Exponential Hedging with Optimal Stopping and Application to ESO Valuation

Transcription

1 Exponenial Hedging wih Opimal Sopping and Applicaion o ESO Valuaion Tim Leung Ronnie Sircar November 3, 2008 Absrac We sudy he problem of hedging early exercise American) opions wih respec o exponenial uiliy wihin a general incomplee marke model. This leads us o consruc a dualiy formula involving relaive enropy minimizaion and opimal sopping. We furher consider claims wih muliple exercises, and saic-dynamic hedges of American claims wih oher European and American opions. The problem is imporan for accurae valuaion of Employee Sock Opions ESOs), and we demonsrae his in a sandard diffusion model. We find ha incorporaing saic hedges wih marke-raded opions induces he holder o delay exercises, and increases he ESO cos o he firm. 1 Inroducion Many applicaions of Financial Mahemaics involve he opimizaion of expeced uiliy of wealh, combined wih one or many opimal sopping decisions, over a finie ime horizon. Typical of hese are hedging, indifference valuaion, or asse managemen of porfolios conaining American early exercise) derivaive securiies. A he same ime, he exponenial uiliy funcion has become popular because of is basis for he enropic convex risk measure, which has convenien dynamic and analyic properies ha make i amenable for compuaions. A very general dualiy heory wih problems of relaive enropy minimizaion has been developed by Frielli 2000); Rouge and El Karoui 2000); Delbaen e al. 2002); Kabanov and Sricker 2002) and Becherer 2001), among ohers, for exponenial hedging of claims wih no early exercise feaure. Our goal here is o develop he analogous dualiy formula for exponenial hedging of American claims, under minimal assumpions on he underlying price process. This is hen applied o he problem of employee sock opion ESO) valuaion, illusraed wihin a sandard diffusion-based financial model. A key feaure of he problems we are ineresed in is he finie ime horizon ha corresponds naurally o he expiraion dae of he American claim. Allowing for cash flows a opimally chosen) prior imes requires specificaion of how he payoff of he exercised American opion is invesed hereafer. Here he ime-consisency of exponenial uiliy or he self-generaing propery of is associaed Meron funcion) is crucial for racabiliy. We refer o Musiela and Zariphopoulou 2007) for a discussion and alernaive specificaions. There is, of course, an enormous lieraure on uiliy maximizaion problems of mixed opimal sopping/conrol ype, over an infinie horizon, wih uiliy funcions defined on R +, where sopping represens he decision o ge ou or reire from invesing. We refer o Karazas and Wang 2000) for deails and references. Super-hedging of American claims under porfolio consrains was sudied in Karazas and Kou 1998). Deparmen of Applied Mahemaics and Saisics, Johns Hopkins Universiy, Balimore MD 21218; imleung@jhu.edu. Work suppored by NSF gran DMS and a Charloe Elizabeh Procer Fellowship. ORFE Deparmen, Princeon Universiy, E-Quad, Princeon NJ 08544; sircar@princeon.edu. Work suppored by NSF grans DMS and DMS

2 Valuaion of employee sock opions has become imporan since he Financial Accouning Sandards Board FASB) required heir inclusion in firms accouning saemens since 2005, where previously hey had been exemped. The mehod of valuaion is a conroversial opic, wih a radeoff beween simpliciy and realism, coupled wih a poliical lobby o push for whaever proposal resuls in booking a lower value. The major concern in ESO valuaion ha moves he problem away from sandard no-arbirage pricing mehods is he hedging resricion: employees canno ake shor posiions in heir firm s sock. However here are oher possibiliies ha can be aken ino accoun. In he approach we analyze here, he employee is able o dynamically inves in he marke index, and ake saic posiions in marke-raded vanilla opions wrien on he firm s sock. This is a naural advance from earlier uiliy-based approaches using jus cerainy equivalen wih no marke rading for example, Huddar 1994)), hen incorporaing dynamic hedging wih a correlaed index Henderson 2005); Grasselli 2005); Leung and Sircar 2009); Carpener e al. 2008)), and now incorporaing marke opions daa for more accurae calibraion. However, high ransacion coss discourage frequen opion rades, so recen work for example Carr e al. 1998); İlhan and Sircar 2005)) has focused on saic hedging wih opions, which involves purchasing a porfolio of sandard opions a iniiaion and no rades aferwards. The combinaion of a dynamic rading sraegy and saic posiions, which is referred o as saic-dynamic hedging, leads us o sudy how marke prices of raded pu opions affec he employee s opimal exercising sraegy. As shown in Secions 4.1 and 4.2, he opimal saic posiion is found from he Fenchel-Legendre ransform of he employee s indifference price as a funcion of he number of pus, evaluaed a he marke price. In Proposiions 5.2 and 5.3, we find ha saic hedges wih pu opions induce he employee o delay exercises, which in general leads o a higher ESO cos. The paper is srucured as follows. In Secion 2, we invesigae he case of single American claim in a semimaringale framework. This allows us, in Secion 3, o exend our resuls o he case of American claims wih muliple exercising righs. In Secion 4, we sudy incorporae European and American pu opions ino he invesor s hedging sraegy. We solve for he opimal exercising sraegy along wih he opimal saic hedge. In Secion 5, we apply our mehodology o ESO valuaion in a diffusion framework. We sudy early exercises hrough he examinaion of he employee s opimal exercise boundaries, and illusrae heir impac on he ESO cos. 2 Dynamic Hedging of American Claims wih Single Exercise We begin wih he problem of exponenial hedging of an American opion using he underlying asse, wihin a general incomplee marke model. We derive he dual problem in Proposiion 2.4. o characerize he opimal exercise ime, we inroduce he indifference price in Secion 2.3. Finally, we give resuls on large and small risk aversion limis as well as quaniy asympoics in Secion Noaion and Assumpions In he background, we fix an invesmen horizon wih a finie erminal ime T, which is chosen o coincide wih he expiraion dae of all securiies in our model. We work on a probabiliy space Ω, F, P ) wih a filraion F = F ) 0 T, which saisfies he usual condiions of righ coninuiy and compleeness. Hence, all processes are assumed o have righ coninuous pahs wih lef limis. We adop he shorhand for aking condiional expecaions: IE = IE F. The basic rading asses consis of a riskless asse e.g. he bank accoun) ha pays ineres a consan rae r 0, and a risky asse e.g. he marke index) whose discouned price process is a non-negaive F-locally bounded semimaringale S ) 0 T. We denoe by X ) 0 T he discouned rading wealh process wih a dynamic rading sraegy θ which represens he number of shares held a ime. The se of admissible sraegies is defined in 2.1) below. Wih iniial capial X a 2

