Questions tagged [stochastic-control]
Stochastic control is widely used in finance since it is about decision taking in a random environment. Option replication, portfolio construction, intraday trading are typical applications of stochastic control.
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Understanding the calibration of High-frequency trading in a limit order book
I am trying understand and replicate this thesis, which is based on, High-frequency trading in a limit order book by (Avellaneda and Stoikov, 2008) and Optimal market making, by Olivier Gueant, 2017, ...
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Is stochastic control with the HJB equation used in market making/algo trading at institutions?
In chapter 5 of https://www.maths.ed.ac.uk/~dsiska/LecNotesSCDAA.pdf, they use stochastic control and the Hamiltonian Jacobi Bellman (HJB) equation in attempt to measure bid-ask spreads and optimal ...
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Beta Weighting Deltas: What happens to the non-correlation part?
At various informational websites about option trading, it is often mentioned that in order to compare different underlyings in an apples-to-apples comparison, it is useful to beta-weight the deltas. ...
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American option pricing formulation
Assuming the usual setup of:
$\left(\Omega, \mathcal{S}, \mathbb{P}\right)$ our probability space endowed with a filtration $\mathbb{F}=\left(\mathcal{F}_t\right)_{t\in[0,T]}$,
$T>0$ denoting the ...
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How do your solve for trader's optimal demand in market similar to Kyle's model?
Suppose that $(\Omega,\mathcal{F},\mathbb{P})$ is a standard probability space and $Z_t=(Z_t^1,Z_t^2)$ is a two dimensional Brownian motion with the filtration $\mathcal{F}^Z_{t}$ and $Z_t^1$, $Z_t^2$ ...
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How to derive this HJB equation?
I'm reading the paper by J.Gatheral and A.Schied (2012) - "Optimal Trade Execution under
Geometric Brownian Motion in the Almgren and Chriss Framework".
On page 6, the authors provide a ...
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optimal log growth under a path dependent GBM
Consider an extension to the (one-dimensional) geometric Brownian motion model,
$$dS_t = \mu(t,S_.)S_t dt + \sigma(t, S_.)S_t dB_t,$$
where $\mu$ and $\sigma$ are previsible path functionals, i.e. ...
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non-Markovian/path-dependent optimal log utility and HJB-PDE
Basic question:
Can we generalize the HJB PDE to apply to optimal controls of non-Markovian/path-dependent SDEs? Specifically, how do we generalize the log-optimal portfolio to path-dependent ...
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References for Stochastic Control for finance
What are some good references to study Stochastic Control with applications to Finance, like the Merton problem and other variants? Books or review papers?
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Solving option market making problem
I am currently working on a paper for quoting option as a market maker from Bastien Baldacci , Philippe Bergault & Olivier Guéant
Without dwelling on details on how to obtain the HJB equation for ...
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Game theory and stochastic calculus
Does anybody know any details of game theory literature combined with stochastic calculus in finance? If yes, please recommend some papers of any authors who are doing exceptional work on the filed. ...
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Ansatz and HJB equation
Suppose we have an HJB equation of the form
$$
\frac{\partial v}{\partial t}+\frac{1}{2}\sigma^{2}\frac{\partial^{2}v}{\partial s^{2}}+max_{\delta^{a}}\left\{ \lambda^{a}(\delta^{a})\left[v(t,s,x+s+\...
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Help in Bernoulli's differential equation
I want to solve the following Bernoulli differential equation:
$$A'(t)=A^2(t)[-2\sigma +1]-2aA(t)$$
where $\sigma$ and $a$ are real numbers.
Until now I have divided both sides of the equation with $A^...
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Stochastic optimization and mean field games : textbooks
Which textbooks and online courses would you recommend to learn :
stochastic optimization
mean field games
applied to quantitative finance.
My goal would be to read research articles like the ones ...
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Question on derivation step in portfolio replication under different borrowing and lending rates
I'm currently trying to understand the derivation of a pricing PDE on a european claim that considers stock lending fees: https://cs.uwaterloo.ca/~paforsyt/hjb.pdf
In Appendix A.2, the author talks ...
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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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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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Interchange Expectation and Supremum in Snell Envelope/American Options
I had a question about the properties of a snell envelope, $\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$, which came to me while studying American options.
I know that in general,...
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Proving Flow Property of Stochastic Differential Equation
I am trying to show that $X_t^{s,x} = X_t^{r, X_r^{s,x}}$ for $0 \leq s \leq r \leq t$, $x \in \mathbb{R}^n$ is a given initial condition for time $s$, for some SDE:
\begin{equation*}
d X(u)=b(X(u))d ...
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Bounded solution for a SDE
I have this SDE
$$
dX(t) = [X(t)(u(t)(\delta-r)+r-\beta(t))+\theta(t)(1-\alpha(t))]dt+X(t)u(t)\sigma dW(t), t \in [0,T] \\
X(0) = X_0(1-\alpha(0))
$$
I've checked some books and I find the solution ...
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Pre-requisites for Finance Mathematics
I would like to pursue research in the areas of Financial Mathematics. Hoping to look into Operations Research, Risk Management and Stochastic Modeling. Anyone got some suggestions on useful resources ...
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Model of asset substitution/risk shifting in continuous time
Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion:
$$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$
where $r>0$ is the risk-free ...
