Lecture 11: The Greeks and Risk Management

Transcription

1 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. Portfolio Hedging B. Delta Hedging C. Gamma Hedging D. Simultaneous Delta and Gamma Hedging E. Theta, Vega, and Rho Hedging VII. The Cost of Greeks VIII. Other Risk Management Approaches

2 I. Motivation Risk Management with Options 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 10- week, 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 Page 2 Robert Novy-Marx

3 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 Page 3 Robert Novy-Marx

4 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 Page 4 Robert Novy-Marx

5 The Greeks Risk Management with Options 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: ƒ Delta ds C ƒ Gamma (ds) ƒ Theta ƒ Vega ƒ Rho dr, or, using the symbols,,, and, dc D c ds C 1 2 c (ds) 2 C c dt C c d C c dr. These Greeks characterize the market risk associated with the option. Bus Page 5 Robert Novy-Marx

6 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,,, and for a stock? S t D 1 S t D 0 S S S D 0 D 0 D 0. Bus Page 6 Robert Novy-Marx

7 What about a bond? B t,t D e r(t t), so B t D 0 B t D 0 B B B D rb t,t D 0 D (T t)b t,t. B D rb t,t > 0 ) Bond becomes more valuable as time passes. B D (T t)b t,t < 0 ) Bond looses value when interest rates rise. Note: B D D B B t,t. Bus Page 7 Robert Novy-Marx

8 What about a forward contract? Replication ) f t D S t K e r(t t), so f t D 1 f t D 0 f D rke r(t t) f D 0 f D (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 Page 8 Robert Novy-Marx

9 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. D N(d 1) > 0 D N( d 1) < 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 Page 9 Robert Novy-Marx

10 What does c look like? Delta vs. Underlying price T = 1 week, 1 month, and 1 quarter Delta Spot S 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 (dashed line) and T D 0.25 (dotted line)]. Figure depicts the case when K D 100, D 0.56 and r D Bus Page 10 Robert Novy-Marx

11 How about c and moneyness? Delta vs. Time-to-Expiration ITM, ATM, OTM Delta Time T 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 Bus Page 11 Robert Novy-Marx

12 Gamma ( ) Risk Management with Options Gamma measures a derivative s convexity. c N 0 (d 1 ) D N 0 (d 1 ) S p T t > 0 p d 1 N 0 ( d 1 ) 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 Page 12 Robert Novy-Marx

13 What does c look like? Gamma vs. Underlying price T = 1 week, 1 month, and 1 quarter Gamma Spot S 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 (dashed line) and T D 0.25 (dotted line)]. Figure depicts the case when K D 100, D 0.56 and r D Bus Page 13 Robert Novy-Marx

14 How about c and moneyness? Gamma vs. Time-to-Expiration ATM, OTM, ITM Gamma Time T 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 Bus Page 14 Robert Novy-Marx

15 Theta () Risk Management with Options Theta measures the derivative s sensitivity to the passage of time. It captures time-decay. t) 1 ) Ke r(t t) N(d 2 ) D t) C Ke r(t 2 t) rke r(t t) N(d 2 ) This can be simplified using SN 0 (d 1 ) D Se (d p 2C T t) 2 =2 p 2 D SN 0 (d 2 )e ( p T t)d 2 2 (T t)=2 D Ke r(t t) N 0 (d 2 ) (d 1 d 2 t) p T t) D 2 p T t Bus Page 15 Robert Novy-Marx

16 Taken together, these imply c D N 0 (d 1 )S 2 p (T t) rke r(t t) N(d 2 ) < 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 Page 16 Robert Novy-Marx

17 By put-call parity, P D C f, we have p f D t) D N 0 (d 1 ) S 2 p (T t) C rke r(t t) N( d 2 ). 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 Page 17 Robert Novy-Marx

18 What does c look like? Theta vs. Underlying price T = 1 week, 1 month, and 1 quarter Theta Spot S 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 (dashed line) and T D 0.25 (dotted line)]. Figure depicts the case when K D 100, D 0.56 and r D Bus Page 18 Robert Novy-Marx

19 How about c and moneyness? Theta vs. Time-to-Expiration ATM, OTM, ITM Theta Time T 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 Bus Page 19 Robert Novy-Marx

20 Vega () Risk Management with Options Vega measures the derivative s sensitivity to the volatility of the underlying security. You ll occasionally see it called Kappa (). D S p T t N 0 (d 1 ) > 0 D c Note that: 0 for S K r(t t) is largest for S K e 0 for S K Important: Vega is important to traders who worry about changes in the volatility of the underlying security. Bus Page 20 Robert Novy-Marx

21 What does c look like? Vega vs. Underlying price T = 1 week, 1 month, and 1 quarter Vega Ν Spot S 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 (dashed line) and T D 0.25 (dotted line)]. Figure depicts the case when K D 100, D 0.56 and r D Bus Page 21 Robert Novy-Marx

22 How about c and moneyness? Vega vs. Time-to-Expiration ATM, OTM, ITM Vega Ν Time T 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 Bus Page 22 Robert Novy-Marx

