Optimal exercise of an executive stock option by an insider

Transcription

1 Opimal exercise of an execuive sock opion by an insider Michael Monoyios Mahemaical Insiue Universiy of Oxford Andrew Ng Mahemaical Insiue Universiy of Oxford Augus 6, 2 Absrac We consider an opimal sopping problem arising in connecion wih he exercise of an execuive sock opion by an agen wih inside informaion. The agen is assumed o have noisy informaion on he erminal value of he sock, does no rade he sock or ouside securiies, and maximises he expeced discouned payoff over all sopping imes wih regard o an enlarged filraion which includes he inside informaion. This leads o a sopping problem governed by a ime-inhomogeneous diffusion and a call-ype reward. We esablish condiions under which he opion value exhibis ime decay, and derive he smooh fi condiion for he soluion o he free boundary problem governing he maximum expeced reward, and derive he early exercise decomposiion of he value funcion. The resuling inegral equaion for he unknown exercise boundary is solved numerically and his shows ha he insider may exercise he opion before mauriy, in siuaions when an agen wihou he privileged informaion may no. Hence we show ha early exercise may arise due o he agen having inside informaion on he fuure sock price. Inroducion In his paper we model he exercise decision of an insider who is graned an execuive sock opion (ESO). The agen is an employee who is graned a single American-syle opion on a sock of his employing company. This execuive is barred from rading he sock, does no rade oher securiies eiher, and has some inside informaion on he fuure evoluion of he sock a he erminal dae of he opion. The execuive is modelled as risk-neural, so has a linear uiliy funcion, and hence maximises he discouned expecaion under he physical measure P of he opion payoff a he exercise ime. We do no endow he agen wih risk-averse preferences as we wish o focus exclusively on he role of inside informaion on he exercise decision, and his is also he reason for assuming away all oher rading opporuniies, as well as oher conracual complicaions ha are common in ESOs, such as a vesing period, reseing of srikes, parial exercise opporuniies, job erminaion, and so on. The exercise ime is a sopping ime wih respec o an enlarged filraion formed by augmening a filraion F wih he σ-algebra of a random variable L, which corresponds o noisy informaion on he value of he sock a he opion mauriy ime T. Wriing he sock dynamics under he enlarged filraion F L, he sock price is governed by a ime-inhomogeneous diffusion wih sae and ime-dependen drif and consan volailiy, and we are faced wih an opimal sopping problem governed by he ime-inhomogeneous diffusion. The enlargemen of filraion, leading o a sae and ime-dependen drif, leads o he heoreical possibiliy of early exercise. We esablish ha early exercise can occur and provide a numerical compuaion of he early exercise boundary. We esablish he equivalence beween he opimal sopping problem and a free boundary PDE. We furher esablish ha he value funcion governing he maximum expeced reward Corresponding auhor. We hank Peer Bank, Kasper Larsen, Goran Peskir, Marin Schweizer, Mihalis Zervos, wo anonymous referees and paricipans a he Workshop on Foundaions of Mahemaical Finance (Fields Insiue, 2) for helpful suggesions.

2 exhibis ime decay for suiably low realisaions of L, and for zero ineres rae, regardless of L. In hese cases we prove ha he value funcion saisfies he smooh-fi condiion a he free boundary, and from his we derive he early exercise decomposiion of he value funcion ino a European-syle payoff and an early exercise premium. This leads o an inegral equaion characerising he free boundary, which is solved numerically. The numerical resuls show ha he insider can indeed exercise he ESO prior o mauriy in siuaions in which an execuive wihou he inside informaion would no. Hence, we show ha privileged informaion can also be a facor conribuing o he early exercise of ESOs. The res of he paper is as follows. Secion 2 describes he model and he opimal sopping problems for an insider and a non-insider. Secion 3 conains our main resuls. We analyse he value funcion of he insider s discouned ESO value as a funcion of log-sock price. We use ideas of sochasic flows o esablish convexiy and monooniciy of he value funcion, derive he free boundary PDE, condiions under which he ESO value exhibis ime decay, and he smooh fi condiion a he exercise boundary. We use hese resuls o derive an early exercise decomposiion for he ESO value, and a resuling inegral equaion for he exercise boundary. The properies are well-known in sandard American opion problems wih consan parameers under a maringale measure, bu our problem is raher non-sandard, conaining a ime-inhomogeneous drif erm no equal o he ineres rae, since valuaion is performed under he physical measure. We solve he inegral equaion numerically and presen resuls which show ha he insider can be induced o early exercise by possessing privileged informaion. Secion 4 concludes. 2 The model We have a single sock price process S = (S ) T defined on a complee probabiliy space (Ω, F, P ) equipped wih a filraion F = (F ) T ha saisfies he usual condiions of righconinuiy and compleeness. A (P, F)-Brownian moion B = (B ) T drives he sock price, which follows he geomeric Brownian moion ds = µs d + σs db, where µ and σ > are known consans. There is a consan risk-free ineres rae r. I will be someimes be more convenien o work wih he log-sock price X := log S, saisfying X = X + γ + σb, T, () where γ := µ σ 2 /2. Our financial agen (an execuive) is an employee of he firm whose sock price is S, and is awarded a ime zero a single American-syle call opion on he sock wih mauriy T. We assume ha he agen is barred from rading S or ouside securiies, ha here is no opporuniy for parial exercise of he opion, and we ake he agen s preferences o be risk-neural, so he maximises he expeced discouned payoff under he objecive probabiliy measure P. Grasselli and Henderson [6 or Rogers and Scheinkman [6 focus on he effecs of risk aversion and ouside rading on early and block exercise. We do no inroduce conracual complicaions ha someimes feaure in ESOs, as done by Sircar and co-auhors [2, 3, 8. We exclude he above feaures of ESOs in order o focus exclusively on he impac of inside informaion on he agen s opimal sopping problem of when o exercise he opion. In paricular, we can examine a case in which he absence of inside informaion resuls in no early exercise, and we can hen show ha he inroducion of addiional informaion can lead o early exercise, and we compue he early exercise boundary numerically. Even wih he simplificaions ha we make, we shall see ha we are neverheless faced wih a non-sandard American problem wih a imeinhomogeneous diffusion for he sock, so ha many sandard properies of he value funcion are no known a priori and have o be esablished. These properies include monooniciy and convexiy in he log-sock price, ime decay, and he smooh fi condiion (coninuiy of he firs spaial derivaive) a he opimal exercise boundary. 2

