• C++ Programming for Financial Engineering
    Highly recommended by thousands of MFE students. Covers essential C++ topics with applications to financial engineering. Learn more Join!
    Python for Finance with Intro to Data Science
    Gain practical understanding of Python to read, understand, and write professional Python code for your first day on the job. Learn more Join!
    An Intuition-Based Options Primer for FE
    Ideal for entry level positions interviews and graduate studies, specializing in options trading arbitrage and options valuation models. Learn more Join!

Black Scholes PDE

Mtm

Joined
3/1/11
Messages
5
Points
11
Hello everyone,

I have been reading a great book about the mathematics that are used in asset pricing.
One of the famous quantitative tools that are used for pricing options is the Black Scholes PDE.
And the formula goes like this:

2f8339883088e41aa40a457fc6b7bea5.png



Where C(S,t) is the value of the call option,N(d1) is the delta for call on non-dividend paying asset and N(d2) the risk-neutral probability of the call option to expire in-the-money.

But most importantly,
b10af632e794204ade814c84f49b15fa.png
6f76322966dfa96d44a0c3a02273950e.png


Here is where everything gets a little complicated...what are the formal names of d1 and d2? How were they computed/calculated? They are obviously solutions to some kind of differential equations but what are those?
Thank you!!!

 
I've heard the term "market eye" for N(d2) (if I'm not mistaken).
N(d1) is the delta hedge (if I'm not mistaken again) :)
 
Mtm said:
Yes but what do these quotients represent?? what were they computing to get these quotients??because we obviously need them to use in the normal distribution formula.
Click to expand...

If you try to derive the black scholes price (in your case of a european call option) it becomes evident what d1 and d2 represent - they are the bounds on the integral that remain after substitution. You get to them by figuring out the relationship between your assumed evolution of your price and the normal distribution.

I would suggest looking at the chapter dedicated to Black Scholes in this book: http://www.amazon.com/Primer-Mathem...=sr_1_1?ie=UTF8&s=books&qid=1302213438&sr=8-1 for further insight.
 
If you try to derive the black scholes price (in your case of a european call option) it becomes evident what d1 and d2 represent - they are the bounds on the integral that remain after substitution.
Click to expand...

Yes it is clear in sense what it represents but we are seeking "one name" which I'm not sure if there exists for d1 and d2. You can call it something once having extracted it's role from the formula.
 
Alexei Smirnov said:
If you try to derive the black scholes price (in your case of a european call option) it becomes evident what d1 and d2 represent - they are the bounds on the integral that remain after substitution.
Click to expand...
They clearly represent the upper bounds....but the formulas for d1 and d2 did not just happen to be, Scholes was clearly looking for something when he computed these bounds.
You don't get them by
figuring out the relationship between your assumed evolution of your price and the normal distribution
Click to expand...
like you said....it is only after you have d1 and d2 that you use the normal distribution function
2032b6d5a33c78400a94e7e39df45c37.png
....
 
As Alexei said, the factors come out easily using the risk neutral derivation. I don't think the original paper used the risk neutral derivation as most people do it today.
 
The full derivation would take me an eternity to latex up. But you assume that your asset follows the following distribution:
\[ln(S_T/S_0)=(r-q-\sigma^2/2)T+\sigma\sqrt(T)Z\]

... do you see where at least one of d1 and d2 comes from?
 
amanda.jayne said:
N(d2) is the risk neutral probability of exercize
Click to expand...
To elaborate further, both \(N(d_1)\) and \(N(d_2)\) are risk-neutral probabilities of exercise. In the familiar money-market numeraire, this is a call option and \(K e^{-r \tau}\) is the quantity of the numeraire asset that must be paid on exercise. Since our numeraire asset never "changes value"--all other values change in relation to it--the coefficient \(N(d_2)\) is evidently the probability that you will have to make that payment.

If you think of the stock as the numeraire, then this option is a put on the money market asset, and \(S e^{-q \tau}\) is in essence also the quantity of the numeraire asset that will be received on exercise (more specifically, 1--but expressed in currency today). By similar reasoning, \(N(d_1)\) must be the probability weight that describes how likely it is you will receive this amount. If you think of the payoff as a put payoff you can show with a bit of algebra that what we call \(N(d_1)\) in the call option (mm numeraire) setup is also equal to \(N(-d_2)\) in the put option (stock numeraire) setup.

It's a bit odd for people initially to think of the stock never changing value and the value of the money-market asset being the thing that's volatile, so it's usually easier for people to see put-call duality in the context of FX options by switching around the "foreign" and "domestic" numeraires. Indeed, in FX a call on AUD that requires payment in USD is quoted explicitly as an "AUD call USD put." Nevertheless, the same idea applies even to the familiar case.

This is yet another illustration of the fact that the interpretation of these quantities as "probabilities" can be a bit misleading: They are probabilities under a specific set of required conditions. You can think of these quantities also as arising out of general solutions to the heat equation, but at that point--for me anyway--they lose all their intuitive meaning and simply become mechanical necessities to solve the PDE subject to a certain set of boundary conditions.
 
Thank you so much everybody, I think I have figured out how to compute d1 and d2. Yaaay!!!
 
Back
Top