23 The Gamma-Vega Relationship Remember: c D N 0 (d 1 ) S p T t So: c D S p T t N 0 (d 1 ) 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 Page 23 Robert Novy-Marx

24 Rho ( c ) Risk Management with Options Rho measures the derivative s sensitivity to the risk-free interest rate. SN(d 1 ) Ke r(t t) N(d 2 ) D 1) Ke r(t C (T t)ke r(t t) N(d 2 ). Now 1 N 0 (d p T N 0 (d 2 ) N 0 (d 2 ) SN 0 (d 1 ) D Ke r(t t) N 0 (d 2 ). Bus Page 24 Robert Novy-Marx

25 So c D (T t)ke r(t t) N(d 2 ) > 0. Put-call parity gives p S C Ke r(t t) D c (T t)ke r(t t) D (T t)ke r(t t) N( d 2 ) < 0. The value of the call always goes up with the interest rate. The PV of S(T ) is always S(t). The PV of K drops. The opposite is true for puts. The value of a put falls with the interest rate. Bus Page 25 Robert Novy-Marx

26 What does c look like? Rho vs. Underlying price T = 1 week, 1 month, and 1 quarter Rho Ρ Spot S 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 (dashed line) and T D 0.25 (dotted line)]. Figure depicts the case when K D 100, D 0.56 and r D Bus Page 26 Robert Novy-Marx

27 How about c and moneyness? Rho vs. Time-to-Expiration ATM, OTM, ITM Rho Ρ Time T 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 Bus Page 27 Robert Novy-Marx

28 Other Greeks Much less common; just mentioning their existence Lambda Delta per dollar invested Volga (or Vega-Gamma) c D c C Second-order sensitivity to volatility 2 C Sensitivity of Delta to 2 Moneyness changes with underlying price, and implied volatilities change with moneyness Bus Page 28 Robert Novy-Marx

29 IV. The Gamma-Theta Relationship Remember the Black-Scholes partial differential equation: S 2 C @t r C D 0. Using the Greeks we can rewrite this as S 2 c C rs c 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 Page 29 Robert Novy-Marx

30 First, rewrite the constraint as rc D rs c C S 2 c C c. Now remember: rc 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 Page 30 Robert Novy-Marx

31 rc D rs c C S 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: S 2 c C c D S 2 c 0 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 Page 31 Robert Novy-Marx

32 It also gives us another way to understand this: Theta vs. Time-to-Expiration ATM, OTM, ITM Theta Time T 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 How so? Well what happens to Gamma ATM, ITM, and OTM as T! 0? Bus Page 32 Robert Novy-Marx

33 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. C=f(S,t) Straddle: ATM CALL + ATM PUT py g ( ) y S = 100 K = 100 t = 1 r = 1.15 d = 1.00 =. 3 (BUY ATM $18.84) (BUY ATM $5.80) Profit Future Asset Price Straddle Value = $ $5.80 = $ Loss Strategy: Believe volatility of of asset asset price price will will be be high, high, but but have have no no clue clue about about direction. Bus Page 33 Robert Novy-Marx

34 You re not buying Delta. At least not very much: pcc D c C p D N(d 1 ) N( d 1 ) 0. pcc D 0 if d 1 D 0, K D Se (rc2 =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 Page 34 Robert Novy-Marx

35 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 $ Spot S Straddle prices (P 100 CC 100 ) as a function of the spot price of the underlying, for three different time-to-expirations [T D (solid line), T D 0.02 (dashed line) and T D (dotted line)]. Figure depicts the case when D 0.56 and r D Bus Page 35 Robert Novy-Marx

36 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 Gamma Spot S Spot S That is, K 1 2 ( K ı C KCı ) K > 1 2 ( K ı C KCı ), so BFS D K ı 2 K C KCı 0 BFS D K ı 2 K C KCı < 0. Bus Page 36 Robert Novy-Marx

37 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 $ Spot S Butterfly prices (C 90 2C 100 C C 110 ) as a function of the spot price of the underlying, for three different time-toexpirations [T D (solid line), T D 0.02 (dashed line) and T D (dotted line)]. Figure depicts the case when D 0.56 and r D Bus Page 37 Robert Novy-Marx

38 VI. Risk Management Risk Management with Options 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 n 1 A 1 C n 2 A 2 C n 3 A 3 where: V is the value of the portfolio, n i is the number of shares of asset i, and A i is the market value of one share of asset i. Then the sensitivity of the portfolio to some arbitrary factor, D n C n C n Bus Page 38 Robert Novy-Marx

39 The objective of x-hedging is to pick the n i such that the value of the portfolio stays constant when x changes. That is, pick n 1, n 2, and n 3 D n C n C n D 0. Then the value of the portfolio stays approximately constant when x changes by a small amount: dx 0. 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 Page 39 Robert Novy-Marx

40 B. Delta Hedging Risk Management with Options 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: D n C n C n 3 D n 1 1 C n 2 2 C n 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 portfolio ds D 0. Bus Page 40 Robert Novy-Marx

