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Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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Expectation of average, conditional on terminal value

Silly question, but for some reason I'm a bit uncertain about this this trivial example perhaps: I have the following simple BS model $$ S_T = S_t \exp \left\{ -\frac12 \sigma^2 (T-t) + \sigma (W_T - ...
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Constant cancellation model for volume of LOB queues

I have been reading Jean-Philippe Bouchaud's book on stochastic models of LOB queues in Chapter 5, which starts with the simplest model. In this model, market/limit/cancel orders are assumed to be of ...
pierce's user avatar
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ARCH-Vasicek model solution

I understand how we can obtain the solution of Vasicek model $dr_t=\alpha(\mu-r_t)dt+\sigma dW_t$: $$ r_t=r_0e^{-\alpha t}+\mu(1-e^{-\alpha t})+\sigma\int_0^te^{-\alpha(t-s)dW_{s}} $$ This easily ...
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Parameters bounds for Heston model calibration

Still working on my master thesis and I have a question I have been looking at for some time but can't find a good reason. I am looking to follow the steps of Horvath et al. (2019) in order to ...
sxminho's user avatar
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Heston Model Sensitivity Qualitative Property

Consider the following Heston model: $$\begin{aligned} \mathrm{d}S_t&=rS_t\mathrm{d}t+\sqrt{v_t}S_t\mathrm{d}B_{1,t}\\ \mathrm{d}v_t&=-\kappa(v_t-\bar{v})\mathrm{d}t+\sigma_v\sqrt{v_t}\mathrm{...
Ben's user avatar
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What is the meaning of Fr 0,20

In Louis Bacheliers The Theory of Speculation he often uses “Fr number”. I have no idea what it means.
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Weak stationarity of continuous ARMA process from Brockwell

I am currently working on Brockwell "Levy-driven CARMA processes" (2001) and I am stuck in the introduction. So we have a continuous AR process (CAR(p)) \begin{align*} X_t=e^{At}X_0+\...
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The partial derivative of a call option with respect to $t$ [closed]

In Black-Scholes related computations, why do we not treat the stock price $S$ as a function of $t$ when taking partial derivatives with respect to $t$? For example, if $$c(t,T)=SN(d_1)-Ke^{-r(T-t)}N(...
user81883's user avatar
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About Hedging of One-touch Options

The pricing of American Digital Call (one-touch Calls) has the following formulas, taken from P13, the textbook \begin{aligned} C_{\mathrm{d}}^{\mathrm{Am}}(S, t ; E) & =\left(\frac{S}{E}\right)^{\...
newbiesolidty's user avatar
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Orthogonalizing brownian path

I want to improve the stability of my SDE sample (statistical properties do not change much when using a different seed). I am using a sobol brownian bridge to generate the brownian path increments dw....
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Model for markets with friction

Is there a stochastic model for describing how equities behave in markets with trading fees, and if so what model is most commonly used? I'm envisioning something similar to the Black Scholes model, ...
djr's user avatar
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Geometric Brownian motion with volatility as function of time

With the following process: $$dS_t = r S_t dt + σ(t) St dW_t \tag1$$ and $$ \sigma (t) = 0.1 \ \ \ if \ \ t < 0.5 \\ \sigma (t) = 0.21 \ \ \ otherwise$$ I know the general solution should be : $$...
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Moments of the integral of the exponential of Brownian motion/Normal random variable

I'm studying arithmetic Asian options and there is integral of the following form: $$X_T=\int_0^T e^{\sigma W_t+\left(r-\frac{\sigma^2}{2}\right)t}dt,$$ where $W_t$ is a Brownian motion/Wiener process....
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Approximation of an Itô integral with python

Exercise 3.11 (Approximation of an Itô Integral). In this example, the stochastic integral $\int^t_0tW(t)dW(t)$ is considered. The expected value of the integral and the expected value of the square ...
Jessie's user avatar
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Distribution of Geometric Brownian with time-dependant volatility

The process $S(t) =\exp\left(\mu.t + \int_0^t\sigma(s) \text{d}W(s) - \int_0^t \frac{1}{2}\sigma^2(s)\text{d}s\right)$ where $\sigma(s) = 0.03s$ is log-normally distributed, but i'm not sure about the ...
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Analytic Hull White model with correlated stochastic processes

I am trying to price a path dependent option which uses two underlyings (a stock index and an interest rate index). I am using Hull White model for interest rate modelling and local vol for stock ...
Madhuresh's user avatar
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134 views

Price a contingent claim with payoff $(S_{1T}-S_{2T})^+$ at time $T$

Two stocks are modelled as follows: $$dS_{1t}=S_{1t}(\mu_1dt+\sigma_{11}dW_{1t}+\sigma_{12}dW_{2t})$$ $$dS_{2t}=S_{2t}(\mu_2dt+\sigma_{21}dW_{1t}+\sigma_{22}dW_{2t})$$ with $dW_{1t}dW_{2t}=\rho dt$....
Mr. Ivan's user avatar
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Why is Feynman-Kac formula applicable in Burgard-Kjaers PDE paper?