3 The agen has inside knowledge a ime zero of an F-measurable random variable L, corresponding o noisy knowledge of he erminal log-sock price X T. We shall someimes refer o his agen as he insider or he execuive. We shall also consider an agen whose informaion is represened by he filraion F, so who does no have he privileged informaion. This agen will someimes be referred o as he regular agen or as he non-execuive. The random variable L will be given by L = ax T + ( a)ɛ, < a <, (2) where ɛ is a sandard normal random variable independen of F. Hence, he execuive s informaion flow is represened by he enlarged filraion F L = (F L ) T, defined by F L := F σ(l), T. See Danilova, Monoyios and Ng [ for similar examples of such inside informaion wihin he conex of parial informaion invesmen problems. The dynamics of he log-sock price wih respec o he enlarged filraion are given by classical enlargemen of filraion mehods (see Yor [9) in he following lemma. Lemma. Wih respec o he enlarged filraion F L, he dynamics of he log-sock price are dx = ( γ(t a T ) + L ) T a a X d + σdb L, where B L is an F L -Brownian moion and T a is he modulaed ime defined by ( ) 2 a T a := T +, < a <. (3) aσ Proof. Classical enlargemen of filraion resuls (Theorem 2. in Yor [9) imply ha he F-Brownian B has a semi-maringale decomposiion wih respec o F L of he form B = B L + ν(l, u)du, T, wih B L an F L -Brownian moion, and he process (ν(l, )) T, is called he informaion drif, given by he logarihmic derivaive of he condiional densiy of L given F. This resuls in B = B L + and combining his wih (), he lemma follows. L a(x + γt + σb u ) du, T, (4) aσ(t a u) 2. The opimal sopping problems Denoe by T he se of all sopping imes wih respec o he filraion F, and by T L he se of all sopping imes wih respec o he enlarged filraion F L. Inroduce he following subses of T and T L : T,T := {τ T P (τ [, T ) = }, T <, T,T L := {τ T L P (τ [, T ) = }, T <. Of course, we have T,T T and T,T L T L. The execuive sock opion is an American call wih srike K. If exercised a ime τ [, T, he discouned payoff a ime zero is Y τ, where Y = (Y ) T is he reward process, defined by Y := e r (e X K) +, T, 3

4 assumed o saisfy [ E sup Y <. The insider s (ha is, he execuive s) opimal sopping problem is o find a sopping ime τ T L o achieve he maximal expeced reward V (L) := sup τ T L E[Y τ F L. Noice ha he supremum is over sopping imes of he enlarged filraion F L, so we emphasise his wih he condiioning on he enlarged iniial σ-field F L. The maximal expeced reward V (L) is hus an F L -measurable random variable (hough from he perspecive of he insider, L is a known consan) and he relevan dynamics of he sae variable X are given by Lemma. When no confusion arises we suppress he dependence on L of V. The non-execuive faces a similar opimal sopping problem, bu over F-sopping imes, so in his case X is given by (). We denoe he non-execuive s maximal expeced reward a ime zero by V := sup E[Y τ, where he expecaion is condiional upon he (assumed rivial) σ-algebra F. 2.2 Benchmark case: µ r and no inside informaion τ T For µ r, he reward process Y is a (P, F)-submaringale, so he regular agen s value for he American ESO coincides wih he European value: V = E[Y T. In paricular, he exercise ime τ = T is opimal for he regular agen. This slighly arificial conclusion derives from he fac ha here are no rading opporuniies for he regular agen and also ha he agen has a linear uiliy funcion. This resul serves as a useful benchmark for us. Given he same rading opporuniies (none) and he same preferences for he insider as he regular agen, our main goal is o show ha inside informaion on he sock can resul in early exercise, because he drif of he sock becomes ime and price-dependen. 2.3 The insider s problem In his secion we analyse he opimal sopping problem for he insider. The log-sock price follows he ime-inhomogeneous diffusion of Lemma, which we wrie as where β(, x) β(, x; L) is given by dx = β(, X )d + σdb L, (5) β(, x) = C x T a, C := γ(t a T ) + L a. (6) Given an iniial condiion X = x R, for [, T, he soluion o (5) is he Gaussian process (X u ) u T given by X u = x + β(, x)(u ) + σ(t a u) u T a ρ dbl ρ, u T. (7) In paricular, he ransiion densiy p(, x; u, y) for moving from X = x o X u dy a u is given explicily by [ p(, x; u, y) = Σ(; u) 2π exp ( ) 2 y m(, x; u), x, y R, u T, (8) 2 Σ(; u) where m(, x; u) and Σ(; u) > are given by m(, x; u) = x + β(, x)(u ), ( ) Σ 2 (; u) = σ 2 Ta u (u ). (9) T a 4