3 ime [0, T ], he discouned wealh a a laer ime u [, T ] is given by X u = X + G,u θ), wih G,u θ) := u θ s ds s. The sochasic inegral G,u θ) is he discouned capial gains or losses from rading wih sraegy θ from ime o u. We wrie Gθ) for he process G 0, θ)) 0 T. We denoe by T he se of all sopping imes wih respec o F aking values in [0, T ]. This will be he collecion of all admissible exercise imes for American claims considered in his paper. For any sopping imes s, u T wih s u, we se T s,u := τ T : s τ u. The se of absoluely coninuous equivalen) local maringale measures for S wih respec o P are denoed by P a P ) := Q P S is a local Q, F)-maringale, P e P ) := Q P S is a local Q, F)-maringale. For any measure Q, he relaive enropy of Q wih respec o P is given by IE Q log dq HQ P ) := dp, Q P, +, oherwise. We inroduce P f P ), he se of measures in P a P ) wih finie relaive enropy wih respec o P, and define he se of admissible sraegies as ΘP ) := θ LS) Gθ) is a Q, F) maringale for all Q P f P ), 2.1) where LS) is he se of F-predicable S-inegrable R-valued processes. For noaional simpliciy, we wrie respecively P a, P e, P f, Θ for P a P ), P e P ), P f P ), ΘP ) when no ambiguiy arises. When we specify he rading horizon [s, u], we wrie Θ s,u o denoe he se of admissible sraegies over he period [s, u]. Throughou, we assume ha here exiss some equivalen local maringale measure wih finie relaive enropy wih respec o P. Assumpion 2.1 P f P e. By Theorem 2.1 and 2.2 of Frielli 2000), his assumpion ensures he exisence of a unique measure Q E P f P e ha minimizes he relaive enropy wih respec o P over all measures in P f, ha is, Q E = arg min Q P f HQ P ). This measure is called he minimal enropy maringale measure MEMM). By Theorem 2.3 of Frielli 2000) and Theorem 2.1 of Kabanov and Sricker 2002), i has a densiy of he form dq E dp = c E exp G 0,T θ E ) ), 2.2) for some θ E Θ, and log c E = HQ E P ) <. We can derive from 2.2) he densiy process of Q E wih respec o P dq Z E E := IE = c E IE e G 0,T θ E ). 2.3) dp In general, for any wo measures Q a, Q b such ha Q a Q b, we wrie he densiy process of Q a wih respec o Q b as Z Qa,Q b dq a := IE Qb dq b, [0, T ]. I is well-known ha exponenial uiliy opimizaion is closely linked o he minimizaion of relaive enropy. The dynamic version of he problem involves minimizing he condiional relaive enropy wih respec o P, so we sae here is definiion in our noaion. 3

4 Definiion 2.2 For any [0, T ] and Q P f P ), he condiional relaive enropy of Q wih respec o P a ime is H T Q P ) := IE Q log ZQ,P T Z Q,P. 2.4) Remark 2.3 For any [0, T ] and Q P f P ), he random variable log Z Q,P is Q-inegrable see Lemma 3.3 of Delbaen e al. 2002)), so he condiional relaive enropy is well-defined. As is well-known, an applicaion of Jensen s Inequaliy yields H T Q P ) 0. By Proposiion 4.1 of Kabanov and Sricker 2002), he MEMM Q E also minimizes he condiional relaive enropy H T τ Q P ) a any τ T. Tha is, ess inf Q P f P ) HT τ Q P ) = Hτ T Q E P ), τ T. 2.5) Throughou, we consider a risk-averse invesor whose risk preferences are described by he exponenial uiliy funcion U : R R defined by Ux) = e γx, x R, where γ > 0 is he coefficien of absolue risk aversion. Precisely, Ux) is he invesor s uiliy for having discouned) wealh x a ime T. As discussed in he inroducion, exponenial uiliy is a convenien convex bu no coheren) risk measure up o a log and change in sign), as sudied in Föllmer and Schied 2004), wih good dynamic ime-consisency) properies Klöppel and Schweizer 2006)), and i is he only nonlinear) uiliy funcion of his kind Cheridio and Kupper 2005)). 2.2 Exponenial Hedging of an American Opion In order o formulae he problem of hedging an American claim, we firs need o consider an invesmen problem in which he risk-averse invesor wihou any claims dynamically rades in he riskless and risky asses hroughou he horizon [0, T ]. This is a well-sudied problem firs inroduced by Meron 1969). For an invesor wih saring wealh X a ime [0, T ], his maximal expeced uiliy from erminal wealh is M, X ) := ess sup θ Θ,T IE UX T ). 2.6) We consider an American claim, denoed by A, wih a payoff process A ) 0 T which we assume o be bounded and adaped o F. We will use he boundedness of A in he proof of Proposiion 2.4. More relaxed assumpions on he payoff for European-syle coningen claims wih no early exercise) can be found in Becherer 2001) and Delbaen e al. 2002), bu hese sill exclude European call opions wih geomeric Brownian moion. We do no aemp o relax he assumpions on he American claim payoff A in his paper. The holder of he claim maximizes his expeced uiliy by choosing he opimal exercise policy and dynamic rading sraegy. Upon exercise, he immediaely reinvess he conrac proceeds, if any, ino he rading porfolio, and coninues o rade up o ime T. This means ha he invesor will face he Meron problem afer exercising he claim. A ime [0, T ], he holder s value funcion is given by V, X ; A) := ess sup ess sup IE M τ, X τ + A τ ). 2.7) τ T,T θ Θ,τ The value funcion in 2.7) is he primal problem, for which we will derive he dual in Proposiion 2.4. For exponenial uiliy, Delbaen e al. 2002) derive a dualiy resul for invesmen including a European-syle coningen claim wih no early exercise), which involves opimizing he expeced 4

5 payoff over a se of maringale measures bu penalized by an enropy disance from he hisorical measure. Since our opimal hedging problem includes an American claim wih he possibiliy of early exercise), he dual value funcion also involves finding he holder s opimal exercise ime in addiion o he opimal measure. The nex proposiion is he main resul in his secion. Proposiion 2.4 The dual of he value funcion is given by V, X ; A) = UX ) exp ess sup ess inf IE Q γa τ + H τ Q P ) + IE Q τ T,T Q P f P ) HT τ Q E P ) )). 2.8) In developing his dualiy resul, we shall make use of some useful properies of he Meron problem Properies of he Meron Funcion Firs, we observe a separaion of variables and a dualiy formula for he Meron funcion. Proposiion 2.5 For an invesor wih saring wealh X τ a τ T, he Meron funcion admis a separaion of variables Mτ, X τ ) = UX τ ) IE τ e G τ,t θ E ), 2.9) and is dual is given by ) Mτ, X τ ) = UX τ ) exp ess inf Q P f P ) HT τ Q P ). 2.10) Moreover, he opimal rading sraegy is θ := θe γ, and he opimal measure in 2.10) is QE. Proof. Recall ha X T = X τ + G τ,t θ), and so by 2.6), Mτ, X τ ) = ess sup IE τ UX T ) = UX τ ) ess inf IE τ e γg τ,t θ). 2.11) θ Θ τ,t θ Θ τ,t For any θ Θ τ,t, we can apply a change of measure from P o Q E using he densiy in 2.3) o obain IE τ e γg τ,t θ) = IE τ e G τ,t θ E ) IE QE τ e G τ,t γθ+θ E ) IE τ e G τ,t θ E ), where he las inequaliy follows from Jensen s inequaliy and ha Gγθ + θ E ) is a Q E -maringale. The inequaliy becomes an equaliy when θ = θe γ, so he infimum in 2.11) is aained and 2.9) follows. On he oher hand, we observe from 2.5) ha he righ-hand side of 2.10) is UX τ ) exp Hτ T Q E P )). By he definiions of Z E in 2.3) and condiional relaive enropy in 2.4), we can wrie H T τ Q E P ) = IE QE τ e G 0,T θ E ) log IE τ e G 0,T θ E) = log IE τ e G τ,t θ E), 2.12) where we have used he fac ha Gθ E ) is a Q E -maringale. Hence, by exponeniaing 2.12) and comparing i wih 2.9), we obain 2.10). Nex, we show ha he Meron funcion saisfies he following dynamic programming propery. I is called he self-generaing condiion in Musiela and Zariphopoulou 2007), and horizon-unbiased condiion in Henderson and Hobson 2007). 5

