Faculty of Business and Economics University of Hong Kong Dr. Huiyan Qiu

1. MFIN6003 Derivative Securities Lecture Note Three Faculty of Business and Economics University of Hong Kong Dr. Huiyan Qiu

2. Outline • Options as basic insurance strategies • Options and views on direction and volatility • Spreads and collars: bull and bear spreads; ratio spreads; collars; box spreads • Speculating on volatility: straddles; strangles; butterfly spreads; asymmetric butterfly spreads

3. Basic Insurance Strategies • Insurance strategies using options: • Insure a long asset position • Buy put options • Insure a short asset position • Buy call options • Written against asset positions (selling insurance) • Covered call • Covered put

4. Insuring a Long Position • A long position in the underlying asset combined with a put option • Goal: to insure against a fall in the price of the underlying asset • At time 0 • Buy one stock at cost S0(long position in the asset) • Buy a put on the stock with a premium p • An insured long position (buy an asset and a put) looks like a call!

5. Example: S&R index and a S&R put option with a strike price of $1,000 together

6. Combined Payoff / Profit

7. Protective Puts • The portfolio consisting of a long asset position and a long put position is often called “Protective Put”. • Protective puts are the classic “insurance” use of options. • The protective put in the portfolio ensures a floor value (strike price of put) for the portfolio. That is, the asset can be sold for at least the strike price at expiration. • Varying the strike price varies the insurance cost.

8. Insuring a Short Position • A call option is combined with a short position in the underlying asset • Goal: to insure against an increase in the price of the underlying asset • At time 0 • Short one stock at priceS0 • Buy a call on the stock with a premium c • An insured short position (short an asset and buy a call) looks like a put

9. Selling Insurance • For every insurance buyer there must be an insurance seller • Naked writing is writing an option when the writer does not have a position in the asset • Covered writing is writing an option when there is a corresponding position in the underlying asset • Write a call and long the asset • Write a put and short the asset

10. Covered Writing • Covered calls: write a call option and hold the underlying asset. (The long asset position “covers” the writer of the call if the option is exercised.) • A covered call looks like a short put • Covered puts: write a put option and short the underlying asset • A covered put looks like a short call

11. Covered Writing: Covered Calls • Example:holding the S&R index and writing a S&R call option with a strike price of $1,000

12. Combined Payoff / Profit Writing a covered call generates the same profit as selling a put!

13. Insurance vs. Pure Option Position • Buying an asset and a put generates the same profit as buying a call • Short-selling an asset and buying a call generates the same profit as buying a put • Writing a covered call generates the same profit as selling a put • Writing a covered put generates the same profit as selling a call How to make the positions equivalent?

14. Insurance vs. Pure Option Position To make positions equivalent, borrowing or lending has to be involved. Following table summarizes the equivalent positions.

15. Options and Directional Views • Call option and put option can also be used for speculation. • The implied market view of option position on the direction of price movement:

16. Options and Volatility • Volatility is a measure of uncertainty in price movements; roughly, more volatility means that larger price swings may occur. • Options react to volatility! Option values depend on how much uncertainty one expects in the price of the underlying over the life of the option. • Both long call and long put benefit from volatility.

17. Options and Volatility (cont’d) • Example: consider a call option on a stock with strike price K = 10. • Case 1: ST is 11 or 9 with equal probability • Case 2: ST is 12 or 8 with equal probability • The option payoffs are: • Case 1: CT is 1 or 0 with equal probability • Case 2: CT is 2 or 0 with equal probability • Stock price in case 2 has the same mean but greater volatility. Option buyer would be willing to pay more for the higher payoff.

18. Pure Option Strategies • Each option position corresponds to a unique combination of views on market direction and market volatility  pure option strategy

19. More Option Strategies • Combined option positions can be taken to speculate on price direction or on volatility. • Speculating on direction: bull and bear spreads; ratio spreads; collars • Speculating on volatility: straddles; strangles; butterfly spreads; asymmetric butterfly spreads • Synthetic forward; Box spread

20. Underlying Asset and Options • Underlying asset: XYZ stock with current stock price of $40 • 8% continuous compounding annual interest rate • Prices of XYZ stock options with 91 days to expiration:

21. Bull Spreads • A bull spread is a position with the following profit shape. • It is a bet that the price of the underlying asset will increase, but not too much

22. Bull Spreads (cont’d) • A bull spread is to buy a call/put and sell an otherwise identical call/put with a higher strike price • Bull spread using call options: • Long a call with no downside risk, and • Short a call with higher strike price to eliminate the upside potential • Bull spread using put options: • Short a put to sacrifice upward potential, and • Long a put with lower strike price to eliminate the downside risk

23. K2-K1 Bull Spread with Calls Long a call (strike price K1, premium c1) Value = 0 when ST≤ K1 = – K1 + [1]ST when ST> K1 Value K2 Short a call (K2 > K1, c2 < c1) Value = 0 when ST≤ K2 = K2 + [-1]STwhen ST > K2 Portfolio Value = 0 when ST ≤ K1 = –K1+ [1]STwhen K1<ST ≤ K2 = K2 – K1 when ST>K2 K1 K2 FV(-c1+c2) ST -K1 c1 > c2 Initial cash flows = – c1 + c2 <0

24. Bear Spreads • A bear spread is a position in which one sells a call (or a put) and buys an otherwise identical call (or put) with a higher strike price. Opposite of a bull spread. • Example: short 40-strike call and long 45-strike put • It is a bet that the price of the underlying asset will decrease, but not too much • Option traders trading bear spreads are moderately bearish on the underlying asset

25. Ratio Spreads • A ratio spread is constructed by buying a number of calls ( puts) and selling a different number of calls (puts) with different strike price Figure: profit diagram of a ratio spread constructed by buying a low-strike call and selling two higher-strike calls. Limited profit and unlimited risk. To bet that the stock will experience little volatility.

