Week 13 Introduction to the Greeks and Portfolio Management:

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1 Week 13 Introduction to the Greeks and Portfolio Management: Hull, Ch. 17; Poitras, Ch.9: I, IIA, IIB, III. 1

2 Introduction to the Greeks and Portfolio Management Objective: To explain how derivative portfolios can be managed. 2

3 Introduction to the Greeks and Portfolio Management 1. Introduction. 2. Naked vs. Covered Positions. 3. Stop-Loss Strategy. 4. Delta Hedging. 5. Theta. 6. Gamma. 7. Relationship Between Delta, Theta and Gamma. 8. Vega. 9. Rho. 10. Managing the Greeks. 11. Synthetic Options. 12. Hedging in Practice. 3

4 1. Introduction 4

5 What are the Greeks? The Greeks are quantities that reflect different risk dimensions in an option position. They can be managed so that all risks remain at an acceptable level. The Greeks also give us an indication of how to hedge an option by replication. 5

6 Hedging the Risk of an Option How to hedge the risk of selling an option? Take the opposite position in an exchange traded option. What if the option is not exchange traded? Create a synthetic option by using the Greeks. We will focus on delta hedging here! 6

7 Example A financial institution has sold a European call option on 100,000 shares of a non-dividend paying stock for $300,000. Option Parameters: S0 49 K 50 r T BS price = $240,000. How to hedge this position? 7 Jorge Cruz Lopez - Bus 316: Derivative Securities

8 Hedging Alternatives Naked Positions Covered Positions Stop-Loss Strategies Dynamic Portfolio Hedging 8

9 2. Naked vs. Covered Positions 9

10 Naked Positions Naked positions involve a position in an option without a position in the underlying asset. A naked position is effective as long as the option is not exercised. In our example, a naked position would work if the stock price is below K=50 at the end of 20 weeks. Then the profit of the financial institution is the premium ($300,000). A naked position is not very effective if the option is exercised. If S T > K, then the option is exercised and the cost to the financial institution is (S T K) * 100,

11 Covered Position Covered positions involve a position in an option covered by a position in the underlying asset. In our example, a covered position would involve buying 100,000 shares as soon as the option is sold. A covered position is effective if the option is exercised. If S T >K The financial institution delivers the stocks and collects the strike price. Total income = (K - S 0 )*100,000 A covered position can lead to losses if the option is not exercised. Cost to the financial institution = (S 0 S T )*100,000 11

12 3. Stop-Loss Strategy 12

13 Stop-Loss Strategy It is a combination of naked and covered positions. Have no position in the underlying asset (i.e. have a naked position) as long as the option is OTM. Take a covered position as soon as the option is ITM. In our example, a stop-loss strategy would involve buying 100,000 shares a soon as S>K, and then sell these shares as soon as S<K. 13

14 Stop-Loss Strategy It seems that the cost of writing and hedging the option is: Q=max(S 0 K). However, this equation is wrong because: It ignores the time value of money. We cannot buy and sell exactly at K all the time! 14

15 4. Delta Hedging 15

16 Delta Delta measures the sensitivity of the value of a portfolio of derivatives to the price of the underlying asset. where S Π is the value of the derivative portfolio (or derivative security). S is the price of the underlying asset. The delta of a portfolio is the weighted average of the deltas of its components: n i 1 i i 16 Jorge Cruz Lopez - Bus 316: Derivative Securities

17 Delta Delta can be seen as the slope of the curve obtained when the portfolio value is plotted against the stock price. Delta is a good approximation of the change in the value of the portfolio given a small stock price movement. 17

18 Delta Option price B Slope = A Stock price 18

19 Delta Hedging The objective of delta hedging is to obtain a delta neutral portfolio. That is, an overall position with a delta value of zero. Generally, delta changes over time; thus, one needs to continuously rebalance the portfolio to maintain a delta neutral position. 19

20 Delta Hedging Example S 0 =$100 Call price = $10 Size = 100 shares Delta of the call = 0.6 An investor sells 20 call options on the stock. Portfolio 1: short 20 calls. Delta 1 = (-20)(100)(0.6) = -1,200. In order to hedge her position, the investor would have to get an asset with a delta of +1,200. Since the delta of a single stock is 1, the investor could buy 1,200 shares and add them to portfolio 1. Portfolio 2: Portfolio 1 plus 1200 shares. Delta 2 = -1, ,200(1) = 0 20 Jorge Cruz Lopez - Bus 316: Derivative Securities

21 Delta Hedging Example Portfolio 2 is a delta neutral portfolio. That means that the gain (loss) in the option position will be offset by the loss (gain) on the stock position. So for example if the stock price increases by 1, the stock position increases by 1,200, but the option position would tend to decrease by 1,200 too! (20 х 100 х 0.6). 21