6 Proposiion 2.6 Wih saring wealh X a [0, T ], he Meron funcion saisfies M, X ) = ess sup IE Mτ, X τ ), τ T,T. θ Θ,τ Proof. Recall ha X τ = X + G,τ θ). By he separaion of variables formula 2.9) and a change of measure from P o Q E, we have ess sup IE Mτ, X τ ) = ess inf IE UX + G,τ θ)) IE τ e G τ,t θ E) θ Θ,τ θ Θ,τ = UX ) IE e G,T θ E) ess inf IE QE e γg,τ θ) e G,τ θ E ) θ Θ,τ = M, X ) ess inf IE QE e G,τ γθ+θ E ). 2.13) θ Θ,τ Nex, applying Jensen s inequaliy and he fac ha Gγθ + θ E ) is a Q E -maringale, we obain for every θ Θ,τ ha IE QE e G,τ γθ+θ E ) 1, and equaliy is aained a θ = θe γ. This implies ha θ aains he infimum in 2.13) and he infimum is 1. This complees he proof Proof of Proposiion 2.4 To prove Proposiion 2.4, we shall also need he following lemma regarding enropy minimizaion. Lemma 2.7 For any [0, T ], and τ T,τ, we have ess inf Q P f P ) HT Q P ) = ess inf H τ Q P ) + IE Q Q P f P ) ) ess inf Q P f P ) HT τ Q P ) Proof. The definiion of condiional relaive enropy gives he simple equaliy. 2.14) H T Q P ) = H τ Q P ) + IE Q H T τ Q P ). Taking infimum on boh sides, we easily deduce he inequaliy H τ Q P ) + IE Q ess inf Q P f P ) HT Q P ) ess inf Q P f P ) ess inf Q P f P ) HT τ Q P ) ). 2.15) Nex, on he righ-hand side of 2.14), we observe from 2.5) ha Q E solves he inner minimizaion. The ouer minimizaion depends on he densiy process Z Q,P only over he sochasic inerval [, τ], so i is unchanged if we minimize over he se P τ := Q P f : Z Q,P u = Z E u, for u τ. Hence, we wrie he righ-hand side of 2.14) ess inf H τ Q P ) + IE Q H T τ Q E P ) ) = ess inf Q P τ Q P τ H T Q P ) ess inf Q P f P ) HT Q P ). This gives he reverse inequaliy o 2.15) and hus complees he proof. To complee he proof of Proposiion 2.4, for any [0, T ], and τ T,τ, we define he measure P A by dp A dp := c A e γaτ, wih c 1 A = IEe γaτ. 2.16) 6

7 Given ha A is bounded, we have c A 0, ) and P A P. Nex, we wan o show ha Indeed, for a measure Q P, he relaion E Q log dq = E Q log dq dp dp A and 2.16) imply P f P ) = P f P A ). 2.17) + E Q log dp A dp HQ P ) = HQ P A ) + log c A γie Q A τ. For Q P f P ), HQ P ) is finie. Since he las wo erms on he righ-hand side are also finie due o he boundedness of A), we conclude ha HQ P A ) is also finie, and herefore, P f P ) P f P A ). The reverse inclusion can be shown using similar argumens. Furhermore, using and ha Z P A,P u = Z P A,P τ A log ZQ,P ν Z Q,P A s = log ZQ,P ν Zs Q,P for u [τ, T ], we conclude ha, A log ZP,P ν, s, ν T, s ν, 2.18) Z P A,P s H T τ Q P ) = H T τ Q P A ). 2.19) We apply he dualiy formula 2.10) wih saring wealh X τ + A τ ) along wih a change of measure from P o P A, and hen use 2.17) and 2.19) o wrie he value funcion ) V, X ; A) = ess sup ess sup IE UX τ ) e γaτ exp ess inf τ T,T Θ,τ P ) Q P f P ) HT τ Q P ) ) = ess sup c 1 A A ZP,P ess sup IE P A τ T,T Θ,τ P A ) UX τ ) exp ess inf H Q P f P A τ T Q P A ) ). Nex, applying Proposiion 2.6 and Lemma 2.7 wih he prior measure being P A, we have ) V, X ; A) = ess sup c 1 τ T,T = ess sup c 1 τ T,T = ess sup c 1 τ T,T A A ZP,P A A ZP,P A A ZP,P UX ) exp UX ) exp UX ) exp ess inf Q P f P A ) ess inf Q P f P A ) H T Q P A ) H τ Q P A ) + IE Q 2.20) ess inf H Q P f P A τ T Q P A ) ) ess inf H Q P f P A τ Q P A ) + IE Q HT τ Q E P )) ), ) )) 2.21) where he las equaliy follows from 2.17) and 2.19) and ha Q E is he enropy-minimizing measure. Lasly, we use definiion 2.16) and he equaliy 2.18) o wrie he condiional relaive enropy of Q wih respec o P A in erms of is enropy wih respec o P H τ Q P A ) = IE Q log ZQ,P τ Z Q,P + γa τ Subsiuing his ino 2.21), we immediaely obain 2.8). 7 log c A + log Z P A,P.

8 2.3 The Indifference Price I is ofen beer for inuiive purposes o characerize opimal exercising sraegies in erms of indifference prices, which we inroduce nex. A holder s indifference price of an American claim is defined as he reducion in wealh such ha he holder s value funcion V is he same as he Meron funcion M from invesmen wihou he claim. For ime [0, T ], denoe p p A) as he indifference price of claim A. I is defined by he equaion Nex, we give some general expressions for he indifference price. Proposiion 2.8 The indifference price can be wrien as and p = ess sup τ T,T ess inf Q P f P ) p = 1 γ log ess sup τ T,T V, X p ; A) = M, X ). 2.22) ess sup IE QE θ Θ,τ e γg,τ θ)+a τ ) ), 2.23) IE Q A τ + 1 γ Hτ Q P ) + 1 ) γ IEQ HT τ Q E P ) 1 γ HT Q E P ). 2.24) If P f P ) = P f Q E ), hen he las represenaion can be simplified as p = ess sup ess inf IE Q τ T,T Q P f Q E A τ + 1 ) ) γ Hτ Q Q E ). 2.25) Proof. Applying he separaion of variables formula 2.9) and a change of measure from P o Q E using he densiy in 2.3), we wrie V, X ; A) = ess sup ess inf IE UX + G,τ θ) + γa τ ) IE τ e G τ,t θ E) τ T,T θ Θ,τ = UX ) IE e G,T θ E) ess sup ess sup IE QE e γa τ e γg,τ θ) e G,τ θ E ) τ T,T θ Θ,τ e γaτ e γg,τ θ+ θeγ ) = M, X ) ess sup τ T,T = M, X ) ess sup τ T,T ess sup IE QE θ Θ,τ ess sup IE QE U A τ + G,τ θ)). 2.26) θ Θ,τ Since i follows from 2.22) ha V, X ; A) M, X ) = Up ), 2.27) subsiuion of 2.26) ino 2.27) yields 2.23). The second expression 2.24) follows from he dual 2.8) and he definiion 2.22). To show 2.25), we use he simple equaliy H τ Q P ) = H τ Q Q E ) + IE Q log ZE τ Z E, Q P f P ), 2.28) where, by 2.2), he las erm can be wrien as IE Q log ZE τ Z E = IE Q HT τ Q E P ) + H T Q E P ) + IE Q G,τ θ E ). 2.29) The assumpion P f P ) = P f Q E ) implies ha he las erm is zero. To complee he proof, we subsiue 2.28) and 2.29) ino 2.24). Remark 2.9 A sufficien condiion for P f P ) = P f Q E ) is dqe dp İlhan and Sircar 2005). In general, we have P f P ) P f Q E ). L2 P ) as shown in Lemma 2 of 8