5 For a saring ime [, T he maximal expeced discouned payoff is given by he F L - adaped process V V (L) := ess sup E[e r(τ ) (e Xτ τ T,T L We are hus led o consider he F L -adaped process U defined by K) + F L = e r ess sup E[Y τ F L, T. τ T,T L U U (L) := ess sup E[Y τ F L, T, () τ T,T L saisfying U = e r V a.s., for any [, T. Classical opimal sopping heory (Appendix D of Karazas and Shreve [) characerises he soluion o he opimal sopping problem () as follows. There exiss a non-negaive càdlàg (P, F L )-supermaringale U = (U ) T, he Snell envelope of Y, such ha U is he smalles (P, F L )-supermaringale ha dominaes Y, wih U T = Y T a.s. A sopping ime τ T L is opimal for he problem () saring a ime zero if and only if U τ = Y τ a.s., and he sopped supermaringale U τ defined by U τ := U τ, T, is a (P, F L )-maringale. The smalles opimal sopping ime in T,T L for he problem () is τ, defined by 3 The value funcion τ := inf{ρ [, T U ρ = Y ρ } T, T. We are ineresed in he opimal sopping problem wih reward process Y = e r (e X K) + =: f(, X ), where f : [, T R R + is he coninuous non-negaive funcion given by f(, x) := e r (e x K) +, and where (X u ) u T is he soluion (7) of (5) wih X = x deerminisic, for [, T. For a fixed value of he random variable L, say L = l R, we define he value funcion F : [, T R R + by F (, x) F (, x; l) := sup E[f(τ, X τ ) X = x, L = l. () τ T,T L Then, in a very general coninuous-ime Markov seing, F is a coninuous funcion and he process U = (F (, X )) T is he Snell envelope of Y = (f(, X )) T (see for insance El Karoui, Lepelier and Mille [5). The insider s value process for he ESO is (V (, S ; L)) T, where V : [, T R + R + is given by V (, s) V (, s; l) := sup E[e r(τ ) (S τ K) + S = s, L = l, τ T,T L and we suppress dependence on L when no confusion arises. Hence he value funcions F and V are relaed according o e r V (, s(x)) = F (, x), wih s(x) := e x. The (smalles) opimal sopping ime for he problem () saring a ime [, T wih X = x is τ (, x) given by τ (, x) = inf{ρ [, T F (ρ, X ρ ) = f(ρ, X ρ )} = inf{ρ [, T V (ρ, S ρ ) = (S ρ K) + }. The coninuaion region C is defined by C := {(, x) [, T ) R F (, x) > f(, x)} = {(, s) [, T ) R + V (, s) > (s K) + }. 5

6 Since F, V are coninuous, C is open. This suggess (and we show below) ha here is funcion x : [, T R (respecively, s : [, T R + ), he criical log-sock price (respecively, criical sock price) or opimal early exercise boundary, such ha he opion is exercised he firs ime he log-sock price exceeds x (). Since i is never opimal o exercise if he sock is below he srike K, we mus have x () log K for all [, T. We shall characerise he early exercise boundary and he value funcion F as a soluion o a free boundary problem, and we also esablish he smooh fi condiion a he boundary ha is common in many opimal sopping problems. This is no guaraneed and in general needs o be verified on a case by case basis. This is he siuaion we are faced wih here, as we are dealing wih a non-sandard American opion problem involving a sock wih a sae and ime-dependen drif. 3. Convexiy and monooniciy of he value funcion in x We wish o show ha he value funcion F is increasing and convex in x. Alhough hese properies do no necessarily imply similar properies for he ESO value V in he sock price, hey will be sufficien o allow us o characerise he exercise boundary and esablish bounds on he derivaive F x, which are ingrediens we need o obain he free boundary PDE and he smooh fi condiion saisfied by F. These hen lead easily o a corresponding free boundary PDE and smooh fi condiion for V. We shall uilise ideas of sochasic flows applied o he log-sock price. We wrie X(x) for he log-sock price wih iniial condiion X = x, considered as he soluion o a diffusion SDE wih ime and sae-dependen drif. In Lemma 2 we show ha he map x X(x) is non-decreasing, and give a condiion on he drif β of X for his map o be convex in x. This condiion is indeed saisfied in our specific model. From he properies of x X(x) we deduce he corresponding properies for he map x F (, x). Noe ha for a diffusion wih ime and sae dependen drif, properies such as monooniciy and convexiy in he iniial condiion are no auomaic, so he obvious properies of he map x X(x) under F do indeed need o be shown o hold under F L. An alernaive o our approach would be o use a echnique due o El Karoui e al [4. They prove convexiy of sandard American opion prices wih respec o sock price (so evaluaed under a maringale measure) in diffusion models wih deerminisic ineres rae. They also employ ideas of sochasic flows, firs o show he propery for European prices, hen, adaping an ieraive procedure found in El Karoui [3, hey exend he resul o American prices. This approach can be shown o work in our model, since he European opion value can be wrien as an inegral wih respec o he ransiion densiy of X, given in (8). Indeed, we adap his echnique laer for par of our analysis of he ime decay propery of he American ESO value: see he proof of Theorem 3. In principle one migh ry o use our echniques o prove convexiy and monooniciy of ESO value funcion V in he saring sock price S = s, for any [, T. This does no appear o be sraighforward using our mehods, because i does no appear easy o prove ha he map s S(s) is increasing and convex for a general diffusion. Indeed, we shall see in Remark ha, when we use he explici soluion (7) for X(x), he map x X(x) is indeed increasing and convex, bu ha he map s S(s) is increasing bu no convex. I is well known ha convexiy of American opion prices wih respec o sock price does no immediaely follow from he convexiy of he payoff process when he reurn disribuion of he sock depends on he sock price, as shown by Meron [4 (Theorem and he couner-example in Appendix ). Oher auhors have also analysed convexiy of American opion values wih respec o sock price. Eksrom [2 used sochasic ime changes and a limiing argumen based on approximaing American opion by a Bermudan opion, and Hobson [7 uilised coupling mehods. Similar o [4, hese papers consider sandard American pricing problems under a risk-neural measure, wih a deerminisic rae of ineres. We have a raher non-sandard problem where he sock price drif is no he ineres rae, and in addiion is boh sae and We hank an anonymous referee for poining his ou. 6

