[Z^2 (t) ] -...
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1answer
140 views

Stochastic volatility Levy models

Hey I have some questions about stochastic volatility for Levy processes. If I understand correctly, if we change the time in Levy's process by CIR process, the newly received process is not Levy's ...
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73 views

Continuous option pricing: Brownian Bridge

I have a question on the proof of the formula of Sup(S) between 2 simulation points. Do you know how the prove the following formula? Thanks
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41 views

American put option with $r=0$ [duplicate]

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
69 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
125 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
129 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
135 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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77 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 ...
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75 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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39 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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38 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. ...
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1answer
318 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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452 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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0answers
75 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
81 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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41 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_{...
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2answers
201 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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70 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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49 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-...
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1answer
61 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
209 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
119 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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28 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 ...
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1answer
97 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
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0answers
69 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 $f(t,x)$...

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