Mathematical Modeling in Finance with Stochastic Processes

240 CHAPTER 7. THE BLACK-SCHOLES MODEL

and then present the solution as an algorithm putting the pieces together to
obtain the final answer. Specifically, let

d 1 =

log(S/K) + (r+σ^2 /2)(T−t)

σ

√

T−t

d 2 =

log(S/K) + (r−σ^2 /2)(T−t)

σ

√

T−t

so that

VP(S,t) =S(Φ (d 1 )−1)−Ke−r(T−t)(Φ (d 2 )−1).

Using the symmetry properties of the c.d.f. Φ, we obtain

VP(S,t) =Ke−r(T−t)Φ (−d 2 )−SΦ (−d 1 ).

Graphical Views of the Put Option Value

For graphical illustration letPbe the value of a put option with strike price
K= 100. The risk-free interest rate per year, continuously compounded is
12%, sor= 0.12, the time to expiration isT = 1 measured in years, and
the standard deviation per year on the return of the stock, or the volatility
isσ= 0.10. The value of the put option at maturity plotted over a range of
stock prices 0≤S≤150 surrounding the strike price is illustrated below:
Now we use the Black-Scholes formula to compute the value of the option
prior to expiration. With the same parameters as above the value of the put
option is plotted over a range of stock prices 0≤S≤150 at time remaining
to expirationt= 1 (red),t= 0.8, (orange),t= 0.6 (yellow),t= 0.4 (green),
t= 0.2 (blue) and at expirationt= 0 (black).
Notice a couple of trends in the value from this graph:

1. As the stock price increases, for a fixed time the option value decreases,


2. As the time to expiration decreases, for a fixed stock value price lass
   than the strike price the value of the option increases to the value at
   expiration.

We can also plot the value of the put option as a function of security price
and the time to expiration as value surface: