Risk Management or Derivatives he Greeks are coming the Greeks are coming! Managing risk is important to a large number o iniviuals an institutions he most unamental aspect o business is a process where we invest, take on risk an in exchange earn a compensatory return he key to success in this process is to manage your riskreturn trae-o Managing risk is a nice concept but the iiculty is oten measuring risk here is a saying what gets measure gets manage o alter this slightly, What cannot be measure cannot be manage Hence risk management always requires some measure o risk Risk in the most general context reers to how much the price o a security changes or a given change in some actor In the context o Equities, Beta is a requently use measure o risk Beta measures the relative risk o an asset High Beta stocks or portolios have more variable returns relative to the overall market than low Beta assets I a Beta o 00 means the asset has the same risk characteristics as the market, then a portolio with a Beta great than one will be more volatile than the market portolio an consequently is more risky with higher expecte returns Conversely assets with a Beta less than 00 are less risky than average an have lower expecte returns Portolio managers use Beta to measure their risk-return trae-o I they are willing to take on more risk (an return), they increase the Beta o their portolio an i they are looking or lower risk they ajust the Beta o their portolio accoringly In a CAPM ramework, Beta or market risk is the only relevant risk or portolios For Bons, the most important source o risk is changes in interest rates Interest rate changes irectly aect bon prices Moiie Duration is the most requently use measure o how bon prices change relative to a change in interest rates Relatively higher Moiie Duration means more price volatility or a given change in interest rates For both Bons an Equities, risk can be istille own to a single risk actor For Bons it is Moiie Duration an or equities it is Beta In each case, the risk can be measure an ajuste or manage to suit ones risk tolerance For a iscussion o Beta please see, Reerence or Duration see, Duration an Convexity, (UVA-F-38) or R Brealy an S Myers: Principles o Corporate Finance, 7 th eition, McGraw-Hill, Boston, 003, pages 674-678 his note was prepare by Proessor Robert M Conroy Copyright 003 by the University o Virginia Daren School Founation, Charlottesville, VA All rights reserve o orer copies, sen an e-mail to arencases@virginiaeu No part o this publication may be reprouce, store in a retrieval system, use in a spreasheet, or transmitte in any orm or by any means electronic, mechanical, photocopying, recoring, or otherwise without the permission o the Daren School Founation
-- In each o these cases, inancial theory provies a measure o risk Using these risk measures, holers o either bons or equities can ajust or manage the risk level o the securities that they hol What is nice about these asset categories is that they have a single measure o risk Derivative securities are more challenging Risk Measures or Derivatives In the iscussion which ollows we will eine risk as the sensitivity o price to changes in actors that aect an asset s value More price sensitivity will be interprete as more risk he basic option pricing moel A simple European Call option can be value using the Black-Scholes Moel 3 : r European Call Option Value UAV N e N( ) Where N Cumulative Stanar Normal Function, () ( UAV ) ln + r + σ σ, σ, UAV Unerlying Asset Value, ime to maturity, σ UAV volatility Exercise (Strike) Price Risk-ree rate r his very basic option pricing moel emonstrates that the value o a European call option epens on the value o ive actors: Unerlying Asset Value, Risk-ree rate, volatility, ime to maturity, an Exercise price I the value o any one o these actors shoul change, then the value o the option woul change O the ive inputs or actors, one is ixe (the exercise price), another is eterministic (time to maturity), an the other three change ranomly over time (unerlying asset value, volatility an the risk-ree rate) Hence the risk associate with holing a European Call option is that the unerlying asset value coul change, the volatility coul change, or the risk-ree rate coul change Any change in the value o these things woul change the value o the call option 3 Reerence or Black-Scholes Moel see, R Brealy an S Myers: Principles o Corporate Finance, 7 th eition, McGraw-Hill, Boston, 003, pages 60-607
