Black Scholes PDE

[Mtm]

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:

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,

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!!!

 

[Tsotne]

http://www.dougvestal.com/blackscholespde.pdf

As for d1 and d2, I don't think they are called something formally, they are just d1 and d2. Shortened notation for those quotients.

 

[ExSan]

Hello everyone,
I have been reading a great book about the mathematics that are used in asset pricing.

... and the book is :

 

[Mtm]

http://www.dougvestal.com/blackscholespde.pdf

As for d1 and d2, I don't think they are called something formally, they are just d1 and d2. Shortened notation for those quotients.

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.

 

[roni]

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)

 

[amanda.jayne]

N(d2) is the risk neutral probability of exercize

 

[Lyosha]

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.

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.

 

[Tsotne]

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.

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.

 

[Mtm]

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.

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

like you said....it is only after you have d1 and d2 that you use the normal distribution function

....

 

[Eugene Krel]

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.

 

[Lyosha]

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?

 

[myampol]

There is a different paper on Mr. Doug Vestal's website which illustrates exactly where d1 and d2 come from:

http://www.dougvestal.com/europeancallprice.pdf

 

[bob]

N(d2) is the risk neutral probability of exercize

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.

 

[Mtm]

Thank you so much everybody, I think I have figured out how to compute d1 and d2. Yaaay!!!