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Questions tagged [differential-equations]

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4
votes
1answer
159 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 $dX_t = t\,dS_t$.
1
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0answers
33 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 ...
2
votes
0answers
54 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
0answers
106 views

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 ...
1
vote
0answers
96 views

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 ...
1
vote
1answer
39 views

Finding B(t) in the Vasicek model relating to the bond equation, more specifcally from the initial condition

In the Vasicek model for derving bond prices, we have the ODE $$\frac{dB}{dt}=\gamma B-1$$ which gives rise to the general solution $$B(t)=C_1 e^{\gamma t}+C_2$$My problem is that we have the "initial"...
1
vote
0answers
53 views

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{\...
1
vote
0answers
44 views

Functional Analysis or Ordinary Differential Equations? [closed]

I am a current undergraduate and will be looking to apply to Quant Programs next year. This semester I have the choice between selecting Functional Analysis and Ordinary Differential Equations. I have ...
1
vote
1answer
137 views

Black-Scholes to Diffusion Initial Condition

I'm having troubles with the transformation from the Black-Scholes PDE and transforming it to the diffusion equation. I read this other stackexchange post (Here) and I understand most of the process, ...
-1
votes
1answer
70 views

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
votes
1answer
701 views

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
0answers
41 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 $$ ...
1
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0answers
50 views

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 ...
1
vote
0answers
53 views

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
votes
1answer
242 views

Feynman Kac Terminal value problem two variables

So, I need some help to move forward with this problem. $$ \begin{cases} \frac{\partial F(t,x,y)}{\partial t}+\frac{1}{2}\frac{\partial^2 F(t,x,y)}{\partial x^2}+\frac{9}{2}\frac{\partial^2 F(t,x,y)}...
0
votes
0answers
97 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.
2
votes
0answers
323 views

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 ...
2
votes
1answer
295 views

Riccati Equation in spot rate model

Given that $dr=(\eta-\gamma r)dt+\sqrt{\alpha r+\beta}dW$ Let $Z(r,t)=e^{A(t;T)-rB(t;T)}$, \begin{matrix} \frac{dA}{dt}=\eta B-\frac{1}{2}\beta {{B}^{2}} \\ \frac{dB}{dt}=\frac{1}{2}\alpha {{...
4
votes
1answer
299 views

Pricing the Passport option

Suppose underlying asset $S$ $$dS = \mu Sdt + \sigma Sd W$$ our portfolio $\pi$ consist with $q(t)$ stock $S$ and cash $\pi - qS$...
2
votes
1answer
2k views

How to understand the market price of risk

Consider the stochastic vol: $$dS = \mu Sdt + \sigma SdW_1$$ $$d\sigma = p(\sigma,S,t)dt + q(\sigma,S,t)dW_2$$ $$dW_1dW_2 = \rho dt$$ We want to obtain the price of option $V(\sigma,S,t),$ we use the ...
1
vote
0answers
54 views

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 ...
2
votes
1answer
207 views

How to price the American style Asian option with recent N day average

How to price the American style Asian option with recent N day average, for example, we exercise at t day, then the payment is $$...
3
votes
2answers
196 views

The PDE of caplet and floors

I know following PDE is the continuous payment case, but a caplet pays as rate: $\max(r - r^*,0),$ use the hedge portfolio $\Pi = V- \Delta Z$ $$d\Pi = dV- \...
2
votes
2answers
2k views

why futures contract has no value

Can any one tell me, why futures contract has no value? We know that the value of future(Maybe I confuse the concept of ...
1
vote
0answers
29 views

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. ...
2
votes
2answers
202 views

The PDE of the probability hitting the barrier before T

Suppose: $$d S=\mu S dt+\sigma Sd W$$ $Q(t,S)$ is the probability that $S$ hit the barrier $B(S_t<B)$ before $T,$ then $Q$ satisfies following PDE $$Q_t+\dfrac{1}...
1
vote
1answer
1k 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 ...
3
votes
0answers
74 views

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 $\...
4
votes
1answer
710 views

How to solve this PDE using Feynman-Kac?

I have the following problem right now: solve $$F_t(t,x) + rxF_x(t,x) + \frac{\sigma^2}{2}F_{xx}(t,x) = rF(t,x), \\ F(T,x) = (x - K)^2.$$ How do I solve this? There exists a theorem to solve this, ...
1
vote
1answer
417 views

PDE for Pricing Interest Rate Derivatives

Suppose that interest rate $r(t)$ follows some short-rate models, say Vasicek, so that$dr = a(b-r) dt + \sigma dZ$, with constants $a,b,\sigma$. It is well known that the price of zero-coupon bond $...
7
votes
1answer
374 views

Modelling EUR/USD with Ornstein-Uhlenbeck + jumps?