-3- Our original einition o risk was; How much oes the value o a call option change given a change in the value o the unerlying actors? Base on this there nee to be ierent risk measures or each actor hese risk measures are the Greeks he inclue Delta, Gamma, Vega 4 an Rho he next sections will eal with each o these he process is to examine how the Black-Scholes moel value changes in response to a change in one o the inputs shown above Delta Delta is a measure o how much the value o an option, orwar or a utures contract values will change over a very short interval o time or a given change in the asset price he simplest is the Delta or a long position in one share o stock Since a $ change results in a $ change in value, the Delta is A one ollar change in the stock results in a one ollar change in the value o the long position Once we know how much the value o a position in stock, options or utures contracts will change or a given change in the price o the unerlying asset (Delta), we can then use this inormation to hege the unerlying price risk his heging o price risk is oten reerre to as Delta Heging Calculating Delta or Call Options an Put Options he most obvious actor that aects an option s value is the value o the unerlying asset Using the basic Black-Scholes moel, we can take the partial erivative with respect to UVA ( ) r UAV N e N Call Call UAV It turns out that the solution 5 to this is ( ) where ln ( UAV ) + r + σ ( ) Call N, σ his is a general result It is the relationship between the value o the call option an the value o UAV 4 Vega is not actually a Greek letter but we can be a bit generous with our terminology 5 It is important to note that taking the partial erivative o the ull Black-Scholes moel is more complicate than it appears when you look at the solution You can see this by noting that the value o also inclues the value UAV However, conveniently everything cancels out an you get the rather simple expression shown
-4- In practice, most people want to know the relationship between the call value an the stock price an this is what is usually calle the Delta o a call option Delta Call Call It is important to note that the value o Delta or a particular call will epen on how UAV is eine he simplest case is or call options on non-ivien paying stocks In this case, UAV stock price As such, UAV an N N( ) Delta Call Call N Consier a European Call option on stock YZ with a strike price o $50 an a time to maturity o 90 ays he YZ s stock price toay is $55, the estimate volatility is 35 an the current 90 ay risk-ree rate is 5% Given these parameters, the Black-Scholes value is UAV $55 Maturity 90ays Call $6877 $50 an the Delta woul be R σ 5% 35 ( S ) ln( 55 + r + ) + 05 + ( 35) ( 90 σ ) ln Delta N N σ 50 35 he Call price is $6877 an Delta o this call option is 749 90 365 365 749 Given a Delta o 749, we woul expect that a $00 change in the stock price will result in an increase in the value o the call o $0749 Changing the UAV rom $55 to $550 in the Black-Scholes moel above gives a call option price o $695 an an actual change o $00749 I the stock price were to increase to $56, then the preicte price change given the Delta woul be $0749 an the actual Black-Scholes calculate price change woul be $0765 hey are not exactly the same an this inicates that Delta is really only accurately measures the option price change or small changes in the unerlying stock price is the cumulative stanar normal istribution Given this einition 6 o Delta, it has a minimum o 0 an a maximum o 00 Call In this case, Delta N( ), where N( ) 6 Since it is a cumulative istribution, it can only take on values between 0 an one
