Question

Company XYZ’s stock is currently trading at $22 and it is forecasted to either increase by 20% or fall by 15% in 3 months. The 3-month effective interest rate is 1.5%. Assuming there is no active options market. What will an investor do in order to buy a call option on XYZ stock with an exercise price of $20 to achieve the same payoff? Detail all necessary steps and your reasoning.

Answer 1

Call option gives the right but not the obligation to buy back the asset at a predetermined price and at a pretermined date.

Here, we will use Binomial option Valuation Model to solve the problem

Firstly we will calculate delta,the no of shares you should buy so that the payoff is same.

General form of Binomial Option model is given in the screenshot.

In our case,

u = 1 + 20% = 1.2             

d = 1-.15 = .85

In other words, the price might become 1.2 times current or .85 tme current price in next 3 months.

For No arbitrage opportunity.

Payoff+ is the payoff u get   if price rises and payoff- is when prices fall.

solving the given equation for delta, we get

Here,

= 22*1.2 = 26.4

= 22*0.85 = 18.7

= option is in the money = 22*1.2 - 22 = 4.4

= option is out of money = 0

Substituting in above,

= 0.5714

So, the investor should buy 0.5714 shares of XYZ.

Lets calculate the payoff and check if they are same using

=26.4*0.5714 - 4.4 = 10.685

= 18.7*0.5714 - 0= 10.685

As you can see, payoff+ = payoff- if the investore buys amount of shares

We can also find c that is call option value at the start at t=0

PV is the present value

r = 1.5%

c = 22*0.5714 - (10.685/1.015) = 2.04

P.S. Please let me know in the comment if any part of the solution is not clear