1. 1 Strategies with Options Chapter 10

2. 2 Strategies with Options No slides Use of board

3. 3 ASSUMPTIONS: 1.The market is frictionless: No transaction cost nor taxes exist. Trading are executed instantly. There exists no restrictions to short selling. 2.Market prices are synchronous across assets. If a strategy requires the purchase or sale of several assets in different markets, the prices in these markets are simultaneous. Moreover, no bid-ask spread exist; only one trading price.

4. 4 3.Risk-free borrowing and lending exists at the unique risk-free rate. Risk-free borrowing is done by selling T-bills short and risk-free lending is done by purchasing T-bills. 4.There exist no arbitrage opportunities in the options market

5. 5 NOTATIONS: t= the current date. S t = the market price of the underlying asset. K= the option’s exercise (strike) price. T= the option’s expiration date. T-t= the time remaining to expiration. r= the annual risk-free rate.  = the annual standard deviation of the returns on the underlying asset. D= cash dividend per share. q = The annual dividend payout ratio.

6. 6 NOTATIONS: C t = the market premium of an American call. c t = the market premium of an European call. P t = the market premium of an American put. p t = the market premium of an European put. In general, we express the premiums as functions of the following variables: C t, c t = c{S t, K, T-t, r, , D }, P t, p t = p{S t, K, T-t, r, , D }.

7. 7 Options Risk-Return Tradeoffs PROFIT PROFILE OF A STRATEGY A graph of the profit/loss as a function of all possible market values of the underlying asset We will begin with profit profiles at the option’s expiration; I.e., an instant before the option expires.

8. 8 Options Risk-Return Tradeoffs At Expiration 1. Only at expiry; T. 2. No time value; T-t = 0 CALL is:exercised if S > K expires worthless if S  K Cash Flow = Max{0, S – K} PUT is: exercised if S < K expires worthless if S ≥ K Cash Flow = Max{0, K – S}

9. 9 3.All legs of the strategy remain open till expiry. 4. A Table Format Every row is one leg of the strategy. Every row is analyzed separately. The total strategy is the vertical sum of the rows.The profit is the cash flow at expiration plus the initial cash flows of the strategy, disregarding the time value of money.

10. 10 6.A Graph of the profit/loss profile The profit/loss from the strategy as a function of all possible prices of the underlying asset at expiration.

11. 11 The algebraic expressions of profit/loss at expiration: Profit/Loss: Long stock: – S t + S T Short stock:S t - S T Long call:-c t + MaX{0, S T -K} Short call: c t + Min{0, K- S T } Long put:-p t + MaX{0, K- S T } Short put: p t + Min{0, S T -K}

12. 12 Borrowing and Lending: In many strategies with lending or borrowing capital at the risk-free rate, the amount borrowed or lent is the discounted value of the option’s exercise price: Ke -r(T-t). The strategy’s holder can buy T-bills (lend) or sell short T-bills (borrow) for this amount. At the option’s expiration, the lender receives K. If borrowed, the borrower will pay K, namely, a cash flow of – K.