-5- options that are eep out o the money will have a Delta o approximately 0 his relects that act that i the option is way out o the money, its value is very small an small changes in the stock price will not materially impact on the value o the call option Hence a small change in stock price will not aect the call s value an the Delta woul be close to zero From the example above, i the stock price were $30 an not $55, the call value woul be $0004 an the Delta 000 he low option value relects the very small probability that the stock price will rise above $50 in the next 90 ays he small Delta relects that act that even i the stock price goes to $3 that is not materially going to increase the probability o the price rising above $50 an as such, there will be very little change in the value o the call option On the han, or call options which are eep in the money, the Delta woul be 00 I the stock price were $70, the call value woul be $040 an the Delta 980 Here the Delta is almost 00 because a $00 change in price gets incorporate into the intrinsic value an there is only a very small negative change in the time value 7 Exhibit shows the call option value or this option given ierent stock prices Here you get the amiliar result that the relationship between stock prices an call prices are lat at low stock prices an linear or high prices In between, they are convex Delta starts at zero an increases as the stock price increases he maximum value o Delta is 00 Exhibit European Call Option ($50, σ 35, 90 ays, r 5%) Call Option Values & Delta 50000 000 00000 0000 Exercise Price 08000 Call Option Value 50000 00000 06000 Delta Call Value Delta 04000 50000 0000-30 3 34 36 38 40 4 44 46 48 50 5 54 56 58 60 6 64 66 68 70 Stock Price - 7 For call options, when the stock price is below the exercise price, the time value relects the probability that the stock price will be above the exercise price at maturity As such, as the stock price gets closer to the exercise price the time value increases When the stock price is above the exercise price, the time value relects the probability that the stock price will be below the exercise price he value o an option over owning the stock is the act that at maturity you o not have to buy it i the stock price is below the exercise price Hence the higher the stock price relative to the exercise price the less the valuable the option is relative to the stock hat is the time value ecreases as the stock price increases Consequently a $ change in stock price increases the intrinsic value by $ but the time value actually ecreases an the net change in value o the call option is less than $
-6- he Call value can be ivie into the intrinsic value an a time value Exhibit shows the intrinsic value an time values base on the call option values in Exhibit Beginning at a price below the exercise price the call option value is all time value As the stock price increases the time value oes not increase at a rate o one or one In act on the ar let o the exhibit the time value is essentially lat an Delta 0 ime value increases at an increasing rate as we move towar the exercise price In act, time value is greatest when the stock price is equal to the exercise price As the stock price increases beyon the exercise price, the intrinsic value increases on a one or one basis but the time value actually begins to ecrease his results in a Delta less than 00 until we get to the ar right o the exhibit where the time value line goes to zero an is again essentially lat Here the Delta is 00 Exhibit European Call Option ($50, σ 35, 90 ays, r 50% Call Option Values Call Value Intrinsic Value + ime Value 500 000 Exercise Price Value (Intrinsic & ime) 500 000 Intrinsic Value ime Value 500 000 30 3 34 36 38 40 4 44 46 48 50 5 54 56 58 60 6 64 66 68 70 Stock price In orer to hege price risk, we estimate the Delta o the security an then take a position in another asset with a negative Delta Value For example i the Stock price were $55 an we were the holer (buyer) o the call option above, then Delta749 o hege this position we coul use woul nee to in another call option on the same stock For example, another call option, with the same maturity but an exercise price o $45, woul have a price o $0769 an a Delta o 899 I we write (sell) 833 o this call option, the Delta woul be -749 (833 x 899 749) Combining the long position in the call an the short position in the secon call, results in a net Delta o 0 Call option Delta or ivien paying stocks he Delta or a ivien paying stock presents a slightly ierent challenge For ivien paying stocks, the UAV oes not equal the stock price In the known ivien approach to valuing call option, the Unerlying Asset Value is UVA S (Present Value o Diviens) In this ormulation, since the present value o the iviens oes not epen on S, UAV an
