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Lecture 11: The Greeks and Risk Management This lecture studies market risk management from the perspective of an options trader. First, we show how to describe the risk characteristics of derivatives. Then, we construct portfolios that eliminate these risks. I. Motivation II. Partial Derivatives of Simple Securities III. Partial Derivatives of European Options IV. The Gamma-Theta Relationship V. Popular Options Strategies and the Greeks VI. Risk Management A. B. C. D. E. Portfolio Hedging Delta Hedging Gamma Hedging Simultaneous Delta and Gamma Hedging Theta, Vega, and Rho Hedging VII. The Cost of Greeks VIII. Other Risk Management Approaches Risk Management with Options I. Motivation Traders who write derivatives must hedge their risk exposure. We’d like to simply characterize the main risks associated with a complicated portfolio of positions on an underlying. – Ultimately, that’s where we’re headed. Example: Suppose you’re trading options for Goldman-Sachs, and you just wrote, for $5, a 10week, ATM European call. The underlying’s trading at $50, and D 50%. The risk-free rate is 3.0%. Black-Scholes tells you that the call option is worth $4.50. How can you make the profit of $0.50 per option without risk? Buy the same option for $4.50 elsewhere. Spend $4.50 on a replicating portfolio (i.e., buy a synthetic option) that has the same payoff. Bus 35100 Page 2 Robert Novy-Marx Risk Management with Options Question 1 : can you really perfectly replicate the option’s payoff? That is, can you perfectly hedge away all of the risk associated with the call you wrote? You can if both: – The binomial tree model perfectly describes the stock price dynamics. – You can trade without transaction costs. You can if both: – The log-normal model perfectly describes the stock price dynamics. – You can trade continuously and without transaction costs. But in the real world: We can’t trade continuously. Transaction costs can be substantial. The volatility of the underlying and the risk-free rate aren’t constant. Bus 35100 Page 3 Robert Novy-Marx Risk Management with Options Question 2: So what should you do, if you can’t perfectly hedge the risk of the call you’ve written? 1. Identify the different sources of risk. The value of the call changes if any of the following factors changes: S D Stock Price t D Time D Volatility r D Interest Rate 2. Form an approximate replicating portfolio for the written call option. The value of this portfolio should change by about the same amount as that of the option. – At least for small changes in the factors. But how do we figure out how sensitive the option is to the factors? Bus 35100 Page 4 Robert Novy-Marx Risk Management with Options The “Greeks” To construct the approximate replicating portfolio, we have to know how much the value of the option changes as the various factors change. That is, the sensitivity of the call to each factor. Using calculus (i.e., a linear approximation), for small changes in the factors, the value of the call option changes by: 1 @2 C @C @C @C @C 2 (dS ) dS C C dt C d C dr, dC D 2 @S 2 @S @t @ @r „ƒ‚… „ƒ‚… „ƒ‚… „ƒ‚… „ƒ‚… Delta Theta Gamma or, using the symbols , dC D c dS C 12 2 c (dS ) Vega Rho , , and , C c dt C c d C c dr. – These “Greeks” characterize the market risk associated with the option. Bus 35100 Page 5 Robert Novy-Marx Risk Management with Options II. Example: Stocks, Bonds, and Forwards Before we