26. Collars • A collar is along put combined with a short call with higher strike price • To bet that the price of the underlying asset will decrease significantly • A zero-cost collarcan be created when the premiums of the call and put exactly offset one another • Long 40-strike put and short 45-strike call

27. Speculating on Volatility • Non-directional speculations: • Straddles • Strangles • Butterfly spreads • Asymmetric butterfly spreads • Who would use non-directional positions? • Investors who have a view on volatility but are neutral on price direction • Speculating on volatility

28. Straddles • Buying a call and a put with the same strike price and time to expiration • A straddle is a bet that volatility will be high relative to the market’s assessment Figure Combined profit diagram for a purchased 40-strike straddle.

29. Strangles • Buying an out-of-the-money call and put with the same time to expiration • A strangle can be used to reduce the high premium cost, associated with a straddle Figure 40-strike straddle and strangle composed of 35-strike put and 45-strike call.

30. Written Straddles • Selling a call and put with the same strike price and time to maturity • A written straddle is a bet that volatility will be low relative to the market’s assessment FigureProfit at expiration from a written straddle: selling a 40-strike call and a 40-strike put.

31. Butterfly Spreads • A butterfly spread is = write a straddle + add a strangle = insured written straddle FigureWritten 40-strike straddle, purchased 45-strike call, and purchased 35-strike put.

32. Butterfly Spread Value Sell a call (Strike price K2, premium c2) Sell a put (Strike price K2, premium p2) Written straddle Value = –K2 + [1]ST when ST≤ K2 = K2 + [-1]ST when ST>K2 Long a put (Strike price K1<K2,premium p1) K3 K1 0 Long a call (Strike price K3>K2>k1, c3) K2 ST Long strangle Value = K1 + [-1]ST when ST≤K1 = 0 when K1<ST ≤ K3 = –K3 + [1]ST when ST>K3 Butterfly spread (K3 – K2 = K2 – K1) Value = –K2+K1 when ST≤K1 = –K2 + [1]ST when K1<ST ≤ K2 = K2 + [-1]ST when K2<ST≤K3 = –K3 + K2 when ST>K3

33. Butterfly Spread Value Initial option costs = c2 + p2 – p1 – c3 = (c2 – c3) + (p2 – p1) > 0 K1 K3 A butterfly spread is an insured written straddle. Can be used to bet for low volatility. 0 K2

34. Asymmetric Butterfly Spreads • By trading unequal units of options

35. Summary of Various Strategies Option strategy positions driven by the view on the stock price and volatility directions.

36. Synthetic Forwards • Underlying asset: S&R Index, spot price = $1,000 • 6-month Forward: forward price = $1,020 • 6-month 1,000-strike call: call premium = $93.81 • 6-month 1,000-strike put: put premium = $74.20 • Effective interest rate over 6 month = 2% • Positions: long call + short put • Time-0 cash flow: – 93.81 + 74.20 = – 19.61 • What happens 6 months later?

37. Long Call + Short Put • Outcome at expiration: pay the strike price of $1,000 and own the asset

38. Synthetic Forwards • A synthetic long forward contract: buying a call and selling a put on the same underlying asset, with each option having the same strike price and time to expiration • Example:buy the $1,000-strike S&R call and sell the $1,000-strike S&R put, each with 6 months to expiration

39. Synthetic Forwards (cont’d) • Both synthetic long forward contract and actual forward contract result in owning the asset at the expiration. • Differences • The forward contract has a zero premium, while the synthetic forward requires that we pay the net option premium • With the forward contract, we pay the forward price, while with the synthetic forward we pay the strike price

40. Put-Call Parity • The net cost of buying the index using options (synthetic forward contract) must equal the net cost of buying the index using a forward contract – C + P – PV(K) = – PV(F) • Call (K, T) – Put (K, T) = PV (F0,T – K) • One of the most important relations in options! – K – C + P Synthetic Forward – F 0 Actual Forward

41. Off-Market Forward • Forward by definition has a zero premium. • A forward contract with a nonzero premium must have a forward price which is “off the market (forward) price”. Thus, it is sometimes called an off-market forward. • Unless the strike price equals the forward price, buying a call and selling a put creates an off-market forward.

42. Box Spreads • A box spreadis accomplished by using options to create a synthetic long forward at one price (long call and short put with strike price K1) and a synthetic short forward at a different price (short call and long put with strike price K2 ≠ K1). • At time 0, cash flow: • – C(K1, T) + P(K1, T) + C(K2, T) – P(K2, T) • At expiration, • Synthetic long forward: pay K1 to buy asset • Synthetic short forward: sell asset for K2 •  fixed cash flow K2 – K1

43. Box Spreads • A box spread is a means of borrowing or lending money. It has no stock price risk! • Box spread can be a source of funds. • However, it works usually for option market-makers only because they have relatively low transaction costs. • Before 1993, box spreads also provided a tax benefit for some investors in the US stock market.

44. End of the Notes!