41 More concretely: Remember the call you wrote for Goldman- Sachs: S D 50 K D 50 T t D D 0.50 r D Questions: how many share of the stock should we buy to Delta hedge the option? We re short the call, and c D S of a share is one, so we buy n S shares such that: n S D 0. So, we buy n S D S D shares of the stock. Bus Page 41 Robert Novy-Marx

42 C. Gamma Hedging A portfolio is Gamma neutral (i.e., hedged) if the of the portfolio is zero. Gamma Take the portfolio of three assets and let x D S: portfolio 2 D n C n C n 3 D n 1 1 C n 2 2 C n 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 Page 42 Robert Novy-Marx

43 Example continued... We just learned that the portfolio 1. Short the ATM call, and 2. Long 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 C(51, 50, 10, 0.50, 0.03) D Then 0.554(51 50) ( ) D 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 Page 43 Robert Novy-Marx

44 Large change in S: Suppose S increases from 50 to 60. C(50, 50, 10, 0.50, 0.03) D C(60, 50, 10, 0.50, 0.03) D Then 0.554(60 50) ( ) D 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 Page 44 Robert Novy-Marx

45 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 1 2 c(ds) 2 D ds C (ds) 2. Buying 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 Page 45 Robert Novy-Marx

46 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 and 1 D 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 n S S C n C55 C55 C n C50 C50 D 0. Gamma-hedging requires that n S S C n C55 C 55 C n C50 C 50 D 0. Bus Page 46 Robert Novy-Marx

47 We already know C50 D C 50 D And S D 1, S D 0. Black-Scholes gives C55 D C 55 D Finally, n C50 D 1, so need to buy n S shares of the stock and n C55 calls at 55 such that: n S C n C D 0 0 C n C D 0. Solving these yields n S D and n C55 D Bus Page 47 Robert Novy-Marx

48 How stable is the value of this portfolio to changes in S? Small change in S: Suppose S increases from 50 to 51. C 50 (51) C 50 (50) D D C 55 (51) C 55 (50) D D and C D The value of the portfolio changes (increases) by less than 0.1 cent. That s pretty good hedging. Bus Page 48 Robert Novy-Marx

49 Large change in S: Suppose S increases from 50 to 60. C 50 (60) C 50 (50) D D C 55 (60) C 55 (50) D D and C D 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 Page 49 Robert Novy-Marx

50 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 Page 50 Robert Novy-Marx

51 VII. The Cost of Greeks Risk Management with Options 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 Page 51 Robert Novy-Marx

52 Rho is also easy to price What does duration cost? Nothing! Remember, for a bond B D B D B D 0, and B B D rb t,t D (T t)b t,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 (T l t) C (T s t) D (T l T s ) So P D 0 Bus Page 52 Robert Novy-Marx

53 Now it s easy to price theta Again, for a bond B D B D B D 0, and B B D rb t,t D (T t)b t,t Rho is free So buy a bond, at a cost of B t,t Hedge the Rho risk using a zero-cost portfolio of bonds Hedging Rho is free Your pure theta exposure of rb t,t cost you B t,t Unit price is (price paid) / (total exposure), so P D B t,t rb t,t D 1 r Bus Page 53 Robert Novy-Marx

54 Gamma and Vega are slightly harder Let s start by summarizing what we know Cost of,, and : P D S P D 0 P D 1=r C D SN(d 1 ) Ke r(t t) N(d 2 ) and c D N(d 1 ) c D N 0 (d 1 ) S p T t c D N 0 (d 1 )S 2 p (T t) rke r(t t) N(d 2 ) c D S 2 (T t) c c D (T t)ke r(t t) N(d 2 ) Cost of a delta/rho/theta-hedged call? Bus Page 54 Robert Novy-Marx

55 Cost and Greeks of the hedged call: P C P C P C P C D SN(d 1 ) Ke r(t t) N(d 2 ) SN(d 1 ) 1 r N 0 (d 1 )S 2 p (T t) rke r(t t) N(d 2 )! D N 0 (d 1 )S 2r p (T t) and P D 0 P D N 0 (d 1 ) S p T t P D 0 P D S 2 (T t) c P D 0 Bus Page 55 Robert Novy-Marx

56 Now note that P P D N 0 (d 1 )S p 2r (T t) N 0 (d 1 ) S p T t D 2 S 2 2r i.e., P D 2 S 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= 2 S 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 D P D 2 S 2 P 2r Bus Page 56 Robert Novy-Marx

57 Summarizing: Risk Management with Options P D S P D 2 S 2 2r P 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 D P C P C P C P C P D S C C r 1 2 S 2 2r We can also write this as rv D rsc S C S 2 C SS C C t The Black-Scholes PDE! Bus Page 57 Robert Novy-Marx

58 VIII. Other Risk Management Approaches Stop-Loss Rules. Scenario Analysis: 1. Monte Carlo Simulations. 2. Stress Testing. 3. Value at Risk (VAR). Bus Page 58 Robert Novy-Marx