Question

A stock price is currently $50. It is known that at the end of two months it will be either $53 or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a two-month European call option with a strikeprice of $49? Use no-arbitrage arguments.

At the end of two months the value of the option will be either $4 (if the stock price is $53) or $0 (if the stock price is $48). Consider a portfolio consisting of:

+Δ : shares

-1 : option

The value of the portfolio is either 48Δ or 53Δ - 4 in two months. If

48Δ=53Δ-4

i.e., Δ=0.8

the value of the portfolio is certain to be 38.4.

I want to ask why 0.8 is related with 38.4? Thank you

Answer 1

* Any doubt please comment.

0.8 is net change in the value of the portfolio and for this value it is riskless. The portfolio value is riskless for the  Δ chosen so that final value of the portfolio is same for both the alternatives.

If the stock price moves up to $53 , the value of the portfolio = 53* 0.8 = 42.4

If the stock price moves down to $48 , the value of the portfolio = 48* 0.8 = 38.4

Hence the value of the portfolio will not fall below $38.4 given a change of 0.8 i.e. it is the least value that the portfolio can have.