7 ime-dependen. For hese reasons, we canno direcly read off he required properies of he value funcion from hese papers. For simpliciy consider a saring ime =. The same ideas apply o any saring ime [, T. Consider he log-sock price wih iniial condiion X = x, and wrie X X(x), following X (x) = x + β(u, X u (x))du + σb L, T. (2) which for each [, T and each ω Ω are diffeo- We may choose versions of (X (x)) T morphisms in x from R R. Tha is, he map x X(x) is smooh. Define b(, x) := β(, x), x D (x) := x X (x). (3) Lemma 2. The map x X(x) is increasing, and if β xx (, y), also convex. Proof. We have so x X(x) is increasing. Define c(, x) := b x (, x) = β xx (, x). Then ( ) D (x) = exp b(u, X u (x))du >, x D (x) = D (x) c(u, X u (x))d u (x)du, which is non-negaive if c(, x) for all (, x) [, T R. Then x X(x) is convex. Remark. Lemma 2 holds for a general diffusion wih a ime and sae-dependen drif. Alernaively, wih he explici soluion (7) we can direcly compue D (x) = T a >, x D (x) =, which direcly shows ha x X(x) is increasing and convex. The same mehod applied o he map s S(s) (he sock price wih iniial condiion S = s > ) gives s S (s) = S ( (s) ) >, s T a 2 s 2 S (s) = S ( (s) s 2 ) <, T a T a so ha s S(s) is increasing, bu no convex (hough his does no necessarily imply ha s V (, s) is no convex). Theorem. The map x F (, x) is increasing for any [, T. Suppose β xx (, x). Then he map x F (, x) is convex for any [, T. Proof. We se = wihou loss of generaliy. Then F (x) F (, x) is given by F (x) = sup τ T L E[e rτ (exp(x τ (x)) K) + F L, (4) and where X (x) = x. Le τ (x) T L denoe he opimal sopping ime for he problem in (4). Then we may wrie F (x) = E[e rτ (x) (exp(x τ (x)(x)) K) +, where for breviy we have suppressed he condiioning on F L. Since he map x X(x) is increasing, we have, for x < x, (exp(x τ (x )(x )) K) + < (exp(x τ (x )(x )) K) +. 7

8 Muliply boh sides by e rτ (x ), ake expecaions, and use he fac ha τ (x ) is sub-opimal for he saring sae x, o obain F (x ) < E[e rτ (x ) (exp(x τ (x )(x )) K) + F (x ), which shows ha x F (x) is non-decreasing. To esablish convexiy, define x λ := λx + ( λ)x for x < x and λ [,. Using he propery ha x X(x) is convex, we have ha x (exp(x(x)) K) + is also convex. Hence (exp(x τ (x λ )(x λ )) K) + λ(exp(x τ (x λ )(x )) K) + + ( λ)(exp(x τ (x λ )(x )) K) +. Muliplying by exp( rτ (x λ )), aking expecaions and using he fac ha τ (x λ ) is sub-opimal for he saring saes x i, i =,, we obain so x F (x) is convex. F (x λ ) λf (x ) + ( λ)f (x ), 3.2 Free boundary problem for he value funcion As F is increasing and convex, he exercise boundary x () divides he domain of F ino he coninuaion region C and he sopping region S, given by C = {(, x) [, T ) R x < log x ()} = {(, s) [, T ) R + s < s ()}, wih S = C c. Define he exended generaor L of X by Lg(, x) := g (, x) + β(, x)g x (, x) + 2 σ2 g xx (, x). Denoe he closure of he coninuaion region by C. Theorem 2. The value funcion F in () solves, in C, he free boundary problem LF (, x) =, (, x) C, F (, x) > e r (e x K), (, x) C, F (, x ()) = e r (e x () K), T, F (T, x) = e rt (e x K) +, x R. Proof. This is by sandard mehods (Theorem in Karazas and Shreve [). 3.3 The exercise boundary is non-increasing We now analyse he ime decay of he ESO value, ha is, ha he map V (, s) is nonincreasing, for any [, T and s R +. This propery will imply ha he exercise boundary is a non-increasing funcion of ime. Recall he F L -measurable random variable C in (6). Theorem 3.. If C log K hen he map V (, s) is non-increasing. 2. If he ineres rae is zero, hen V (, s) is non-increasing for any value of C. Time decay for American-syle claims canno be expeced o hold in general when he price dynamics are governed by a ime-inhomogeneous process, as poined ou by Eksrom [2. He describes a drasic couner-example, in which volailiy can jump from zero o a posiive value a some fuure ime. Time decay is ofen aken for graned, as longer-daed opions have all he exercise opporuniies of shorer-daed claims, so holds in ime-homogeneous models. For his reason, here seems o be very lile analysis of his propery in he lieraure. 8

