Management des systèmes d information

Transcription

1 Cahier de la Recherche de l ISC Paris CRISC N 19 : Management des systèmes d information 1 er trimestre 2008 ISBN

2 Conseil scientifique Liste des membres : BRESSON Yoland, Professeur d'économie, ancien doyen, Université Paris - Val de Marne Paris XII CUMENAL Didier, Directeur de la recherche, professeur de Management des Systèmes d'information, Doctorat ès sciences de gestion ESCH Louis, Professeur de Finance, Directeur académique d'hec Liège, Université de Liège HETZEL Patrick, Professeur à l Université de Limoges KUZNIK Florian, Recteur, économiste, Université d'economie de Katowice (Pologne) MORIN Marc, Professeur en management des ressources humaines, Doctorat d'etat PARIENTE Georges, Doyen de la recherche, professeur d'économie, Doctorat d'etat PESQUEUX Yvon, Professeur titulaire de la chaire Développement des Systèmes d'organisation au CNAM PORTNOFF André-Yves, Directeur de l'observatoire de la Révolution de l'intelligence à Futuribles REDSLOB Alain, Professeur d'économie, ancien doyen de la faculté des Sciences Economiques de Paris II ZEFFERI Bruno, Directeur Cegos Dirigeants 2

3 Comité de lecture Liste des membres : AGARWAL Aman, Professor of Finance and Director of Indian Institute of Finance, Editor of Finance India CHEN Kevin C., California State University, Editor, International Journal of Business CLARK Ephraïm, University of Middlesex, U.K. DESPRES Charles, Directeur de l International Institute of Management du Conservatoire National des Arts et Métiers, Paris DOMINGUEZ Juan Luis, Professeur titulaire de la Chaire Economie Financière et Comptabilité, Faculté d économie et sciences de l entreprise, Université de Barcelone, Espagne JÂGER Johannes, Doyen de University of Applied Sciences, Vienne (Autriche), Lecturer Fachochschule des bfi Wien Gesellschaft m.b.h. KUMAR Andrej Professor, Holder of Chair Jean Monnet, Faculty of Economics, University of Ljubljlna, Slovenia PARLEANI Didier, Professeur de droit à l Université de Paris 1 Panthéon- Sorbonne PRIGENT Jean-Luc, Professeur de finance à l Université de Paris Cergy RYAN Joan, Professor of Global Banking and Finance at the European Business School, London, Grande-Bretagne SCHEINWBERGER Albert G., Professeur à l Université de Constance, Allemagne 3

4 CRISC déjà parus Cahier n 1 : Finance (Edité en avril 2002) Cahier n 2 : Marketing (Edité en septembre 2002) Cahier n 3 : Economie (Edité en mars 2003) Cahier n 4 : Contrôle de gestion (Edité en décembre 2003) Cahier n 5 : Droit (Edité en mai 2004) Cahier n 6 : Ressources humaines (Edité en juin 2004) Cahier n 7 : Les NTIC (Edité en septembre 2004) Cahier n 8 : Microstructures et marchés financiers (Edité en janvier 2005) CRISC hors série Actes de la 3 ème Conférence Internationale de Finance IFC 3 (mars 2005) Cahier hors série n 1 Finance Cahier hors série n 2 Bourse Cahier hors série n 3 Formalisation et Modélisation 4

5 Cahier n 9 : International (Edité en mai 2005) CRISC déjà parus (suite) Cahier n 10 : Marketing : études et décisions managériales (Edité en septembre 2005) Cahier n 11 : Actes du colloque de ressources humaines du 24 novembre 2005 «La responsabilité sociétale de l entreprise : quel avenir pour la fonction RH?» (Edité en janvier 2006) Cahier n 12 : Stratégie (Edité en mars 2006) Cahier n 13 : Normes IFRS (Edité en juillet 2006) Cahier n 14 : Corporate Governance (Edité en octobre 2006) Cahier n 15 : Dynamique des organisations (Edité au 1 er trimestre 2007) Cahier n 16 : Actes du colloque IFC 4 (Parution 2 ème trimestre 2007) Cahier n 17 : Actes du colloque : «Entrepreneuriat, nouveaux défis, nouveaux comportements» (Parution 3 ème trimestre 2007) Cahier n 18 : Outils d analyse stratégiques et opérationnels en marketing (Parution 4 ème trimestre 2007) 5

6 CRISC prochainement disponibles Cahier n 20 : Economie-Finance (Parution 2 trimestre 2008) Cahier n 21 : Logistique (Parution 3 trimestre 2008) 6

7 Sommaire PARIENTE Georges Doyen de la Recherche de l ISC Paris Editorial p 8 BELLALAH Mondher Management of information systems : p 10 the case of trading derivatives CUMENAL Didier Pourquoi les Systèmes Interactifs d Aide à la décision p 47 («SIAD») ne nous dispensent pas d être intelligents? LIOTTIER Miguel Optimisation des transactions boursières électroniques p 80 par une approche phénoménologique et performance dans l usage d un workflow MORIN Marc Les enjeux de la circulation d informations entre les p 119 Comités d entreprise européens et les sommets stratégiques des groupes (enseignements d une analyse de terrain des pratiques allemandes et françaises) Les articles sont classés par ordre alphabétique des noms d auteurs 7

