Questions tagged [black-scholes-pde]
The black-scholes-pde tag has no usage guidance.
110
questions
0
votes
1
answer
62
views
When to use total derivative and when not to?
as I was trying to teach myself financial mathematics, I came across this topic on transforming black scholes pde to a heat equation. I had the exat same question as this post Black Scholes to Heat ...
- 1
0
votes
1
answer
155
views
Black Scholes PDE explicit Scheme
I am currently working on the implementation of classic schemes to solve the BS PDE and it seems that I make a mistake in my code because the result looks far from the result of the BS formula.
Here ...
3
votes
1
answer
165
views
What are the parallels between the Black-Scholes equation and the heat equation?
I'm trying to understand the analogy between the Black-Scholes equation (1) and the heat partial differential equation (2). I understand that (1) can be written in the form of (2) mathematically, but ...
- 31
1
vote
0
answers
68
views
Finite Difference Application
We all know that the traditional BS equation is:
$$\frac{\partial \mathrm V}{ \partial \mathrm t } + \frac{1}{2}\sigma^{2} \mathrm S^{2} \frac{\partial^{2} \mathrm V}{\partial \mathrm S^2}
+ \...
2
votes
1
answer
299
views
How to find IV from market prices accodring to Bergomi
I was conviced to read Bergomis book on stochasic volatility to learn how options are traded in practice. He basically writes that the probabilisitc side is rather useless and that one only uses the ...
- 25
3
votes
1
answer
218
views
Since $S = e^{(\mu-\frac{\sigma^2}{2})t+\sigma W_t}$, why treat it as a constant when calculating the greek Theta (dC/dt) for a European call option?
In a nutshell, if S is dependent on 't', why treat it as a constant when calculating the partial derivative $\frac{dC}{dt}$?
The equation for $\frac{dC}{dt}$ in a European call option is:
$\frac{SN'(...
1
vote
0
answers
95
views
Feymann Kac for multidimensional pde
I Have to solve the following PDE:
\begin{equation}
\begin{cases}
\dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{\...
- 163
1
vote
0
answers
92
views
Technical difficulties with degenerate PDEs
Crossposted at Mathematics SE
I have seen lot of discussions in Math.stackexchange platform about 'degenerate partial differential equations'. But I still unclear about the 'technical difficulty' ...
- 111
0
votes
0
answers
98
views
Locally riskless
Most derivations of the Black-Scholes formula end up with the following dynamics of some (hedged) portfolio:
$$
\int_{t=0}^{T} \left(\frac{\partial f}{\partial \tau}(S(t),t)+\frac{1}{2}\cdot\frac{\...
- 33
2
votes
2
answers
202
views
Black-Scholes PDE derivation gap
Most derivations of the Black-Scholes formula end up with the following dynamics of some (hedged) portfolio:
$$
\int_{t=0}^{T} \left(\frac{\partial f}{\partial \tau}(S(t),t)+\frac{1}{2}\cdot\frac{\...
- 33
4
votes
1
answer
328
views
Boundary condition issues for Black-Scholes PDE using finite-differences
I have been implementing an, in my opinion, interesting finite difference method (Runge-Kutta-Legendre of second order) to price American options in the standard Black-Scholes model (see "...
- 661
0
votes
1
answer
86
views
Black-Scholes PDE transformation
From "Mathematics of Financial Derivatives" by Wilmott, Howison and Dewynne, section 5.4, p76.
How do I start making the transformations to get to the dimensionless equation? I.e. we start ...
- 152
0
votes
1
answer
160
views
Black-Scholes differential equation rewritten [closed]
I have seen that the Black-Scholes equation
$$\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$$
can also be written in the ...
- 3
0
votes
1
answer
118
views
Power option's PDE
I am looking to understand the PDE of Power Options in Paull Willmot on Quantitative Finance (2nd Ed), Ch. 8.9 - Formulae for Power Options (p. 149).
Suppose the payoff depends on the asset price at ...
- 152
1
vote
1
answer
411
views
Any book which is intro to PDEs but prioritises techniques useful for solving Black-Scholes?
Summary:
Can you recommend any book which is:
Intro/first course in PDEs
Covers solution methods useful for Black-Scholes model?
Background
I have just started learning about PDEs (after studying ...
- 152
0
votes
0
answers
67
views
Option pricing with risk-neutral approach
Problem
Given $Y_t$ price of a stock (no-dividents), and a derivative paying $Y_T^2$ at maturity $T$, evaluate the price of the instrument now using risk-neutral approach and check that it satisfies ...
- 101
1
vote
2
answers
323
views
Black-Scholes Portfolio
In the black-scholes model, the hedging portfolio is given (in some textbooks) by
$$\Pi_t = V_t - \Delta S_t,$$
i.e., the portfolio consits of a long position in the option $V$ and $\Delta$ units of ...
