Questions tagged [differential-equations]
The differential-equations tag has no usage guidance.
92
questions
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Differential product Correlated processes
I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now:
We have the following two stochastic processes:
$dX_t= \mu_t dt +\sigma_t dW_t$...
0
votes
0
answers
29
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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 ...
2
votes
0
answers
85
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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*}
...
0
votes
1
answer
100
views
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, ...
4
votes
2
answers
314
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Heston Riccati equation
Let
$$
\begin{align*}
dY_{t} &= \left(r - \frac{1}{2} V_{t}\right) dt + \sqrt{V_{t}}dW_{t}\\
dV_{t} &= \kappa(\theta - V_{t}) dt + \rho \sigma \sqrt{V_{t}}dW_{t} + \sigma\sqrt{1-\rho^{2}}\sqrt{...
5
votes
1
answer
823
views
Book/reference to practice stochastic calculus and PDE for interviews
I will be going through interview processes in next months.
I would like to have a book/reference to practice the manipulation of PDE, and stochastic calculus questions.
For example, I get a bit ...
9
votes
1
answer
582
views
Prove $E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$ given $Y_t$ is a martingale
Edit years later: No idea why I'm upvoted. I actually am not sure how I'm correct. But maybe I haven't forgotten conditional expectation as much as I thought I have.
We are given a filtered ...
2
votes
1
answer
228
views
How to solve numerically the IDE of GUILBAUD & PHAM model?
By the Guilbaud & Pham model (Optimal high frequency trading with limit and market orders, 2011), the authors said that integro-differential-equation (IDE)
can be easily solved by numerical method....
2
votes
1
answer
220
views
Conditional Expectation of Integral of Squared Brownian Motion - PDE Approach
I am looking to compute the following using Ito's formula.
$$u(t,\beta_t) = \mathbb{E}(\int_t^T\beta_s^2ds|\beta_t)$$
Knowing the properties of brownian motion, it is rather easy to show that the ...
2
votes
0
answers
97
views
Perpetual Option Paying Chooser Option
A perpetual option solves the ODE
$$rSV_S+\frac{1}{2}\sigma^2S^2V_{SS}-rV=0$$
The general solution is $$V(S)=aS+bS^{\gamma}$$ where $\gamma=-\frac{2r}{\sigma^2}<0$.
For an American put option with ...
4
votes
2
answers
473
views
Transformation of local volatility model
Assume we have an SDE
$$dX_t=\mu(X_t)dt + \sigma(X_t)dW_t$$
where $\sigma>0$ and $W_t$ is a Wiener process. Is there a transformation $y(X_t)$ that will make the dynamics of the transformed process ...
2
votes
0
answers
177
views
Implied Volatility is the harmonic average of Local Volatility
I am trying to demonstrate the famous result that states that when $T \rightarrow 0$, the Implied Volatility is the harmonic average of Local Volatility.
I am st the final stage, and I have the ...
0
votes
1
answer
167
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Black-Scholes differential equation rewritten [closed]
I have seen that the Black-Scholes equation
$$\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+
rS\frac{\partial V}{\partial S}-rV=0$$
can also be written in the ...
1
vote
1
answer
154
views
To gamble or not to gamble! (solving a system of ODEs maybe?)
Assume we have some money. At every point in time $0\le t \le T$, we can take either action 1 that is to keep our money until $T$ say in a bank and have an expected return of $f(t)$ or take action 2 ...
0
votes
1
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112
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Closed form formula of asset that incorporates another asset's interest rate on top of its own
I'm trying to find a closed form formula for the price of an asset $D$ that has the following properties:
The asset grows by some interest rate $\mu$ at every instant.
Another asset's ($B$) interest ...
1
vote
1
answer
245
views
Using the risk neutral version of the First Fundamental Theorem of Asset Pricing to derive a partial differential equation
I have to use the risk neutral version of the First Fundamental Theorem of Asset Pricing to
derive a partial differential equation (PDE) that the price/value process, $V_t = F(t,S_t)$,
of a self-...
0
votes
1
answer
88
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Proof verification : risk free rate [closed]
I want to prove that
$$r_t = \theta + (r_0 -\theta)e^{-kt}$$
satisfies $$dr_t = k(\theta-r_t)dt, \ r(0) = r_0$$
I have
\begin{split}\frac{1}{\theta - r_t} dr_t = kdt \Rightarrow & \int_0^t \frac{1}...
3
votes
1
answer
4k
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General way to solve Partial differential equation using Feynman kac representation
Consider the following PDE on interval [0,T]
$\left(\frac{\partial F}{\partial t}(t,x)+\mu (t,x)\frac{\partial F}{\partial x}+\frac{1}{2}\sigma^2(t,x)\frac{\partial^2F}{\partial x^2}(t,x)=rF(t,x)\...
