1. Copyright K.Cuthbertson, D.Nitzsche. 1 Version 11/9/2001 Lecture Options Pricing

2. Copyright K.Cuthbertson, D.Nitzsche. 2 Valuation and Pricing (Black Scholes) Speculation Delta Hedging Topics

3. Copyright K.Cuthbertson, D.Nitzsche. 3 Valuation/Pricing of Options

4. Copyright K.Cuthbertson, D.Nitzsche. 4 Strike Price K ___ Expiry Month. Strike Price K ___ Expiry Month. October Jan April October Jan April 360 36.5 50 57.5 390 21.5 35.5 44 Current Share Price, S = 376 How do the call premia vary with the time to maturity ? How do the call premia vary with the strike price? Could you make a profit from immediately exercising these options ? Oct 360 call: Intrinsic Value = $ amount if immediately exercised = 376-360 = S t - K = 16 (In-the-money) So, Time Value = C - 41 = 20.5 (payment for possibility of price rise in the future T7.3 Price Quotes:Call Premia on BA: 27th July 2000 (pence)

5. Copyright K.Cuthbertson, D.Nitzsche. 5 Call Premium (price) C depends on: Time to maturity, T (+) Current Spot price, relative to strike price, S /K (+) Volatility spot price (+) Interest rate r (+) Put Premium (price) P depends on: Time to maturity, T (+ or -) Current Spot price, relative to strike price, S /K ( - ) Volatility spot price (+) Interest rate r (+ or - ) Value of Calls and Puts Prior to Expiry

6. Copyright K.Cuthbertson, D.Nitzsche. 6 PV = present value of the strike price = K e -rT Black-Scholes (Merton)

7. Copyright K.Cuthbertson, D.Nitzsche. Stock Price K 0 Value of call prior to expiry B A Payoff from call at expiry. C 0 = 9.6 C 1 = 10 S 1 = 101S 0 = 100 C.. D StSt CtCt CD=Intrinsic value BC=Time value Figure 21.5 : Black-Scholes Call Premium

8. Copyright K.Cuthbertson, D.Nitzsche. Key results are CALLS Call premium increases as stock price increases (but less than one-for-one) Call premium increases if the volatility of the stock increases PUTS Put premium falls as stock price increases (but less than one-for-one) Put premium increases if the volatility of the stock increases Black-Scholes

9. Copyright K.Cuthbertson, D.Nitzsche. 9 Speculation with Options

10. Copyright K.Cuthbertson, D.Nitzsche. 10 Buy low and sell high Expect a bull market - then buy a call and close out the position before maturity ( that is, sell the call at a higher call premium) Profit (before maturity) = C 1 - C 0 = 5.2 - 5.0 Your net position of zero is noted by the Clearing House which sends you the $0.2 (and cancels any delivery to you) Sell high and buy low Expect a bear market then sell a call at C 0 = 10 and close out the position by buying back a call at C 1 = 9.8 (prior to maturity) Speculation with Options (Before Maturity)

11. Copyright K.Cuthbertson, D.Nitzsche. 11 Delta Hedging with Options

12. Copyright K.Cuthbertson, D.Nitzsche. 12 Problem: You currently hold shares but you fear high volatility of stock prices over the next next month. You want to protect the current value of your stock position until the market returns to normal. Can you hedge your stock position using options ? A call option on the share is available with a delta of 0.4 which implies: When S increases by +$1 (e.g. from 100 to 101), then C increases by $0.4 (e.g. from 9.6 to 10) Delta Hedging with Options

13. Copyright K.Cuthbertson, D.Nitzsche. 13 Note: The contract size for one call option contract is for 100 shares But the price of the call option C is quoted as if there was only one share underlying the option (i.e. we need to multiply C by 100 to get the invoice price of the option) Delta Hedging with Options

14. Copyright K.Cuthbertson, D.Nitzsche. Stock Price K 0 B A Payoff from call at expiry. C 0 = 9.6 C 1 = 10 S 1 = 101S 0 = 100 C.. D StSt CtCt CD=Intrinsic value BC=Time value Delta is the slope of this curve Delta of a Call Call premium

15. Copyright K.Cuthbertson, D.Nitzsche. 15 Consider the following portfolio 40-shares plus 1 written (sold) call option (at C 0 = 10) Suppose S falls by $1 over the next month THEN fall in C is 0.4 ( = delta of the call)So C falls to C 1 = 9.6 To close out you must now buy back at C 1 = 9.6 Loss on 40 shares = $40 Gain on initial written call = 100 (C 0 - C 1 )= 100(0.4) = $40 DELTA HEDGING YOUR 40 SHARES WITH 1 WRITTEN CALL OPTION MEANS THAT THE VALUE OF YOUR POSITION WILL BE UNCHANGED. Delta Hedging with Options

16. Copyright K.Cuthbertson, D.Nitzsche. Call Premium Stock Price 100 0. 110 B A. Delta Hedging: Rebalancing As S changes then so does delta, so you have to rebalance your portfolio. E.g. delta = 0.5, then hold 50 stocks for every written call.

17. Copyright K.Cuthbertson, D.Nitzsche. 17 END OF SLIDES