I'm trying to simulate a process as close as possible to EUR/USD of the ten past years. I've used a Ornstein-Uhlenbeck process: $$d X_t = -\theta (X_t - \mu) d t + \sigma d B_t$$ with the ...
1
vote
1answer
272 views

Prove that $E[g(X_T)|\mathscr F_t] = E[g(X_T)|X_t]$

Let $T > 0$. Let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \sigma(W_u, u \in [0,t])$ where $W_t$ is standard Brownian ...
2
votes
1answer
166 views

Prove $E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$ given $Y_t$ is a martingale

We are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $\mathbb ...
1
vote
1answer
151 views

Prove uniqueness, and prove $Y_t$ is a martingale by considering $dZ_t$ and $dL_t$

Suppose we are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $...
3
votes
1answer
406 views

Solving a backwards heat equation using stochastic calculus

Given the PDE $$\frac{\partial F}{\partial t} + \frac{1}{2}\sigma^2 \frac{\partial^2 F}{\partial x^2} = 0$$ with condition $F(T,x) = x^2$, one can use the Feynman-Kac formula to arrive at $$F(t,x) =...
2
votes
1answer
154 views

Regarding “Two Singular Diffusion Problems” by William Feller

I'm currently reading the research paper, Two Singular Diffusion Problems, by William Feller (1950). However, I don't understand how Feller derived the solution $(3.5)$ given equation $(3.4)$ in his ...
2
votes
2answers
209 views

Transformation into Martingale

If $f$ is some function of BV on $\mathbb{R}$ and $dZ_t = f(W_t)dW_t + \mu_t dt$ ($W_t$ is a $1$-dimensional standard Brownian Motion), then what choice of real valued function $F$ makes: \begin{...
2
votes
1answer
445 views

How to apply the chain rule for partial derivatives to transformations?

I'm currently working to solve the Black-Scholes model partial differential equation (it's a model for a.o. stock option prices). The Black-Scholes equation for a calloption C(S,t) is given by $ \...
3
votes
0answers
108 views

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 ...
3
votes
0answers
273 views

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 $V(S,T)=f(S)$, ...
3
votes
1answer
283 views

How to get Black Scholes' Geometric Brownian Motion differential form form the closed form?

My instructor has mostly self contained notes, where our textbook is mostly a reference. She has it written that: $$S_t = S_0e^{(\mu - \frac{\sigma^2}{2})t + \sigma W_t} \iff dS_t = S_t(\mu dt + \...
3
votes
1answer
2k views

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)\...
3
votes
2answers
1k views

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: ~...
29
votes
4answers
12k views

Is there an intuitive explanation for the Feynman-Kac-Theorem?

The Feynman-Kac theorem states that for an Ito-process of the form $$dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dW_t$$ there is a measurable function $g$ such that $$g_t(t,x) + g_x(t, x) \mu(t,x) + \frac{1}{...
9
votes
2answers
457 views

Why does Black-Scholes equation hold on continuation region of American Option?

Explanation for Put Option: $$ \frac{\partial V}{\partial t}+ \mathcal{L}_{BS} (V) = 0, $$ where $\mathcal{L}_{BS} (V) = \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-q) S \frac{\...
8
votes
1answer
863 views

Connections between random walk and heat equation (Material for ~)

I am preparing an undergraduate lecture in quantitative finance and I am looking for material that combines the topics: random walk and heat equation The material should be accessible (intuitive!), ...
7
votes
2answers
274 views

How to get an analytic result for option price based on this model?

I defined such a model for stock price (1).... $$dS = \mu\ S\ dt + \sigma\ S\ dW + \rho\ S(dH - \mu) $$ , where $H$ is a so-called "resettable poisson process" defined as (2).... $$dH(t) = dN_{\...
5
votes
2answers
371 views

Can we explain physical similarities between Black Scholes PDE and the Mass Balance PDE (e.g. Advection-Diffusion equation)?

Both the Black-Scholes PDE and the Mass/Material Balance PDE have similar mathematical form of the PDE which is evident from the fact that on change of variables from Black-Scholes PDE we derive the ...
11
votes
3answers
2k views

What tools are used to numerically solve differential equations in Quantitative Finance?

There are a lot of Quantitative Finance models (e.g. Black-Scholes) which are formulated in terms of partial differential equations. What is a standard approach in Quantitative Finance to solve these ...
5
votes
1answer
362 views

An equation for European options

So, any European type option we can characterize with a payoff function $P(S)$ where $S$ is a price of an underlying at the maturity. Let us consider some model $M$ such that within this model $V(S,\...