- Option Pricing 181
Theorem 8.5 (Cox–Ross–Rubinstein Formula)
In the binomial model the price of a European call and put option with strike
priceXto be exercised afterNtimestepsisgivenby
CE(0) =S(0) [1−Φ(m− 1 ,N,q)]−(1 +r)−NX[1−Φ(m− 1 ,N,p∗)],
PE(0) =−S(0)Φ(m− 1 ,N,q)+(1+r)−NXΦ(m− 1 ,N,p∗).The initial replicating portfoliox(1),y(1) is given by
x(1) y(1)
for a call 1 −Φ(m− 1 ,N,q) −(1 +r)−NX[1−Φ(m− 1 ,N,p∗)]
for a put −Φ(m− 1 ,N,q) (1 +r)−NXΦ(m− 1 ,N,p∗)Exercise 8.7
LetS(0) = 50 dollars,r=5%,u=0.3andd=− 0 .1. Find the price of
a European call and put with strike priceX= 60 dollars to be exercised
afterN= 3 time steps.Exercise 8.8
LetS(0) = 50 dollars,r=0.5%,u=0.01 andd=− 0 .01. Findm, x(1),
and the priceCE(0) of a European call option with strikeX= 60 dollars
to be exercised afterN= 50 time steps.Exercise 8.9
Consider the scenario in which stock goes up at each step. At which step
will the delta of a European call become 1?8.2 American Options in the Binomial Tree Model ...............
Even the formulation of a precise mathematical definition of an American type
contingent claim presents some difficulties. Nevertheless, the informal descrip-
tion is simple: The option can be exercised at any time stepnsuch that
0 ≤n≤N,withpayofff(S(n)). Of course, it can be exercised only once.
The price of an American option at timenwill be denoted byDA(n).