22 Delta in the BS Model The BS model values options by setting up a riskless portfolio that can be discounted at the risk free rate. This riskless position is obtained by creating a delta neutral portfolio. In the BS model the delta of a call is N(d 1 ). The delta of a put is N(d 1 )-1. For call options on dividend paying stocks, delta is e -qt N(d 1 ), where d 1 is defined in the dividend paying stock formula. For put options on dividend paying stocks, the delta is e -qt [N(d 1 )-1], where d 1 is defined in the dividend paying stock formula. 22

23 Delta of a Call Option K = 50, t=0.4, r=0.6, σ=0.3 23

24 Delta of a Call Option K = 50, r=0.6, σ=0.3 S=55 S=50 S=45 24

25 Delta of a Put Option K = 50, t=0.4, r=0.6, σ=0.3 25

26 Delta of a Put Option K = 50, r=0.6, σ=0.3 S=55 S=50 S=45 26

27 Delta in the BS Model Also see: Source: 27

28 Delta of a Forward Contract The value of a forward contract is: f = S 0 e -qt Ke -rt Therefore, the delta of a forward contract on one share of stock is e -qt. When the stock does not pay any dividends, the delta of the forward contract is 1. 28

29 Delta of a Futures Contract Futures contracts are market to market daily based on the new futures price. The CF from a long futures contract at the end of the day is: F K = S 0 e (r-q)t K Therefore, the delta of a futures contract is e (r-q)t. 29

30 Delta of a Futures Contract We can use futures contracts instead of the underlying asset to create delta neutral positions by following: where H F = e -(r-q)t H A H F is the alternative position in the futures contract. H A is the position that you would take in the underlying asset. 30

31 The Cost of Delta Hedging Delta hedging requires continuous rebalancing. For example, if you have a short position in a call option, you need to buy stocks to create a delta neutral portfolio. If the price of the stock increases, you need to buy more stocks because the delta of the call option increases (it approaches to 1 as S increases). 31

32 The Cost of Delta Hedging If the price of the stock decreases, you need to sell stocks because the delta of the call option decreases (it approaches to 0 as S decreases). Therefore, it can be seen that we re buying high and selling low! If one ignores transactions cost, the cost of delta hedging converges to the Black- Scholes price of the option. 32

33 5. Theta 33

34 Theta Theta (Q) measures the sensitivity of the value of a portfolio to the passage of time. Q t Theta is also refered as the time decay of the portfolio. 34

35 Theta of a Call K = 50, t=0.4, r=0.6, σ=0.3 35

36 Theta of a Call K = 50, r=0.6, σ=0.3 S=45 S=50 S=55 36

37 Theta of a Put K = 50, t=0.4, r=0.6, σ=0.3 37

38 Theta of a Put K = 50, r=0.6, σ=0.3 S=45 S=50 S=55 38

39 6. Gamma 39

40 Gamma Gamma (G) is the rate of change of delta ( ) with respect to the price of the underlying asset. G 2 S 2 S 40

41 Gamma Hedging Gamma addresses delta hedging errors caused by curvature. Call price C C C S S Stock price 41 Jorge Cruz Lopez - Bus 316: Derivative Securities

42 Gamma of a Call K = 50, t=0.4, r=0.6, σ=0.3 42

43 Gamma of a Call K = 50, r=0.6, σ=0.3 S=45 S=50 S=55 43

44 Gamma of a Put K = 50, t=0.4, r=0.6, σ=0.3 44

45 Gamma of a Put K = 50, r=0.6, σ=0.3 S=45 S=50 S=55 45

46 7. Relationship Between Delta, Theta and Gamma 46

47 Relationship Between Delta, Theta and Gamma For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q: 1 Q ( r q) S 2 S 2 G r 2 47

48 8. Vega 48

49 Vega Vega (n) measures the sensitivity of the value of a portfolio to its volatility. V 49

50 Vega of a Call Option K = 50, t=0.4, r=0.6, σ=0.3 50

51 Vega of a Call Option K = 50, r=0.6, σ=0.3 51

52 9. Rho 52

53 Rho Rho measures the sensitivity of the value of a portfolio to changes in the interest rate. For currency options there are 2 rhos. Rho r 53

54 Rho of a Call Option K = 50, t=0.4, r=0.6, σ=0.3 54

55 Rho of a Call Option K = 50, r=0.6, σ=0.3 S=55 S=50 S=45 55

56 10. Managing the Greeks 56

57 Managing Delta, Gamma and Vega Delta,, can be changed by taking a position in the underlying asset. To adjust gamma, G, and vega, n, it is necessary to take a position in an option or other derivative. 57

58 11. Synthetic Options 58

59 Hedging vs Creation of an Option Synthetically When we are hedging we take positions that offset, G, n, etc. When we create an option synthetically we take positions that match, G, & n 59

60 12. Hedging in Practice 60

61 Hedging in Practice Traders usually ensure that their portfolios are delta-neutral at least once a day. Whenever the opportunity arises, they improve gamma and vega. As portfolio becomes larger hedging becomes less expensive. 61

62 Scenario Analysis A scenario analysis involves testing the effect on the value of a portfolio of different assumptions concerning asset prices and their volatilities. 62

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