Question

Shares of ABC will sell for either $180 or $120 in three months, with probabilities 0.75 and 0.25 respectively. A European call on ABC with an exercise price of $150 sells for $20 today, and an identical put sells for $4. Both options mature in three months. What is the price of a three-month zero-coupon bond with a face value of $100

Answer 1

We know probability of upmove=((1+r)^t-Sd/S0)/(Su/S0-Sd/S0)
=>((1+r)^t-120/S0)/(180/S0-120/S0)=0.75
S0*(1+r)^t-120=0.75*(180-120)
=>S0*(1+r)^t=165

Delta of call=(MAX(180-150,0)-MAX(120-150,0))/(180-120)=0.5

Replicating portfolio is sell 1 call buy 0.5 shares of stock

Cost today=0.5*S0-C

Present value of payoff=(0.5*180-MAX(180-150,0))/(1+r)^t

Hence,
(0.5*180-MAX(180-150,0))/(1+r)^t=0.5*S0-C
(0.5*180-MAX(180-150,0))/(1+r)^t=0.5*S0-20
=>60/(1+r)^t=0.5*S0-20
=>S0=(60/(1+r)^t+20)*2
=>S0=120/(1+r)^t+40
=>S0*(1+r)^t=120+40*(1+r)^t

Hence,
120+40*(1+r)^t=165
=>(1+r)^t=45/40

Using put call parity
X/(1+r)^t=S0+P-C
Price of zero coupon bond with face value of 150=150/(45/40)=133.3333333