Question

The stock of Company ABC is currently trading at a price of $50. According to analyst forecasts, the share price will be either $45 or $55 at the end of six months. Suppose that the risk-free interest rate is 10% per annum with continuous compounding. What is the value of a six-month European put option with a strike price of $50 using the Delta-hedging method as discussed in the lecture notes?

Answer 1

For DELTA HEDGING,

By the principle of replication, we need to create a portfolio with Stock and Bonds which gives the same payoff as that of the Put option, Suppose, we combine Δ stocks with $B invested in Bonds to create the same portfolio

Then , in case of upside Δ stocks value would be Δ*55 and $B invested in bonds would be valued at B*exp(0.10*6/12), and Put options value would be max (50-55,0) =0

Therefore, Δ*55+B*exp(0.05)=0

Similarly. for downside, Δ*45+B*exp(0.05)= 5

Solving these for Δ and B , we get, Δ = -5/10= -0.5 and B = $26.16

This means, a hypothetical portfolio with Short position in 0.5 stocks and $26.16 invested in Riskfree Bonds will give us the same value as that of the Put.

The DELTA of the Put option is -0.5

Hence, the value of the option today should be the same as that of constructing the portfolio today

which is -0.5*50 +26.16 = $1.16

So, the value of the European Put option is $1.16