Questions tagged [differential-equations]
The differential-equations tag has no usage guidance.
34
questions with no upvoted or accepted answers
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ODE Solution in Carr's Randomized American Put
In Carr's 1998 paper Randomization and the American Put, he sets up the following ODE for the value of an American put with expiration given by the first jump time of a Poisson process with rate $\...
- 2,828
3
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0
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71
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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 ...
- 141
3
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0
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131
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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 ...
3
votes
0
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719
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Differential Sortino Ratio
I'm attempting to optimize a reinforcement learning system to maximize risk adjusted returns. I have currently defined the reward as the differential Sharpe ratio at each step: the influence of the ...
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0
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122
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Dixit & Pindyck (1993) Chapter 4, equation 13
Starting with the Bellman equation for the optimal stopping problem:
$$F(x,t)=max\{\Omega(x,t), \pi(x,t)+(1+\rho dt)^{-1} E[F(x+dx, t+dt)|x]\}$$
In the continuation region where the second term is the ...
- 153
3
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0
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301
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PDE and Black Scholes problem
Consider Black Scholes problem $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$ with boundary condition $V(S,T)=f(S)$, ...
- 131
2
votes
0
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86
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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*}
...
- 135
2
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0
answers
97
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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 ...
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2
votes
0
answers
178
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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 ...
- 863
2
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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 ...
- 2,055
2
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0
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52
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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 ...
- 21
2
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0
answers
278
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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 ...
- 157
2
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0
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548
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Need to solve the stochastic differential equation of Vasicek Model
How to solve the stochastic differential equation of the Vasicek model for the analysis of credit risk? I search in the article "The Distribution of loan portfolio value" (Vasicek) but he doesn't ...
- 121
2
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0
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131
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Milstein discretization of the CIR process
Given the CIR process $\ dX_t = (a − bX_t ) dt +
\sigma \sqrt{X_t}dW_t$ - I want to show that its Milstein scheme is $\ X_{i+1} - X_i = ((a − bX_i) - 0.25\sigma^2)\Delta + \sigma\sqrt{X_i}\sqrt{\...
- 342
1
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0
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116
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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 ...
- 11
1
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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 ...
- 437
1
vote
0
answers
71
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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^...
- 129
1
vote
0
answers
88
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:
$$\frac12 \sigma^2 V^2 F_{vv}(V,t) + \...
- 327
1
vote
0
answers
78
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Value of continuously rebalanced stock portfolio
I'm thinking about what a theoretical continuously re-balanced stock portfolio could look like, in which the portfolio is uniformly distributed over a selection of stocks at all times.
For example, if ...
- 177
1
vote
0
answers
236
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Replicating portfolio in the Heston model
Given the Heston model $$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}$$
how should the replicating portfolio $V_t$ for the derivative $F_t$ be composed?
I ...
- 223
1
vote
0
answers
67
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A Cauchy problem 2: How can I find the following solution?
Suppose that we have the following time-dependent partial differential equation:
\begin{equation}
\frac{\partial V(t, x)}{\partial t} = \frac{1}{2}\sigma^2 x\frac{\partial^2 V(t, x)}{\partial x^2}+ \...
- 419
1
vote
0
answers
111
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Generalized Black Scholes PDE in a Two Factor model
I'm reading the book of Clewlow and Strickland on Energy derivatives. In the section about the two-factor model, an equation, similar to B&S PDE is presented, but the proof is not presented.
Spot ...
- 11
1
vote
0
answers
170
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Implicit Scheme for Cox-Ingersoll-Ross Model PDE
I am considering the PDE for the price of a bond $V(r,t)$ with maturity $T$ under the Cox-Ingersoll-Ross model,
$$V_t+\frac12\sigma^2rV_{rr}+\nu(\theta-r)V_r-rV=0\quad r>0, t\in(0,1)$$
with ...
- 609
1
vote
0
answers
697
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CIR model. Is there a closed-form solution or even a good proxy of analytical solution?
Is there a closed-form (analytical) solution for the Cox-Ingersoll-Ross SDE
\begin{equation}
dr_t=k_r(\theta_r-r_t)dt+\sigma_r\sqrt{r_t}dW_t\tag{1}
\end{equation}
?
Notice that $\{r_t\}$ is our ...
1
vote
0
answers
63
views
Differential Equation of Type Ricatti as part of Short Rate Model
I currently despair of the following solution of a differiental equation (Ricatti Type) as part of a short rate model:
$$
B_t=\frac{1}{2}aB^2+bB-1
$$
First I am "guessing" a particular solution
$$ ...
- 11
1
vote
0
answers
57
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Computing squared returns given differential equation for prices
I am looking for general advice on how to start tackling the problem below. My background in math is fairly bad when it comes to stochastic differential equations, but if you have any recommendations ...
- 133
1
vote
0
answers
81
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Probability distributions as solutions to differential equations
As far as what I can tell, the popularity of the Black-Scholes-Merton model partly stems from the fact that it formulates the value of a derivative in a differential form in which the solution has a ...
- 2,995
1
vote
0
answers
118
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Boundary condition of lookback option
This is a well know conclusion of the boundary condition of lookback option. Here
$$\dfrac{d S_t}{S_t} = (\mu - D)dt + \sigma ...
- 1,243
1
vote
0
answers
33
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Some confusion on american put pde
Suppose
$$L(v) = \dfrac{\partial v}{\partial t} + rS\dfrac{\partial v}{\partial S} + \dfrac{1}{2}\sigma^2S^2$\dfrac{\partial^2 v}{\partial S^2} -rv$$
is Black-Scholes operator.
...
- 1,243
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29
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State space equation of CARMA(p,q) processes
Thanks for visting my question:)
I am currently working on Carma(p,q) processes and do not understand how to derive the state equation. So the CARMA(p,q) process is defined by:
for $p>q$
the ...
- 135
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votes
0
answers
89
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Is there any method/module/library to directly solve an SDE in python? Especially if it's just geometric brownian motion
Now, I'm given an SDE $$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. ...
- 101
0
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1
answer
316
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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:
$dX_t= \mu_t dt +\sigma_t dW_t$...
- 11
0
votes
0
answers
246
views
Why do we have to use discretization methods for SDE?
I haven't found the answer for the question above in google. Why can't we just discretize the equation instead of using methods like euler or milstein for the discretization.
