Mathematics of Financial Derivatives

Transcription

1 Mathematics of Financial Derivatives Lecture 8 Solesne Bourguin bourguin@math.bu.edu Boston University Department of Mathematics and Statistics

2 Table of contents 1. The Greek letters (continued) 2. Volatility smiles 3. Trading strategies involving options 1

3 The Greek letters (continued)

4 The Greeks: vega Definition of vega Up to now, we have assumed that the volatility of the underlying asset is constant. In practice, it changes over time. So the value of a derivative is liable to change because of movements in volatility. The vega of a portfolio of derivatives, ν, is the rate of change of the value of the portfolio with respect to the volatility of the underlying asset. It is the derivative of the portfolio with respect to σ. ν Π = Π σ. From the BSM formulas, it can be shown that ν(call) = ν(put) = S 0 T N (d 1 ). 2

5 The Greeks: vega Properties of vega If vega is highly positive or negative, the portfolio s value is very sensitive to small changes in volatility. If it is close to zero, small changes in volatility have very little impact on the value of the portfolio. A position in the underlying has zero vega. If ν is the vega of a portfolio and ν T is the vega of a traded option, the portfolio can be made vega neutral by adding ν/ν T options to the portfolio. However, a portfolio that is gamma neutral will not be vega neutral and vice versa. 3

6 The Greeks: vega Vega as a function of the stock price Vega of a call and put 0 K Stock price 4

7 The Greeks: rho Definition of rho The rho of a portfolio of options is the rate of change of the value of the portfolio with respect to the interest rate. It measures the sensitivity of the value of a portfolio to a change in the interest rate when all else remains the same. ρ Π = Π r. From the BSM formulas, it can be shown that ρ(call) = KTe rt N(d 2 ); ρ(put) = KTe rt N( d 2 ). 5

8 The Greeks in practice The realities of hedging In an ideal world, traders would like to be able to rebalance their portfolios very frequently in order to maintain all Greeks equal to zero. This is not possible in practice. Traders make their portfolios delta-neutral once a day. A zero gamma and a zero vega are less easy to achieve because it is difficult to find options that can be traded in the required volume at competitive prices. 6

9 Volatility smiles

10 Volatility smiles Definition of a volatility smile In practice, traders use the BSM model, but not in the way it was intended to be used: they allow the volatility to change. They have to do so, because observed prices of traded options reflect this when computing the implied volatilities depending on the strike. A volatility smile is a plot of the implied volatility of an option with a certain maturity as a function of its strike. 7

11 Volatility smiles An example of a volatility smile Implied volatility 0 Strike 8

12 Volatility smiles Properties of volatility smiles The smile for a put or a call is the same. This comes from the call put parity formula. In general, it is not the same for different types of options. Depending on the underlying asset (stocks, foreign currencies, interest rates), the smile takes various forms. 9

13 Volatility smiles Reasons for volatility smiles: assumptions of BSM The main reason is that the actual probability distribution of returns of the underlying asset IS NOT lognormal. This has the effect that an artificial change of volatility has to be implemented in order to match prices with the BSM formulas (where it is assumed that the returns are lognormal). Another reason is that the assumption that the volatility is constant is false in practice, and that the price of the underlying asset sometimes changes abruptly (with jumps) as opposed to smoothly. 10

14 Volatility smiles Reasons for volatility smiles: leverage effect A more concrete reason for the existence of a smile of volatility is the leverage effect. When a stock decreases in value, holding that stock becomes more risky (because of the risk of bankruptcy), and volatility increases. When a stock increases in value, holding that stock becomes less risky and volatility decreases. So volatility could be seen as a decreasing function of the stock price, which is exactly what the smile plot shows. 11

15 Trading strategies involving options

16 Trading an option and the underlying asset Covered call: long stock + short call. The long stock position protects the investor from the payoff on the short call. Protective put: long stock + long put. This is equivalent to a long call plus some cash. Keep in mind the call-put parity relation, which can be used to understand what trading options and the underlying is equivalent to. p + S 0 = c + Ke rt. 12

17 Covered call Protective put Profit Long stock Profit Long stock 0 K S T 0 K S T Long put Short call 13

18 Spreads A spread trading strategy involves taking a position in two or more options of the same type (i.e., two or more calls or two or more puts). Bull spreads A bull spread is created by buying a call with a certain strike and selling a call with a higher strike. Both options are on the same stock with the same maturity. The payoff of a bull spread is given in the following table. Stock price range Payoff from long call option Payoff from short call option Total payoff S T K K 1 < S T < K 2 S T K 1 0 S T K 1 S T K 2 S T K 1 (S T K 2 ) K 2 K 1 14