9 Theorem 3 saes ha he ime decay propery always holds for zero ineres rae. The same holds for sandard American pricing problems (under a maringale measure) in diffusion models (see [2). Regardless of he ineres rae, ime decay for he ESO holds for suiably low realisaions of he random variable L. Indeed, C < log K corresponds (modulo he noise in he inside informaion, governed by he parameer a (, )) o knowledge ha he sock price will end up below he srike. Wih his knowledge, i is inuiively plausible ha he insider would exercise he opion early, knowing ha i will end up ou of he money, and his would make he ESO less valuable as ime progresses. We shall use his propery below o esablish ha he exercise boundary is non-increasing, which is an ingredien in our subsequen proof of he smooh pasing condiion. An alernaive o our approach would be o use an ieraive procedure due o Muhuraman [5, which seeks o solve American opion problems using a sequence of problems each wih known exercise boundary, and wih successively beer approximaions o he rue boundary. This would be a good opic for fuure research, and migh be able o show ha he smooh pasing condiion holds. 2 In paricular, his would imply ha in fac he exercise boundary is non-increasing and ha he ime decay propery is valid. Proof of Theorem 3. The dynamics of he sock price wih respec o he enlarged filraion F L are ds = S [α(, S )d + σdb L, α(, s) = β(, log s) + 2 σ2. (5) Using he Tanaka-Meyer formula (Jeanblanc e al [9, Chaper 4) applied o he semi-maringale S, we have e ru (S u K) + = e r (S K) + r + 2 u u e rρ (S ρ K) + dρ + e rρ dl K ρ (S), u T, u e rρ {Sρ>K}dS ρ where L K (S) denoes he local ime of S a level K. We ake expecaion given S = s (and of course, implicily, given L = l, wih his dependence suppressed). I is no hard o verify ha he sochasic inegral is a (P, F L ) maringale, and we obain, on using he dynamics (5), E[e ru (S u K) + S = s = e r (s K) + [ u + E e rρ [(α(ρ, S ρ ) r)s ρ + rk {Sρ>K} dρ S = s + [ u 2 E e rρ dl K ρ (S) S = s, u T. (6) We proceed formally for he momen, and indicae furher below how o make he following argumen rigorous. The local ime may be represened as L K (S) = δ(s ρ K)d S ρ, T, where δ( ) is he Dirac dela funcion. We shall give meaning o his heurisic expression furher below. Using his represenaion of L K (S) we conver (6) ino [ u E[e r(u ) (S u K) + S = s (s K) + = E e r(ρ ) A(ρ, S ρ )dρ S = s, u T, where A(, s) := [(α(, s) r)s + rk {s>k} + 2 σ2 s s δ(s K), T, s R +. 2 We hank an anonymous referee for poining us owards his reference. (7) 9

10 Jacka and Lynn [8 use a similar consrucion o (7), bu for smooh payoff funcions, o analyse ime decay of opimal sopping problems governed by diffusions. Now consider wo imes, saisfying < T. Suppose ha (, s) C. Le τ (, s) denoe he opimal sopping ime for saring sae (, s) and define v by τ (, s) =: + v. Applying (7) beween and + v, we obain [ +v < V (, s) (s K) + = E e r(ρ ) A(ρ, S ρ )dρ S = s. (8) Since + v is in general sub-opimal for he saring sae (, s), he same argumen applied over [, + v gives [ +v V (, s) (s K) + E e r(ρ ) A(ρ, S ρ )dρ S From (8) and (9) we see ha if A(, s) is non-increasing in, hen we will have V (, s) (s K) + V (, s) (s K) + >, implying ha value funcion is non-increasing in ime. The condiion ha A(, s) is non-increasing in ranslaes o (C log s) (T a ) 2 {s>k}. = s. (9) This condiion is clearly saisfied whenever s K. When s > K (which is he case whenever (, s) C) i will always be saisfied for C log K, and wih he ouline below of how o make he above argumens fully rigorous, his proves he firs saemen in he heorem. To be fully rigorous, one mus give precise meaning o he represenaion of he local ime in erms of he Dirac dela funcion. This can be done in he classical manner in which he generalised Iô formula for convex funcions is esablished, by approximaing he Dirac dela funcion δ(x) by a sequence of probabiliy densiies wih increasing concenraion a he origin. This ype of argumen can be found in Secions 3.6 and 3.7 of Karazas and Shreve [ and is oulined below. One defines a sequence of probabiliy densiy funcions (or mollifiers, posiive C funcions wih compac suppor ha inegrae o ) (ϕ n (x)) n N as well as a sequence of funcions (u n (x)) n N, given by u n (x) := x y such ha he following limiing relaions hold: ϕ n (z K)dzdy, x R, n, lim u n(x) = (x K) +, n lim n u n(x) = lim n x ϕ n (z K)dz = {x>k}, as well as lim n R u n(x)g(x)dx = lim ϕ n (x K)g(x)dx δ(x K)g(x)dx = g(k), n R R for any Borel funcion g( ). Thus, in he limi as n, he funcion ϕ n ( ) akes on he same properies as he Dirac dela. One now applies he same argumens ha led o (7) wih u n (x) in place of (x K) +, so we are able o use he Iô formula because he u n are C 2. This gives [ u E[e r(u ) u n (S u ) S = s u n (s) = E e r(ρ ) A n (ρ, S ρ )dρ S = s,