-7- Delta Call Delta N Delta N ( ) Call N, where ln ( UAV ) + r ( S PVoDiviens) + σ ln σ σ r + + σ y However, i we use the ivien yiel approach, then UAV S e, where y is the ivien yiel an here the ajustment or the ivien oes epen on S As such, UAV y e Hence or call value base on using the ivien yiel moel, Delta Call Call y Delta e Delta N e y where, ln y ( UAV ) ( S e ) r + + σ ln σ r + σ + σ European put options Delta For European Put Options we can use put-call parity to estimate Delta he Put-Call parity no arbitrage relationship or European options on non-ivien paying stocks is, Stock + Put Call + Bon I we take the partial erivative o the put-call parity relationship with respect to stock price we get ( Stock) ( Bon ) + Put Call + + Put Call + 0 Put Call Put N( ) Delta Put Delta Call
-8- Calculating Deltas or Forwar an Futures Contracts Consier a Futures contract or 000 bushels o corn with contract price K an elivery at time I at the en o the ay the price o elivery at time is F, where F r Spot e hen because the contract is settle aily, the value o the Futures contract is Value o Futures Contract VF VF F K Spot e r K he Delta o the value o a Futures Contract is VF pot F K S e ( ) Delta Futures ( ) r r K e his tells us that or every $ change in the spot price the value o the utures contract r changes by e Hence i we wishe to hege the spot price o 000 bushels o corn overnight using this particular Futures contract we woul use utures contract or a total r o e 000 bushels Forwar Contracts Delta Here consier you have the Buy-sie o a Forwar contract or 000 bushels o corn with contract price K an elivery at time I at the en o the ay the orwar price 8 o r elivery at time is F, where F Spot e > K, then to realize the value o the contract you woul have to take the sell sie o a contract that matures at time with orwar price F Because these are orwar contracts, the value o the payo is Value o Forwar Contract VFW VFW r r r r ( Spot e K ) e Spot K e ( F K) e he Delta o the value o a Forwar contract is just VFW VF F Spot pot K S K e ( ) Delta Forwar ( ) 00 his tells us that or every $ change in the spot price the value o the orwar contract changes by $00 Hence i we wishe to hege the spot price o 000 bushels o corn overnight using this particular Futures contract we woul use utures contract or a total o 000 bushels r 8 In this simple example we will ignore storage costs, spoilage an convenience yiels
-9- GAMMA (Γ) From Exhibit we see that or options Delta changes as the stock price changes I we set up a hege using Delta, as the stock price changes, Delta changes an as such we nee to ajust hege ratio How much Delta changes as the stock price changes is important he rate o change in Delta is calle Gamma Calculating Gamma or Call an Put options o capture this, we calculate Gamma in the ollowing way, Delta UAV Delta Γ ( ) N UAV S S UAV σ Again this is a very general ormulation that inclues UAV However, since Gamma like Delta is eine in terms o change in Delta relative to a change in stock price, the tem is important For non-ivien paying stocks, the UVAS, UAV, an the unction or Gamma reuces to Γ N, S σ ( ) where N ( ) e A call option with an exercise price o $50, r 5%, 90 π ays, σ 35, an stock price o $55 has a Gamma 00333 Exhibit 3 shows the relationship between Delta an Gamma as the stock price changes or this call option he largest Gamma is when the stock price is just below the exercise price Exhibit 3 Delta an Gamma Call Option ($50, risk-ree rate5%, time 90 ays an volatility 35) 005 0045 Exercise Price 004 0035 08 003 Gamma 005 06 Gamma Delta 00 005 04 00 0 0005 0 0 30 3 34 36 38 40 4 44 46 48 50 5 54 56 58 60 6 64 66 68 70 7 74 76 78 80 8 84 86 stock price High values o Gamma inicate that small changes in the stock price result in relatively large changes in Delta As such, iniviuals or institutions oing elta heging or
-0- options where the current stock price is close to the exercise price have to ajust the hege more requently Because UAV or ivien paying stocks oes not equal the stock price, Gamma or options on ivien paying stocks is slightly ierent I we are using a known ivien moel, then Gamma is Γ Delta UAV ( Delta) UAV N UAV S Γ N where UAV σ ln ( UAV ) + r + σ σ I we are using a ivien yiel moel, then Gamma is UAV σ UAV an UVA S Present Value o Diviens Γ Delta UAV ( Delta) UAV N UAV S Γ N Γ N e y S σ y ( S e ) e y σ e y e y UAV σ UAV where ln S e y r + σ + σ Note 9 that since the Delta Put Delta Call, the Gamma o a Put option is equal to DeltaPut Delta Call an the Gamma o a Put option is equal to the Gamma o a call option with the same characteristics Vega (ν) Vega is the relationship between the option value an changes in volatility Again we use the Black-Scholes moel an ask the question o all other things equal, how oes the value o a call option change with a change in volatility? he answer is Vega Vega Call UAV N ( ) ν σ 9 Please see the section on Delta s