14
votes
3
answers
2k
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Deterministic interpretation of stochastic differential equation
In Paul Wilmott on Quantitative Finance Sec. Ed. in vol. 3 on p. 784 and p. 809 the following stochastic differential equation: $$dS=\mu\ S\ dt\ +\sigma \ S\ dX$$ is approximated in discrete time by $$...
1
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0
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114
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Finite difference methods with discontinuity in the payoff function
I have implemented a finite difference scheme for pricing options using a Black-Scholes-like model. I tested my implementation on a call option, and found that it gave extremely inaccurate results. I ...
2
votes
1
answer
157
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Proof about discounted zero coupon bond
Hey guys I am having trouble finishing this proof:
Proposition 5.1 Under the above assumptions, the process $r$ satisfies under $\mathbb{Q}$
$$
d r(t)=\left(b(t)+\sigma(t) \gamma(t)^{\top}\right) d t+\...
11
votes
1
answer
2k
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Market Making Strategies Found by Hamilton-Jacobi-Bellman Equation
Im working my way through the book "Algorithmic and High-Frequency Trading" (AHFT) by Cartea, Jaimungal and Penalva and i'm curious to see how the market making model with an exponential utility ...
1
vote
0
answers
70
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Dealing with the ru term in an ADI Finite Difference Scheme
I'm trying to code up the algorithm from this paper. The paper presents an ADI algorithm for pricing options in the Heston-Hull-White model.
The starting point is the Heston-Hull-White PDE, given ...
1
vote
0
answers
70
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Help in Bernoulli's differential equation
I want to solve the following Bernoulli differential equation:
$$A'(t)=A^2(t)[-2\sigma +1]-2aA(t)$$
where $\sigma$ and $a$ are real numbers.
Until now I have divided both sides of the equation with $A^...
1
vote
1
answer
2k
views
The solution to arithmetic brownian motion
I would like to obtain an explicit solution to $X$ when it satisfies
$$dX_t = \mu X_t dt + \sigma dW_t, X_S = x$$
Here, $S > 0$, and we want an explicit solution for $X_T$, $T > S$.
I am not ...
4
votes
1
answer
114
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Cauchy-Euler ODE with indicator function in coefficient
Consider the following Cauchy-Euler ODE, which is in particular the asset pricing equation for a (perpetual coupon defaultable) bond:
$$\frac12 \sigma^2 V^2 F_{vv}(V,t) + \mu V F_{v}(V,t) - r F(V,t) + ...
3
votes
1
answer
128
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Justification for substituting "Itô differentials"
I'm reading Shreve's Stochastic Calculus for Finance, Volume II. In it, he uses the stochastic differential notation. For example, he may write
$$\mathrm{d}X(t) = \sigma(t)\mathrm{d}W(t)+\alpha(t)\...
1
vote
0
answers
86
views
Derivation defaultable bond price in Leland 1994 (Merton)
Consider the model in Leland (Journal of Finance, 1994).
The partial differential equation that describes the price of the (perpetual coupon defaultable) bond is:
$$\frac12 \sigma^2 V^2 F_{vv}(V,t) + \...
1
vote
0
answers
78
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Value of continuously rebalanced stock portfolio
I'm thinking about what a theoretical continuously re-balanced stock portfolio could look like, in which the portfolio is uniformly distributed over a selection of stocks at all times.
For example, if ...
1
vote
0
answers
232
views
Replicating portfolio in the Heston model
Given the Heston model $$dS_t=\mu S_tdt+\sqrt{\nu_t}S_tdB_{1,t}\\ d\nu_t=k(\theta-\nu_t)dt+\eta\sqrt\nu_tB_{2,t}$$
how should the replicating portfolio $V_t$ for the derivative $F_t$ be composed?
I ...
1
vote
0
answers
67
views
A Cauchy problem 2: How can I find the following solution?
Suppose that we have the following time-dependent partial differential equation:
\begin{equation}
\frac{\partial V(t, x)}{\partial t} = \frac{1}{2}\sigma^2 x\frac{\partial^2 V(t, x)}{\partial x^2}+ \...
0
votes
0
answers
89
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Is there any method/module/library to directly solve an SDE in python? Especially if it's just geometric brownian motion
Now, I'm given an SDE $$dS_t = 2S_t\,dt + 4 S_t\,dW_t$$ which I need to find the solution of. I have the solution on paper, but I want to know if there's any way I can solve this directly in python. ...
3
votes
0
answers
71
views
Numerical approximation of SPDE
I've already posted this question on MSE, but I'm not quite sure if it's the right community so I'm posting it here as well.
Background
I want to approximate an SPDE of adensity process $V_t$. The ...
1
vote
0
answers
107
views
Generalized Black Scholes PDE in a Two Factor model
I'm reading the book of Clewlow and Strickland on Energy derivatives. In the section about the two-factor model, an equation, similar to B&S PDE is presented, but the proof is not presented.