11 where A n (, s) := α(, s)su n(s) + 2 σ2 s s u n(s) ru n (s), T, s R +. Wih his is place one looks for condiions such ha A n (, s) is non-increasing in, and finally akes he limi as n, drawing he same conclusions as before. To prove he second par of he heorem, we need o esablish ha when C > log K, hen ime decay holds provided r =, since we already know ha ime decay is valid for C log K, regardless of r. We do his by adaping a procedure found in El Karoui e al [4, firs considering he ime decay of a European ESO, and hen invoking a varian of an ieraive procedure originally due o El Karoui [3 which allows one o infer ha he American ESO will inheri whaever ime decay propery holds for he European ESO. The European ESO value for saring sae (, s) [, T R + and mauriy u T is given by [ V E (, s; u) = E e r(u ) (S u K) + S = s, where he dependence on a given value of L is suppressed as usual. A sraighforward compuaion using he ransiion densiy (8) gives [ V E (, s; u) = e r(u ) e b(,s;u) Φ(z(, s; u)) KΦ(z(, s; u) Σ(; u)), where Φ( ) is he sandard cumulaive normal disribuion funcion and b(, s; u) = m(, log s; u) + 2 Σ2 (; u), z(, s; u) = Σ(; u) + m(, log s; u) log K, Σ(; u) wih m, Σ defined in (9). Differeniaion wih respec o gives V E [( (, s; u) = e r(u ) r + m ) + Σ Σ e b Φ(z) rkφ(z Σ) + K Σ Φ (z Σ), where we have suppressed argumens of funcions for breviy. Since Φ and Φ are posiive and Σ/ is negaive, he las wo erms on he righ hand side are negaive, so he European ESO value will be guaraneed o be non-increasing wih ime provided ha This condiion ranslaes o r + m (, log s; u) + Σ(; u) Σ(; u). C log s r (T a ) 2 T a u 2 σ2 (T a u). Suppose r =. We are ulimaely ineresed in when he American ESO value will exhibi ime decay, and since V (, s) = for s K, we only consider he case when s > K. Then, for he European ESO value o exhibi ime decay in he region s > K we require C log s 2 σ2 (T a u), when s > K. (2) Since he righ hand side is negaive, he condiion will be guaraneed if C log K. Hence we conclude ha for r = and s > K, V E / (, s; u) if C log K. To complee he proof we now invoke he ieraive procedure of El Karoui e al [4 o infer a propery for he American opion from he corresponding propery for he European value. Denoe he payoff of he opion by h(s) = (s K) +. Denoe by (S u (, s)) u T he sock price process given iniial condiion S = s, for [, T. Recall ha he American ESO value is given by [ V (, s) = sup τ T, L E e r(τ ) h(s τ (, s)), T, s >.

12 For fixed (, s) he process (e r(u ) V (u, S u (, s))) u T is he smalles supermaringale ha dominaes (e r(u ) h(s u (, s))) u T. We now consruc V by an ieraive procedure found in [4, adaped o he siuaion in hand. For any coninuous Borel funcion g : [, T R + R, we define [ (R u g)(, s) := E e r(u ) g(u, S u (, s)), u T, s >. So, in paricular, we have V E (, s; u) = (R u h)(, s) and his is decreasing in for r =, s > K and C log K. Define he operaor (Kg)(, s) := sup (R u g)(, s), T, s >. u [,T I is sraighforward o see ha Kh also exhibis ime decay. Moreover, for r = and s > K, Kh h if C log K, by virue of he Jensen inequaliy, since we have for any u [, T : (Kh)(, s) (R u h)(, s) = e r(u ) E [ (S u (, s) K) + Then, if r =, we see ha (Kh)(, s) h(s) provided ha e r(u ) (E[S u (, s) K) + [ = e (s r(u ) exp β(, log s)(u ) Σ2 (; u) K). β(, log s)(u ) + 2 Σ2 (; u). This is (2), so for s > K will hold whenever C log K. Since Kh h, we have K n+ h K n h, where K n denoes he n-fold ierae of K. We can hus define w := lim n Kn h = sup K n h. I is easy o see ha w inheris he properies of Kh, so w also exhibis ime decay. The remainder of he proof follows he same reasoning as Theorem 9.4 in El Karoui e al [4, o esablish ha w is he smalles fixed poin of K dominaing h, and hence ha w coincides wih V, so ha V also displays ime decay when r = and C log K. Since V displays ime decay when C log K, we conclude ha ime decay holds for all values of C when r =. This ends he proof. For compleeness, here is he argumen. We have w K n+ w = K(K n w). Leing n, we obain w Kw. The reverse inequaliy is rivial. If u is a fixed poin of K dominaing h, hen u = K n u K n h. Leing n, we obain u w. Fix (, s) and consider Z u = e r(u ) w(u, S u (, s)). For u u 2 T, we have n N E[Z u2 F L u = e r(u ) E[e r(u2 u) w(u 2, S u2 (, s)) F L u = e r(u ) (R u2 w)(u, S u (, s)) e r(u ) (Kw)(u, S u (, s)) = Z u. Thus, Z is a supermaringale dominaing e r(u ) h(s u (, s)), and so mus dominae e r(u ) V (u, S u (, s)) as well. In paricular, w(, s) = Z V (, s). For he reverse inequaliy, we observe from he supermaringale propery for e r(u ) V (u, S u (, s)) ha (R u V )(, s) V (, s), and hence KV V. Therefore, V is a fixed poin of K, and being a fixed poin of K, V mus dominae w. Hence, V and w coincide, and so V inheris he properies of w, and we are done. 2