In the paper Partial Differential Equation Representation of Derivatives with Bilateral Counterparty Risk and Funding Costs by Burgard and Kjaer, they say we may formally apply the Feynman-Kac theorem ...
zoom's user avatar
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State space equation of CARMA(p,q) processes

Thanks for visting my question:) I am currently working on Carma(p,q) processes and do not understand how to derive the state equation. So the CARMA(p,q) process is defined by: for $p>q$ the ...
Valentin's user avatar
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How to understand Short Gamma and Long Volatility for Leveraged ETFs?

In the book Leveraged Exchange-Traded Funds: Price Dynamics and Options Valuation, it describes a static delta-hedged long volatility position by simultaneously shorting regular/inverse leveraged ETFs ...
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Ito formula and confusion with the differential operator $d$

Thanks for visiting my question. Im am currently working on this paper (https://arxiv.org/abs/2305.02523) and I am stuck at page 21 (Theorem 14 proof). First these SDE's were defined: \begin{align*} ...
Valentin's user avatar
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Kalman Filtering to estimate parameters of G2++ Model

I'm trying to use Kalman Filtering to estimate the parameters of the G2++ short rate model. For this, I've been using Implementing Short Rate Models: A Practical Guide by F.C. Park. For reference, he ...
Pudge Superior's user avatar
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213 views

Balland - SABR goes normal

To summarise this very long post : please help me understand the undetailed proof of the quoted paper. I am not comfortable using a result I do not fully understand. I am reading Balland & Tran ...
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What’s the Ito’s lemma of compound Poisson process with two-sided jump and mean-reverting jump size?

In the book Cartea and Jaimungal (Algorithmic and High Frequency Trading, page 249.), the midprice dynamics follows $ dS_t=\sigma dW_t+\epsilon^+ dM^+_t-\epsilon^- dM^-_t$ (10.22) where $M^+_t$ and $M^...
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143 views

Proof: Deterministic Ito Integral (Thomas Mikosh Chapter 2)

I'm referencing Elementary Stochastic Calculus with Finance in View by Thomas Mikosch between chapters of Shreve's Volume II text. In one section Mikoshch text makes the following claim without proof: ...
James Bender's user avatar
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105 views

Construction of Ito Integral

I am self-learning basic stochastic calculus. In my book, the author first defines the Ito integral for simple step adapted processes and then extends it to a larger class $\mathcal{L}_{c}^{2}(T)$ of ...
Quasar's user avatar
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Why is the stochastic process of the volatility of a stock price square integrable?

I am taking a course in financial mathematics(Ito-Integrals, Black-Scholes,...) and there is something that is not immediately clear to me. When constructing our stock price model, the integral $\...
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Solving Equation for estimation risk averse parameter

Let the portfolio value follow the SDE: $$V_t=(\mu w+r(1-w))\cdot V_t\cdot dt +\sigma \cdot w\cdot V_t \cdot dB_t $$ where $\mu$ = drift of the portfolio, $\sigma$=standard deviation of the portfolio, ...
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Is homogeneity preserved under change of measure?

In a paper, Joshi proves that the call (or put) price function is homogeneous of degree 1 if the density of the terminal stock price is a function of $S_T/S_t$. In the paper I think Joshi is silently ...
Frido's user avatar
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Integrated Brownian motion

I occasionally see a post here: Integral of brownian motion wrt. time over [t;T]. This post has the conclusion that $\int_t^T W_s ds = \int_t^T (T-s)dB_s$. However, here is my derivation which is ...
Wang Jing's user avatar
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1 answer
210 views

Time-shifted power law in path dependent volatility

I can't understand a function which is part of a volatility model. This is all explained in an open access paper titled "Volatility is (mostly) path-dependent" by Guyon and Lekeufack. My ...
s5s's user avatar
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Can the PDE of Black and Scholes really be derived from the CAPM?

Black and Scholes (1973) argue that their option pricing formula can directly be derived from the CAPM. Apparently, this was the original approach through which Fischer Black derived the PDE, although ...
MMFdW's user avatar
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Necessary conditions to ensure that stochastic integral is a normal variable

Let $\left(W_t\right)_{t\geq 0}$ be a Brownian motion with respect to filtration $\mathbb{F}=\left(\mathcal{F}_t\right)_{t\geq 0}$. Let $\left(\alpha_t\right)_{t\geq 0}$ be an $\mathbb{F}$-adapted ...
fwd_T's user avatar
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Characteristic function of Gamma-OU process

Consider the Gamma-Ornstein-Uhlenbeck process defined in the way Barndorff-Nielsen does, but consider a different long running mean $b$ which may be bigger than zero: $$dX(t) = \eta(b - X(t))dt + dZ(t)...
Tom's user avatar
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3 answers
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Three mathematical mistakes in Black-Scholes-Merton option pricing?

