Question

An investment bank is offering a new financial instrument called a “happy call.” The happy call has a payoff at the end of the year equal to max [0.5S1, S1−$10], where S1 is the unknown price of a stock at the end of the year. You always get something with the happy call. The market price of the stock today is$11. There are two ordinary call options with strike prices of $10 and $20 being traded in the market. The market prices of these two calls today are $1.5 and $0.5, respectively.

Q1) Illustrate graphically the value of a long position in the happy call at the end of the year.

Q2) What would be the fair price of the happy call today?

Answer 1

Q1. by simply putting 1 to 10 we will get the unknown price of stock at the end of the year.

Year end price of stock	Pay off of strike price $10	Payoff of strike price with $20
0.5	-1.5	- 0.5
1.0	-1.5	- 0.5
1.5	-1.5	- 0.5
2.0	-1.5	- 0.5
2.5	-1.5	- 0.5
3.0	-1.5	- 0.5
3.5	-1.5	- 0.5
4.0	-1.5	- 0.5
4.5	-1.5	- 0.5
5.0	-1.5	- 0.5

b. fair price of option by black scholes formula

C=St​N(d1​)−Ke−rtN(d2​)     where:    d1​=​lnSt/ K ​​+ (r+σv2​​/2) t​    /σs * √ ​ t    and             d2​=d1​−σ √ s​t

​C=Call option price, S=Current stock (or other underlying) priceK=Strike pricer=Risk-free interest ratet=Time to maturityN=A normal distribution​

by putting values in formula we are able to calculate fair value but risk free rate & standard deviation is not given in the question.so i am no able to calculate it.