Question

You wish to create a financial instrument that has a payoff in 6 months’ time equal to the maximum value of $2,000 and $2,000 + $0.5*(S&P Index in 6 months’ time – 3,200). The 6-month call and put options with strike price 3,200 is trading at 120 and 110, respectively. What is the cost of your instrument?

Answer 1

lets use Put-call parity method here
C + Pv (x) = S + P
even written as
C + X/(1+r)t = S+ P
here,
C is call premium and P is put premium & X is strike price & r is interest & t is time period & S is initial price
hence,
C + X/(1+r)t = S+ P
3200/(1+r)t = 3200 + 110 – 120
(1+r)t = 3200 / (3200+110-120)
(1+r)t = 3200 / 3190
(1+r)t = 1.0031348
here,
Payment is = Max [2000, spot price at expiration – X ]
= Max [2000,2000+0.5 x (St-3200)
= 2000 + Max(0,0.5 x (St-3200))
= 2000 + 0.5 x Max(0,St-3200))
as mentioned payment looks like executed of 2000+0.5xC
while, C shows the call choice with x of 3200
therfore,amount = 2000 / (1+r)t+ (0.5 x 120)
put the value of (1+r)t = 1.0031348 in the mentioned formula
= 2000 / 1.0031348 + 60
= 1993.74999 + 60
= 2053.75.