Question

A New York firm is offering a new financial instrument called a "happy call”. It has a payoff function at time T equal to max{0.5ST, ST – K}, where St is the price of the stock at T and K is a fixed strike price. You always get something with a happy call. Let S be the price of the stock at time 0 and let C1 and C2 be the prices of ordinary calls with strike prices K and 2K, respectively. The fair price of the happy call is of the form Ch = aS + BC1 + 7C2. Find the constants a, B and y.

Answer 1

Happy payoff calloption payoff at time T = max(0.5ST, ST-K)

Payoff for call option C1 = max(0, ST-K) because strike price is K and the option will be worthless if stockprice is < K

Payoff for call option C2 = max(0, ST-2K) because strike price is 2K and the option will be worthless if stockprice is < 2K

fair price of happy call  

CH=aS + BC1 + C2

Since we have three variables, we will need three equations to solve and determine the values of the variables.

We will use three conditions, ST<K, ST=1.5K (value between K and 2K, for convenience sake we will use 1.5K) and ST=2.5K ( a value greater than 2K, for convenience sake we will use 2.5K), on the stock price at maturity to create the three equations

condition 1: ST < K

Because the stock price is less than K, the payoff from happy call will be 0.5ST, payoff from C1 is max(0,value less than K-K)= max (0, negative number) = 0 and payoff from C2 is max (0,value less than K-2K) = max (0, negative number) = 0.

After substituting these values in the equation we get,

0.5ST =

0.5 ST = a St + 8 0+*0= OST

Hence alpha = 0.5

condition 2: ST =1.5 K (value between K and 2K, for convenience sake we will use 1.5K)

Because the stock price is equal to K, the payoff from happy call will be max (0.5K, 1.5K-K) = max(0.5K,0.5K) = 0.5K, payoff from C1 is max(0,1.5K-K) = 0.5K and payoff from C2 is max (0,1.5K-2K) = max (0,-0.5K) = 0.

After substituting these values in the equation we get,

0.5K = 0.5*(1.5K) + 8 +0.5K +7*0=0.75K +*0.5K

0.5K - 0.75K =

B* 0.5K

-0.25K =

B* 0.5K

Hence beta = -0.5

condition 3: ST = 2.5K ( a value greater than 2K, for convenience sake we will use 2.5K)

Because the stock price is equal to K, the payoff from happy call CH is max (0.5K, 2.5K-K) = max(0.5K,1.5K) = 1.5K, payoff from C1 is max(0,2.5K-K) = 1.5K and payoff from C2 is max (0,2.5K-2K) = max (0,0.5K) = 0.5K

After substituting these values in the equation we get,

1.5K = a* 2.5K + B * 1.5K +7* 0.5K

Substituting the values of alpha and beta we get:

1.5K = 0.5*2.5K + (-0.5) *1.5K + gamma*0.5K = 1.25K-0.75K +gamma *0.5K = 0.5K + gamma * 0.5K

hence gamma = (1.5K - 0.5K)/0.5K = 1K/0.5K = 2

Answers:

alpha = 0.5,

beta = -0.5

gamma = 2