Implied volatility phenomena as market's aversion to risk

Transcription

1 Implied volailiy phenomena as marke's aversion o risk S.Siyanko, Universiy College London, Gower Sree, London WC1E 6B, UK, M. Ponier, Insiu Mahémaiques de oulouse, France June 27, 2017 Absrac In his aricle, we show how o price derivaives in he presence of jumps and, in paricular, when jumps are direced owards, wha marke paricipans believe, is he rue, inrinsic value of an asse dened by is company fundamenals. Geomeric Brownian moion is considered as a model for he asse price dynamics and jumps are inroduced owards a deerminisic exponenial funcion. he funcion represens he fundamenal value, and jumps are here o model price correcions in response o a new informaion available o he marke. he objecive is no only o inroduce an alernaive pricing formula bu, more imporanly, o have an algorihm ha explicily prices and measures he risk of price correcions assumed by he marke. Keyword: derivaives pricing, implied volailiy, mean reversion hrough jumps, risk of price correcions. 1 Inroducion he Black-Scholes formula is one of he mos renowned expressions in nancial mahemaics [2]. I is used by praciioners o calculae prices of sandard call or pu opions which are respecively conracs o buy or sell an asse a a predened price and a a predened ime. he formula requires six inpus - volailiy of he underlying asse, is spo price, ime o mauriy, srike, risk-free ineres rae, and dividend yield. A rs sigh, i may be somewha of a surprise ha opion prices only depend on he spo price bu no on any oher informaion abou he underlying such as is expeced reurn. Afer all, one migh argue ha such informaion should clearly inuence buy decisions because nobody would wish o purchase an opion ha is likely o be ou of he money a is mauriy. his in urn should inuence he demand bu somehow i does no happen in he Black-Scholes analysis. Of course, hose adep in nancial mahemaics would noe ha i is no he expeced value of he underlying bu raher he possibiliy o replicae he opion's nal payo ha ses he price. I happens ha one can sell an opion and hen mee one's obligaions a mauriy by acively managing a self-nancing porfolio consruced from he underlying asse and a money marke accoun. I is hen he iniial cos of his porfolio ha ses he price. Indeed, if his was no he case, i would be possible o creae an arbirage sraegy by aking opposie posiions in he replicaing porfolio and he opion (you would need o shor he opion if i is overpriced or buy i if i is underpriced). I is hen he willingness of large insiuions o do arbirage ha will fuel demand and supply in he direcion of he Black-Scholes formula and make he opion price indieren o he expeced rae of reurn on he underlying asse. One of he key assumpions behind he Black-Scholes model is ha perfec replicaion is possible or, a leas, we can replicae he payo on average. his is usually referred o as 1