9 2.4 The Opimal Exercise Time In his secion, we provide a characerizaion for he holder s opimal exercise ime by analyzing he indifference price, which is expressed in 2.23) and 2.24) as wo join sochasic conrol and opimal sopping problems. Firs, we re-wrie 2.23) as Up ) = ess sup ess sup IE QE UG,τ θ) + A τ ), 2.30) τ T,T θ Θ,τ which can be regarded as a special example of a cooperaive sochasic game. The second expression yields a non-cooperaive sochasic game: p = ess sup τ T,T ess inf Q P f P ) IEQ Aτ + l Q,γ,τ, 2.31) where he penaly erm l Q,γ,τ := 1 γ [H Tτ Q E P ) + log ZQ,P τ Z Q,P ] H T Q E P ). 2.32) Their srucures are very similar o he sochasic games sudied in Karazas and Zamfirescu 2005), and we will apply some of heir resuls here. In he heory of opimal sopping, i is common o require quasi-lef-coninuiy for he associaed processes see, among ohers, El Karoui 1981), Shiryaev 1978), and Thompson 1971)). This assumpion is quie general, and i allows for he processes commonly used in finance, including diffusion processes and Lévy processes. In fac, all sandard Markov processes are quasi-lef-coninuous. Guasoni 2002) sudies opimal invesmen wih quasi-lef-coninuous asse prices subjec o ransacion coss. Definiion 2.10 A process Y ) 0 T is P -quasi-lef-coninuous if for any increasing sequence of sopping imes τ n ) n N T and wih τ := lim n τ n T, we have lim n Y τn = Y τ, P a.s. Remark 2.11 If Y is P -quasi-lef-coninuous, hen i is also quasi-lef-coninuous wih respec o any Q P f P ). This also means ha Y is P f P )-quasi-lef-coninuous in he sense of Definiion 2.9 of Karazas and Zamfirescu 2005). Assumpion 2.12 The processes A ) 0 T, S ) 0 T, and l Q,γ 0, ) 0 T are quasi-lef-coninuous wih respec o every Q P f P ). Proposiion 2.13 For any [0, T ], he opimal exercise ime for 2.23) or 2.30) is given by τ = inf u T : p u = A u 2.33) so ha p = 1 γ log ess sup θ Θ,τ IE QE e γg,τ θ)+a τ )). Proof. Since S is P -quasi-lef-coninuous, so is he process Gθ) for any feasible θ Θ. By Remark 2.11, he process UG θ) + A )) 0 T is Q E -quasi-lef-coninuous. I is also bounded above by zero. Then, by Theorem 2.10 of Karazas and Zamfirescu 2005), we have for any [0, T ] ha wih τ given by 2.33). Up ) = ess sup IE QE θ Θ,τ e γg,τ θ)+a τ ), 9

10 The opimal exercise ime 2.33) is he firs ime when he indifference price equals he payoff from immediae exercise. This is highly inuiive because he indifference price is he minimum amoun of money he holder demands in order o forgo he claim. A he opimal exercise ime, he claim payoff is sufficien o induce he holder o exercise. Turning our aenion o expression 2.31), we wan o show ha he order of choosing he opimal exercise ime via essenial supremum) and opimal measure via essenial infimum) does no aler he problem. Proposiion 2.14 For any [0, T ], define he upper value process by p := ess inf ess sup IE Q A τ + l Q,γ,τ, Q P f P ) τ T,T and he sopping ime τ := inf u T : p u = A u. Then, for any Q P f P ), and τ [, τ ], we have p IE Q p τ + l Q,γ,τ. 2.34) In paricular, when τ = τ, we have he equaliy p = ess inf Q P f P ) IEQ A τ + l Q,γ, τ. 2.35) The proof is given in he Appendix A.1. The las equaliy indicaes ha τ is opimal for he upper value process p. The nex proposiion, which follows easily from Proposiion 2.14, shows ha he upper value process is he same as he indifference price process, and τ is in fac equal o τ. Proposiion 2.15 For any [0, T ], he indifference price and upper value processes are he same, i.e. p = p. Consequenly, he corresponding opimal sopping imes are idenical, i.e. τ = τ. Proof. We always have p p. The preceding proposiion gives he oher direcion: p = ess inf Q P f P ) IEQ A τ + l Q,γ, τ ess sup ess inf Q P f P ) IEQ τ T,T Aτ + l Q,γ,τ = p. 2.5 Risk Aversion and Volume Asympoics We now analyze he effecs of risk aversion and holding volume on indifference prices and he corresponding opimal exercise imes. To his end, le us consider an invesor wih risk aversion parameer γ who holds α > 0 unis of an American claim A, and suppose ha all α unis have o be exercised simulaneously. By definiion 2.7), he holder s value funcion is given by V, X ; αa). The corresponding indifference price, denoed by p α, γ), is defined by he equaion V, X ; αa) = M, X + p α, γ)). The opimal exercise ime is he firs ime ha he indifference price reaches he payoff from exercising all claims: τ α, γ) = inf u T : p u α, γ) = αa u. 2.36) The following resul esablishes monooniciy in he risk-aversion coefficien. Proposiion 2.16 Le γ 2 γ 1 > 0. Then, for any [0, T ] and α 0, we have p γ 2, α) p γ 1, α), and τ α, γ 2 ) τ α, γ 1 ). Tha is, a higher risk aversion implies an earlier opimal exercise ime of he American claims. 10

