Links to all tutorial articles (same as those on the Exam pages)Quick primer on Black Scholes
The conceptual idea behind Black Scholes is rather simple – but as the argument advances beyond the initial idea, things become more complex with differential equations, risk-neutrality and log returns stepping in. For the PRMIA exam, you will not be asked for a derivation of Black Scholes, so it may suffice to know just a couple of things. This brief write-up aims to summarize just those few things.
The key idea behind Black Scholes:Consider a long position in a call option on a stock. For a small movement in the price of the underlying, the value of the option changes by a certain amount (the delta). The movement can be up or down, but we are only looking at a small (ie ‘local’) movement in the price of the underlying. It is possible to sell a certain amount of stock (the delta, again) and hold a small short position in the stock so that any change in the value of the option will be offset by the change in the value of the short stock holding. For the exam, it is important to break out the first equation above identifying its component parts. Here it is:
Value of putIn the same way, we can get the value of a put as well. We know from the put-call parity (read here if you wish to recall it again) that We substitute the value of call from the above, we know the PV of the exercise price to be Ke^(-rt), and the spot price is S. Therefore
The Black Scholes PDE:The Black Scholes PDE is derived by combining the results of a stock price following a Weiner process with a mathematical result known as Ito's lemma and holds true for all derivatives whose prices depend upon S, the price of the underlying and upon t, which indicates time. For a fuller explanation and mathematical derivation, refer to John Hull’s book on options and other derivatives.
this can also be written as
Here is a spreadsheet that models the Black-Scholes – have a look, and try out a few values. The spreadsheet actually has a custom function (called ‘bs’ – no witty comments please!). Enter the five parameters, and see the call and put values on the right, together with the Greeks. |