Question

Suppose the current stock price S is greater than the strike K, interest rates and dividends are zero. We buy 100 shares of stock at price S, and instruct our broker to sell the stock if its price drops to K and place the proceeds in default-free bonds. We also borrow K dollars at the risk-free rate. Naively, one might think this “stop-loss order” would provide the same kind of insurance as a call. Explain why this is naive, by explaining in detail all of the possible ways in which the payouts from this setup could differ from the payouts of owning a call with strike K.

Answer 1

If the strike price is greater than K, then the return from the set-up = ( S - K)

If the strike price is at K, then the return from the set-up = 0

If the strike price is below K, then the return from the set-up = 0

Reason for Difference:

1. No. of Periods is fixed in European Call Option & Variable in Stop-Loss Order :

Consider a 3-period case:

Period:	0	1	2
Stock Price	S1 > K	S2<K	S3>K
Action	Long Stock and Short bond	Short Stock and Long Bond	No Action

As seen from above that in a 3-period scenario, we find that the return from this set-up = 0

Now, consider a 3-period call option (European) on the same stock:

Period:	0	1	2
Stock Price	S1 > K	S2<K	S3>K
Action	Long Call	No Action	Short Call

As seen from above that the payout from a Call = S₃ - K > 0

2. Payment of Premium in Call option:

Considering only 1- period ( i.e. that price of the stock only changes once and at that very moment the call option expires)

Pay-out from the Stop-loss is = Max(S-K, 0)

Pay-out from the Call option =Max (S-K) - Premium