consider the sensitivity of an option’s price to each of the factors that determine an option’s value, we’ll do some simpler securities. – To get a feel. First, what are , D @S t D 1 @S t S D @S D 0 @S t S D @S t D 0 @t D @S t D 0 @ D @S t D 0. @r S S S Bus 35100 , , and for a stock? Page 6 Robert Novy-Marx Risk Management with Options What about a bond? – Bt,T D e r (T t) , so D @Bt D 0 @S t B D @B D 0 @S t B D @Bt D rBt,T @t D @Bt D 0 @ D @Bt D @r B B B (T t )Bt,T . B D rBt,T > 0 ) Bond becomes more valuable as time passes. B D (T t )Bt,T < 0 ) Bond looses value when interest rates rise. Note: B D Bus 35100 DB Bt,T . Page 7 Robert Novy-Marx Risk Management with Options What about a forward contract? – Replication ) f t D S t Ke f D @f t D 1 @S t f D @f @S t f D @f t D @t f D @f t D 0 @ D @f t D (T @r f r (T t) , so D 0 rKe r (T t) t )Ke r (T t) . A long forward position is worth more (ceteris paribus) – If the underlying goes up. – If interest rates rise. – With more time to maturity. Note: f D 1 and f D 0 explains why we can replicate a forward statically. Bus 35100 Page 8 Robert Novy-Marx Risk Management with Options III. The Greeks for European Options How does the price of a derivative change when we vary one factor and hold all others fixed? Delta () Delta measures a derivative’s sensitivity to the price of the underlying security. c D @C @S p D @P D @S D N(d1) > 0 N( d1 ) < 0. – Note that: c ! 0 as S ! 0 c ! 1 as S ! 1 Important: In the Black-Scholes model, delta tells us how many shares of the stock to buy for the replicating portfolio. Bus 35100 Page 9 Robert Novy-Marx Risk Management with Options What does c look like? Delta vs. Underlying price T = 1 week, 1 month, and 1 quarter Delta HDL 1 0.8 0.6 0.4 0.2 Spot HSL 60 80 100 120 140 160 Delta as a function of the spot price of the underlying, for three different time-to-expirations [T D 0.02 (solid line), T D 0.0833 (dashed line) and T D 0.25 (dotted line)]. Figure depicts the case when K D 100, D 0.56 and r D 0.05. Bus 35100 Page 10 Robert Novy-Marx Risk Management with Options How about c and “moneyness”? Delta vs. Time-to-Expiration ITM, ATM, OTM Delta HDL 1 0.8 0.6 0.4 0.2 Time HTL 0.1 0.2 0.3 0.4 0.5 Delta as a function of time-to-maturity, for three different levels of “moneyness” [K D 100 (solid line, ATM), K D 90 (dashed line, ITM) and K D 110 (dotted line, OTM)]. Figure depicts the case when S D 100, D 0.56 and r D 0.05. Bus 35100 Page 11 Robert Novy-Marx Risk Management with Options Gamma ( ) Gamma measures a derivative’s convexity. c D D p D @c @S @d1 @S N 0 (d1) N (d1) D p > 0 S T t 0 @( d1) @S N 0 ( d1) D c. – Note that: ! 0 as S ! 0 ! 0 as S ! 1 is high when S K. – What does gamma tell us? It tells us how much we gain as the underlying rises. It also tells us how quickly a delta-hedged derivative becomes unhedged. Bus 35100 Page 12 Robert Novy-Marx Risk Management with Options What does c look like? Gamma vs. Underlying price T = 1 week, 1 month, and 1 quarter Gamma HGL 0.05 0.04 0.03 0.02 0.01 Spot HSL 80 100 120 140 Gamma as a function of the spot price of the underlying, for three different time-to-expirations [T D 0.02 (solid line), T D 0.0833 (dashed line) and T D 0.25 (dotted line)]. Figure depicts the case when K D 100, D 0.56 and r D 0.05. Bus 35100 Page 13 Robert Novy-Marx Risk Management with Options How about c and “moneyness”? Gamma vs. Time-to-Expiration ATM, OTM, ITM Gamma HGL 0.035 0.03 0.025 0.02 0.015 0.01 0.005 Time HTL 