8 Editorial Georges PARIENTE Docteur ès Sciences Economiques Doyen de la recherche à l ISC Paris Les CRISC de l année 2008 seront consacrés à une approche plus pointue des techniques utilisées en gestion, alors que les cinq années précédentes nous avions proposé des présentations plus transversales. Ce premier numéro s intéresse au management des systèmes d information abordé à partir de quatre thématiques différentes mais complémentaires. Le premier article, par Mondher BELLALAH, Docteur en Sciences de Gestion, professeur de finance associé à l ISC Paris, s applique aux produits dérivés. Il s agit de mesurer les différents risques des paramètres inclus dans les formules d options en fonction de la manière dont ces options sont gérées. La connaissance des modifications de ces paramètres de risque est nécessaire pour gérer une position d option et pour déterminer les profits et pertes associés à cette position. Cela permet d ajuster la position de façon continue en fonction des changements dans les conditions du marché. Les Systèmes Interactifs d Aide à la Décision sont aujourd hui réservés à des professionnels explique Didier CUMENAL, Docteur en Sciences de Gestion et Directeur de la Recherche à l ISC Paris. Le motif de cette diffusion restreinte n est pas seulement technique mais plutôt cognitif et organisationnel. C est pourquoi cet article s appuie sur l apport des sciences cognitives pour construire un prototype de système destiné à suggérer aux décideurs les indicateurs pertinents dans une démarche interactive. Miguel LIOTTIER, Docteur en Sciences de Gestion et professeur de Management des Systèmes d Information à l ISC Paris applique la méthode du «workflow» (gestion électronique des procédés) à 8

9 l optimisation des transactions boursières sur Euronext. Les outils phénoménologiques, épistémologiques et philosophiques sont utilisés pour obtenir la méthodologie la plus rationnelle et la plus efficace. Marc MORIN, Docteur en Sciences Economiques et en Sciences des Organisations, responsable du «PREMA», pôle de recherche entrepreneuriat, management à l ISC Paris, analyse les enjeux de la circulation des informations pour les acteurs sociaux européens. Les comités d entreprises d une part, les dirigeants stratégiques des groupes de l autre se trouvent dans une situation asymétrique qui peut expliquer certains comportements de ces acteurs. Comme d habitude nous restons à l écoute de vos critiques et suggestions. Georges PARIENTE Doyen de la Recherche recherche@groupeisc.com 9

10 BELLALAH Monder Docteur en Sciences de Gestion Professeur associé de Finance à l ISC Paris Management of information systems : the case of trading derivatives Summary This chapter presents the main Greek letter risk measures, i.e. the delta, the gamma, the theta and the vega in the context of the European analytical models. The risk measures are simulated for different parameters which enter the option formulas. Then the magnitude of these risk measures is evaluated in connection with the management and the monitoring of an option position. The knowledge of the changes in these risk parameters is necessary for the management of an option position and for the determination of the profits and losses associated with the position. To put it differently, the pricing of a European call option can be viewed as requiring inputs (the underlying asset and the treasury bill) and a production technology (the hedge portfolio and the Greek letter risk measures). In a B-S world, by tracking continuously the hedge ratio (being delta neutral), the investor makes sure that the duplicating portfolio does mimic the call option, namely does produce the option. In the course of doing so, the investor controls his production costs and protects his mark-up on the option. However, these risk measures depend on the theoretical model used for the valuation and the management of the option position. There is what one could call a technological risk. This position must be adjusted nearly continuously in response to the changes in market conditions. 10

11 Introduction to banks information systems in OTC trading Banks and market trading systems account for an information systems that reflects risks in real time. It is rather important to monitor the variations in a derivative asset price with respect to its determinants or the parameters which enter the option formula. These variations are often named Greek-letter risk measures. The most widely used measures indicating are known as the delta, charm, gamma, speed, colour, theta, vega, Rho and elasticity. The delta measures the absolute change in the option price with respect to a small variation in the underlying asset price. Charm corresponds to the partial derivative of the delta with respect to time. The gamma gives the change in the delta, or in the hedge ratio as the underlying asset price changes. Colour corresponds to the gamma's derivative with respect to the time remaining to maturity. The theta measures the change in the option price as time elapses, namely the time decay of the option. The vega or lambda is a measure of the change in the option price for a small change in the underlying asset's volatility. In this paper, we show how to use information risk systems in risk management. We calculate some of these parameters within the context of each analytical model presented in the previous chapter. Also, we develop examples to show how to use the Greek-letter risk measures in the monitoring and the management of an option position in response to changing market conditions. This paper is organised as follows. 1. In section 1, information systems regarding option price sensitivities are presented and applied. 2. In section 2, the risk Management matrix represented by Greekletter risk measures are simulated for different parameters. The issue of monitoring and managing in real time is studied. 3. In section 3, some of the characteristics of volatility spreads are presented. Risk management and Information systems in Banking The sensitivity parameters are important in managing an option position. 11

