Here, we want to describe the logic behind the Black-Scholes Option Pricing Formula which looks like this:
>What do all those symbols ...?
^{2},
has an Expected Value of
which we recognize as the square of the
Standard Deviation.
>We do? >Just what's the purpose of all this? We know that, at expiry of a Call Option, if S is the stock price and K is the Strike Price then the Option is worth S - K provided S is greater than K ... or it's worth nothing (if S is less than K). In other words, the Option Price (at expiry) is the maximum of S - K or 0, namely:
(1)
Ah, but that's at expiry of the Option.
Suppose that, t years in the future, the stock price is The Expected Value of this Quantity, assuming some distribution of yearly (or monthly or weekly) stock returns, is then
(2) which we recognize as = where we integrate from K since Max(x - K,0) is zero when x < K. >If that's the Expected value of the Option at expiry, then what's the expected value now, today, this very minute? r.
That means it's worth
(3) C = e or
(4) C =
e where F is the cumulative distribution. >Where did that e
{1+r} as the annual growth factor, we
use {e^{r}} and so {1+r}^{-t} becomes
{e^{r}} ^{-t} which is e^{-rt}, so the present Value
is written as C_{t} e^{-rt} ... instead of C_{t}
= C{1+r}^{-t}.
>Sounds like mumbo-jumbo to me. The math is much, much nicer. Besides, this calculation of present value is what one
means by "risk-neutral": the value of an asset at time t discounted to its present value
using the risk-free rate.
The two pieces in Equation (4) will give rise to the two pieces of the Black-Scholes formula in Figure 1. Now we stare at the stock price at time t, namely S _{t} (which, as a random variable,
we're calling x). If the returns over each time period (a year, a month, a week) are
r_{1}, r_{2}, r_{3}, etc. then we write
1 + r_{k} = exp(g_{k}), where exp(x) means = e^{x}.
The cumulative gain over t time periods, namely
(1+r_{1})(1+r_{2})...(1+r_{t})
can now be written more simply as:
_{1})exp(g_{2})exp(g_{3})...exp(g_{t}) =
exp(g_{1}+g_{2}+...+g_{t}) =
exp(Σg_{k}) =
exp({(1/t)Σg_{k} } t) =
exp(Mt) _{k} }
is the Mean value of the g's (over t time periods).
Of course, the set of g's (namely g Here's where we make a simplifying assumption: we assume that these
g's are Normally distributed and, since 1+r = e >zzzZZZ
Don't worry, I don't intend to evaluate any integrals. It's much too scary for me. |