13 Lemma 3. Suppose he map V (, s) is non-increasing. Then he exercise boundary s () is non-increasing, for [, T. Proof. Choose (, s) C for some s R + and consider saisfying < T. assumpion, V (, s) V (, s), and herefore By V (, s) (s K) + V (, s) (s K) + >, T, so ha (, s) is also in C. Tha is, for <, we have ha s < s () necessarily implies ha s < s ( ), and his can only be rue if s ( ) is a leas as big as s (), ha is, s ( ) s (). This lemma implies ha x () = log s () is also non-increasing. 3.4 Smooh fi condiion In his subsecion we esablish he smooh-fi condiion for F. There are hree ingrediens in he proof: convexiy F in x (Theorem ), a regulariy propery of he exercise boundary x (Lemma 4) and a resul (Lemma 5) which allows us o esablish a lower bound for F x jus below he exercise boundary. The smooh-fi condiion for F is as follows. The proof is given a he end of his subsecion, afer esablishing some auxiliary lemmas. Theorem 4. Suppose he exercise boundary is non-increasing. Then he value funcion saisfies he smooh fi condiion, ha is F x (, x ()) = e r e x () V s (, s ()) =, for all [, T ). When he exercise boundary is non-increasing, we have he regulariy resul below characerising he boundary. I saes ha if he log-sock price process sars arbirarily close o he boundary, hen i will hi he boundary in he nex insan. This is in he spiri of he definiion of a regular boundary poin in he conex of he Dirichle problem (see Definiion and Theorem in [). Lemma 4. Suppose he exercise boundary is non-increasing. Denoe by τ (, x) he opimal sopping ime for F (, x), for some (, x) [, T ) R. Then, we have lim τ (, x () ɛ), a.s., T. ɛ Proof. Wihou loss of generaliy, se he saring ime o zero, wrie X(x) X(, x) for he value of he log-sock price given X = x, as well as τ (x) τ (, x) and x () x. For ɛ >, since he exercise boundary is non-increasing we have τ (x ɛ) inf {ρ [, T ) X ρ (x ɛ) x }. (2) From he soluion (7) for X(x) and he Law of he Ieraed Logarihm for Brownian moion (Secion I.6 of Rogers and Williams [7), we have sup X u (x) > x, u ρ a.s., for every ρ >. Hence here exiss a sufficienly small ɛ > such ha sup X u (x ɛ) x, u ρ a.s. Hence he righ hand side of (2) ends o zero as ɛ, and his complees he proof. 3

14 The nex ingredien we need for he proof of smooh fi is he following lemma. Lemma 5. Le (, x) [, T ) R and denoe by X(, x) he log-sock price wih iniial condiion X = x. Denoe by (τ ɛ ) ɛ> a family of T,T L -sopping imes converging o almos surely as ɛ. Then X(, x) saisfies lim ɛ ɛ (exp(x τ ɛ (, x)) exp(x τɛ (, x ɛ))) = e x, Proof. Wihou loss of generaliy, consider a saring ime =. The same ideas apply o any saring ime [, T ). Wrie X(x) X(, x) for he log-sock price wih iniial condiion X = x R. For ɛ >, define u (ɛ) := ɛ (β(u, X u(x)) β(u, X u (x ɛ))), u < T. (22) Using (2), we have (e Xτɛ (x) e Xτɛ (x ɛ)) = [ { ( τɛ )} (x ɛ) ɛ ɛ exτɛ exp ɛ + u (ɛ)du. Using Taylor s expansion, we ge (e Xτɛ (x) e (x ɛ)) ( Xτɛ = e Xτɛ (x ɛ) + ɛ τɛ where O(ɛ) denoes erms of order ɛ or higher. Observe ha a.s. ) u (ɛ)du + O(ɛ), lim u(ɛ) = b(u, X u (x))d u (x), u T, a.s., ɛ where b, D are defined in (3). Then, using he fac ha lim ɛ τ ɛ = (since we have se = ) and e Xτɛ (x ɛ) = e x a.s. complees he proof. Noe ha if he drif β was consan or a deerminisic funcion of ime, hen he lemma would follow direcly from he fac ha in (22) is equal o zero. We now prove he smooh fi condiion. Proof of Theorem 4 (Smooh fi). I enails no loss of generaliy if we se r = and =, bu significanly simplifies noaion. Wrie F (x) F (, x) and x x (). Then F (x) = (e x K) for x x, so F x (x +) = e x. On he oher hand, from Theorem we know ha he mapping x F (x) is increasing and convex, so F x (x ) e x. Hence i suffices o show ha F x (x ) e x. As before le τ (x) denoe he opimal sopping ime for saring sae x. Since τ (x ɛ) is subopimal for F (x), we have [ (exp(xτ F (x) F (x ɛ) E (x ɛ)(x)) K ) + ( exp(xτ (x ɛ)(x ɛ)) K ) + [( ) E e X τ (x ɛ)(x) e X τ (x ɛ)(x ɛ) {Xτ (x ɛ) (x ɛ) log K}, (23) where we have use he fac ha x X(x) is increasing. By Lemma 4 and he fac ha i is never opimal o exercise below he srike, we have Also, by Lemma 5, we have lim {X ɛ τ(x ɛ) (x ɛ) log K} = {X(x ) log K} =, a.s. (24) lim ɛ ɛ ( e X τ(x ɛ)(x ) e X τ(x ɛ)(x ɛ) ) = e x, a.s. (25) 4