- 21
3
votes
3
answers
614
views
How to derive a pricing PDE for an asset that follows a mean-reverting process?
I want to derive a Black-Scholes type partial differential equation to price options on an asset that follows a mean-reverting process (Schwartz model).
My attempt follows the methodology of deriving ...
- 31
1
vote
0
answers
66
views
Finding the PDE and replicating strategy of a european contigent claim [duplicate]
Suppose that we have the Black and Scholes model where the interest rate and the volatility are time varying:
$dB(t)=r(t)B(t)dt$ and
$dS(t)=S(t)b(t)dt+S(t)\sigma(t)dW(t), S(0)=s>0$
where $r,b,\...
- 129
2
votes
2
answers
2k
views
Boundary Conditions for Call Option in Black Scholes Model
Let $C(t,S)$ be the value function of a call option. I want to price that option using (explicit) finite differences and the Black Scholes PDE. I consider the grid $0=t_0<t_1<...<t_{N-1}<...
- 688
1
vote
0
answers
100
views
Generalized Black Scholes PDE in a Two Factor model
I'm reading the book of Clewlow and Strickland on Energy derivatives. In the section about the two-factor model, an equation, similar to B&S PDE is presented, but the proof is not presented.
Spot ...
- 11
0
votes
3
answers
270
views
Intepreting European call option when expiration approaches to infinity
Assume that dividend = 0, then the price of call option is
$$
C = S\cdot P_{s}[S(T) > K] - e^{-rT}K\cdot P_F[S(T) > K] = SN(d_1)-e^{-rT}KN(d_2)
$$
where
$P_s[S(T) > K]$ = Probability of ITM ...
- 65
0
votes
0
answers
62
views
Derivative Pricing of an Asset
The Stochastic Differential Equation that models the change in an asset price is
$$
dS = (12S-sin(S))dt+\frac{\sigma S}{S^2+1}dX
$$
where dX's are random variables drawn from standard normal ...
1
vote
0
answers
56
views
Is there a relation between the so-called volatility drag and the sigma term in Black-Scholes' model? [duplicate]
The closed-form solution of Black Scholes Dynamics
$dS_t=S_t(\mu dt +\sigma dW_t$) is
$$S_t=S_0 e^{(\mu -\sigma ^2/2) t+\sigma dW_t}.$$
The $-\sigma^2/2$ term is quite similar to the volatility drag ...
- 341
3
votes
0
answers
102
views
Operator splitting method on three assets black scholes equation
Currently I am studying finite difference method on derivatives with three (or more) underlyings and little bit confused on operator splitting method because two papers have different result.
For the ...
2
votes
2
answers
298
views
Implicit finite difference method always guarantees positive and stable price of derivative?
For the following black scholes pde
$$
f_t + rSf_S+\frac{1}{2}\sigma^2S^2f_{SS} = rf
$$
By denoting $f_{i}^{n} = $ Price of derivative at price node $i$ and time node $n$ and assume uniform grid, the ...
- 65
1
vote
2
answers
2k
views
May someone please explain the intuition behind the Black-Scholes Equation?
Consider the Black-Scholes equation for a European Call Option,
\begin{equation}
\begin{cases}\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r\frac{\...
1
vote
2
answers
569
views
Black Scholes PDE boundary conditions
So I'm trying to solve the black scholes equation using a finite difference model, but I'm getting a answer that's off and I'm having trouble understanding why.
This is the result for a option with K ...
- 111
1
vote
1
answer
157
views
Question About Converting Black Scholes Differential Equation to Heat Equation
I'm reading a book about converting Black Scholes equation to heat equation and I highlighted in bold for those I have doubts, and really appreciate your advice on it.
Let $S$,$T$,$V$ denote ...
- 627
1
vote
1
answer
433
views
Nonlinear Black-Scholes model Vs linear Black-Scholes
I am working on a project related to Nonlinear BS partial differential equation, with terms for transaction costs and/or discrete hedging.
I have two questions:
Is there any exact solution to the ...
- 53
1
vote
1
answer
250
views
Black Scholes PDE
I seen two variations of the Black-Scholes PDE with either $+{\frac {\partial V}{\partial t}}$ or $-{\frac {\partial V}{\partial t}}$, and wanted to ask why that is?
a) https://en.wikipedia.org/wiki/...
- 5,749
3
votes
1
answer
327
views
Forward price vs. futures price - Wilmott
I am reading Paul Wilmott's book PWOQF2, and there is something I don't get in his derivation of the convexity adjustment between forward and futures prices (chap. 30).
He models $S$ and $r$ ...
- 2,570
3
votes
1
answer
355
views
Black-Scholes equation Variational / Weak form
I am having difficulty deriving the weak formulation of the Black-Scholes Equation.
I have multiplied it with a test function phi and integrated over Omega. But results on the internet suggest ...