12 The delta measures the absolute change in the option price with respect to a small change in the price of the underlying asset. It is given by the option's partial derivative with respect to the underlying asset price. It represents the hedge ratio, or the number of options to write or to buy in order to create a risk-free portfolio. Call buying involves the sale of a quantity delta of the underlying asset in order to form the hedging portfolio. Call selling involves the purchase of a quantity delta of the asset to create the hedging portfolio. Put buying requires the purchase of a quantity delta of the underlying asset to hedge a portfolio. Put selling involves the sale of delta stocks to create a hedged portfolio. The delta varies from zero for deep out of the money options to one for deep in the money calls. This is not surprising, since by definition the delta is given by the first partial derivative of the option price with respect to the underlying asset. For example the value of a deep in the money call is nearly equal to the intrinsic value (S-K), for which the first partial derivative with respect to S is one. Charm is a risk measure that clarifies the concept of carry in financial instruments. The concept of carry refers to the expenses due to the financing of a deferred delivery of commodities, currencies or other assets in financial contracts. Charm is given by the derivative of the delta with respect to time. Even though charm is used by market participants as an ad-hoc measure of how delta may change overnight, it is an important measure of risk since it divides the theta into its asset-based constituents (See appendix 3). The gamma measures the change in delta, or in the hedge ratio as the underlying asset price changes. The gamma is greatest for at the money options. It is nearly zero for deep in and deep out of the money options. Roughly speaking, the gamma is to the delta what convexity is to duration. The gamma is given by the derivative of the hedge ratio with respect to the underlying asset price. As such, its is an indication of the vulnerability of the hedge ratio. The gamma is very important in the management and the monitoring of an option position. It gives rise to two other measures of risk : speed and colour. Speed is given by the gamma's derivative with respect to the underlying asset price. Colour is given by the gamma's derivative with respect to the time remaining to maturity. The theta measures the change in the option price as time elapses since the passage of time has a negative impact on option values. 12

13 Indeed, options are wasting assets. Theta is given by the first partial derivative of the option premium with respect to time. The vega or lambda measures the change in the option price for a change in the underling s asset volatility. It is given by the first derivative of the option premium with respect to the volatility parameter. The knowledge of the true option price is not sufficient for the monitoring and the management of an option position. Therefore, it is important to know the option price sensitivities with respect to the parameters entering the option formula. We begin our discussion with the delta. The delta The call s delta The delta is given by the option's first partial derivative with respect to the underlying asset price. It represents the hedge ratio in the context of the B-S model. Example When the underlying asset S=18, the strike price K=15, the short term interest rate r=10%, the maturity date T = 0.25 and the volatility =15%, the option's delta is given by : c = N(d 1) Applying this formula needs the calculation of d 1 : d & Ln$ % S K # &! + $ r + '! T " % 2 " ' T 1 2 # 2 & 18 # & 1 Ln$! + $ 0.1+ (0.15) % 15 " % #! 0.25 " 1 = = = Hence, the delta is c = N(2.8017)=0.997 This delta value means that the hedge of the purchase of a call requires the sale of units of the underlying asset. When the underlying asset price rises by 1 unit, from 18 to 19, the option price rises from to ( ), or When the asset price falls by one unit, the option price changes from to ( ), or

14 The put s delta The put's delta has the same meaning as the call's delta. It is given also by the option's first derivative with respect to the underlying asset price. When selling (buying) a put option, the hedge requires selling (buying) delta units of the underlying asset. The put's delta is given by : p = c 1= = The hedge ratio is When the underlying asset price goes from 18 to 19, the put price changes from to ( ), or When it falls from 18 to 17, the put price rises from to ( ), or The gamma The call s gamma The option's gamma corresponds to the option second partial derivative with respect to the underlying asset or to the delta partial derivative with respect to the asset price. In the B-S model, the call's gamma is given by : "# c 1! c = = n ( d1) " S S$ T with 1 1 n( d1) = " d1 2 % $! ' 2( exp # 2 & Using the same data as in the example : 1 2! 0. 5( ) n( d1) = e = ! c = ( ) = (0. 5) When the underlying asset price is 18 and its delta is 0.997, a fall in the asset price by one unit yields a change in the delta from to ( ), or Also, a rise in the asset price from 18 to 19, yields a change in the delta from to ( ), or 1. This means that the option is deeply in the money, and its value is 14

15 given by its intrinsic value (S K). The same arguments apply to put options. The call and the put have the same gamma. The put s gamma The put's gamma is given by : "# p 1! p = = n ( d1) " S S$ T or 1! p = ( ) = ( 0. 5) When the asset price changes by one unit, the put price changes by the delta amount and the delta changes by an amount equals to the gamma. The theta The call s theta The option's theta is given by the option's first partial derivative with respect to the time remaining to maturity. In the B-S model, the theta is given by : ( ) # c S$ n d1 " rt! c = = " + rke N( d2) # T 2 T or using the same data as in the example above : c = = When the option's time to maturity is reduced by one day, the option prices decreases by units. When the time to maturity is shortened by de 1% year, the call's price decreases by 0.01 (1.1918), or and its price changes from to ( ), or The put s theta In the B-S model, the put's Theta is given by : ( ) # p S$ n d1 " rt! p = = " + rke N " d2 # T 2 T ( ) 15