19 Bull spread Profit Long call, strike K 1 0 K 1 K 2 S T Short call, strike K 2 15

20 Bull spreads (continued) An investor who enters into a bull spread is hoping the stock price will increase. A bull spread strategy limits the investor s upside as well as downside risk. Three types of bull spreads can be distinguished. 1. Both calls are initially out of the money (very cheap and low chance of high payoff: aggressive). 2. One call is initially in the money, and the other one is out of the money (more conservative). 3. Both calls are in the money (most conservative). Remark Bull spreads can also be created by buying a put with a low strike and selling a put with a high strike. 16

21 Bear spreads A bear spread is created by buying a put with a certain strike and selling a put with a smaller strike. Both options are on the same stock with the same maturity. The payoff of a bear spread is given in the following table. Stock price range Payoff from long put option Payoff from short put option Total payoff S T K 1 K 2 S T (K 1 S T ) K 2 K 1 K 1 < S T < K 2 K 2 S T 0 K 2 S T S T K

22 Bear spread Profit Short put, strike K 1 0 K 1 K 2 S T Long put, strike K 2 18

23 Bear spreads (continued) An investor who enters into a bear spread is hoping that the stock price will decline. Bear spreads limit the investor s upside as well as downside risk. Remark Bear spreads can also be created by buying a call with a high strike and selling a call with a low strike. 19

24 Box spreads A box spread is a combination of a bull call spread with strikes K 1 and K 2 and a bear put spread with the same two strikes. The payoff of a box spread is always K 2 K 1. The value of a box spread is hence always (K 2 K 1 )e rt. If it has a different value, there is an arbitrage opportunity (find it!). Stock price range Payoff from bull call spread Payoff from bear put spread Total payoff S T K 1 0 K 2 K 1 K 2 K 1 K 1 < S T < K 2 S T K 1 K 2 S T K 2 K 1 S T K 2 K 2 K 1 0 K 2 K 1 20

25 Box spread Profit Bull call spread 0 K 1 K 2 S T Bear put spread 21

26 Butterfly spreads A butterfly spread involves positions in options with three different strikes. It can be created by buying a call with a low strike K 1, buying a call with a high strike K 3, and selling two calls with a strike K 2 that is halfway between K 1 and K 3. Generally, K 2 is close to the current stock price. The payoff of a butterfly spread is given in the following table (assuming K 2 = 0.5(K 1 + K 3 )). Stock price range Payoff from first long call Payoff from second long call Payoff from short calls Total payoff S T K K 1 < S T K 2 S T K S T K 1 K 2 < S T < K 3 S T K 1 0 2(S T K 2) K 3 S T S T K 3 S T K 1 S T K 3 2(S T K 2) 0 22

27 Butterfly spread Profit Long call, strike K 1 Long call, strike K 3 0 K 1 K 2 K 3 S T Short 2 calls, strike K 2 23

28 Butterfly spreads (continued) A butterfly spread leads to a profit if the stock price stays close to K 2, but gives a small loss if there is a significant stock price move in either direction. It is therefore an appropriate strategy for an investor who feels that large stock price moves are unlikely. It requires a small investment initially. Remark Butterfly spreads can also be created using put options. The investor buys two puts, one with a low strike and one with a high strike, and sells two puts with an intermediate strike. 24

29 Combinations A combination is an option trading strategy that involves taking a position in both calls and puts on the same stock. Some combinations are straddles, strips, straps, and strangles. Straddle A straddle is created by buying a call and a put with the same strike and maturity. The payoff of a straddle is given in the following table. Stock price range Payoff from call Payoff from put Total payoff S T K 0 K S T K S T S T > K S T K 0 S T K 25

30 Straddle Profit Long call, strike K 0 K S T Long put, strike K 26

31 Straddle (continued) If the stock price is close to the strike at maturity, the straddle leads to a loss. However, if there is a sufficiently large move in either direction, a significant profit will result. A straddle is appropriate when an investor is expecting a large move in a stock price but does not know in which direction the move will be. 27

32 Strips and straps A strip consists of a long position in one call and two puts with the same strike and maturity. Strip (1 call + 2 puts) Strap (2 calls + 1 put) Profit Profit 0 K S T 0 K S T In a strip, the investor is betting that there will be a big stock price move and considers a decrease more likely than an increase. In a strap, it s the same, but the investor considers that an increase is more likely than an increase. 28

33 Strangle A strangle is created by buying a call and a put with the same maturity but with different strikes. The payoff of a strangle is given in the following table. Stock price range Payoff from call Payoff from put Total payoff S T K 1 0 K 1 S T K 1 S T K 1 < S T < K S T K 2 S T K 2 0 S T K 2 29

34 Strangle Profit Long call, strike K 2 0 K 1 K 2 S T Long put, strike K 1 30

35 Strangles (continued) A strangle is a similar strategy to a straddle. The investor is betting that there will be a large price move, but is uncertain whether it will be an increase or a decrease. We see that the stock price has to move farther in a strangle than in a straddle for the investor to make a profit. However, the downside risk if the stock price ends up at a central value is less with a strangle. 31