Questions tagged [black-scholes-pde]

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12
votes
2answers
801 views

The greeks: where do they come from?

I’m studying the BSM model and having a look at the greeks. I was reading Derivatives, by Paul Wilmott, and he gives the closed form solutions without making the reader see where these solutions come ...
10
votes
3answers
2k views

Black--Scholes hedging argument

I'm trying to understand the standard hedging argument to derive the Black--Scholes PDE. There's one aspect of the derivation which I can't get passed and I'd be very grateful for some clarification ...
7
votes
4answers
828 views

Why is $C(t,S_t)/B_t$ a martingale?

In the derivation of the Black-Scholes formula given by Joshi (extract below), he says $C(t,S_t)/B_t$ is a martingale. Why? I understand this can be deduced from the Black-Scholes PDE since the drift ...
7
votes
1answer
1k views

Solving Black-Scholes PDE using Laplace transform

I'm trying to obtain the Laplace transform of Call option price with repect to time to maturity under the CEV process. The well known Black scholes PDE is given by $$ \frac{1}{2}\sigma(x)^2x^2\frac{\...
7
votes
1answer
782 views

generalized black scholes

I understand how to derive the black scholes solution if $dS_t$ = $\mu S_tdt$ + $\sigma S_tdW_t$ and r is constant. The solution is c(t, x) = $xN(d_{+}(T - t), x))$ - K$e^{-r(T - t)}N(d\_(T - t), x))$ ...
6
votes
1answer
392 views

Heat/Diffusion Equation

I am working on a problem where I have successfully reduced a version of Black Scholes to the Heat Equation and then shown the solution to be: $$u(x,t)=\frac{1}{2\sqrt{t\pi}}\int_{-\infty}^\infty{f(\...
4
votes
2answers
147 views

Why does it make sense that $S$ and $e^{rt}$ are solutions to the Black-Scholes PDE?

It's readily verified mathematically that $V=S$ and $V=e^{rt}$ are solutions to the Black-Scholes PDE $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2} \frac{\partial^2 V}{\partial S^2} + r S \...
4
votes
1answer
331 views

Closed form solution of PDE of Option Price

Let $V=V(S_t,t)$ be the option price and \begin{align} V_t+\mu\,S\,V_S+\frac{1}{2}\sigma^2\,S^2\,V_{SS}=0\\ V(S_T,T)=\ln (S_T)^{2}. \end{align} My question: How can I obtain a closed form solution of ...
4
votes
1answer
637 views

Feynman Kac Formula for path-dependent options

Consier geometric Brownian motion: $dS_t/S_t=\mu dt+\sigma dW_t$ Feynman Kac theorem tells us that the conditional expectation $v(t,x)=E[ e^{-rT}\Psi(S_T) | S_t=x]$ can be computed by solving the ...
4
votes
1answer
261 views

Finite Difference Method for Black-Scholes-Formula

Using finite difference method for the Black-Scholes-Partial Differential Equation one need to impose some boundary conditions on the edge of the grid, i.e for a Grid on $D=[a,b]\times R^+$ one need ...
4
votes
1answer
322 views

Solve Black scholes PDE without using any transformation

I know that one of the methods of solving the black scholes PDE given by : $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV ...
4
votes
2answers
576 views

Solution for american perpetual put

I have been attempting an exercise in which I have to determine the value of an american perpetual put, $P$ in terms of the asset value $S$. The solution to the exercise says: When $S>S_f$ (the ...
4
votes
1answer
201 views

How to price up-out-call by solving heat equation like down-out-call

We know that by changing the variables we can obtain the Black-Scholes formula of vanilla call through solving the ...
4
votes
1answer
828 views

What's the intuition behind the transformation of Black-Scholes into Heat equation?

