1 The Greek Letters Chapter 17

2 The Greeks are coming! Parameters of SENSITIVITY Delta =  Theta =  Gamma =  Vega =  Rho = 

3 c = SN(d 1 ) – Ke –r(T – t) N(d 2 ) p = Ke –r)T –t) N(-d 2 ) – SN(-d 1 ) Notationally: c = c(S; K; T-t; r; σ) p = p(S; K; T-t; r; σ) Once c and p are calculated, WHAT IF?

4 The GREEKS are measures of sensitivity. The question is how sensitive a position’s value is to changes in any of the variables that contribute to the position’s market value.These variables are: S, K, T-t, r and . Each one of the Greek measures indicates the change in the value of the position as a result of a “small” change in the corresponding variable. Formally, the Greeks are partial derivatives.

5 Delta =  In mathematical terms DELTA is the first derivative of the option’s premium with respect to S. As such, Delta carries the units of the option’s price; I.e., $ per share. For a Call:  (c)=  c/  S For a Put:  (p)=  p/  S Results:  (p) =  (c) - 1 For the  (S) =  S/  S = 1

6 THETA  Theta measures are given by:  (c)=  c/  (T-t)  (p)=  p/  (T-t)  s are positive but the they are reported as negative values. The negative sign only indicates that as time passes, t increases, time to expiration, T – t, diminishes and so does the option’s value, ceteris paribus. This loss of value is labeled the option’s “time decay.” Also,  (S) = 0.

7 GAMMA  Gamma measures the change in delta when the price of the underlying asset changes. Gamma is the second derivative of the option’s price with respect to the underlying price.  (c) =  (c)/  S =  2 c/  S 2  (p) =  (p)/  S =  2 p/  S 2 Results:  (c) =  (p)  (S) = 0.

8 VEGA  Vega measures the sensitivity of the option’s market price to “small” changes in the volatility of the underlying asset’s return.  (c)=  c/   (p)=  p/  Thus, Vega is in terms of $/1% change in .  (S)= 0.

9 RHO  Rho measures the sensitivity of the option’s price to “small” changes in the rate of interest.  (c)=  c/  r  (p)=  p/  r Rho is in terms of $/%change of r.  (S) = 0.

10 Example: S=100;K = 100; r = 8%;T-t =180 days;  = 30%.CallPut Premium$ $ The Greeks: Delta =  Theta =  Gamma =  Vega =  Rho = 

11 Again. the Delta of any position measures the $ change/share in the position’s value that ensues a “small” change in the value of the underlying.  (c)=  (p) =

12 Call Delta (See Figure 15.2, page 345) Delta is the rate of change of the call price with respect to the underlying Call price A C Slope =  Stock price

13 THETA  Theta measures the sensitivity of the option’s price to a “small” change in the time remaining to expiration:  (c)=  c/  (T-t)  (p)=  p/  (T-t) Theta is given in terms is $/1 year.  (c) = - $ /year  if time to expiration increases (decreases) by one year, the call price will increase (decrease) by $ Or, /365 = 3.35 cent per day.

14 GAMMA  Gamma measures the change in delta when the price of the underlying asset changes.  =  (c) =.6151;  (p) = If the stock price increases to $101:  (c) increases to.6332  (p) increases to If the stock price decreases to $99:  (c) decreases to.5970  (p) decreases to

15 VEGA  Vega measures the sensitivity of the option’s market price to “small” changes in the volatility of the underlying asset’s return. = (Check on Computer)

16 RHO  Rho measures the sensitivity of the option’s price to “small changes in the rate of interest. Rho =  CallPut Rho is in terms of $/%change of r. (check on computer)

17 DELTA-NEUTRAL POSITIONS A market maker wrote n(c) calls and wishes to protect the revenue against possible adverse move of the underlying asset price. To do so, he/she uses shares of the underlying asset in a quantity that GUARANTEES that a small price change will not have any impact on the call-shares position. Definition: A portfolio is Delta-neutral if  (portfolio) = 0

18 DELTA neutral position in the simple case of call-stock portfolios. V portfolio = Sn(S) + cn(c;S)  (portfolio) =  (S)n(S) +  (c)n(c;S)  ( portfolio ) = 0  n(S) +  (c)n(c;S) = 0. n(S) = - n(c;S)  (c). The call delta is positive. Thus, the negative sign indicates that the calls and the shares of the underlying asset must be held in opposite direction.

19 EXAMPLE: call - stock portfolio We just sold 10 CBOE calls whose delta is $.54/shares. Each call covers 100 shares. n(S) = - n(c;S)  (c).  (c) = 0.54 and n(c) = -10. n(c;S) = - 1,000 shares. n(s) = - [ - 1,000(0.54)] = 540. The DELTA-neutral position consists of the 10 short calls and 540 long shares.