16 p= = Using the same reasoning, the put price changes from to ( ), or The vega The call s vega The option's vega is given by the option price derivative with respect to the volatility parameter. In the B-S model, the call's vega is given by : " c! c = = S T n( d ) "# 1 or using the above data :! c = ( ) = Hence, when the volatility rises by 1 point, the call price increases by The increase in volatility by 1% changes the option price from to [ %( )], or In the same context the put price changes from to [ % ( )], or When the volatility falls by 1% the call's price changes from to [ %( )], or In the same way, the put price is modified from to [ %( )], or zero since option prices cannot be negative. The put s vega In the B-S model, the put's vega is given by : " p! p = = S T n ( d ) "# 1 or ( )! p = = and it has the same meaning as the call's vega. The Rho The call s Rho 16

17 The option's Rho is given by the option first partial derivative with respect to interest rates. In the B-S model, the call's Rho is given by : c Rhoc = # K( t t) ( d ) r = "! " -rt e n 2 # where t is the current time and t* is the maturity date. Using the above data : Rho c = e 0.1(0.25) (0.996)(0.25)=3.64 The Rho does not affect call and put prices in the same way. In fact, a rise in the interest rate yields higher call price (positive Rho) and reduces the put price (negative Rho). The put s Rho In the B-S model, the put's Rho is given by : p Rhop = # K( t t) ( d ) r =! "! " -rt e n 2 # or 5e 0.1(0.25) (0.996)(0.25)=3.64 Elasticity Rho p = 1 The call s elasticity For a call option, this measure is given by : Elasticity =! S c S ( ) c = c N d 1 or using the above data : 18 Elasticity = = The elasticity shows the change in the option price when the underlying asset price varies by 1%. Hence a rise of the asset price by 1%, i.e. 0.18, induces an increase in the call price by 5,33%. The put price changes by 12%. Hence, when the asset price changes from 18 to 18.18, the call's price is modified from to [ (1+5.33%)], or In the same way, the put price changes from to [ (1+12%)], or

18 The put s elasticity The put's elasticity is given by : Elasticity = " S p S [ ( ) ] c = p N d 1! 1 18 Elasticity = [ ! 1] = The knowledge of the variations in these parameters is fundamental for the monitoring and the management of an option position. Monitoring and Managing Information systems in real time in banking Since option prices change in an unpredictable way in response to the changes in market conditions, traders, market makers and all options users must rely upon some model to monitor the evolutions of their profit and loss accounts. Such a model allows them to quantify the variations in option price sensitivities and their risk exposure. With such quantities the monitoring and the management of option positions are more easily achieved. We will illustrate the management of an option position in real time using the model proposed indirectly in Merton (1973) and derived afterwards in Black (1975) and BAW (1987) for the valuation of European futures options. First, option prices are simulated and the sensitivity parameters are calculated. Second, we study the risk management problem in real time with respect to option price sensitivities. Simulation and analysis of option price sensitivities Recall that the commodity call price and the commodity futures call price in the context of Merton's (1973) and BAW's (1987) model is given by : c Se ( b! r ) T! rt = N( d1) + Ke N( d2 ) where b stands for the cost of carrying the underlying commodity. By the put-call parity relationship or by a direct derivation, the put's value is given in the same context by :! p Ke rt ( b! r N ) T = (! d2)! Se N(! d1 ) 18

19 where all the parameters have the same meaning as before. For a-non-dividend paying asset, b=r. For a dividend-paying asset, b = r d where d stands for the dividend yield. For a currency option, b = r r * where r * stands for the foreign riskless rate. The following tables simulate option values and sensitivity parameters for calls and puts in the context of the above model. Sensitivity parameters for call options Table 1 gives call prices, delta, gamma, theta and vega when the underlying commodity price varies from 75 to 110 by a step of $5. For example, when the volatility =20%, r=8%, b=10%, T=3 months (0.25 year). The price of an at-the money call for K=90 is Table 1 Changes in underlying asset prices, at the money =0.2 r=0.08 b=0.10 T=0.25 K=90 Call Asset Delta Gamm Theta Vega a The call has a delta of 0.64, a gamma of 0.04, a theta of 0.17 and a vega of 1,94. Note that an at-the-money option has more theta and vega than in-and out of the money calls. 19

20 Table 2 Changes in underlying asset prices, in the money =0.2 r=0.08 b=0.10 T=0.25 K=100 Call Asset Delta Gamm Theta Vega a Using the same data except for the strike price which is modified from 90 to 100. Erreur! Source du renvoi introuvable. shows that an at the money call has more gamma, theta and vega than in and out of the money calls. Table 3 Changes in time to maturity, at the money =0.2 r=0.08 b=0.1 S=100 K=90 0 Call Maturity Delta Gamm Vega Theta a / Erreur! Source du renvoi introuvable. shows call prices and sensitivity parameters for in the money calls (S>K) when the time to maturity varies from 0.05 to 0.75 year. Note that the call price, the vega and the theta increase with the time to maturity. However, the delta falls when the time to maturity is longer. 20