0.1 0.2 0.3 0.4 0.5 Gamma as a function of time-to-maturity, for three different levels of “moneyness” [K D 100 (solid line, ATM), K D 80 (dashed line, ITM) and K D 120 (dotted line, OTM)]. Figure depicts the case when S D 100, D 0.56 and r D 0.05. Bus 35100 Page 14 Robert Novy-Marx Risk Management with Options Theta () Theta measures the derivative’s sensitivity to the passage of time. It captures time-decay. @C D @t c @ D @(T t) SN(d1) Ke @N(d1 ) S @(T C Ke t) D r (T t) N(d2) r (T t) @N(d2 ) @(T t) rKe r (T t) N(d2) This can be simplified using 0 SN (d1) D p (d2 C T t)2 =2 Se p 2 0 D SN (d2)e D Ke r (T t) p ( T t )d2 2 (T t)=2 N 0 (d2) and @ (d1 @(T Bus 35100 d2) t) D p @ T t D p @(T t ) 2 T Page 15 t Robert Novy-Marx Risk Management with Options Taken together, these imply c D N 0(d1 )S p 2 (T t ) rKe r (T t) N(d2) < 0. c < 0 ) the value of a call decreases as time goes by, ceteris paribus. As time-to-expiration decreases: – The variance of the stock price at maturity decreases. Less value in the right to not exercise. – The time discounting of the exercise price decreases. Expected cost of exercise is higher. Important: the fact that c is negative does not imply that the call price is expected to fall. – Remember, the stock price, on average, rises over time. Bus 35100 Page 16 Robert Novy-Marx Risk Management with Options By put-call parity, P D C f, we have p D @(C f) @t @f D c C @(T t ) D N 0 (d1) S p C rKe 2 (T t ) r (T t) N( d2 ). The first term is again because the variance of the stock price at maturity decreases as time-tomaturity decreases. The second term is positive. – The PV of the strike grows over time. I.e., with less time-to-maturity. – Put receive the strike, so this tends to make the put more valuable as time passes. Bus 35100 Page 17 Robert Novy-Marx Risk Management with Options What does c look like? Theta vs. Underlying price T = 1 week, 1 month, and 1 quarter Theta HQL Spot HSL 80 100 120 140 160 180 -20 -40 -60 -80 Theta as a function of the spot price of the underlying, for three different time-to-expirations [T D 0.02 (solid line), T D 0.0833 (dashed line) and T D 0.25 (dotted line)]. Figure depicts the case when K D 100, D 0.56 and r D 0.05. Bus 35100 Page 18 Robert Novy-Marx Risk Management with Options How about c and “moneyness”? Theta vs. Time-to-Expiration ATM, OTM, ITM Theta HQL Time HTL 0.1 0.2 0.3 0.4 0.5 -10 -20 -30 -40 -50 Theta as a function of time-to-maturity, for three different levels of “moneyness” [K D 100 (solid line, ATM), K D 80 (dashed line, ITM) and K D 120 (dotted line, OTM)]. Figure depicts the case when S D 100, D 0.56 and r D 0.05. Bus 35100 Page 19 Robert Novy-Marx Risk Management with Options Vega () Vega measures the derivative’s sensitivity to the volatility of the underlying security. – You’ll occasionally see it called “Kappa” (). p @C c D DS T @ p D t N 0 (d1) > 0 @P D c @ Note that: 0 for S K is largest for S K e r (T t) 0 for S K Important: Vega is important to traders who worry about changes in the volatility of the underlying security. Bus 35100 Page 20 Robert Novy-Marx Risk Management with Options What does c look like? Vega vs. Underlying price T = 1 week, 1 month, and 1 quarter Vega HΝL 20 15 10 5 Spot HSL 80 100 120 140 160 180 200 Vega as a function of the spot price of the underlying, for three different time-to-expirations [T D 0.02 (solid line), T D 0.0833 (dashed line) and T D 0.25 (dotted line)]. Figure depicts the case when K D 100, D 0.56 