3
votes
0
answers
168
views
Alternative derivation of Black Scholes by Merton
I am currently reading the Theory of Rational Option Pricing (1973) by Robert Merton. In the paper, I encountered a section under the title "An Alternative Derivation of the Black- Scholes Model". I ...
1
vote
1
answer
680
views
Linear Or nonlinear Black Scholes Equation
I have been going through the analytical solutions of black scholes equation which transforms it to a heat equation.
$$u_{t}=\frac{1}{2}\sigma^{2}u_{xx}$$
Now if the volatility is constant , then its ...
- 53
1
vote
1
answer
286
views
Delta hedging: theoretical value vs actual price
One way to derive the Black-Scholes PDE is via the Delta-hedging argument:
Suppose that $V_t = V(t, S_t)$, for some function $V: [0,T] \times \mathbb{R} \to \mathbb{R}$. We construct a portfolio by ...
- 642
3
votes
2
answers
621
views
What class of derivatives satisfy the Black-Scholes PDE?
The title pretty much sums up the question, but I will provide some context.
There is a large class of derivatives—such as those the payoffs from which depend only on the share price at maturity—...
- 613
1
vote
0
answers
123
views
Numerical Solutions to PDEs with Financial Applications
I am reading a paper by Richard White, Opengamma named Numerical Solutions to PDEs with Financial Applications. There is an implementation codes as stated in paper hosted at https://opengamma.com/...
- 533
2
votes
1
answer
213
views
Zero-coupon bond pricing equation derivation
I'm trying to understand how in Chawla's paper that I've linked below, how he obtains equation (2.5) for the zero coupon bond pricing equation?
The equation is:
$\frac{\partial B}{\partial t} + \...
- 55
1
vote
1
answer
107
views
Sensitivity Approximation - Crank Nicolson
I am looking into a new method of calculating sensitivities starting off with a proof of concept with Black Scholes PDE. Suppose I want to calculate Rho and take the derivative of the PDE (heresy!!) ...
- 11
1
vote
1
answer
393
views
Modifying Basic Black Scholes Equation For Time Dependent Variables - Per Wilmott?
I am reading Wilmott's book and don't understand why he makes the following step to re-write the PDE. I get equation 8.4, that's just the typical PDE for a dividend yielding stock where r(t), D(t) ...
- 267
2
votes
0
answers
122
views
What is the recipe for deriving a PDE for the price of an option?
In the Black Scholes setting, here is how my understanding is of how we derive the PDE for the value of an option.
We assume that the price of the option is Markovian in our state variable $S_t$. ...
- 21
2
votes
1
answer
157
views
Is it possible to transform arithmetic-average strike continuous sampling Asian Black-Scholes equation to a heat equation?
By Transformation from the Black-Scholes differential equation to the diffusion equation - and back, we are able to transform vanilla European option into a heat equation.
And we know that the ...
- 23
2
votes
1
answer
102
views
Numerical Solution to 3 Dimensional Backward BS PDE
I have a three dimensional backward BS PDE.
$$ \frac{\partial V}{\partial t} + a(t) S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma(t, S)^2 \frac{\partial^2 V}{\partial S^2} + b(t, M) \frac{\...
- 29
3
votes
1
answer
288
views
Why does Black Scholes formula give inconsistent dimensional analysis result?
For example, distance = speed * time, m = m/s * s.
But this technique gives wrong answer on the Black Scholes formula. The square root in the denominator gives wrong unit inside of the culumulative ...
user40979
1
vote
0
answers
726
views
Pricing Knock Out Barrier Options by solving Black Scholes PDE (MATLAB)
This question is based on MATLAB functions.
Suppose there is a stock S following the process
$dS_t=(r-q)S_tdt+\sigma(S_t,t)dW_t$
r - risk-free rate, q - dividend yield, W - Weiner process
The ...
- 11
2
votes
0
answers
716
views
Black-Scholes equation to Heat equation .(Boundary conditions)
I have been given a problem to code the heat equation which is transformed from B-S equation (European call option) .
Now the boundary conditions are for European call option:
$$C(S,T)=\max(S-K,0)$$...
- 53
3
votes
1
answer
446
views
Errors on Finite Differences + Implicit Scheme + Black & Scholes
I'm solving the classical Black & Scholes (BS) PDE for a European option using finite difference and the implicit scheme. In other words, I'm trying to solve
$\displaystyle\frac{\partial V}{\...
- 334
0
votes
0
answers
142
views
If the value of a call option is not dependent on the drift of the stock, why does a higher stock price mean a higher call option price [duplicate]
I have read that the price of an option is not affected by the drift of the stock since the drift term doesn't appear in the Black Scholes PDE. I become confused because to me, this implies that the ...
- 235
0
votes
1
answer
94
views
Why does a higher stock value imply a higher call option value [closed]
This may seem like a very dumb question, but if the underlying stock price is greater, then why should a call option be worth more.
My reasoning is that, if the option price is not affected by the ...
- 235