Spot ...
2
votes
3
answers
964
views
Approximating Sharpe and Sortino ratios from Exponential moving averages
So I've been studying the paper "Learning To Trade via Direct Reinforcement" Moody and Saffell (2001) which describes in detail how to use exponential moving estimates (EMAs) of returns at ...
1
vote
0
answers
167
views
Implicit Scheme for Cox-Ingersoll-Ross Model PDE
I am considering the PDE for the price of a bond $V(r,t)$ with maturity $T$ under the Cox-Ingersoll-Ross model,
$$V_t+\frac12\sigma^2rV_{rr}+\nu(\theta-r)V_r-rV=0\quad r>0, t\in(0,1)$$
with ...
-4
votes
1
answer
325
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Using geometric brownian motion for stock price forecasting [closed]
I am doing a dissertation in finance on a maths degree. I wanted to forecast stock prices using artifcial neural networks but none of my tutors are able to supervise so I'm having to do something else....
1
vote
1
answer
332
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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 ...
1
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0
answers
686
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CIR model. Is there a closed-form solution or even a good proxy of analytical solution?
Is there a closed-form (analytical) solution for the Cox-Ingersoll-Ross SDE
\begin{equation}
dr_t=k_r(\theta_r-r_t)dt+\sigma_r\sqrt{r_t}dW_t\tag{1}
\end{equation}
?
Notice that $\{r_t\}$ is our ...
3
votes
0
answers
128
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Infinitesimal Generators and Expectation of First Hitting Time as Solution of Differential Equation
I've been learning about Linear Diffusions and how their infinitesimal generators can be used to relate expectations and deterministic differential equations.
Let $X$ be an one-dimensional diffusion ...
2
votes
0
answers
94
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Finding a PDE for an option $V(t,r(t),S(t))$
I have 2 approaches in my mind for finding a pde of an option that depends both on the short rate as well as the stock price- $V(t,r(t),S(t)$. Are these equivalent?
Find a hedging portfolio by ...
2
votes
3
answers
1k
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What are the advantages and limitations of predicting future stock prices using stochastic differential equations?
Recently I came across the following stochastic differential equation that "predicts" the value of a given stock:
\begin{equation}
dS_t = \mu S_t dt + \sigma S_tdW_t \\
S_t(0) =S_0
\end{...
4
votes
1
answer
292
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Differential of time over Browninan motion
I know that $\frac{dW_t}{dt}$, with $W_t$ a brownian motion, does not exist. However, does $\frac{dt}{dW_t}$ exists? Or does it even make sense? I am trying to calculate the quotient of two ...
2
votes
1
answer
931
views
Idea of using logarithm for solving SDE in Black-Scholes model
In the Black-Scholes model they consider that the stock follows this stochastic differential equation: $$ dS = \mu S dt + \sigma S\ dW $$
I was wondering, was it common at the time they work on this ...
3
votes
2
answers
2k
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SVCJ (SVJJ) Duffie et. al Model implementation in Matlab
I'm attempting to implement aforementioned SVCJ model by Duffie et al in MATLAB. so far without success. It's supposed to price vanilla (european) calls . parameters provided, the expected price is: ~...
5
votes
1
answer
241
views
Evaluating the SDE $dX_t = t\,dS_t$
The process $S$ is a geometric Brownian motion with an SDE: $dS_t = S_t(\sigma\, dB_t + \mu\, dt)$. I'm stuck evaluating $E(X_t)$ and $V(X_t)$, where $dX_t = t\,dS_t$.
2
votes
0
answers
52
views
B-S derivative with another boundary condition
I want to use the derivation of BS for another type of derivative, not an option. Known the derivation of the Black-Scholes differential equation, is it possible to use in the same equation when my ...
28
votes
2
answers
31k
views
Transformation from the Black-Scholes differential equation to the diffusion equation - and back
I know the derivation of the Black-Scholes differential equation and I understand (most of) the solution of the diffusion equation. What I am missing is the transformation from the Black-Scholes ...
2
votes
0
answers
274
views
Hull White Equation Derivation
Hello I need your help.
I found the formula for deriving $A(t,T)$ and $B(t,T)$ in Hull White paper is like this
$BB_{tT} - B_{t}B_{T} - B_{T} = 0$ and
$ABA_{tT} - BA_{t}A_{T} - AA_{t}B_{T} + \frac{1}...
2
votes
0
answers
288
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Term structure equation in the Vasicek model
Consider the SDE $$dr_t = (b-ar_t)dt +\sigma dW_t, \text{with } a; b > 0.$$ Let $$F(t; r) = E(\exp(-\int_{t}^{T}r_sds)| r_t = r).$$ (F can be interpreted as price of a zero coupon bond with ...