Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
777
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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 $dW_{1t}dW_{2t}=\rho dt$....
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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 ...
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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 ...
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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*}
...
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50
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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 ...
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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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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:
...
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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 ...
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173
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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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94
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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 ...
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310
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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 ...
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1
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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 ...
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310
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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 ...
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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 ...
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120
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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:
$$dX(t) = \eta(b - X(t))dt + dZ(t)...
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Three mathematical mistakes in Black-Scholes-Merton option pricing?
In this preprint on arXiv (a revised version of the one discussed in a post here) we show that there are three mathematical mistakes in the option pricing framework of Black, Scholes and Merton. As a ...
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multivariate geometric brownian motion equivalent martingale measure
Suppose $W$ is a $\mathbb{P}$-Brownian motion and the process $S$ follows a geometric $\mathbb{P}$-Brownian motion model with respect to $W$. $S$ is given by
\begin{equation}
dS(t) = S(t)\big((\mu - ...
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271
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Did I derive the Kelly criterion correctly?
$$\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 ...
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25
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integral of adapted process with respect to semimartingale is a martingale
Fix $T > 0$ a finite time horizon. Let $H$ be an adapted (or progressively measurable, if needed) continuous process and S be a continuous semi martingale, both on $[0,T]$. Under what conditions is ...
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64
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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 ...
4
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1
answer
156
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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,...
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0
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72
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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 ...
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210
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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 ...
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159
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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 ...
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255
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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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737
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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 ...
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130
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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[ \...
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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]
$...
2
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1
answer
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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 ...
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429
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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) ...
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259
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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 ...
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124
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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 ...
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68
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Volatility of the product of two correlated asset following a log normal distribution [duplicate]
I am trying to solve the problem: Given two assets X and Y that follow a log normal distribution with volatility $\sigma_1$ and $\sigma_2$ respectively and with correlation $\rho$, what is the ...
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72
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Affine Jump Diffusion
I'm currently looking into affine jump-diffusions. I would like to get to know the literature better and I know the paper by Duffie, Pan, and Singleton (2000) is a very celebrated paper. Although I ...
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253
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On first and last zeros before t in a Brownian Motion
Suppose we have the following random variables, given a fixed $t$ we define the last zero before $t$ and the first zero after $t$:
\begin{align*}
\alpha_t &= \sup\left\{ s\leq t: B(s) = 0 \...
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How is variance derived in BS?
The realized variance under classical Black Scholes where the stock price process follows a GBM is given as
$$V_T = \frac1T\int_0^T\sigma_s^2ds\qquad (1)$$
however, the texts I have been reading do ...
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How calculate expectation and variation of stochastic integral Based on Heston model?
I was calculated Heston volatility model. But I think it is wrong.
$dS_t = \mu dt + \sqrt V_t dW_t^s$
$dV_t = k(\theta - V_t)dt + \sigma \sqrt V_t dW_t^v$.
$dW^s_t dW^v_t = \rho dt$
take integral to ...
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229
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What is the PDE for this interest rate derivative?
We have the following model for the short rate $r_t$under $\mathbb{Q}$:
$$dr_t=(2\%-r_t)dt+\sqrt{r_t+\sigma_t}dW^1_t\\d\sigma_t=(5\%-\sigma_t)dt+\sqrt{\sigma_t}dW^2_t$$
What is the PDE of which the ...
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230
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Volatility swaps hedging
I have heard that traders use a straddle to hedge volatility swaps (in the FX context), although I could not figure out the specifics. Is this type of hedge used in practice? And if yes, how does it ...
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208
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Smile wings and varswap pricing
Is it true that far wings of the volatility smile have an outsized influence on the price of a variance swap? Is there a mathematical argument demonstrating this idea? What do we generally refer as ...
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130
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how to calculate pdf and cdf for an Ornstein-Uhlenbeck process
I have the
Task. For Ornstein-Uhlenbeck process generate a path and plot a)
cumulative distribution (cdf), b) density function (pdf), c) calculate the 95%-quantile.
My solution.
From the literature we ...
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Transform non-linear HJB PDE into system of linear ODEs [closed]
I am reading this market making paper, and am trying to understand the transformation presented on page 6. A good resource for background relevant to the transformation is this other market-making ...
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263
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Is this the correct discretisation of the Hull-White SDE for building a python model?
I've tried to build a basic one-factor Hull-White model using python, which I've done by trying to discretise the characteristic SDE.
According to my notes, the Hull-White SDE is
$$
dr_t = \alpha (\mu(...
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0
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69
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Mixing formula for SVJ models
I am trying to understand the mixing formula (Hull and White formula) for stochastic volatility models with jumps in the asset price. One article which discusses this is Lewis, The mixing approach to ...
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0
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What happens trying to price derivatives starting from a non-geometric brownian motion?
To get a better understanding, I tried going through BSM-model starting from a non-geometric brownian motion. However, during the derivation I got stuck, which led me to a specific question.
The set-...
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224
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Calibrating Hull-White model using historical data
I'm in search of a way to calibrate a very simple Hull-White model with a constant volatility and a constant mean-reversion speed, purely based on historical zero rates.
$$dr(t) = (\theta(t) - \alpha ...
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2
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599
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Is Local Volatility a function of the Strike or the Underlying price?
Long story cut short: I am asking why the Local Volatility function can be thought of as a function of the underlying, when in fact it appears to be a function of the strike.
Additionally, I wonder ...
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