21 Table 4 Changes in time to maturity, in the money =0.2 r=0.08 b=0.1 S=100 K=100 0 Call Maturity Delta Gamm Vega Theta a / / / Table 4 gives call prices and sensitivity parameters for at-the-money calls when (S = K = 100) and the time to maturity varies from 0.05 to a year. Note that the call price, its delta and vega are increasing functions of the time to maturity. However, the gamma and the theta are more important for near maturities. Table 5 Changes in volatility, at the money T=0.25 r=0.08 b=0.1 S=100 K=90 0 Call Volatility Delta Gamm Vega Theta a Table 5 gives in the money call prices (S=100, K=90) for different levels of the volatility parameter. When the delta is equal to one, the gamma is nearly equal to zero. Also, the vega is nearly nil and the theta is weak. 21

22 Table 6 Changes in volatility, in the money T=0.25 r=0.08 b=0.1 S=100 K=100 0 Call Volatility Delta Gamm Vega Theta a / Table 6 gives the same information as table 2.5, except that calculations are done for at-the-money options (S = K = 100). Sensitivity parameters for put options Table 7 gives put prices and the sensitivity parameters when : =0.2, r=0.08, b=0.1, T=0.25, K=90. For an-at-the money put, the gamma and the vega are important. The put theta increases when the option tends to parity and decreases afterwards. The same behaviour applies for the put gamma and vega. Table 7 Changes in underlying asset price, at the money r=0.08 b=0.10 T=0.25 K=90 Put Asset Delta Gamma =0.2 Theta

23 Table 8 Changes in underlying asset price, in the money r=0.08 b=0.10 T=0.25 K=100 Put Asset Delta Gamm =0.2 Theta a Table 8 gives the same information as Table 7, except for the strike price which is changed from 90 to 100. For in the money puts, the delta approaches-1, the gamma and vega are not important and the theta is very weak. Table 9 Changes in time to maturity, at the money =0.2 r=0.08 b=0.1 S=100 K=90 0 Put Maturity Delta Gamm Vega Theta a / Table 9 gives out of the money put prices (S=100, K=90) and the put price sensitivities when the time to maturity varies from 0.05 to a year. The values of the delta and gamma are weak. However, the vega and theta are greater for longer maturities. 23

24 Table 10 Changes in time to maturity, at the money s=0.2 r=0.08 b=0.10 S=100 K=100 Put Maturity Delta Gamm Vega Theta a / /24 Table 10 shows the same information for at-the-money put prices (S = K = 100). Note that the deltas and gammas are decreasing functions of the time to maturity. For an at-the-money put, there is more theta on short maturities and more vega on longer maturities. Table 11 Changes in volatility, at the money T=0.25 r=0.08 b=0.10 S=100 K=90 Put Volatility Delta Gamm Vega Theta a Table 11 gives out-of-the money put prices when the volatility varies from 0.05 to 0.4. When the volatility is respectively equal to 0.05 et 0.1 the put price is nil and the sensitivity parameters as well. 24

25 Table 12 Changes in volatility, in the money T=0.25 r=0.08 b=0.1 S=100 K=100 0 Put Volatility Delta Gamm Vega Theta a / Table 12 shows at-the money put prices (S = K = 100) for different levels of the volatility parameter. Note that the put price, the delta (in absolute value), the vega and the theta are increasing functions of the volatility parameter. Monitoring and adjusting the option position in real time Monitoring and managing the delta The call delta lies between zero and one and the put delta between zero and minus one. When the delta is 0.5, the call price rises (falls) by 0.5 point for each increase (decrease) in the asset price by 1 point. A delta of 0.5 corresponds to an at-the-money call. For a deep-in-the money call, the variation in the asset price by one unit implies an equivalent variation in the option price. The delta of an out-of the money call is almost zero and of a deep-in the money call is almost one. The following graphics are drawn according to the data in the preceding tables. 25

26 Figure 1 The call values by underlying asset price and time to maturity Parameters K=100 r=8% b=10%!=25% 25 BAW-Call S : Underlying asset price T : Time to maturity Since the put delta lies in the interval minus one and zero, a rise in the asset price implies a fall in the put price and vice-versa. The delta is 1/2 for an-at-the money put,-1 for a-deep-in the money put and zero for a-deep-out of the money put. Using the data in the preceding tables, the following figure shows the put price as a function of the asset price. The call delta is often assimilated to the hedge ratio. Since the underlying asset delta is one and that of an-at the money call is 0.5, the hedge ratio is 1/0.5 or 2/1. Hence, we need two at-the-money calls to hedge the sale of the underlying asset. The delta corresponds also to the probability that the option finishes in the money at the option's maturity date. Hence, when the delta is 0.6 for a call ( 0.6 for a put), this means that there is 60% chance that the option finishes in the money at maturity. Note that the delta is calculated in practice using the observed volatility or the implied volatility. 26