A sequence of transformations can be used to turn the Black-Scholes PDE into the heat equation. Let $C(S, t)$ be the price of a vanilla European option at time $t$, maturing at time $T$, where the ...
3
votes
1answer
109 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}{\...
3
votes
1answer
231 views

Boundary Conditions for Call Spread

I was just wondering if someone could verify whether these are the two boundary conditions for a Call Spread Black-Scholes PDE. The first one I have is: $max(S_{T} - K_{1}, 0) - max(S_{T}-K_{2},0)$ ...
3
votes
1answer
167 views

Alternative derivation of the Black Scholes formula

I encountered the following derivation of the Black Scholes formula for call price. It may very well be an established method but I had never seen it before so I called it an alternative derivation. ...
3
votes
1answer
279 views

Deriving Black Scholes PDE under stock as a numeraire

There are many ways to derive the Black Scholes PDE. The Martingale way would be to demand the option price is driftless according to particular measures. Below I derive the correct PDE using the bank ...
3
votes
1answer
795 views

Does Implied Volatility always exist?

I am considering a simple Heston Model Market with one risky and one riskless asset. The dynamics of the riskless asset is simply $dB_t=r*B_t*dt$ The dynamics of the risky asset is as follows, $ ...
3
votes
4answers
246 views

The reason behind the selection of a 1 standard deviation movement for self financing delta hedge

I'm learning this material and I can follow the derivation of the BSM PDE fairly well. The only problem I have is there is an assumption in the derivation (that I am reading) that a stock price ...
3
votes
2answers
121 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—...
3
votes
2answers
309 views

von Neumann boundary in the transformed PDE

I have transformed the BSM PDE $$\frac{\partial V}{\partial t} + \frac{\sigma^2}{2}S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV = 0 $$ to $u(\tau,x) = V(T-\tau,S_{0}...
3
votes
2answers
1k views

Black-Scholes Equation - Riskless portfolio derivation

The following is a summary of the derivation of the Black-Scholes equation as given on wikipedia (http://en.wikipedia.org/wiki/Black-Scholes_equation#Derivation) - I have a question regarding the ...
3
votes
1answer
416 views

Boundary conditions: Dirichlet vs Neumann

I'm thinking about the interplay of Dirichlet and Neumann BCs in a FDM scheme. Let's assume a simple Black-Scholes call option problem, with BS PDE with constant coefficients, i.e. instead of $S$, in ...
3
votes
0answers
548 views

Black-Scholes PDE - Change of Variables

In the derivation below, I cannot figure out how to solve for "Step 3". Can anyone help me walk through the steps in detail? Derivation:
3
votes
0answers
75 views

Time discretisations, FDM vs FEM

I am interested in adaptive mesh methods for numerical solution of PDEs with applications to finance. As part of a school project, I have been pricing vanilla European call and put options using 2D ...
3
votes
0answers
271 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 $V(S,T)=f(S)$, ...
2
votes
1answer
120 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 ...
2
votes
3answers
391 views

Analytical soluton to the Black-Scholes equation with a modified European Call Option

Please consider the following modified European Call Option where $ 0 < a \leq 1$. When $a = 1$ the modified European call option is reduced to the standard European call option. Transforming ...
2
votes
1answer
442 views

Brennan-Schwartz algorithm for pricing American options

I'm reading Pricing American Options using LU decomposition by Ikonen and Toivanen (IT). They reference The valuation of American put options by Brennan and Schwartz, and cast it as method that uses ...
2
votes
2answers
192 views

Implications of Black Scholes Plot

I'm pretty new to finances, but I'm heavily into scientific computation. For my scientific computations class, I need to have at least a basic understanding of finances for the presentation I'm going ...
2
votes
2answers
199 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}...
2
votes
1answer
2k views

How to use the Feymann-Kac formula to solve the Black-Scholes equation

I have the Black-Scholes equation for European option with maturity $T$ and strike $K$ $$\begin{cases}\frac{\partial u}{\partial t} = ru - \frac{1}{2} \sigma^2 x^2 \frac{\partial^2 u}{\partial x^2}-r ...
2
votes
1answer
135 views

What is the domain of the Black-Scholes operator?