and r D 0.05. Bus 35100 Page 21 Robert Novy-Marx Risk Management with Options How about c and “moneyness”? Vega vs. Time-to-Expiration ATM, OTM, ITM Vega HΝL 17.5 15 12.5 10 7.5 5 2.5 Time HTL 0.05 0.1 0.15 0.2 Vega as a function of time-to-maturity, for three different levels of “moneyness” [K D 100 (solid line, ATM), K D 80 (dashed line, ITM) and K D 120 (dotted line, OTM)]. Figure depicts the case when S D 100, D 0.56 and r D 0.05. Bus 35100 Page 22 Robert Novy-Marx Risk Management with Options The Gamma-Vega Relationship Remember: c D N 0 (d1) p S T t So: c p D S T t N 0 (d1) D S 2(T t) c or c c D S 2(T t) – They always come together Closely related; sensitivities to expected and realized volatilities – Come in fixed proportion, for a given series – Calender spreads allow you to bet more on one than the other Bus 35100 Page 23 Robert Novy-Marx Risk Management with Options Rho (c ) Rho measures the derivative’s sensitivity to the risk-free interest rate. c D @ @r SN(d1) 1) D S @N(d @r C (T Ke Ke r (T t) N(d2) r (T t) @N(d2 ) @r t )Ke r (T t) N(d2). Now @N(d1 ) @r @N(d2 ) @r D D D @d1 0 N (d1) @r p @(d1 T t) 0 N (d2 ) @r @d1 0 N (d2) @r and SN 0(d1) D Ke Bus 35100 Page 24 r (T t) N 0 (d2). Robert Novy-Marx Risk Management with Options So c D (T t )Ke r (T t) N(d2) > 0. Put-call parity gives p D @(C D c D (T S C Ke @r (T t )Ke t )Ke r (T t) ) r (T t) r (T t) N( d2 ) < 0. The value of the call always goes up with the interest rate. – The P V of S (T ) is always S (t ). – The P V of K drops. The opposite is true for puts. – The value of a put falls with the interest rate. Bus 35100 Page 25 Robert Novy-Marx Risk Management with Options What does c look like? Rho vs. Underlying price T = 1 week, 1 month, and 1 quarter Rho HΡL 20 15 10 5 Spot HSL 80 100 120 140 160 180 Rho as a function of the spot price of the underlying, for three different time-to-expirations [T D 0.02 (solid line), T D 0.0833 (dashed line) and T D 0.25 (dotted line)]. Figure depicts the case when K D 100, D 0.56 and r D 0.05. Bus 35100 Page 26 Robert Novy-Marx Risk Management with Options How about c and “moneyness”? Rho vs. Time-to-Expiration ATM, OTM, ITM Rho HΡL 25 20 15 10 5 Time HTL 0.1 0.2 0.3 0.4 0.5 Rho as a function of time-to-maturity, for three different levels of “moneyness” [K D 100 (solid line, ATM), K D 80 (dashed line, ITM) and K D 120 (dotted line, OTM)]. Figure depicts the case when S D 100, D 0.56 and r D 0.05. Bus 35100 Page 27 Robert Novy-Marx Risk Management with Options Other “Greeks” Much less common; just mentioning their existence Lambda – Delta per dollar invested c D c C Volga (or Vega-Gamma) – Second-order sensitivity to volatility @2 C @ 2 D @c @ Vanna – Sensitivity of Delta to volatility @2 C @ @S D @c @ – Moneyness changes with underlying price, and implied volatilities change with moneyness Bus 35100 Page 28 Robert Novy-Marx Risk Management with Options IV. The Gamma-Theta Relationship Remember the Black-Scholes partial differential equation: 1 2 2 @2 C S @S 2 2 C rS @C @S C @C @t rC D 0. Using the Greeks we can rewrite this as 1 2 2 S c 2 C rSc C c rC D 0. That is, the Black-Scholes PDE implies a relation between C , , , and for a European call option. – They are not determined independently. – This is true in general, not just for calls. How can we interpret this constraint? Bus 35100 Page 29 Robert Novy-Marx Risk Management with Options First, rewrite the constraint as rC D rSc C 12 2S 2 c C c . Now remember: