Relations between Call and Put

Question

I am trying to solve a question in finance but I am pretty much stuck and would need your help :)

Suppose you know the following information about a market:

Future is at 66 70 strike straddle is trading at 27 50-60 put spread is at 2.5 50-60-70 put fly is at .2 Assume volatility is constant across strikes Using the prices given and relationships between options of various strikes, what are the fair values for the 80 Call, 60 Straddle, and 40 Put? Assume we had a volatility smile among the curve, how would this make your markets different?

I started by the following equations: C(70) - P(70) = 66 C(70) + P(70) = 27 P(50) - P(60) = 2.5 P(50) - 2P(60) + P(70) = 0.2

Thanks!

Answer 1

You haven't written down your equations correctly. Ignoring discounting, the equations should be: C(70)-P(70)= -4 (not 66), from put-call parity. Also, C(70) + P(70)= 27;
from these two we get C(70)= 11.5 and P(70)=15.5

Also P(60)-P(50)= 2.5 and P(70)-2P(60)+P(50)=0.2 from which P(70)-P(60)=2.7, hence P(60)=12.8 and P(50)=10.3 so now we know all the option prices for 50, 60, and 70 strikes

So C(60)-P(60)= 6 from put-call parity, giving C(60) = 18.8, and therefore straddle (60)= 31.6

we cannot exactly know where the 80 call and 40 put are, without making a distributional assumption. all we can sa.y is that C(80)<=C(70)= 11.5 and P(40)<=P(50)=10.3