15 Using (24) and (25) and noing ha all erms inside he expecaion in (23) are uniformly inegrable, Theorem II.2.2 in Rogers & Williams [7 gives which complees he proof. lim ɛ ɛ [F (x ) F (x ɛ) e x, 3.5 The early exercise decomposiion We now ransform he sae space from log-sock price o sock price in order o sae he early exercise decomposiion for he ESO value funcion V, given by e r V (, s) = F (, log s). Wih his change of variable he smooh fi condiion becomes V s (, s ()) = and he PDE for F in he coninuaion region ransforms o L S V rv =, where L S is he exended generaor of S, given by L S = + α(, s)s s + 2 σ2 s 2 2 s 2, α(, s) = β(, log s) + 2 σ2. (26) We hen have he following decomposiion for V. Theorem 5. The value funcion V of an execuive sock opion wih srike K and mauriy T has he following decomposiion ino a European opion value and an early exercise premium: where V (, s) = e r(t ) E[(S T K) + S = s + p(, s), (, s) [, T R +, (27) p(, s) := T is he early exercise premium. e r(u ) E[((r α(u, S u ))S u rk) {Su>s (u)} S = sdu, (28) Proof. The smooh fi condiion implies ha F x is coninuous. We have ha F xx is coninuous in he coninuaion region and equal o e r+x in he sopping region. Though he second derivaive migh no be coninuous across he exercise boundary we may neverheless apply he generalised Iô formula for convex funcions o F, o obain he Doob-Meyer decomposiion of he Snell envelope: + T F (T, X T ) = F (, X ) + σ F x (u, X u )dbu L T e [(β(u, ru X u ) + ) 2 σ2 r e Xu + rk {Xu>x (u)}du, T, where we have used F (, x) = e r (e x K) for x > x (). Now ake expecaion condiional on X = x (and of course given a known value of L), change variables from X o S, and (28) follows. 3.6 Inegral equaion for early exercise boundary The inegral equaion for he early exercise boundary follows by seing s = s () in (28), yielding he following corollary. To be explici we resore he dependence on he random variable L. For L = l, denoe he insider s exercise boundary by s l (). Using V (, s l ()) = s l () K, we obain: Corollary. For L = l R, he insider s exercise boundary s l saisfies, for T, + s l () = K + e r(t ) E[(S T K) + S = s (), L = l T e r(u ) E[ {Su>s l (u)} ((r α(u, S u ))S u rk) S = s l (), L = ldu. 5

16 3.7 Numerical soluion of early exercise boundary equaion The algorihm used o numerically solve he inegral equaion in Corollary is as follows. For a fixed [, T and l R, we rea he compuaions of he expecaions as European opion prices, wih sock price dynamics under F L given by ds u = α(u, S u )S u du + σs u db L u, wih α defined in (26). These are compued by solving he associaed Black-Scholes syle PDE using a cenral difference scheme, for a sufficienly wide range of s l (). We discreise he inerval [, T and use he rapezoidal rule o approximae he inegral, solving he discreised inegral equaion using he fixed poin mehod. The exercise boundary is compued by backward recursion wih he saring value s l (T ) = K. 5 a =.5 a =.6 a =.7 Sock price Time o mauriy Figure : Insider s exercise boundaries for L = a log 8 wih differen values of he noise coefficien a, r =., µ =.2, σ =.2, T =, K = 8. Figure shows he insider s exercise boundaries when he sock appreciaion rae µ is higher han he ineres rae. The insider possesses noisy log-sock price knowledge wih L = a log 8 wih a =.5,.6,.7, so he insider knows ha he ESO is likely o be a-he-money a mauriy wih varying degrees of cerainy. The impac of inside informaion in his case is clear. Recall ha i is no opimal for he regular agen o exercise early when µ r. 3 This conclusion is alered for he insider, who has greaer cerainy han he regular agen ha he opion will expire ou of he money, and his induces early exercise. We also observe ha he exercise boundary is lower as a increases owards, and he privileged informaion becomes less noisy. The insider becomes more cerain ha he opion will expire worhless and early exercise is induced a lower hresholds. Figure 2 shows he regular rader s and insider s exercise boundaries when µ < r. The insider possesses noisy log-sock price knowledge where L =.5 log S T wih S T = 78, 8, 82, 9. 3 Indeed, aemps o solve for he regular agen s exercise boundary numerically when µ r leads o divergence when execuing he fixed poin mehod. 6

17 Regular rader S T = 78 S T = 8 S T = 82 S T = 9 9 Sock price Time o mauriy Figure 2: Regular rader s and insider s exercise boundaries for L =.5 log S T values of S T, r =., µ =, σ =.2, T =, K = 8. wih differen For S T = 78 and S T = 8, he insider has a lower exercise boundary han he regular agen due o his pessimisic inside informaion, in a similar vein o he resuls in Figure. For S T = 82, he insider knows ha he ESO is likely o be in-he-money, ye sill exercises he ESO earlier han he regular agen. Alhough he fac ha he ESO is likely o end up in-he-money ends o delay exercise, here is a compeing effec of a lower variance in he sock price as perceived by he insider, and his induces earlier exercise. For he case S T = 9, he privileged informaion is sufficienly opimisic so ha he insider exercises laer han he regular rader. This suggess ha inside informaion has wo poenially compeing effecs: a reduced variance of he sock price ha hasens exercise and a direcional effec, which can hasen or delay exercise. 4 Conclusions Using an iniial enlargemen of filraion o augmen a Brownian filraion wih noisy informaion on he value of a sock a he mauriy ime of an ESO, we have analysed he sopping decision faced by an insider who does no rade he sock or oher securiies. This shows ha he insider can exercise he ESO before mauriy, in siuaions in which a regular agen would no. This involved esablishing fundamenal properies of he value funcion (noably convexiy, ime decay and smooh fi) when he price process is a ime-inhomogeneous diffusion. This paper has se a framework in which such quesions can be sudied. An ineresing direcion for fuure work is o add rading opporuniies in ouside asses and risk aversion for he agen. References [ A. Danilova, M. Monoyios, and A. Ng, Opimal invesmen wih inside informaion and parameer uncerainy, Mah. Financ. Econ., 3 (2), pp

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