r C is the expected, risk-neutral yield to the call. – Here yield = price rate of return. The equation says that yield comes from three sources. If you own a call you’re: – Long the underlying stock. And earning a return on that. – You’re also “Earning” c . Because you’re “long volatility.” – And you’re “Paying” c . Remember: c < 0. Time-to-expiration “runs backwards.” Bus 35100 Page 30 Robert Novy-Marx Risk Management with Options rC D rSc C 12 2S 2 c C c . This equation also says that if two calls are: 1. Priced the same, and 2. Have the same sensitivity to the underlying, – I.e., if C D C 0 and C D C 0 , then: 1 2 2 S c 2 C c D 1 2 2 S c0 2 C c 0 . That is, the call with the higher Gamma also has a higher Theta. – You “pay for” Gamma by taking on Theta. Traders care about this! – A lot. – Having this answer in an interview is the kind of thing that can get you a job. Bus 35100 Page 31 Robert Novy-Marx Risk Management with Options It also gives us another way to understand this: Theta vs. Time-to-Expiration ATM, OTM, ITM Theta HQL Time HTL 0.1 0.2 0.3 0.4 0.5 -10 -20 -30 -40 -50 Theta as a function of time-to-maturity, for three different levels of “moneyness.” Solid line, ATM: K=S D 1; dashed line, ITM: K=S D 0.8; dotted line, OTM: K=S D 1.2. Other parameters: D 0.56 and r D 0.05. How so? – Well what happens to Gamma ATM, ITM, and OTM as T ! 0? Bus 35100 Page 32 Robert Novy-Marx Risk Management with Options V. Popular Options Strategies and the Greeks We can also recast option portfolio strategies in terms of the Greeks. – Traders tend to think about them this way. For example, what are you “buying” when you buy a straddle? – That is, when you buy both a put and a call with the same strike. py g ( ) y Straddle: C=f(S,t) ATM CALL + ATM PUT S = 100 K = 100 Profit (BUY ATM CALL @ $18.84) (BUY ATM PUT @ $5.80) t =1 r = 1.15 25 d = 1.00 V = . 3 50 75 125 150 Future Asset Price -25 Straddle Value = $18.84 + $5.80 = $24.64 Loss Bus 35100 Page 33 Strategy: Strategy: Believe Believevolatility volatilityof of asset price will be high, asset price will be high,but buthave have no noclue clueabout aboutdirection. direction. Robert Novy-Marx Risk Management with Options You’re not buying Delta. – At least not very much: pCc D c C p D N(d1 ) N( d1) 0. pCc D 0 if d1 D 0 , K D Se (r C 2=2)T . But you’re definitely buying Gamma. pCc D c C p D 2 c. – And the Gamma of the call is high, if it’s near the money. You’re paying for it with Theta. – Strike-discounting isn’t the problem. You’re paying on the put, earning on the call. – But the premia on both options shrinks with time. Each moment, your exposure to Gamma costs you Theta. Bus 35100 Page 34 Robert Novy-Marx Risk Management with Options What happens to the straddle price as (T t ) ! 0? – Assuming everything else is unchanged. Straddle Price T = 1 day, 1 week, and 1 month Price H$L 20 15 10 5 Spot HSL 90 100 110 120 Straddle prices (P100 CC100) as a function of the spot price of the underlying, for three different time-to-expirations [T D 0.004 (solid line), T D 0.02 (dashed line) and T D 0.0833 (dotted line)]. Figure depicts the case when D 0.56 and r D 0.05. Bus 35100 Page 35 Robert Novy-Marx Risk Management with Options Butterfly spreads do more or less the opposite. – BS: Buy one ITM, sell two ATM, buy one OTM. But near the money: – Delta is linear w.r.t. moneyness (more or less). – Gamma is convex w.r.t. moneyness. Delta HDL 1 Gamma HGL 0.8 0.04 0.6 0.03 0.4 0.02 0.2 0.01 0.05 Spot HSL Spot HSL 60 80 100 120 140 160 80 That is, K K 1 2 1 2 > (K ( ı K ı 100 120 140 C KCı ) C KCı ) , so BFS BFS Bus 35100 D K D ı 2K C KCı 0 K ı Page 36 2 K C KCı < 0. Robert Novy-Marx Risk Management with Options What happens to the Butterfly price at the central strike as (T t ) ! 