11 Proof. By definiion 2.32) of he penaly erm, we have ) IE Q lq,γ,τ = γ 1 H τ Q P ) + IE Q HT τ Q E P ) H T Q E P ). I easily follows from Lemma 2.7 ha IE Q lq,γ,τ 0. For γ 2 γ 1 > 0, we have IE Q lq,γ 1,τ IE Q lq,γ 2,τ, and herefore p γ 1, α) p γ 2, α). As he risk aversion parameer γ increases, he indifference price decreases, bu he payoff αa does no depend on γ. This leads o a shorer opimal exercise ime according o 2.36). Nex, o analyze he large risk aversion limi, we recall he sub-hedging price of he American claim A is defined as c := ess inf ess sup IE Q A τ, [0, T ]. Q P ep ) τ T,T See, for example, Karazas and Kou 1998). As an invesor becomes more risk-averse, he price he is willing o pay for A ends o is sub-hedging price. The idea is ha he penaly erm vanishes as risk aversion increases o infiniy. Proposiion 2.17 For any [0, T ], we have lim p α, γ) = αc. γ The proof is given in he Appendix A.2. A consequence of his resul is ha, a he large risk aversion limi, he pricing rule will become linear in he quaniy of claims, and he invesor will exercise all he American claims A a a ime independen of α. As he invesor s risk aversion diminishes o zero, he ends o price he American claim under he MEMM, Q E. This limiing price is he American analogue o Davis-price for European-syle coningen claims see Davis 1997)). Proposiion 2.18 For any [0, T ], we have lim p α, γ) = α ess sup IE QE A τ. γ 0 τ T,T Proof. The proof is a sligh exension of Proposiion of Becherer 2001), and is omied. We observe from 2.24) he volume-scaling propery ha for α > 0, α 1 p α, γ) = p 1, αγ). The simulaneous exercise assumpion is essenial for his propery o hold. As he number of claims held increases, i follows from Proposiion 2.16 ha he average indifference price for holding α unis of A decreases. By 2.36), he opimal exercise ime τ α is he firs ime he average indifference price his he claim payoff, so he holder ends o exercise he claims earlier when he holding volume increases. The risk aversion limis in Proposiions 2.17 and 2.18 can be rewrien as volume limis: p α, γ) p α, γ) lim = c ; lim α α α 0 α = ess sup IE QE A τ. τ T,T Finally, we poin ou ha he indifference price for he claim A lies wihin he no-arbirage price inerval. Tha is, c p 1, γ) sup IE QE τ T,T A τ ess sup ess sup IE Q A τ, Q P e P ) τ T,T where he lef-end is he sub-hedging price and he righ-end is he super-hedging price. The indifference price possibly possesses oher properies of ineres. For insance, Becherer 2001) and Mania and Schweizer 2005) poin ou ha he indifference price for holding β unis of European-syle claims is concave as a funcion of β, and sric concaviy and differeniabiliy wih respec o β is esablished in İlhan e al. 2005). However, in our case wih American claims, i is more complicaed and we do no address i here. 11

12 3 Dynamic Hedging of American Claims wih Muliple Exercises For American claims wih muliple exercise righs, he holder can exercise separaely and has o choose he opimal muliple exercise imes for all claims held. Suppose ha an invesor is dynamically hedging a long posiion in n 2 ineger unis of claim A. For any [0, T ] and i n, we denoe by τ i) he opimal exercise ime of he nex claim when i unis remain unexercised a ime. Afer exercising one claim a ime τ i), he invesor has i 1) unis lef. If he holder exercises muliple unis a he same ime, hen some exercise imes may coincide. Upon exercising any American claims), he invesor immediaely reinvess his conrac proceeds ino his rading porfolio unil ime T. The invesor s value funcion for holding i 2 unis of A a ime is given recursively by V i), X ) = ess sup ess sup IE V i 1) τ i, X τi + A τi ), 3.1) τ i T,T θ Θ,τi wih V 1), X ) = V, X ; A). The opimal exercise imes can be characerized via indifference prices for holding muliple claims, which we define nex. Definiion 3.1 For any [0, T ], he holder s indifference price for holding i n claims wih muliple exercises is defined as he random variable p i) such ha Subsiuing 3.2) ino 3.1), we observe ha he value funcion V i), X p i) ) = M, X ). 3.2) V i), X ) = ess sup ess sup IE V i 1) τ i, X τi + A τi ) τ i T,T θ Θ,τi = ess sup ess sup IE M τ i T,T θ Θ,τi ) τ i, X τi + A τi + p i 1) τ i = V, X ; A + p i 1) ). 3.3) The las equaliy 3.3) provides a crucial connecion beween dynamic hedging for claims wih single exercise and for claims wih muliple exercises. We can derive he dual for V i), X ) and he indifference price p i) by jus replacing A by A + p i 1) in Proposiions 2.4 and 2.8. Nex, we give he following general expressions for he indifference price. Proposiion 3.2 The indifference price p i) can be wrien recursively by p i) = 1 γ log ess sup τ i T,T An alernaive expression for he indifference price is ess sup IE QE e ) γ G,τ i θ)+aτ i +pi 1) τ i ). 3.4) θ Θ,τi p i) = ess sup τ i T,T ess inf Q P f P ) IEQ Aτi + p i 1) τ i + l Q,γ,τ i, 3.5) where l Q,γ,τ i is defined in 2.32). In Proposiion 2.16, we showed ha higher risk aversion reduces he indifference price p α, γ) for American claims wih simulaneous exercise. The same also holds in he case wih muliple exercises. We denoe he indifference price by p i) γ) o indicae he dependence on γ. 12

13 Proposiion 3.3 For any ime [0, T ] and ineger i 1, he indifference price p i) γ) is decreasing wih respec o γ. Proof. As shown in Proposiion 2.16, boh p 1) γ) and IE Q lq,γ,τ for any τ T,T ) decrease wih γ. Suppose he same is rue for p i 1) γ). Then, we have IE Q p i 1) γ) + l Q,γ,τ i decreases wih γ, and herefore p i) γ) is also decreasing wih γ. As a resul, he proposiion holds by inducion. To undersand he impac of holding volume on he exercise imes, we consider wo invesors holding i and i + 1) unis of claims respecively, and compare he opimal exercise imes τ i) and τ i+1). Applying he echniques in Secion 3.3, we have = inf u T : p u i) p i 1) u = A u, 3.6) τ i) and we can address he quesion by showing ha he exercise imes are ordered if and only if he indifference price is a concave funcion of i: p i+1) p i) p i) p i 1). Proposiion 3.4 For [0, T ] and ineger i 1, he opimal exercise imes are ordered in he sense τ i+1) τ i) if and only if he indifference price p i) is a concave funcion of i. Proof. Firs, we define he Snell envelope ˆp i) Q) := ess sup τ T,T Noe ha, p i) = ess inf Q Pf ˆp i) Q-supermaringale. Consequenly, Now, we suppose τ i+1) A + p i 1), i follows ha p i 1) ˆp i) Q) IE Q τ i) IE Q IE Q A τ + p i 1) τ + l Q,γ,τ, for Q P f P ). Q). I is well-known ha he process ˆp i) Q) + l Q,γ 0, ) 0 T is a ˆp i) τ Q) + l Q,γ,τ IE Q p i) τ + l Q,γ,τ. τ i 1). Applying 2.34) of Proposiion 2.14 wih A replaced by p i 1) τ Hence, we have he following inequaliies p i+1) + p i 1) = ess sup τ T,τ i) ess inf + l Q,γ,τ, for τ [, τ i) ], Q P f P ). ess inf Q P f P ) IEQ ess sup Q P f P ) τ T i),τ ess inf Q P f P ) ess sup τ T i),τ ess sup IE Qi) τ T,T = 2p i), IE Q where Q i) = arg min ess sup IE Q Q P f P ) τ T,T herefore he indifference prices are concave in i. p i+1) A τ + p i 1) τ IE Q A τ + p i) τ + l Q,γ,τ + p i 1) A τ + p i 1) + IE Q A τ + p i 1) τ A τ + p i 1) τ + l Qi),γ,τ + l Q,γ,τ p i) τ ) + l Q,γ,τ ) + ˆp i) Q) + ˆp i) Q i) ) + l Q,γ,τ. Hence, we have p i+1) + p i 1) 2p i), and On he oher hand, if he indifference price is a concave funcion of i, hen from he inequaliy p i) p i) p i 1), and he definiion of he opimal exercise in 3.6), we can easily conclude ha τ i+1) τ i). 13