20 The Hedge Ratio   c Definition: Hedge ratio. In the example: Hedge ratio = 540/1,000 =.54 Notice that this is nothing other than  (c).

21 In the numerical example, Slide 9: The hedge ratio:  (c) = With 100 CBOE short calls: n(S) = -  (c) n(c;S). n(c;S) = -10,000. n(S) = -(.6151)[-10,000] = +6,151 shares The value of this portfolio is: V = -10,000($ ) + 6,151($100) V = $512,056

22 Suppose that the stock price rises by $1. S NEW = = $101/share. V =- 10,000($ $.6151) +6,151($101) V = - 10,000($ ) + 6,151($100) - 10,000($.6151) + $1(6,151) V = $512,056 - $6,151 + $6,151 V = $512,056.

23 Suppose that the stock price falls by $1. S NEW = = $99/share. V =- 10,000($ $.6151) +6,151($99) V = - 10,000($ ) + 6,151($100) - 10,000( - $.6151) - $1(6,151) V = $512,056 + $6,151 – $6,151 V = $512,056.

24 In summary: The portfolio consisting of 100 short calls and 6,151 long shares is delta- neutral. Price/share: +$1-$1 shares +$6,151-$6,151 calls +(-$6,151)-(-$6,151) Portfolio $0 $0

25 DELTA neutral position in the simple case of put-stock portfolios. V portfolio = Sn(S) + pn(p;S)  (portfolio) =  (S)n(S) +  (p)n(p;S)  ( portfolio ) = 0  n(S) +  (p)n(p;S) = 0. n(S) = - n(p;S)  (p) Since the put delta is negative, then the negative sign indicates that the puts and the underlying asset must be held in the same direction.

26 EXAMPLE: put – stock portfolio. We just bought 10 CBOE puts whose delta is -$.70/share. Each put covers 100 shares. n(S) = - n(p;S)  (p).  (p) = -.70 and n(p) = 10. n(p;S) = 1,000 shares. n(S) = - 1,000(-.70) = 700. The DELTA-neutral position consists of the 10 long puts and 700 long shares.

27 Portfolio: The portfolio consisting of 10 long puts and 700 long shares is delta- neutral. Price/share: +$1-$1 shares +$700 -$700 puts -$700 $700 Portfolio $0 $0

28 In the numerical example, Slide 9: The hedge ratio:  (p) = The Delta neutral position with 100 CBOE long puts requires the holding of: n(S) = -  (p)n(p;S) n(S) = -(-.3849)[10,000] = +3,849shares The value of this portfolio is: V = 10,000($6.4360) + 3,849($100) V = $449,260.

29 Suppose that the stock price rises by $1. S NEW = = $101/share. V =10,000($ ) +3,849($101) V = 10,000( $6.4360) – 3,849($100) - 10,000(.3849) + $1(3,849) V = $449,260- $3,849 + $3,849 V = $449,260.

30 Suppose that the stock price falls by $1. S NEW = = $99/share. V =10,000($ $.3849) +3,849($99) V = 10,000( $6.4360) + 3,849($100) 10,000($.3849) - $1(3,849) V = $449,260+ $3,849 - $3,849 V = $449,260.

31 In summary: The portfolio consisting of 100 long puts and 6,151 long shares is delta- neutral. Price/share: +$1-$1 shares +$3,849-$3,849 calls +(-$3,849)-(-$3,849) Portfolio $0 $0

32 An extension: calls, puts and the stock position V portfolio = Sn(S) + cn(c;S) + pn(p;S)  portfolio =  (S)n(S) +  (c)n(c;S) +  (p)n(p;S). But  S = 1. Thus, for delta-neutral portfolio:  portfolio = 0 and n(S) = -  (c)n(c;S) -  (p)n(p;S).

33 EXAMPLE: We short 20 calls and 20 puts whose deltas are $.7/share and -$.3/share, respectively. Every call and every put covers 100 shares. How many shares of the underlying stock we must purchase in order to create a delta-neutral position? n(S) = -  (c)n(c;S) + [-  (p)]n(p;S). n(S) = -(.7)(-2,000) – [-.3](-2,000) n(S) = 800.

34 Example continued The portfolio consisting of 20 short calls, 20 short puts and 800 long shares is delta- neutral. Price/share: +$1-$1 shares +$800-$800 calls -$1,400+$1,400 Puts+$600-$600 Portfolio $0 $0

35 EXAMPLES The put-call parity: Long 100 shares of the underlying stock, long one put and short one call on this stock is always delta-neutral:  (position) =  (p)n(p;S) +  (c)n(c;S) = [  (c) – 1](100) +  (c)(-100) = 0.