27 Figure 2 The put values by underlying asset price and time to maturity Parameters K=100 r=8% b=10%!=25% BAW-Put S : Underlying asset price T : Time to maturity Figure 3 The BAW call's delta Parameters: K=100 r=8% b=10%!=25% Call-Delta S : Underlying asset price T : Time to maturity 27

28 Figure 4 The BAW put's delta Parameters K=100 r=8% b=10%!=25% Put-Delta S : Underlying asset price T : Time to maturity Delta-neutral hedging needs the adjustment of the option position according to the variations in the delta. When buying or selling a call (put), the investor must sell or buy (buy or sell) delta units of the underlying asset represent a hedged portfolio. In practice, the hedged portfolio is adjusted nearly continuously to account for the variations in the delta's value. An initially hedged position must be rebalanced by buying and selling the underlying asset as a function of the variations in the delta through time. The delta changes as the volatility, the interest rate and the time to maturity are modified. The following table shows how to adjust a hedged portfolio in order to preserve main characteristics of delta-neutral strategies. 28

29 Table 13 Adjustment of the hedged portfolio as a function of the underlying asset price Options Asset price rises : S! Asset price falls : S! Delta hedging Long a Delta increases : short Call more S Short a Delta increases : buy Call more S Long a Put Delta decreases : sell more S Short a Put Delta decreases : buy more S Delta hedging Delta decreases : buy more S Delta decreases : sell more S Delta increases : buy more S Delta increases : sell more S 29

30 Table 14 Adjustment of a hedged position when the volatility changes Adjustment of a hedged position Options Volatility increases Volatility decreases Long a Call -in-the-money -at-the-money -out-of-the money Delta increases : buy more S Delta non adjusted Delta decreases : sell more S Delta increases : sell more S Delta non adjusted Delta decreases : buy more S Short a Call -in-the-money -at-the-money -out-of-the money Delta decreases : resell of S Delta non adjusted Delta increases : buy more S Delta increases : buy more S Delta non adjusted Delta decreases : sell more S Long a Put -in-the-money -at-the-money -out-of-the money Short a Put -in-the-money -at-the-money -out-of-the money Delta decreases : resell of S Delta non adjusted Delta increases : buy more S Delta increases : buy more S Delta non adjusted Delta decreases : sell more S Delta increases : buy more S Delta non adjusted Delta decreases : sell more S Delta increases : sell more S Delta non adjusted Delta decreases : buy more S Adjustment of a hedged position when the volatility changes It is important to note that delta-neutral hedging strategies do not protect completely the option position against the variations in the volatility parameter. The adjustment of the position when the volatility changes can be done as explained in the following table. 30

31 It is also important to note that delta-neutral hedging does not protect the option position against the variations in the time remaining to maturity. The adjustment of the position when the time to maturity changes can be done as explained in the following table. Table 15 Adjustment of a hedged position as a function of time Options Adjustment of the hedged position as a function of time Long a Call -in-the-money -at-the-money -out-of-the money Short a Call -in-the-money -at-the-money -out-of-the money Long a Put -in-the-money -at-the-money -out-of-the money Short a Put -in-the-money -at-the-money -out-of-the money Delta rises : sell more S Delta non modified Delta decreases : buy more S Delta rises : buy more Delta non modified Delta decreases : sell more S Delta rises : buy more S Delta non modified Delta decreases : sell more S Delta rises : sell more S Delta non modified Delta decreases : buy more S The following figure shows the effect of the volatility parameter. When the volatility rises or the time to maturity is lengthened, options which are out become at parity or in the money and viceversa. 31

32 Figure 5 The call s delta and the volatility Parameters 1.2 K=100 r=8% b=10% T= Delta-Call S : Underlying asset price ! : Volatility Note that the deltas are additive. For example, when buying 2 calls having respectively a delta of 0.2 and 0.7, the investor must sell 0.9 units of the underlying asset in a delta-neutral strategy. Option market-makers implement often delta-neutral hedging strategies in order to maintain a nil delta (in monetary unit). When the delta of an option position is positive, this means that the market maker is long the underlying asset. It can be interpreted as a bullish position. If the asset price rises, he makes a profit since he will be able to sell it at a higher price. However, if the asset price decreases, he will loose money since he will sell the underlying asset at a lower price. When the delta is positive, the investor is over-hedged with respect to delta-neutral strategies. When the delta (in monetary unit) is negative, the investor is short the underlying asset. The investor holds a bearish position. If the underlying asset price rises, the investor loses money since he adjusts his position by buying more units of the underlying asset. However, when the asset price falls, he makes a profit since he pays less for the underlying asset. When a portfolio is constructed by buying (selling) the securities and derivative assets vj, the portfolio value is given by : P=n 1 v 1 + n 2 v 2 + n 3 v n j v j 32