14 4 Saic-Dynamic Hedges for American Claims In addiion o invesing in he riskless and risky asses, he holder of claim A can also reduce risk exposure by purchasing some marke-raded claims. For insance, he holder of an American call opion can remove some downside risk by buying and holding some European or American pus wrien on he same underlying asse. As ransacion coss on derivaive securiies are less negligible han on socks, he saic hedging sraegies considered here involve purchasing a porfolio of marke-raded claims a iniiaion and no rades aferwards. To avoid arbirage, he prices of he marke-raded claims are assumed o lie beween he sub-hedging and super-hedging prices. In his secion, we analyze he combinaion of he saic and dynamic hedging sraegies, called saic-dynamic hedges, for a long posiion in an American claim. An applicaion o ESO valuaion is analyzed in Secion Hedging wih European Opions While holding one uni of American claim A, he invesor purchases β European opions each wih bounded payoff, B F T, expiring a he same dae T for simpliciy. If he invesor exercises claim A a ime τ T, hen immediaely his rading wealh increases by A τ, and only European claims remain. A ha poin, his value funcion is ess sup IE τ UX τ + A τ + G τ,t θ) + βb) = M τ, X τ + A τ + h τ βb)), θ Θ τ,t where h τ βb) is he indifference price for holding β unis of B a ime τ. From his, we define he invesor s value funcion as Ṽ, X ; βb) = ess sup ess sup IE M τ, X τ + A τ + h τ βb)). 4.1) τ T,T θ Θ,τ Proposiion 4.1 For any [0, T ], and β R, he value funcion can be wrien as Ṽ, X ; βb) = V, X ; A + hβb)). 4.2) Therefore, he indifference price for holding claim A and β unis of B is p A + hβb)), and he opimal exercise ime is given by τ β) = inf u T : p u A + hβb)) = A u + h u βb). 4.3) Proof. We observe from 4.1) ha he value funcion, Ṽ, X ; βb), is equivalen o ha wih dynamic hedge bu for a differen claim A + hβb) insead of A. Hence, we have 4.2). This equaliy also allows us o derive he dual of Ṽ and hen he corresponding indifference price, denoed by p A+hβB)), by replacing A τ wih A τ +h τ βb) in Proposiions 2.4 and 2.8 respecively. Then, i easily follows from Proposiion 2.13 ha opimal exercise ime is given by 4.3). If he European claim B coss $π each, hen he invesor s wealh is reduced by he amoun $βπ. Therefore, he value funcion is given by Ṽ, X βπ ; βb), and he indifference price is given by p A + hβb)) βπ. However, for any fixed β, he cos does no affec he opimal exercise ime. The invesor chooses he opimal saic hedge, β, o maximize his value funcion, which urns ou o be equivalen o maximizing he indifference price: β = arg max β Ṽ, X βπ ; βb) = arg max p A + hβb)) βπ. 4.4) β Hence, he opimal quaniy when i exiss) of European opions o purchase is found from he Fenchel-Legendre ransform of he indifference price p A + hβb)) as a funcion of β, evaluaed a he marke price π. The marke price of European pus conrols he opimal saic hedge in 4.4), which indirecly affecs he holder s opimal exercise ime. 14

15 4.2 Hedging wih American Opions The holder of claim A may be resriced o purchasing American opions o form a parial saic hedge. In fac, mos non-index opions on socks are American. We consider an American claim D wih an adaped bounded discouned payoff process D ) 0 T. For insance, D can be he payoff of an American pu wrien on he same underlying asse as A. However, as in Secion 2.5, we shall make he simplificaion ha all he American opions used for he saic hedge have o be exercised simulaneously. Fix a ime T, he invesor, while holding claim A, purchases α unis of claim D from he marke for he price $π each. The invesor needs o decide which claims) o exercise firs and when. To his end, we recall ha V, X ; A) and V, X ; αd) represen he invesor s value funcions, respecively, for holding only A and only α unis of D. We denoe by p A, τ A ) and p αd, τ αd ) he corresponding pairs of indifference prices and opimal exercise imes. The invesor s value funcion is given by ˆV, X ; αd) := ess sup ess sup IE maxv τ, X τ + αd τ ; A), V τ, X τ + A τ ; αd)). τ T,T θ Θ,τ Nex, we simplify he problem, and derive he corresponding opimal exercise imes. Proposiion 4.2 For any [0, T ], and α R, he value funcion can be wrien as ˆV, X ; αd) = V, X ; R α ), 4.5) wih R α := maxαd + p A, A + p αd ). Therefore, he invesor s indifference price for holding A and α unis of D is p R α ), and he opimal exercise ime is given by where ˆτ α) = min τ AD α), τ DA α) ), 4.6) τ AD α) = inf u T : p u R α ) = A u + p αd u, 4.7) τ DA α) = inf u T : p u R α ) = αd u + p A u. 4.8) Proof. By he definiions of indifference prices p αd and p A, we have ˆV, X ; αd) = ess sup τ T,T = ess sup τ T,T = V, X ; R α ). ess sup IE max Mτ, Xτ + αd τ + p A τ ), Mτ, X τ + A τ + p αd τ ) ) θ Θ,τ ess sup IE M τ, Xτ + maxαd τ + p A τ, A τ + p αd τ ) ) θ Θ,τ The las equaliy means ha he invesor s value funcion is reduced o he dynamic hedging case for a single ye complex claim R α, paying R α τ a any exercise ime τ T,T. This observaion allows us o apply he resuls in Secion 2 o derive he indifference price p R α ) by replacing A wih R α in 2.23) and 2.24). For any fixed α, he invesor s opimal ime o exercise he firs claim, which could be eiher A or D, is given by ˆτ α) = inf u T : p u R α ) = maxαd u + p A u, A u + p αd u where τ AD α) and τ DA α) are given by 4.7) and 4.8) respecively. ) = min τ AD α), τ DA α) ), When he marke price of he saic hedge is incorporaed, he invesor s wealh is reduced by he amoun $απ, and he value funcion becomes ˆV, X απ ; αd). The invesor s indifference 15