0? – Assuming everything else is unchanged. Butterfly Price T = 1 day, 1 week, and 1 month Price H$L 10 8 6 4 2 Spot HSL 85 90 95 100 105 110 115 120 Butterfly prices (C90 2C100 C C110) as a function of the spot price of the underlying, for three different time-toexpirations [T D 0.004 (solid line), T D 0.02 (dashed line) and T D 0.0833 (dotted line)]. Figure depicts the case when D 0.56 and r D 0.05. Bus 35100 Page 37 Robert Novy-Marx Risk Management with Options VI. Risk Management A. Portfolio Hedging The basic idea of portfolio hedging is that the value of a portfolio can be made invariant to the factors affecting it, such as S , and r . For example, suppose a portfolio consists of three assets: V D n1 A1 C n2 A2 C n3 A3 where: V is the value of the portfolio, ni is the number of shares of asset i, and Ai is the market value of one share of asset i. Then the sensitivity of the portfolio to some arbitrary factor, x, is @V @x Bus 35100 D n1 @A1 @A2 @A3 C n2 C n3 . @x @x @x Page 38 Robert Novy-Marx Risk Management with Options The objective of x-hedging is to pick the ni such that the value of the portfolio stays constant when x changes. – That is, pick n1, n2, and n3 so that: @V @x D n1 @A1 @A2 @A3 C n2 C n3 D 0. @x @x @x Then the value of the portfolio stays approximately constant when x changes by a small amount: dV D @V dx 0. @x Important: It takes n securities to hedge against n 1 sources of uncertainty. For example, with two assets you can only hedge one risk. – E.g., you could pick the relative weights so that the portfolio is Delta-neutral. – All the other exposures are determined by these weights. Bus 35100 Page 39 Robert Novy-Marx Risk Management with Options B. Delta Hedging A portfolio is Delta neutral (i.e., Delta hedged) if the of the portfolio is zero. For example, take our portfolio of three assets, and let x D S : portfolio D @V @S @A1 @A2 @A3 D n1 C n2 C n3 @S @S @S D n1 1 C n2 2 C n3 3. The portfolio will be Delta-hedged if we pick the ns so that this is zero. Then the portfolio value will be insensitive to small changes in S : dV Bus 35100 portfolio dS D 0. Page 40 Robert Novy-Marx Risk Management with Options More concretely: Remember the call you wrote for GoldmanSachs: T S D 50 K D 50 10 52 t D D 0.50 r D 0.03. Questions: how many share of the stock should we buy to Delta hedge the option? We’re short the call, and c D 0.554. S of a share is one, so we buy nS shares such that: nS 1 0.554 D 0. So, we buy nS D S D 0.554 shares of the stock. Bus 35100 Page 41 Robert Novy-Marx Risk Management with Options C. Gamma Hedging A portfolio is Gamma neutral (i.e., Gamma hedged) if the of the portfolio is zero. Take the portfolio of three assets and let x D S : portfolio D @2 V @S 2 D @portfolio @S @1 @2 @3 D n1 C n2 C n3 @S @S @S D n1 1 C n2 2 C n3 3. Question: If a portfolio is already Delta hedged, so its value stays approximately constant for small changes in S , why do we want to Gamma hedge it? Bus 35100 Page 42 Robert Novy-Marx Risk Management with Options Example continued ... We just learned that the portfolio 1. Short the ATM call, and 2. Long 0.554 shares of the stock, is Delta hedged. How stable is the value of this portfolio if S changes? Small change in S: – Suppose S increases from 50 to 51. C (50, 50, 10 , 0.50, 0.03) D 4.498 52 C (51, 50, 10 , 0.50, 0.03) D 5.070. 