33 where nj stands for the numbers of units of the assets bought or sold. The delta's position or its partial derivative with respect to the securities and derivative assets is : ( v1 " ( v2 " ( v3 " ( v j %!position = " n # $ % ' + ( s & 2 $ # ( s % ' + & n 3 $ # ( s % ' n & j $ ' # ( s & or position=n n n n j j Delta-neutral hedging is convenient for an investor who does not have prior expectations about the market direction. However, if the investor expects a rising market, he can have a positive delta, i.e., long the underlying asset, so he can sell at a higher price when the market effectively rises. If the investor expects a down market, he can have a negative delta, i.e., short the underlying asset, so he can buy it at a lower price when the market effectively goes down. Monitoring and managing the gamma The gamma is given by the second derivative of the option price with respect to the underlying asset price. A high value of gamma (positive or negative) shows a higher risk for an option position. The gamma shows what the option gains (loses) in delta when the underlying asset price rises (falls). For example, when the option's gamma is 4 and the option's delta is zero, an increase by 1 point in the underlying asset price allows the option to gain 4 points in delta, i.e. the delta becomes equal to 4. When the delta is constant, the gamma is zero. The gamma varies when the market conditions change. The following graphics show the relationship between the call's gamma, the underlying asset price, the time to maturity and the option's volatility. The gamma is highest for an at the money option and decreases either side when the option gets in or out of the money. The gamma of an at the money option rises significantly when the volatility decreases and the option approaches its maturity date. When the gamma is positive, an increase in the underlying asset price yields a higher delta. The adjustment of the position entails the sale of more units of the underlying asset. When the asset price falls, the delta decreases and the adjustment of the option position requires the purchase of more units of the underlying asset. Since 33

34 the adjustment is made in the same direction as the changes in the market direction, the monitoring of an option position with a positive gamma is easily done. Figure 6 The Gamma and time to maturity Parameters K=100 r=8% b=10%!=25% Call-Gamma S : Underlying asset price T : Time to maturity Figure 7 The Gamma and volatility Parameters K=100 r=8% b=10% T=0.5 Gamma-Call S : Underlying asset price ! : Volatility 34

35 When the gamma is negative, an increase in the underlying asset price reduces the delta. The adjustment of a delta neutral position implies the purchase of more units of the underlying asset. When the asset price decreases, the delta rises. The adjustment of the option position requires the sale of more units of the underlying asset. Hence, the adjustment of the position implies a rebalancing against the market direction which produces some losses. Table 16 The adjustment of a hedged position and the Gamma Position Gamma Adjustment of the hedged position and the Delta Long options positive Market up : sell more S Market down :buy more S Short options negative Market up : buy more S Market down :sell Effect on the position easy adjustment yields profits difficult adjustment yields losses more S In general, the option gamma is a decreasing function of the time to maturity. The longer the time to maturity, the weaker the gamma and vice-versa. When the option approaches its maturity date, the gamma varies significantly. Table 16 The variations in gamma and the time to maturity Longer Short Near maturity maturity maturity Gamma low high high for at-the money options and very low for out of the money options Effect on the option position low easy adjustment of an option position the delta is very sensitive to the asset price, the gamma is used with care The variations in the gamma for in, at and out of the money options is explained below in the following table. 35

36 Gamma Effect on an option position Table 17 Effect of the Gamma on an option position Out of the At the money money options options gamma is near zero weak the gamma is high for a shorter maturity, it is stable for a longer maturity Gamma is funda-mental to manage the option position In the money options near zero for a near maturity weak The management of an option position with a positive gamma is simple. When the market rises, the investor becomes long and must sell some quantity of the underlying asset to re-establish his deltaneutral position. When the underlying asset decreases, the investor becomes short and must buy more units of the underlying asset to re-establish his delta-neutral position. This yields a profit. The management of an option position with a negative gamma is more difficult when the underlying asset's volatility is high. When the market rises, the investor becomes short and must buy some quantity of the underlying asset to re-establish his delta-neutral position. This produces a loss. When the underlying asset decreases, the investor becomes long and must sell more units of the underlying asset to re-establish his delta-neutral position. This yields a loss. Examples If =30. an increase in the asset price by $1 yields an increase of the delta and produces a gain of $30. If = 30. an increase in the asset price by $1 yields a decrease in the delta and produces a loss of $30. When the underlying asset price S=145, =97 and =32, an increase in S by $1 (S=146) gives a new delta; =, or (97+32), 129. However, a decrease in S by 1$ gives a new delta; 36

37 =, or 65. The delta-neutral strategy implies the sale of 65 units of the underlying asset. When S=360, =135,! = 26, an increase in S by $1 gives a new delta; =, or 109, (135 26). A decrease in S by $1 gives a new delta, =, or (135+26), 161. The delta-neutral strategy implies the sale of 161 units of the underlying asset. It is possible to have a negative delta and a negative gamma. For example, when S=378, = 167, = 8, an increase in S by $1 gives a new delta; =, or ( 167 8), 175. The delta-neutral strategy implies the purchase of 175 units of the underlying asset. A decrease in S by $1 yields a delta, =, or (-167+8), 159. The delta-neutral strategy implies the purchase of 159 units of the underlying asset. In general, it is not a good strategy to adopt a positive gamma since the option position looses from its theta when the market is not volatile. In this context, it is better to have a negative gamma. However, when the market is volatile, a position with a positive gamma allows profits since the adjustment needs buying the underlying asset when the market falls and selling it when the market rises. The gamma of an option position with several assets is : of the position=!" position #!" 1 & #!" 2 & #!" j & = n1% ( + n 2 % ( n j% (! S $! s ' $! s ' $! s ' For delta-neutral strategies, a positive gamma allows profits when market conditions change rapidly and a negative gamma produces losses in the same context. Monitoring and managing the theta The theta is given by the option partial derivative with respect to the remaining time to maturity. As the maturity date approaches, the option looses value. An option is a wasting asset. The theta is often expressed as a function of the number of points lost each day. A theta of 0.4, means that the option looses $0.4 in value when the maturity date is reduced by one day. In general, the gamma and the theta are of opposite signs. A high positive gamma is associated with a high negative theta and vice-versa. 37