16 price for holding A and purchasing α unis of D a price $π each is given by p R α ) απ. In view of 4.4), he invesor s opimal saic hedge, denoed by α if i exiss, is given by α = arg max p R α ) απ. α In his secion, we have formulaed he basic framework for saic-dynamic hedging of American claims wih simulaneous exercise, in which he saic hedging insrumens are all exercised a once. The problem of saic-dynamic hedging American claims wih muliple exercising righs using oher muliple exercising American claims can also be formulaed similarly using he principle of dynamic programming. To wrie down he value funcion, one has o firs consider he maximal expeced uiliies from all possible orders of exercises. As he number of muliple exercising claims increases, he value funcion becomes very edious, hough sraighforward, o wrie down. Moreover, he resuling opimal exercising sraegies will also be oo complex o describe. As an approximaion, one can limi he number of exercise opporuniies o a small finie number, or jus adop he case wih simulaneous exercise. 5 ESO Valuaion We will apply he mechanism of dynamic hedging and saic-dynamic hedging o ESO valuaion. ESOs are American call opions wrien on he firm s sock graned o he employee as a form of compensaion. In mos cases, he ESOs are no exercisable unil a pre-specified vesing period has elapsed. Typically, he erms of an ESO conrac sipulae ha he employee is no allowed o sell or ransfer he opion, shor sell he firm s sock, or ake shor posiions in call opions wrien on he firm s sock. These resricions preven he employee from perfecly hedging his ESO. Hence, he employee faces a consrained invesmen problem in which he has o decide how o opimally hedge and exercise his ESO. Henderson 2005) sudies a valuaion model for a European ESO ha capures he employee s risk aversion and dynamic invesmen in he marke index. Grasselli and Henderson 2008) sudy he case of muliple American ESOs wih infinie mauriy. For ESOs of American ype wih muliple exercise righs and finie mauriy, Leung and Sircar 2009) sudy he combined effec of risk aversion, dynamic hedging, vesing, and job erminaion risk on he opimal exercise policy and he corresponding ESO cos. Here we will only consider he hedging and valuaion of a single ESO. We summarize he resuls on dynamic hedging of an ESO in Secion 5.1. In his case, he employee rades in a marke index and he bank accoun, bu no he firm s sock. Then, in Secions and 5.2.2, we will augmen he employee s rading sraegy by incorporaing, respecively, saic hedges wih European and American pus on he firm s sock. Our goal is o analyze he non-rivial effecs of saic-dynamic hedges on he employee s opimal exercising sraegies Figures 2 and 3), and sudy is impac on he ESO cos o he firm Secion 5.3). 5.1 Dynamic Hedge for one ESO The marke index S and company sock price Y are described by he following SDEs ds = µs d + σs dw 1, dy = ν q)y d + ηy ρ dw 1 + ρ dw 2 ), wih consan parameers µ, σ, ν, q, η > 0, correlaion coefficien ρ 1, 1) and ρ = 1 ρ 2. The wo independen Brownian moions, W 1 and W 2, are defined on he given probabiliy space Ω, F, F ), P), and F ) 0 T is he augmened filraion generaed by hese wo processes. Since he processes are coninuous, Assumpion 2.12 of quasi-lef-coninuiy is saisfied. 16

17 Now suppose, a ime T, he employee holds an ESO wih mauriy T and a vesing period v T. We will assume no vesing in his secion, and address he effec of vesing in Secion A any exercise ime τ, he employee receives he payoff AY τ ) := Y τ N K) +, where N is a very large consan such ha his payoff equals ha of a call opion excep for unrealisically high sock prices. Throughou he period [, T ], he employee can hedge o rade dynamically in he marke index and he bank accoun ha pays ineres a consan rae r 0. A rading sraegy θ u ; u T is he cash amoun invesed in he marke index S, and i is deemed admissible, denoed by θ Θ,T, if i is F u -progressively measurable and saisfies he inegrabiliy condiion IE T θ 2 u du <. The employee s rading wealh evolves according o dx = [θ µ r) + rx] d + θ σ dw 1. As in Secion 2.2, he employee faces he Meron invesmen problem afer exercise. A any ime T wih wealh $x, he employee s maximal expeced uiliy is given by M, x) = sup Θ,T IE e γx T X = x = e γxert ) e µ r)2 2σ 2 T ). The employee s value funcion a ime [0, T ], given ha his wealh X = x and he company sock price Y = y, is V, x, y) = sup τ T,T sup Θ,τ IE Mτ, X τ + AY τ )) X = x, Y = y. 5.1) To faciliae he presenaion, we use he following shorhands for condiional expecaions: IE,x,y = IE X = x, Y = y, IE,y = IE Y = y, and inroduce he differenial operaors: L E u = η2 y 2 2 u r u + ν q ρµ η)y 2 y2 σ y, A ql u = u + LE u ru 1 2 γ1 ρ2 )η 2 y 2 e rt ) u y )2. The operaor L E is he infiniesimal generaor of Y under he minimal enropy maringale measure, Q E, and he second operaor A ql is quasilinear. Due o he exponenial uiliy funcion, he value funcion has a separaion of variables see Oberman and Zariphopoulou 2003)): The funcion H solves a linear free boundary problem. 1 V, x, y) = M, x) H, y) 1 ρ 2 ). 5.2) H + L E H 0, H κ, 5.3) ) ) H + L E H κ H = 0, 5.4) for, y) [0, T ) 0, + ), where κ, y) = e γ1 ρ2 )Ay)e rt ). The erminal condiion is HT, y) = e γ1 ρ2 )Ay). 5.5) 17

18 The employee s indifference price for holding he ESO, denoed by p, y), is defined via he equaion V, x, y) = M, x + p, y)). I saisfies he quasilinear variaional inequaliy A ql p 0, p Ay), 5.6) A ql p Ay) p) = 0, 5.7) for, y) [0, T ) 0, + ), wih pt, y) = Ay). From Proposiion 2.13, he employee s opimal exercise ime is given by τ = inf u T : pu, Y u ) = AY u ). 5.8) In pracice, we numerically solve he free boundary problem for H in 5.3)-5.5)) o obain he employee s exercise boundary, which is he criical sock price a ime. I is given by y ) = inf y 0 : H, y) = κ, y), for [0, T ]. A numerical example of his opimal exercise boundary is shown in Figure 2. Deails of he numerical scheme, verificaion and exisence resuls can be found in our previous analysis of he dynamic hedging case Leung and Sircar 2009)). Remark 5.1 The variaional inequaliy for p in 5.6)-5.7) is conneced wih a refleced BSDE, in which he driver has quadraic growh. On ha fron, Kobylanski e al. 2002) sudy he link beween a quadraic refleced BSDE wih a bounded obsacle and he corresponding variaional inequaliy, and provide an example of pricing an American opion wih exponenial uiliy. For a sudy on pricing European claims wih exponenial uiliy using BSDE, we refer he reader o Rouge and El Karoui 2000) Effecs of Vesing When a vesing period of v years is imposed, he employee canno exercise he ESO during [0, v ), bu he pos-vesing exercising sraegy will be unaffeced. The employee s value funcion a ime [0, T ] is given by F, x, y) = sup sup IE,x,y Mτ, X τ + AY τ )). 5.9) τ T v,t Θ,τ Observe ha F, x, y) V, x, y) due o exercise resricion before v, bu we have F, x, y) = V, x, y) for v. The opimal exercise ime associaed wih F, x, y), denoed by τ F, is simply τ v = τ v, and he employee s pos-vesing exercising sraegy is unaffeced by he vesing provision. Therefore, i is sufficien o solve he employee s hedging problem for no-vesing case, and raise he pre-vesing par of he exercise boundary o infiniy ESO Cos o he Firm In general, he firm is able o hedge or ransfer he ESO liabiliy, so we assume he firm is riskneural, which is in compliance wih financial regulaions 1. By no-arbirage argumens, he firm sock price follows he following SDE under he risk-neural measure Q: where W Q is a Q-Brownian moion. dy u = r q) Y u du + ηy u dw Q u, 5.10) 1 In paragraph A13 of Saemen of Financial Accouning Sandards No.123 revised), he Financial Accouning Sandards Board FASB) approves he use of risk-neural models. 18