52 – Then 0.554(51 50) (5.070 4.498) D 0.018. A loss of less than 2 cents for a $1 increase in the stock price. Not too bad. – But what about bigger moves in the underlying? Bus 35100 Page 43 Robert Novy-Marx Risk Management with Options Large change in S : – Suppose S increases from 50 to 60. C (50, 50, 10 , 0.50, 0.03) D 4.498 52 C (60, 50, 10 , 0.50, 0.03) D 11.541. 52 – Then 0.554(60 50) (11.541 4.498) D 1.54. A loss of $1.54 for a $10 increase in the stock price. Not so good. Our “hedged” position still had an effective 15% exposure the large move in the underlying. Lesson from the Example: Delta hedging works well for small changes in S only. Gamma hedging can improve the quality of the hedge. Bus 35100 Page 44 Robert Novy-Marx Risk Management with Options To understand why, look at the Taylor series expansion for the change in the call price C (S C dS ) C (S ) c dS C 12 2 c (dS ) D 0.554 dS C 0.018 (dS )2. Buying 0.554 shares hedges the first term. We’re still exposed to the second term. – We’d need to Gamma hedge as well to eliminate that exposure. For the large change in S (dS D 10), the second term is $1.80, which explains our $1.55 loss. – Any discrepancy is due to the missing 3rd-, 4th-, and higher- order terms. Questions: Is Gamma hedging alone more effective than Delta hedging alone? Can we use stock to Gamma hedge an option? Bus 35100 Page 45 Robert Novy-Marx Risk Management with Options D. Simultaneous Delta and Gamma Hedging What if we wanted to hedge our writhen call such that our net position was both Delta-neutral and Gamma-neutral? – For that ATM call 1 D 0.554 and 1 D 0.0361. We need another asset – One that has Gamma. So any call should do We’ll consider the call struck at 55. Delta-hedging requires that nS S C nC55 C55 C nC50 C50 D 0. Gamma-hedging requires that nS Bus 35100 S C nC55 C55 Page 46 C nC50 C50 D 0. Robert Novy-Marx Risk Management with Options We already know C50 C50 And S D 1, S D 0.554 D 0.0361. D 0. Black-Scholes gives C55 C55 D 0.382 D 0.0348. Finally, nC50 D 1, so need to buy nS shares of the stock and nC55 calls at 55 such that: nS C 0.382 nC55 0.554 D 0 0 C 0.0348 nC55 0.0361 D 0. Solving these yields nS D 0.158 and nC55 D 1.037. Bus 35100 Page 47 Robert Novy-Marx Risk Management with Options How stable is the value of this portfolio to changes in S ? Small change in S: – Suppose S increases from 50 to 51. C50(51) C50(50) D 5.067 4.498 D 0.569 C55(51) C55(50) D 3.002 2.602 D 0.400 and 0.158 1 C 1.037 0.400 1 0.569 D 0.001. – The value of the portfolio changes (increases) by less than 0.1 cent. That’s pretty good hedging. Bus 35100 Page 48 Robert Novy-Marx Risk Management with Options Large change in S : – Suppose S increases from 50 to 60. C50(60) C50(50) D 11.581 4.498 D 7.084 C55(60) C55(50) D 8.104 2.602 D 5.501 and 0.158 10 C 1.037 5.501 1 7.084 D 0.201. – The value of the portfolio changes (increases) by 20 cents. That’s much less than a change of $1.55 change for the Delta hedged portfolio. Important: We needed three securities to form a portfolio hedged in both Delta and Gamma. In general we need n securities to form a portfolio that is insensitive to small variations in n 1 factors. Bus 35100 Page 49 Robert Novy-Marx Risk Management with Options E. Theta, Vega, and Rho Hedging These don’t