36 EXAMPLES A long STRADDLE: Long 15 puts and long 15 calls (same underlying asset, K and T-t), with:  (c) =.64;  (p) =  (straddle)= 15(100)[.64 + (-.36)] =$420/share. Long 420 shares to delta neutralize this straddle.

37 Results: 1.The deltas of a call and a put on the same underlying asset, (with the same time to expiration and the same exercise price) must satisfy the following equality:  (p) =  (c) Using the Black and Scholes formula:  (c) = N(d 1 )  0 <  (c) < 1  (p) = N(d 1 ) – 1  -1 <  (p) < 0

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42 THETA  Theta measures the sensitivity of the option’s price to a “small” change in the time remaining to expiration:  (c)=  c/  (T-t)  (p)=  p/  (T-t) Theta is given in terms is $/1 year. Thus, if  (c) = - $20/year, it means that if time to expiration increases (decreases) by one year, the call price will increase (decreases) by $20. Or, $20/365 = 5.5 cent per day.

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47 GAMMA  Gamma measures the change in delta when the market price of the underlying asset changes.  (c) =  (c)/  S =  2 c/  S 2  (p) =  (p)/  S =  2 p/  S 2Results:  (c) =  (p)  (S) = 0.

48 GAMMA  In general, the Gamma of any portfolio is the change of the portfolio’s delta due to a “small” change in the underlying asset price. As the second derivative of the option’s price with respect to S, Gamma measures the sensitivity of the option’s price to “large” underlying asset’s price changes.  May be positive or negative.

49 Interpretation of Gamma The delta neutral position with 100 short calls and 6,151 long shares has Γ= -$181 S Negative Gamma means that the position loses value when the stock price moves more and more away from it initial value $512,056 Position value More negative Γ

50 Interpretation of Gamma S Negative Gamma S Positive Gamma

51 Result: The Gammas of a put and a call are equal. Using the Black and Shcoles model:  (c) = n(d 1 ) and  (p) = n(d 1 ) – 1. Clearly, the derivatives of these deltas with respect to S are equal. EXAMPLE:  (c) =.70;  (p) = -.30;  = Holding a long call and a short put has:  =.70 - (-.30) =  =.2345 –.2345 = 0.

52 EXAMPLE:  (c) =.70,  (p) = -.30 and let gamma be Holding the underlying asset long, a long put and a short call yields a portfolio with:  = (-.30) = 0 and  = 0 - 0, ,2345 = 0, simultaneously! This portfolio is delta-gamma-neutral.

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56 VEGA  Vega measures the sensitivity of the option’s market price to “small” changes in the volatility of the underlying asset’s return.  (c)=  c/    (p)=  p/   Thus, Vega is in terms of $/1% change in  

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61 RHO  Rho measures the sensitivity of the option’s price to “small changes in the rate of interest.  (c)=  c/  r  (p)=  p/  r  Rho is in terms of $/%change of r.

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66 SUMMERY OF THE GREEKS Position Delta Gamma VegaTheta Rho LONG STOCK SHORT STOCK LONG CALL SHORT CALL LONG PUT SHORT PUT

67 The sensitivity of portfolios, a summary. 1.A portfolio is a combination of securities and options. 2.All the sensitivity measures are partial derivatives. 3.Theorem(Calculus): The derivative of a linear combination of functions is the combination of the derivatives of these functions. Thus, the sensitivity measure of a portfolio of securities is the portfolio of these securities’ sensitivity measures.

68 Example:The DELTA of a portfolio of 5 long CBOE calls, 5 short puts and 100 shares of the stock long:  (portfolio) =  (500c - 500p + 100S) =  (500c - 500p + 100S)/  S = 500  c/  S  p/  S = 500  c  p This delta reveals the $/share change in the portfolio value as a function of a “small” change in the underlying price

69 Example:S = $48.57/barrel. 1 call = 1,000bbls. CallDeltaGamma A $0.63/bbl$0.22/bbl B$0.45/bbl$0.34/bbl C$0.82/bbl$0.18/bbl Portfolio: Long:3 calls A; 2 calls C; 5,000 barrels. Short: 10 calls B..

70 Example:  =(0.63)3,000+ (0.82)2,000 + (1)5,000 + (0.45)(-10,000)  =(0.22)3,000+ (0.18)2,000 + (0)5,000 + (0.34)(-10,000)  = $4,030.  =- $2,380..