19 The firm s graning cos is given by he no-arbirage price of a barrier-ype call opion wrien on sock Y, wih srike K and mauriy T. The barrier for his opion is he employee s opimal exercise boundary y, and he opion pays as soon as he firm sock reaches he boundary y. Due o vesing, he employee does no exercise before v, and in he regions C =, y) : v < T, 0 y < y ). The cos of a vesed ESO a ime v, given ha he sock price Y is y and he ESO is sill alive, is given by C, y) = IE Q,y e rτ ) AY τ ). 5.11) Then, he cos of an unvesed ESO a ime v, given ha he sock price Y = y, is given by C, y) = IE Q,y e rv ) C v, Y v ). 5.12) Associaed wih C, y) and C, y) are wo PDE problems, which we numerically solve using an implici finie-difference mehod. Nex, we incorporae saic-dynamic hedges ino our valuaion mehodology. 5.2 Saic-Dynamic Hedging wih Pu Opions In addiion o dynamic hedging, he employee can reduce risk exposure by aking saic posiions in oher derivaive securiies. For examples, hey can use synheic insrumens such as a collar conrac which involves simulaneous purchase of a pu and sale of a call) as discussed in Beis e al. 2001), or baske opions wrien on correlaed underlying asses, suggesed by Schizer 1999). In his secion, we incorporae saic hedging wih wo simple derivaives European and American pu opions, which are easily available in he marke o employees for mos publicly raded companies. Our goal is o examine he impac of incorporaing saic hedges wih hese opions on he employee s ESO exercise policy and he corresponding ESO cos Hedging wih European Pus For simpliciy, we assume ha he employee purchases β 0 unis of European pus wih he same mauriy T and srike K, even hough a wide array of European pus wih various srikes and expiraion daes could be available. We ake he marke price of each European pu, denoed by π, as he Black-Scholes price, since we model he company sock price as following a geomeric Brownian moion. Following our formulaion in Secion 4.1, we firs wrie down he indifference price for holding β European pus. By Theorem 3 in Musiela and Zariphopoulou 2004), he indifference price can be wrien as 1 h, y; β) = γ1 ρ 2 )e rt ) log IEQE e γ1 ρ2 )βk Y T ) + Y = y, where IE QE indicaes he expecaion is aken under he minimal enropy maringale measure. The indifference price can be found from solving he quasilinear parial differenial equaion A ql h = 0, for, y) [0, T ) 0, + ), wih ht, y) = βk y) +. Then, by 4.1), he value funcion for holding an ESO and β European pus is given by Ṽ, x, y ; β) = sup τ T,T sup Θ,τ IE,x,y Mτ, X τ + AY τ ) + hτ, Y τ ; β)). 19

20 Incorporaing he cos of European pus, he employee s value funcion becomes Ṽ, x βπ, y ; β). We apply he same ransformaion 5.2) o Ṽ. Tha is, we le Then, H solves he linear variaional inequaliy Ṽ, x βπ, y ; β) = M, x βπ) H, y ; β) 1 1 ρ 2. H + L E H 0, H κ, 5.13) H + L H) E κ H ) = 0, for, y) [0, T ) 0, + ), where κ, y ; β) = e γ1 ρ2 )Ay)+h,y ;β))e rt ). The erminal condiion is HT, y ; β) = e γ1 ρ2 )Ay)+βK y) +). 5.14) In pracice, we apply sandard finie difference mehods o numerically solve 5.13)-5.14) for he opimal exercise boundary, and compue he indifference price for holding an ESO along wih β unis of European pus using he formula: 1 p, y ; β) = γ1 ρ 2 )e rt ) log H, y ; β). 5.15) For ESO hedging, he employee considers only long posiions in he European pus β 0). According o 4.4), he opimal quaniy of European pus o purchase, β, is found from he Fenchel-Legendre ransform of he indifference price p, y ; β) as a funcion of β, evaluaed a he marke price π. Tha is, β = arg max p, y ; β) βπ. 0 β< We illusrae how o deermine β hrough an numerical example in Figure 1. Having deermined β, we use he corresponding exercise boundary o compue he cos of he ESO o he firm, following he seps in Secion Indifference Price : p, y; β) slope=π β = β Figure 1: The opimal saic hedge β is he poin a which he indifference price solid curve) has slope equal o he marke price π. The parameers are K = K = 10, T = 10, r = 5%, q = 0%, ν = 8%, η = 30%, µ r)/σ = 20%, ρ = 30%, γ = 0.3. The Black-Scholes pu price is π = From Proposiion 4.1, he employee s opimal exercise ime, for any fixed β 0, is given by τ β) = inf u T : pu, Y u ; β) = AY u ) + hu, Y u ; β). 5.16) 20

21 The combinaion of risk aversion and saic hedge has a profound impac on he employee s opimal exercising sraegy. In he presence of hedging resricions, i is well-known ha a risk-averse employee may find i opimal o exercise an American opion early even if he underlying sock pays no dividend see, for example, Deemple and Sundaresan 1999)). We see a similar effec in our model, bu we also idenify opposie effecs of risk aversion and saic hedges wih pus on he employee s opimal exercise policy. In he nex proposiion, we compare he opimal exercise imes τ and τ β), and show ha long posiions in European pus will delay he employee s ESO exercise. In essence, he pu opions offer proecion from he sock s downward movemen, which effecively makes he employee less conservaive in exercising his ESO. This effec can be seen in he numerical example in Figure 2, where he employee s opimal exercise boundary shifs upward when European pus are used. 22 Company Sock Price Y ) hold ESO+ β Euro. pus hold ESO only Time in Years Figure 2: The employee who hedges he ESO dynamically will exercise he opion as soon as he firm s sock his he lower dashed boundary. If saic-dynamic hedges wih European pus are used, he employee will exercise he ESO laer a he upper solid boundary. The parameers are he same as hose in Figure 1. Proposiion 5.2 For every β 0, we have p, y ; β) p, y) + h, y ; β), for [0, T ], y R +. From his, i follows ha τ β) τ. Proof. Fix a β 0. We firs observe ha pt, y ; β) = pt, y) + ht, y ; β), for y 0. From 5.15), we can derive he variaional inequaliy for p, y ; β) and express i in he following form M p, y ; β)) := min A ql p, y ; β), p, y ; β) h, y ; β) Ay) = 0. We wan o show ha M p, y) + h, y ; β)) 0 since his will imply p, y ; β) p, y)+h, y ; β) by he comparison principle see Oberman and Zariphopoulou 2003)). To his end, we consider A ql p, y) + h, y ; β)) = A ql p A ql h γ1 ρ2 )η 2 y 2 e rt ) p y h y. Noice ha he second erm is zero, and he las erm is non-posiive because p is non-decreasing wih y i is he indifference price of an American call opion), bu h is non-increasing wih y i is he indifference price of β European pu opions). Hence, we have M p, y) + h, y ; β)) = min A ql p γ1 ρ2 )η 2 y 2 e rt ) p y h y, p, y) Ay) min A ql p, p, y) Ay) = 0, 21