Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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78 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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187 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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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
70 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
104 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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61 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
52 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
128 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
108 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
72 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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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 $f(t,x)$...
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1answer
279 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
141 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
454 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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42 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 ...
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2answers
574 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
98 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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0answers
60 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 ...
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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
52 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
126 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
178 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
107 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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23 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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121 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
64 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
107 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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19 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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114 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
115 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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96 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 $...
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0answers
36 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} ∼ E·Q^p \...
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227 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 ...
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1answer
68 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}(\...
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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/...
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1answer
97 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 ...
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1answer
82 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 ...
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1answer
88 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 ...
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71 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?...

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