-- Again using the call option presente above with UAV$55, 90 ays, σ 35, $50 an r 50%, Vega 870 his implies that a change in the volatility rom 35 to 36 woul result in a $00870 change in the call option price 0 Vega or European call options on non-ivien paying stocks where UAV S is Vega ν Call S N ( ) σ Vega or European call options on ivien paying stocks is either Vega or known ivien moel ν Call ( S PViviens) N ( ), σ or Vega or ivien yiel moel Call ( S e y ν ) N σ eine with the appropriate UAV value in each case Exhibit 4 European Call Option ($50, 90 ays, σ 35, r 50%) Vega, where is 0 Exercise Price 50 8 Vega 6 Vega 4 Rho 0 30 3 34 36 38 40 4 44 46 48 50 5 54 56 58 60 6 64 66 68 70 7 74 76 78 80 8 84 86 Stock Price Rho measures the sensitivity o a call option price to changes in the risk-ree rate It is eine as Rho Call r e r N ( ) 0 he call option price woul change rom $6877 (σ35) to $6965 (σ36), or a change o $00877 Note that I will use the roman letters or rho he Greek letter ρ is very oten use as the symbol or correlation coeicient I have not use it here to avoi conusion
-- he value o Rho or our example, UAV$55, 90 ays, $50, σ 35 an r 5% is 8459 an a change in the risk-ree rate rom 50% to 60% woul result in a change o $0008459 in the call option price Note that shoul inclue the einition o UAV consistent with the no ivien, the known ivien or the ivien yiel moels he Rho or a Put is again erive base on Put-Call Parity, Stock + Put Call + Bon S + Put Call + e Put Call + e r r S an Rho Rho Rho Rho Rho Put Put Put Put Put Put r Put r e e e Call r Call r r r r N + ( ) ( N ( )) N ( ) r ( e ) e e r r r + 0 + S r Delta, Vega an Rho Delta, Vega an Rho represent measures o the three risk actors that one aces when you hol options either iniviually or in portolios Changes in the unerlying asset value, volatility or the risk-ree rate all aect option values heta (Θ) heta measures the change in the value o an option with respect to time his is unamentally ierent than the other actors Delta, Vega an Rho all measure price sensitivity to actors which can change ranomly ime on the other han is eterministic For example consier an option where the exercise price is $50, the current stock price is $55, risk-ree rate is 5%, time to maturity is 3 months an volatility is 35 One month rom now the stock price, risk-ree rate an/or volatility coul be anything but with certainty the time to maturity will be two months an i all the other actors remaine the same we woul know with certainty what the change in the call option price woul be Clearly it woul be less his is known as the time ecay o an option heta measures the rate o time ecay an is equal to heta Call UAV N σ r e r N
-3- For call options on non-ivien paying stocks, UAV S Once again using a call option with $50, 90 ays, σ 35, r 5% an current stock price o $55, the value o heta woul be heta 7033 where S N heta σ r e r N 55 N 35 0550 e 90 365 05 90 365 N an ln ( S ) ln( 55 + r + ) + 05 + (35) ( 90 σ ) σ σ 06708 35 50 90 365 35 04970 90 365 Exhibit 5 shows how heta changes as stock price changes 365 06708 Exhibit 5 European Call Option ($50, 90 ays, σ 35, r 50%) heta 0 30 3 34 36 38 40 4 44 46 48 50 5 54 56 58 60 6 64 66 68 70 7 74 76 78 80 8 84 86 - - -3-4 heta -5-6 -7-8 -9 As we can see heta or time ecay is largest when the option is at at the money It is iicult to interpret heta because oten the time units are not clear In the example about heta -7033 his is expresse in terms o years For example, a ecrease in maturity rom 90 ays to 85 ays is 5 ays or 0037 years With a heta o -7033, this ecrease in maturity shoul ecrease the option price by $00963 (-7033 x 037 -
-4-00963) It is very typical to express heta in terms o calenar ays, -0093 (- 7033/365) or in terms o traing ays, -079 (-7033/5) Bottom line is that you nee to be clear what the time element is or eining heta Summary In this note we examine ive ierent measures o how an option price can change given a change in one o the basic valuation parameters; Delta, Gamma, Vega, heta an Rho