In this preprint on arXiv (a revised version of the one discussed in a post here) we show that there are three mathematical mistakes in the option pricing framework of Black, Scholes and Merton. As a ...
MMFdW's user avatar
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multivariate geometric brownian motion equivalent martingale measure

Suppose $W$ is a $\mathbb{P}$-Brownian motion and the process $S$ follows a geometric $\mathbb{P}$-Brownian motion model with respect to $W$. $S$ is given by \begin{equation} dS(t) = S(t)\big((\mu - ...
yrual's user avatar
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1 answer
335 views

Did I derive the Kelly criterion correctly?

$$\frac{dX_t}{X_t}=\alpha\frac{dS_t}{S_t}+(1-\alpha)\frac{dS^0_t}{S^0_t}$$ where $\alpha$ is proportion of the investment in the risky asset $S_t$ and $S^0_t$ is the risk-free asset. $S_t$ follows a ...
user67303's user avatar
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Equivalent definition of brownian motion

I'm having a question about this characterization of Brownian Motion : Theorem : If a process : $\big( X_t \big)_{t\geq 0}$ satisfies these conditions, $\big( X_t \big)_{t\geq 0}$ is a Gaussian ...
Ahmed EL YOUSEFI's user avatar
4 votes
1 answer
166 views

Deriving an Analytical Expression for Standard Deviation of Log Returns

I am looking to find an expression for the standard deviation log returns of a stock price process. I have a stock price which follows the following dynamics: $dY(t) = Y(t)(r(t)dt + η(t)dW(t))$ Here,...
user67245's user avatar
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0 answers
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Method of conditional expectations for basket

I am reading paper "An analysis of pricing methods for baskets options". Unfortunatly, I can not find the working paper "Beisser, J. (1999): Another Way to Value Basket Options, Working ...
Nick's user avatar
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1 answer
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If the price of a stock follows a Geometric Brownian motion, then does stock return depends on past stock returns? [closed]

Got this question from my homework. I think if past returns are keep raising then current return should also be positive, but the answer is it's not related to past returns, why? I tried to ask ...
nearhome's user avatar
1 vote
1 answer
260 views

Bloomberg FXFM: what is the point of knowing risk neutral probabilities?

Among other things, Bloomberg FXFM function allows you to check risk neutral probabilities for currencies. For instance, you can check the probability of the euro depreciating 5% vs the dollar in 6 ...
Peter's user avatar
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4 votes
1 answer
305 views

Vega hedge of a barrier option

I was re-reading Lorenzo Bergomi's paper Smile Dynamics I. On the first page, he makes the point that it is necessary for a model to match the vanilla smile observed in markets in order to incorporate ...
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5 votes
1 answer
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Book/reference to practice stochastic calculus and PDE for interviews

I will be going through interview processes in next months. I would like to have a book/reference to practice the manipulation of PDE, and stochastic calculus questions. For example, I get a bit ...
Joanna's user avatar
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Expected value and variance of the short rate under the Vasicek model

Would be grateful for any assistance. Below are the expected value and variance of the integral of the short rate under the Vasicek model (https://www.researchgate.net/publication/41448002): $E\left[ \...
user1171853's user avatar
3 votes
0 answers
169 views

Feynman-Kac formula: Ito's lemma for exponentiated integrals $e^{-\int b dr}$

Consider the stochastic process $$ dy = f(y,s)ds + g(y,s)dw $$ where, $w$ is Brownian motion. Now consider the following exponentiated integral $$ z_1(s) = \exp \left[ - \int_t^s b(y(r),r) dr \right] $...
TheTwistedSector's user avatar
2 votes
1 answer
80 views

Mean level of the state variables under the risk-neutral measure in Arbitrage-free Nelson Siegel

I do not understand why mean levels of the state variables under the risk-neutral measure, $\theta^{\mathbb{Q}}$, in Arbitrage-free Nelson-Siegel is set to zero. It should follow from the following ...
Martin N.'s user avatar
1 vote
1 answer
653 views

How to hedge a dual digital option

Let us assume we have two FX rates: $ 1 EUR = S_t^{(1)} USD$ and $ 1 GBP=S_t^{(2)} USD $. Let $K_1>0, K_2>0$ be strictly positive values and a payoff at some time $ T>0 $ (called maturity) ...
fwd_T's user avatar
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2 votes
1 answer
264 views

Question on Merton's self financing derivation

I'm reading Merton's Optimum Consumption and Portfolio Rules in a Continuous-time Model, and don't understand the step where he goes from discrete to continuous time. Specifically, my confusion is ...
user2520938's user avatar
4 votes
1 answer
138 views

Stochastic integral involving Poisson Process

Consider an (inhomogeneous) Poisson process $N_t$ with intensity $\lambda_t$. Then I want to compute the following integral $\mathbb{E} \left(\int f(t,N_{t-}) d\tilde{N}_t\right)^2$ for some smooth ...
Student's user avatar
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