Question

A share now sells for $40.00. This price will either increase by 10% (by a factor of u = 1.10) or decrease by 20% (by a factor of d = 0.80) over a six-month period. A European call option on this share has an exercise price of $42 and a time to expiry of six months. The risk-free rate is 6% per year. Use the one-period binomial option pricing model to find the price of the call option. You should present your calculations and explanations as follows: a. Draw tree-diagrams to show the possible paths of the share price and call price over one six-month period. Note: Show the numbers that are known and use letter(s) for what is unknown in your diagrams. b. Compute the hedge ratio and show clearly how to form a perfect hedge. c. Find the call option price. Explain your calculations clearly. An analyst disagrees with the current share price movement predictions and believes that the share price will either increase by 20% or decrease by 30% over a six-month period. Assume that the European call option has the same exercise price and the same time to expiry. d. Would the analyst price the European call option at, above, or below the original price calculated in Part c? Why? [Maximum: 200 words]

Answer 1

a) Current stock price, S_O = 40

Upside, u = 1.1, downside,d =0.8, riskfree rate,r = 6%per year

Time to expiry, T = 0.5 Years; Strike price, X = 42; let call price be c

Risk neutral probability, p = [(1+6%)^0.5 -d]/[u-d] = (1.029-0.8)/(1.1-0.8) = 0.23/0.3 = 0.7652

In period 1:

If the stock price goes up, S = 1.1*40 = $44,

Value of call option = $44-$42 = $2

If the stock price goes down, S= 0.8*40 = $36

Value of call option = 0 (as the call option will not be execised)

Value of call option at time of expiration = p*2 + (1-p)*0 = 0.7652*2 = $1.53042

Price of call option = PV(Value of call option) = $1.53042/(1.06^0.5) = $1.486

b) At time t=0,

lets make a portfolio of short 1 call option and long on x stock, where x is the hedge ratio and this portfolio is perfectly hedged.

At time of expiry,

Value of portfolio if stock price goes up = x*44 - 2

Value of portfolio if the stock price goes down = x*36

if this is perfectly hedged portfolio, the value of portfolio should not vary whether stock is going up or down.

44x-2 = 36x

x = 2/8 =0.25

c) call option price is calculated above to be $1.486

d)

Current stock price, S_O = 40

Upside, u = 1.2, downside,d =0.7, riskfree rate,r = 6%per year

Time to expiry, T = 0.5 Years; Strike price, X = 42; let call price be c

Risk neutral probability, p = [(1+6%)^0.5 -d]/[u-d] = (1.029-0.7)/(1.2-0.7) = 0.329/0.5 = 0.66

In period 1:

If the stock price goes up, S = 1.2*40 = $48,

Value of call option = $48-$42 = $6

If the stock price goes down, S= 0.7*40 = $28

Value of call option = 0 (as the call option will not be execised)

Value of call option at time of expiration = p*6 + (1-p)*0 = 0.66*6 = $3.96

Price of call option = PV(Value of call option) = $3.96/(1.06^0.5) = $3.84

Hence, the analyst would price the call option higher than the original primarily because of increased volatility of the underlying .