By the Black-Scholes operator I mean the following. $$L_{BS}u(x) = \frac{1}{2}\sigma^2x^2\frac{\partial^2}{\partial x^2}u(x) + rx\frac{\partial}{\partial x}u(x) - ru(x)$$ Obviously, the domain of $...
2
votes
1answer
97 views

Is there a quick way to see why this claim $C(S, t)$ on $S$ does not satisfy the Black-Scholes PDE?

I'm self-studying for an actuarial exam on financial economics and encountered the below practice exam problem. An exam problem should typically take 5-6 minutes to complete, so I'm wondering if ...
2
votes
1answer
80 views

A bug in delta hedging, when for a certain step dS=0

Suppose we are doing a delta hedging simulation according to Black Scholes, where the initial condition are [stockPrice, strike, timeToExpire ,riskFreeRate, dividend, sigma, isCall] = [100, 100, 1, 0, ...
2
votes
1answer
98 views

Why there is some inhomogeneous term in the PDE of fixed income

We consider one factor driving model of fixed income product say short-term interest $r(t)=\lim\limits_{T\rightarrow t} R(t,T),$ $R(t,T)$ is yield i.e $$B(t,T)e^{(T-t)R(t,T)} = 1$$ Then we see ...
2
votes
1answer
360 views

Monte Carlo and PDE results are different for a Call Option!

Okay so this might be a fairly trivial question but I'm having an issue with valuing a call option using both a Monte Carlo method and a PDE method. When I started I first used the parameters: Spot =...
2
votes
1answer
1k views

American Swaption Pricing with Monte-Carlo method

I want to price an American swaption but I am not sure about what I am doing. Tree methods and PDE discretization seem difficult to adapt to a swaption. I am trying a Monte-Carlo approach. (in ...
2
votes
1answer
75 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} + \...
2
votes
1answer
210 views

Black Scholes in the case of dividends

Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a ...
2
votes
1answer
335 views

Rigorous derivation of $d\Pi$ for stock with continuous dividend

Suppose we are holding a replicating portfolio $\Pi_t$ of long an option $f(S,t)$ and short some stock, so $$\Pi_t=f(S_t,t)-\Delta_t S_t$$ Suppose the stock follows geometric Brownian motion and pays ...
2
votes
3answers
582 views

Numerical Solution to BS PDE - Digital Option

Here is a relatively simple question about PDE's pricing. Assume that we are within the BS framework and moreover that interest rate is zero. The price $V(t,S_t)$ of the digital is known to be $\Phi(...
2
votes
0answers
66 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 $S_t$. ...
2
votes
0answers
236 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: $$C(S,T)=\max(S-K,0)$$...
2
votes
0answers
97 views

Theta from Black-Scholes PDE - is it possible to use implied volatility?

There is a need to derive theta $\theta$ of an option out of standard Black-Scholes PDE. In usual notation ($P$ - price of an option, $S$ - underlying spot): $\theta=r_dP−Sr_d\delta−\frac{1}{2}\...
2
votes
0answers
37 views

Transforming and minimisation of the BS PDE

I'm trying a novel numerical substitution/fitting method to solve the BS PDE, but the issue is that due to the large range of magnitude of prices $V(s,t)\in[10^{-20},10^1]$, when I try to minimise the ...
2
votes
0answers
53 views

How does the diameter of the spatial grid affects the solution of a Crank Nicholson algorithm?

this is my first question so I hope I express myself clearly. I'm trying to implement an Implicit and a Crank Nicolson algorithm for the generic PDE $\partial_\tau u(\tau,x)+a \partial_x^2 u(\tau,x) +...
2
votes
0answers
433 views

American Swaption Pricing with PDE discretization

So I am still trying to price an american swaption. (MC approach here: American Swaption Pricing with Monte-Carlo method) I've found in Paul Wilmott, The mathematics of financial derivatives, a PDE ...
1
vote
1answer
823 views

Pricing log-contract with Black-Scholes PDE

I was wondering if someone could help me with a problem, regarding the Merton Black Scholes PDE. I have an exam soon and this question on an old exam has been bothering me and a friend for quite a ...