Question

Suppose you observe a European call option that is priced at less than the value Max[0, S0 - K(1+r)-T].

What type of transaction should I execute to achieve the maximum benefit? How would I create a payoff table showing the outcomes of expiration?

Answer 1

Here,

S₀ = Spot price of asset

K= exercise price of European call option on the asset

r= interest rate

t = time period for the option to mature

If the European Call option has a price C less than Max ( 0, S₀ - K(1+r)^-t ) i.e. C< Max ( 0, S₀ - K(1+r)^-t )

i.e C < S₀ - K(1+r)^-t and S₀ - K(1+r)^-t >0,  because if S₀ - K(1+r)^-t < 0 , price will be 0 (options cannot have negative value) and the question will not arise.

Therefore, S₀ > C+ K(1+r)^-t (in this case)

Also, (S₀ -C) >K(1+r)^-t  

and  (S₀ -C) (1+r)^t > K

then we can benefit by the following activities

1. Borrow a share, Short (Sell) it and get S₀

2. Out of these proceeds, buy the option and invest an amount of S₀ -C (at a rate of r %) for a period t

and wait for the maturity

On maturity ,your invested amount becomes equal to (S₀ -C) (1+r)^t > K

If Spot price at that time S_t < K

then (S₀ -C) (1+r)^t > K > S_t

then the Option's Value is Zero and you can buy the share at a price of S_t with the investment proceeds and return it and you are left with an amount (S₀ -C) (1+r)^t - S_t as profit.

If Spot price at that time S_t > K

then the Option's Value is (S_t - K) and you can buy the share at a price of S_t with the investment proceeds and return it and by exercising your option you get (S_t - K) , So you are left with an amount {(S₀ -C) (1+r)^t + ((S_t - K)} - S_t   = (S₀ -C) (1+r)^t - K as profit which is a positive figure .

The payoff table for the above is shown below

	Value
	Today	in Future
		if St< K	If St >K
Short Stock	-S0	-St	-St
Invest S0-C	S0-C	(S0-C)(1+r)^t	(S0-C)(1+r)^t
Buy Call option	C	0	St-K
Total	0	(S0-C)(1+r)^t -St	(S0-C)(1+r)^t -K

Since (S0-C)(1+r)^t -St > 0 when St< K

and (S0-C)(1+r)^t -K >0 when St > K

we can benefit in all situations.