Put/call Option Relation Exercice

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Hi all,

 

I'm a total beginner and want to learn about trading. I've read some books and websites and found this one; the articles are really helpful!

 

While studying options, I've come across the following question and I don't know how to solve it:

 

Suppose you know the following:



- 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?

 

Here is what I think: let's denote by p(K) and c(K) the put and call with strike K (assume sigma, time to maturity T, risk free r are constant and the same for all options). We know that

 

p(70)+c(70)=27

-p(50)+p(60)=2.5

p(50)-2*p(60)+p(70)=0.2

 

We also know that put-call party holds, i.e. c+K e^(-r*T)=p+S_0 and the price of the forward F=S_0*e^(r*T)=66.

 

From this information how do I compute c(80), p(40) and p(60)+c(60)? What do they mean by "relationships between options of various strikes"?

 

Thanks!