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

Put-call parity equation

C + Present Value (x) = S + P

It can be written as

C + X/(1+r)^t = S+ P

Where,

C = call premium

P = put premium

X = strike price

r = interest

t = time period

S = initial price

Therefore,

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

Now,

Payoff = Max [2000, spot price at expiration – strike price]

= Max [2000,2000+0.5 x (St-3200)

= 2000 + Max(0,0.5 x (St-3200))

= 2000 + 0.5 x Max(0,St-3200))

The above payoff seems like derived from 2000+0.5*C

where,

C indicates call option with strike price of 3200

Hence,

Cost = 2000 / (1+r)^t+ (0.5 x 120)

Substitute the value of (1+r)^t = 1.0031348 in the above equation

= 2000 / 1.0031348 + 60

= 1993.74999 + 60

= 2053.75

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