get worried about as much – They’re still important, but not as important. The mechanics of these hedging strategies are similar to Delta or Gamma hedging. – For example, a portfolio is Theta hedged, or is Theta neutral, if its is zero. Some questions: How does the of a portfolio relate to the ’s of the securities that form the portfolio? Can a bond be used for Theta hedging? Can a stock be used to Theta hedge an option? Bus 35100 Page 50 Robert Novy-Marx Risk Management with Options VII. The Cost of Greeks We can construct pure exposures to individual Greeks – By hedging all the other risks away This allows us to “price” the exposures – Figure what it costs to take on a pure exposure to a given Greek The easiest Greek to price is delta – How do you get a pure exposure to ? – What’s the cost of a unit exposure to ? The underlying is a pure exposure to delta – So cost of a unit exposure: P D S Bus 35100 Page 51 Robert Novy-Marx Risk Management with Options Rho is also easy to price – What does “duration” cost? Nothing! Remember, for a bond B D B B D @Bt D rBt,T @t B D @Bt D @r (T D B D 0, and t )Bt,T So buy $1 of a long bond, sell $1 of a short bond – It’s free – l=s D l=s D l=s D 0, and l=s D r r D 0 l=s D (Tl t ) C (Ts D (Tl Ts ) t) So P D 0 Bus 35100 Page 52 Robert Novy-Marx Risk Management with Options Now it’s easy to price theta Again, for a bond B D B D B D 0, and B D @Bt D rBt,T @t B D @Bt D @r (T t )Bt,T Rho is free So buy a bond, at a cost of Bt,T – Hedge the Rho risk using a zero-cost portfolio of bonds – Hedging Rho is free Your pure theta exposure of rBt,T cost you Bt,T Unit price is (price paid) / (total exposure), so P Bus 35100 D Bt,T rBt,T Page 53 1 D r Robert Novy-Marx Risk Management with Options Gamma and Vega are slightly harder Let’s start by summarizing what we know – Cost of , , and : P D S – C D SN(d1) c c P D 0 P D 1=r r (T t) Ke N(d2 ) and D N(d1) D N 0(d1) p S T t N 0 (d1)S p 2 (T t ) c D c D S 2(T c D (T t) t )Ke rKe r (T t) N(d2) c r (T t) N(d2) Cost of a delta/rho/theta-hedged call? Bus 35100 Page 54 Robert Novy-Marx Risk Management with Options Cost and Greeks of the hedged call: P C D P C SN(d1) 1 r D P C r (T t) Ke 0 N (d1)S p 2 (T t ) P C N(d2) rKe SN(d1) r (T t) ! N(d2) N 0(d1)S p 2r (T t ) and P P Bus 35100 D 0 D N 0 (d1) p S T t P D 0 P D S 2(T P D 0 Page 55 t) c Robert Novy-Marx Risk Management with Options Now note that P D P N 0(d1)S 2r i.e., P p (T t) N 0(d1 ) p S T t D D 2S 2 2r 2S 2 2r P This is independent of all contractual parameters – $1 of any delta/rho/theta-hedged call gives you the same amount of gamma: P =P D 2r= 2S 2 So a long/short portfolio of delta/rho/theta-hedged calls is gamma-neutral – It’s also delta/rho/theta-neutral – It’s not generally Vega-neutral Unless same maturity on both sides I.e., pure Vega-exposure is free ) P D 0 So ame equation gives us the price of gamma: P Bus 35100 D P P Page 56 D 2S 2 2r Robert Novy-Marx Risk Management with Options Summarizing: P D S P P 2S 2 D 2r D 0 P D 0 P D 1=r So the value of any derivative V on S , in terms of its exposures to the risk factors, is V C P C P C P D P C P 2 2 S 1 D S C 2r C r – We can also write this as rV D rS CS C 12 2S 2CS S C C t The Black-Scholes PDE! Bus 35100 Page 57 Robert Novy-Marx Risk Management with Options VIII. Other Risk Management Approaches Stop-Loss Rules. Scenario Analysis: 1. Monte Carlo Simulations. 2. Stress Testing. 3. Value at Risk (VAR). Bus 35100 Page 58 Robert Novy-Marx