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Price of a stochastic game between an agent and the market
In the article Pricing via utility maximization and entropy from Richard Rouge and Nicole El Karoui, they define the value function of the optimization problem as
\begin{align}
V(x,C) = \dfrac{1}{\...
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The duality of the free energy and relative entropy used to deduce deduce the stochastic game between an agent and the market
I am reading the article Pricing via utility maximization and entropy by Richard Rouge and Nicole El Karoui. They talk about the relative entropy of a probability measure $Q$ with respect to the ...
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Reference Request: Control Theory Prerequisites for Quantitative Finance
Right now, even though I have a mathematical background, I did not take up control theory in college. I'm looking for an introductory text on (stochastic?) control theory as applicable to quantitative ...
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Code examples of solving Stochastic Optimal Control problems
I'm currently reading a book demonstrating how Stochastic Optimal Control can solve common optimization problems encountered within quantitative finance. I haven't covered much continuous mathematics ...
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How do you actually solve a stochastic HJB equation in practice?
I've read a number of recent papers on market making. Nearly all of the more recent papers focus on defining the problem in terms of a state and action space, deriving the relevant HJB equations and ...
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How to Implement an optimal Stochastic Control Optimization? [closed]
I'm currently working on an stochastic optimal control problem applied to a portfolio asset allocation. In principle, the problem is to maximize the return of a fixed income portfolio under certain ...
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Flow Variable and Stock Variable
I am new to stochastic control and I need your help! Suppose that we are a trader and we are trading based two sources of signal. One comes from the stock's flow of dividends as well as another trader'...
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How to, from various hypotheses on the P&L, get known models (BS, Heston etc ...)
Usually models in quantitative finance are taught by giving, let's say, stochastic differential equations, initial conditions, and then pricing, under the model, various derivatives written on the ...
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Understanding the HJM drift condition's dimensions
In an HJM model the forward rate dynamics follow
$$
df_t(T) =a_t(f_t(T))dt+b_t(f_t(T))dW_t
$$
where $W_t$ is a $d$-dimensional brownian motion, $b_t$ takes values in $\mathbb{R}^{d\times d}$ and $a_t$ ...
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Merton portfolio allocation problem proportions/weights >1 or <0?
In the classical Merton portfolio problem, lets assume:
$$ dX_t \, = \, \frac{\pi_t X_t}{S_t} S_t(\mu dt +\sigma dW_t)
= \pi_t X_t (\mu dt +\sigma dW_t) $$
ie: zero interest rates for simplicity.
...
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example Hamilton-Jacobi-Bellman Equation - clarification of $dX_t$ derivation using $\pi_t$, $\Pi_t$
I have a market with safe rate r and risky asset S
$$ \frac{dS_t}{S_t}=(r+Y_t)dt+\sigma dW_t \quad \quad (1)$$
$$ dY_t = - \lambda Y_t +dB_t \quad \quad (2)$$
where W, B are Brownian Motions with ...
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optimal strategy problem (using Jensen's inequality)
I have a strategy in Samuelson model with zero safe rate defined as $$Z_t^{\Pi}=\frac{X_t^{\Pi}}{X_t^{\rho}} \quad \quad (1)$$ where
$$\frac{dX_t^{\Pi}}{X_t^{\Pi}} = \mu \pi dt + \sigma \pi \ dW_t \...
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trading strategy problem - initial capital x buys S over time [0,T] at the constant rate of x/T euros per unit of time
I am looking for clarification to the trading strategy problem where the number of stocks is depending on time.
In the Market with zero safe rate and stock dynamics defined as
$$\frac{dS_t}{S_t}=\...
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investor terminal value of portfolio with two risky assets 1) correlated 2)not correlated $\phi_t^1=S^{2}_{t}, \ \phi_t^2=S^{1}_{t}$
I am analyzing a problem where I have two stocks described by the equations
$$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t}$$
$$ \frac{dS^{2}_{t}}{S^{2}_{t}}=\mu_{2} dt + \sigma_{2}...
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Closed-form solution to optimal single assset position sizing with predicted returns
Say that I observe a predictor $w_t \sim N(0,\sigma_1)$ for the returns in a single asset over the next time interval:
$$ r_t = \alpha w_{t-1} + z_t $$
where $z_t \sim N(0,\sigma_2)$ is unobserved ...
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Portfolio insurance strategy with path dependence
I have the following problem.
Let us assume that $S_t$, the stock price follows, geometric Brownian moation with parameters $(\mu,\sigma^2)$. We are given an amount of money $M$ and at each point in ...
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Lipschitz condition in mathematical finance
I am interested in a rigorous explanation on why the Lipschitz condition plays a major part in stochastic calculus, most significantly in mathematical finance.
To be specific, suppose we want to ...
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Is there any theoretical work to find an optimum size for the size of horizon in finite-horizon optimization or control?
we learn a lot about finite and infinite horizon control in dynamic programming. but I was wondering if we want to minimize the cost per time(discrete time) is there any work to find the optimum size ...
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Application of Control Theory in Quantitative Finance
I have recently completed an MSc in Control Systems from a top university. It seems to me that control theory must have an application within quantitative finance. I would like to apply my degree ...
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