Question

A unit of stock is valued at $90. In one year, the stock price will be either $105 or $80 (no probabilities are given). The current (time 0) riskless interest rate is 4%. Use the game theory approach of Section (2.2.4) to price a put option on the stock, where the option expires in one year with a strike price of $110.

Basically I understand the concept of this, but from all of the equations I can find in my notes, it feels like I'm missing some information. I have the solution from the book, but I don't know how to arrive at it, so I'd like to see the work.

Answer 1

Initial Portfolio

A0 = V0-aS0 in next state there are two possibilities either up or down

in case of Up scenario

Au = Vu-aSu

in case of Down scenario

Ad = Vd-aSd

therfore

In Game theory we look for stable equillibrium as We have Same portfolio value in each case

Vu-aSu=Vd-aSd

a=(Vu-Vd)/(Su-Sd)= Change in V/ Change in S

we have

V0- aS0=exp(-rt)*(Vu-aSu)

V0= aS0+exp(-rt)*(Vu-aSu)

V0 is the value of an option

We need to calculate "a"'

Vu is Value of Put Option when Su= 105; Vu=Max(K-Su,0)=Max(100-105,0)=0

Vd=Max(100-80,0)=20

a=(0-20)/(105-80)=-20/25=-0.8

V0= aS0+exp(-rt)*(Vu-aSu)=(-0.8)+exp(0.04)*(0-0.8*105)=-0.8*90+exp(-0.04)*(84)=84*0.9607-0.8*90=80.6988-72=8.6988

Value of Put option is 8.6988