38 By analogy with the gamma, as a high gamma is an indicator of a high risk associated with the underlying asset price, a high theta is an indicator of a high exposure to the passage of time. An at the money option with a short maturity looses value much more than a corresponding option on a longer term. The theta of an at the money option is often higher than that of an equivalent in the money or out of the money option having the same maturity date. Figure 8 The Call price and the time to maturity Parameters K=100 r=8% b=10%!=25% 25 BAW-Call S : Underlying asset price T : Time to maturity 38

39 Figure 9 The Put price and the time to maturity Parameters K=100 r=8% b=10%!=25% BAW-Put S : Underlying asset price T : Time to maturity Figure 10 The Theta Parameters 25 K=100 r=8% b=10%!=25% 20 Call-Theta S : Underlying asset price T : Time to maturity The option buyer looses the theta value and the option writer "gains" the theta value. 39

40 Lost in time value Effect on an option position Table 18 The option value and the theta Longer Shorter Near maturity maturity maturity low high very high needs a passive monitoring profit for the seller, loss for the buyer profit for the seller, loss for the buyer Examples A theta of $1 000 means that the option buyer pays $1 000 each day for the holding of an option position. This amount profits to the option writer. The theta remains until the last day of trading. When a position shows a positive gamma, its theta is negative. In general a high gamma induces a high theta and vice-versa. For example, when = the theta may be $ , i.e. a loss of $ each day for the option position. This loss is compensated by the profits on the positive gamma since the adjustments of the position imply selling buying) more units of the underlying asset when the market rises (falls). When the = for an option position, the theta may be i.e. a gain of $ each day. However, the position implies a loss on the underlying asset since the adjustments are done against the market direction when the gamma is negative. The theta of an option position is : of the "! ( v1 % "! ( v3 % position= n1$ ' + n # 2 $ ' + ( t & # ( t & Monitoring and managing the vega n 3 "! ( v $ # ( t 3 % ' n & j "! ( v $ # ( t The vega is given by the option derivative with respect to the volatility parameter. It shows the induced variation in the option price when the volatility varies by 1%. The vega is always positive for call and put options since the option price is an increasing function of the volatility parameter. A vega of 0.6 means that an increase in the volatility by 1% increases the option price by 0.6. For a fixed time to maturity, the vega of an at the j % ' & 40

41 money option is higher than that of an in the money or out of the money option. Figure 11 The Call's price and the volatility Parameters K=100 r=8% b=10% T= BAW-Call S : Underlying asset price ! : Volatility Figure 12 The Put 's price and the volatility Parameters K=100 r=8% b=10% T= BAW-Put S : Underlying asset price ! : Volatility 41

42 Figure 13 The vega Parameters K=100 r=8% b=10%!=25% 30 Vega-Call S : Underlying asset price T : Time to maturity The graphic 2.17 shows a decrease in the vega when maturity is shortened, i.e. a longer term option is more sensitive to the volatility parameter than an otherwise identical shorter term option. Since all the option pricing parameters are observable, except the volatility, buying (selling) options is equivalent to buying (selling) the volatility. When monitoring an option position, a trade-off must be realised between the gamma and the vega. Buying options and hence having a positive gamma is easy to manage. However, when the implied volatility falls, the investor must adopt one of the two following strategies. He can either preserve a positive gamma if he thinks that the loss due to a decrease in volatility will be compensated by adjusting the gamma in the market direction. He can sell the volatility (options) and re-establish a position with a negative gamma. In this case, the losses due to the adjustments of the delta must be sufficient to compensate for the decrease in volatility. 42

43 Table 19 Effect of the volatility on a portfolio of options Long options Profit (loss) when the Long volatility volatility rises (falls) Short options Loss (profit) when the Short volatility volatility rises (falls) Table 20 Effect of the vega with respect to time to maturity Longer Shorter Near maturity maturity maturity Vega high low the implicit volatility is affected by factors other than time Effect on very little the implicit volatility is the sensitive sensitivity affected by factors other position to to than time volatility volatility The impact of the vega on at the money option is highest and is summarised in the following table. Vega Table 21 Effect of the vega on the option price Out-of the At the money In the money money options Options options depends on the for a given depends on time to maturity maturity, the the time to vega is higher maturity Option low High low position When the vega is $500, this means that a rise in the volatility by 1% produces 43