71  = 4,030  a “small” change of the oil price, say one cent per barrel, will change the value of the above portfolio by $40.30 in the same direction.  = - 2,380  a “small” change in the oil price, say one cent per barrel, will change the delta by $23.80 in the opposite direction. Also, Gamma is negative  when the price per barrel moves away from $48.57, the portfolio value will decrease.

72 A financial institution holds: 5,000 CBOE calls long;delta.4, 6,000 CBOE puts long;delta -.7, 10,000 CBOE puts short; delta -.5, Long 100,000 shares  (portfolio)= (.4)500,000 + (-.7)600,000 +(-.5)[-1,000,000] + 100,000 = $380,000.

73 GREEKS BASED STRATEGIES Greeks based strategies are opened and maintained in order to attain a specific level of sensitivity. Mostly, these strategies are set to attain zero sensitivity. What follows, is an example of strategies that are: 1.Delta-neutral 2.Delta-Gamma-neutral 3.Delta-Gamma-Vega-Rho-neutral

74 EXAMPLE: The underlying asset is the a stock. The options on this stock are European. S = $300; K = $300; T = 1yr;  = 18%; r = 8%; q = 3%. c = $  =.6245  =.0067 =.0109  =.0159

75 DELTA-NEUTRAL Short 100 calls. n 0 = - 10,000; Long n S = 6,245 shares Case A1: S increases from $300 to $301. PortfolioInitial Value New valueChange -100Calls- $282,500- $288,800- $6,300 6,245S $1,873,500 $1,879,745 $6,245 Error:- $55 Case A2:S decreases from $300 to $299. PortfolioInitial valueNew valueChange -100Calls- $282,500- $276,200+ $6,300 6,245S $1,873,500 $1,867,255- $6,245 Error: + $55

76 Case B1:S increases from $300 to $310. PortfolioInitial ValueNew valueChange -100Calls- $282,500- $348,100- $65,600 6,245S $1,873,500 $1,935,950 $62,450 Error: -$3,150 The point here is that Delta has changed significantly and.6245 does not apply any more. S = $300$301$310  = We conclude that the delta-neutral portfolio must be adjusted for “large” changes of the underlying asset price.

77 Call #0Call #1S = $300 K = $300K = $305 T = 1yr T = 90 days  = 18% r = 8% q = 3% c = $28.25c = $10.02  =.6245  =.4952  =.0067  =.0148 =.0109 =.0059  =.0159  =.0034

78 A DELTA-GAMMA-NEUTRAL PORTFOILO  (portfolio) = 0: n S + n 0 (.6245) + n 1 (.4952) = 0 Γ(portfolio)= 0: n 0 (.0067) + n 1 (.0148) = 0 Solution: n 0 = -10,000 n 1 = - (-10,000)(.0067)/.0148 = 4,527 n S = - (-10,000)(.6245) – (4,527)(.4952) = 4,003 Short the initial call :n 0 = -10,000 Long of call #1n 1 = 4,527 Long 4,003 sharesn S = 4,003

79 THE DELTA-GAMMA-NEUTRAL PORTFOLIO Case A1:S increases from $300 to $301. Portfolio Initial value New value Change 0) -10,000 - $282,500 - $288,800-$6,300 1) 4,527 $45,360 $47,657 $2,297 S) 4,003 $1,200,900 $1,204,903 $4,003 Error: 0 Case B1:S increases from $300 to $310. Portfolio Initial value New valueChange 0) -10,000- $282,500 - $348,100- $65,600 1) 4,527 $45,360 $70,930 $25,570 S) 4,003 $1,200,900 $1,240,930 $40,030 Error: 0

Risk ,003 4,003S ,242 4, ,245-10,000 RhoVegaGammaDeltaPortfolio The above numbers reveal that the Delta- Gamma-neutral portfolio is exposed to risk associated with the volatility and the risk-free rate If we examine the exposure level to all parameters, however, we observe that:

81 Case C1: S increases from $300 to $310 and simultaneously, r increases from 8% to 9%. Portfolio Initial value New value Change -10,000 - $282,500 - $330,500 - $48,000 4,527 $45,360 $73,166 $27,806 4,003S $1,200,900 $1,240,930 $40,030 Error: - $10,756

82 Delta-Gamma-Vega-Rho-neutral portfolio CALL K T(days) Volatility 18% 18% 18% 18% r 8% 8% 8% 8% Dividends 3% 3% 3% 3% c $28.25 $10.02 $15.29 $18.59

83 Delta-Gamma-Vega-Rho-neutral portfolio CALL     S

84 The DELTA-GAMMA-VEGA-RHO-NEUTRAL- PORTFOLIO In order to neutralize the portfolio to all risk exposures, following the sale of the initial call, we now determine the portfolio’s holdings such that all the portfolio’s sensitivity parameters are zero simultaneously.  = 0 and  = 0 and  = 0 and  = 0 simultaneously!

