Question

Suppose that p1, p2, and p3 are the prices of European put options with strike prices K1, K2, and K3, respectively, where K3>K2>K1 and K3-K2=K2-K1. All options have the same maturity. Show that p2 <= 0.5(p1+p3) (Hint: Consider a portfolio that is long one option with strike price K1, long one option with strike price K3, and short two options with strike price K2.)

Answer 1

As per Hint Given:-

Portfolio-

Option A :- Long 1 option with Strike price K1 thus paying Option premium of p1

Option B:- Long 1 option with strike price K3 thus paying Option Premium of p3

Option C:- Short 2 option with Strike price K2 thus receiving option premium of 2*p2

Also given that k3 >K2 >K1

and K3-K2= K2 -K1

Since all Options has same maturity i.e Time Value of all Options is same and thus can be ignored

P1, p2,p3 comprises of Intrinsic value and Time Value since time value of all options is same, we ignore that and thus lets consider p1, p2, p3 all are Intrinsic value of options

Intrinsic Value of Put Option = Put Strike Price - Current Stock Price

Option A : k1 - CSP = p1 ----------(1)

Option B: K2 - CSP = p2 ---------(2)

Option C: K3 - CSP = p3 ---------(3)

since CSP is constant thus p3>p2>p1

Comparing 1 and 2

we get k1-k2 = p1 -p2 -------------(4)

Comparing 2 and 3

we get k2-k3 = p2 -p3 ===== -(K3-k2) = p2-p3

and it is given that k3-k2 = k2-k1

thus -(k2-k1) = p2-p3 ===== k1-k2 = p2-p3 -------------(5)

comparing 4 and 5 equation we get

p1-p2 = p2-p3

p1+p3 = 2*p2

p2 = 0.5(p1+p3)

Also we get 2p2 - p1- p3 =0 which is our net proceeds/ payments from the portfolio i.e 2p2 -p1-p3

also since p3>p2>p1

Thus p2 being less p3 and assuming the options we have entered it can never be more than average of p1 and p3 but possibility is there that it can be lower than that

we get 2p2 - p1 -p3

thus P2 < = 0.5(p1+p3)

Happy Learning!