85  = 0 n S +n 0 (.6245)+n 1 (.4954)+n 2 (.6398)+n 3 (.5931)=0  = 0 n 0 (.0067)+n 1 (.0148)+n 2 (.0138)+n 3 (.0100)=0  = 0 n 0 (.0109)+n 1 (.0059)+n 2 (.0055)+n 3 (.0080) =0  = 0 n 0 (.0159)+n 1 (.0034)+n 2 (.0044)+n 3 (.0079) =0

86 The solution is: Exact n Short 100 CBOE calls #0;-10,000 Short 339 calls #1;-33,927 Long 265 calls #2; 26,534 Long 204 calls #3; 20,420 Short 6,234 shares. -6,234

87 Case D:S increases from $300 to $310 r increases from 8% to 9%  increases from 18% to 24% Portfolio Initial Value New value 0) - 10,000 - $282,468 - $428,071 S)- 6,234 - $1,870,200- $1,932,540 1) - 33,927 - $340,023- $664,552 2)26,534 $405,668 $694,062 3)20,420 $379,677 $622,240 TOTAL - $1,707,356 $1,708,861 Error:$1,505 or.088%.

88 DYNAMIC DELTA - HEDGING The market stock price keeps changing all the time. Thus, a static DELTA- neutral hedge is not sufficient. A continuous delta adjustment is not practical. An adjusted Delta-neutral Position: 1.Every day, week, etc. 2.Following a given % price change.

89 DYNAMIC DELTA HEDGING Market makers provide traders with the options they wish to trade. For example, if a trader wishes to long (short) a call, a market maker will write (long) the call. The difference between the buy and sell prices is the market maker’s bid-ask spread. The main problem for a market maker who shorts calls is that the premium received, not only may be lost, but the loss is potentially unlimited.

90 DYNAMIC DELTA - HEDGING Recall: The profit profile of an uncovered call is: At expiration P/L STST c K

91 DYNAMIC DELTA - HEDGING Recall: The profit profile of a covered call is: Strategy IFC At expiration S T < KS T > K Short call c0-(S T – K) Long stock -S t STST STST Total- S t + c S T K P/LS T - S t + cK - S t + c

92 DYNAMIC DELTA - HEDGING Recall: The profit profile of a covered call is: At expiration P/L STST K –S t + c K -S t + c

93 DYNAMIC DELTA - HEDGING The Dynamic Delta hedge is based on the impact of the time decay on the call Delta. Recall that:

94 DYNAMIC DELTA - HEDGING Observe what happens to d 1 when T-t  0. 1.For:S t > Kd 1   and N(d 1 ) =  (c)  1 2.For:S t < Kd 1  -  and N(d 1 ) =  (c)  0

95 DYNAMIC DELTA - HEDGING The Dynamic hedge: 1.Write a call and simultaneously, hedge the call by a long Delta shares of the underlying asset. As time goes by, adjust the number of shares periodically. Result: As the expiration date nears, delta: goes to 0in which case you wind up without any shares. goes to1 in which case you call is fully covered.

96 DYNAMIC DELTA - HEDGING S t :S t K  (c):  0  1 Call:uncoveredfully covered n(S):0 n(c;S) 1

97 Table 15.2 Simulation of Dynamic delta - hedging.(p.364) Cost of Stock Shares shares Cummulative Interest Week przce Delta purchased purchased cost cost ($000) ($000) ($000) ,200 2, , (6,400) (308.0) 2, (5,800) (274.7) 1, , , , , , , (300) (15.9) 3, (6,500) (337.2) 3, (3,200) (164.4)3, , , (23,700) (1,182.2) 2, (13,700) (664.4) 2, , , , , , , (900) (46.7) 3, , , , , , , , , ,263.3

98 Table 15.3 Simulation of dynamic Delta - hedging.(p. 365) Cost of Stock Shares shares Cumulative I nterest Week price Delta purchased purchased cost cost ($000) ($000) ($000) 0 49, ,200 2, , , , , (12,600) (630.0) 2, (12,000) (580.6) 2, (1,600) (77.2) 2, , , , , (12,000) (579.0) 2, (1,000) (48.2) 2, ,800 1, , , , (15,000) (748.2) 2, (400) (20.0) 2, (13,800) (672.7) 2, (16,400) (779.0) 1, , , (19,900) (920.4) , , (17,600) (820.7) (700) (33.7) 256.6