2 marke compleeness. In realiy, he markes are no complee, and spo prices are no he only informaion abou he underlying ha markes respond o. Expeced reurn on an asse can be esimaed by marke analyss hrough so-called fundamenal analysis which akes ino accoun he rm's income generaing abiliy. his clearly is of high imporance for hedge funds and indeed invesors in general. A he same ime, we know ha here is echnical rading going on which can make asses over or undervalued wih respec o heir rue, inrinsic value dened by he fundamenal analysis. When his happens, marke paricipans may expec a correcion in he asse price, and he correcion would be sudden and unpredicable (oherwise, one could easily make money by buying an asse when i is undervalued and selling when i is overvalued). I is worh noing ha he price correcion may no happen a all (afer all, fundamenal analysis is no exacly a precise science) bu i is he expecaion of his correcion ha is imporan. Clearly, opion sellers would see he possibiliy of price correcions as a risk. For example, if marke paricipans expec an asse price o suddenly go up, he wrier of a call opion will be cauious abou selling i cheap because he may no be able o hedge he risk of facing a higher payou by immediaely readjusing posiions in he underlying. his risk would obviously arac some premium, and, since here is a higher demand for his opion (i is more likely o end up in he money) and limied arbirage opporuniies (he marke is no complee), we have a new price ha is higher han he price prediced by he Black-Scholes model. We hypohesize ha a premium for he risk of price correcions is indeed included in opion prices, and i manifess iself hrough marke phenomena called volailiy skew and volailiy smile. While, for he purpose of his research, his is an assumpion, here is some evidence o sugges ha he hypohesis is reasonable. Web sies argeed a invesors ofen encourage hem o inerpre volailiy skews as a sign of a bullish" or bearish" marke in he underlying, and i is clear ha boh behaviours, being speculaive in naure, indicae misalignmen of he sock price wih is fundamenal value. he volailiy smile phenomenon may also appear afer a sharp price correcion in he underlying (e.g. following a nancial crisis) which suppors he hypohesis because one would expec he volailiy smile o appear when he sock price is close o wha markes believe is is rue, fundamenal value. here are, of course, many models ha would have a good o volailiy skews and smiles. Perhaps he mos known and well used in he indusry would be he Heson model [8]. I has wo correlaed Brownian moions driving he square of sochasic volailiy and he price process iself. he sochasic volailiy process is a mean-revering process, and he moivaion comes from he analysis of he saisics of log reurns. I is known ha he physical volailiy of a sock normally exhibis volailiy clusering (small moves are more likely o be followed by small moves and large moves by large) suggesing auocorrelaion and hus moivaing he mean reversion because mean-revering processes would normally exhibi auocorrelaion. In conras, we will have a mean reversion ha does no rely on or describe saisical properies of sock prices. I is fundamenal ha derivaives prices in our model will be formed no by a response o physical sock price movemens bu by a fear ha sharp correcions in he sock price may occur. Having large jumps wih low probabiliy may simply no manifes iself in he saisics of log reurns for years bu i is clear ha such evens may have a signican inuence over derivaives prices. I is also ineresing ha ing volailiy surface o he Heson model means ha i does price risk bu prices i exacly a he same level as everybody else. his is because, if model parameers are no independen of he model, he only way o esimae hem is hrough calibraion o hisoric derivaives prices. If we are o price risks dierenly from he marke, he derivaives would eiher be oo expensive o sell (if we are overpricing risks) or we would expose ourselves o greaer risks han he oher nancial 2

3 insiuions. ha would lead us o conclude ha models ha are pricing risks explicily (or, o ha maer, jus exposing hem) are no going o be very popular in he indusry. hey would simply no serve he business objecive of pricing risks a he same level as everybody else does i. I is only when one wans o measure risks assumed by he marke or/and, perhaps, o price risks in a more responsible way, our model would have is superior uiliy. Mean reversion hrough jumps is used o model sharp correcions in he sock price, and here are many models ha have his feaure as well. his is pariculariy common when we consider asses ha can no be sored, e.g. elecriciy. he non-sorabiliy of elecriciy means ha insananeous shocks o demand or supply (e.g. increase in demand due o a hea wave) are no smoohed by invenories and will resul in shocks o he spo price followed by mean reversion o levels deermined by he marginal coss of generaing i. An example of such model would be [7]. Here, we have jumps and heir direcion (bu no size) depends on upper and lower price hresholds in a way ha leads o mean reversion. here is also a research by [1] which are paricularly useful. I happens ha he problem for derivaives pricing was sudied for a very wide class of diusion processes wih jumps, and he processes are similar o he risk-neural sock price dynamics in our model. However, here are echnical consrains ha ensure ha soluions are of an exponenial ype. he model here can be viewed as an exension o his framework leading o very dieren ype of soluions - soluions of a non-exponenial ype. Finally, here is a model ha has been inroduced by [13]. I is acually a special case of our model bu in he conex of pricing defaul risks - i is he Meron's jump-o-ruin model. If we se he fundamenal value of he sock o zero, he risk of he price correcion is eecively he same as he risk of defaul. While his is slighly ou of conex, [6] does commen on he suiabiliy of his model o describe he lef wing (low srike srucure) volailiy skew and, in paricular, for he socks wih high credi spreads. He has also commened ha he model does no generae he righ wing which is jus no consisen wih he daa. Since we will have boh upside and downside risk of he price correcion, our model will produce boh wings of he volailiy surface, and one can view i jus as a naural exension o he Meron's jump-o-ruin model. We sar by inroducing mean-revering jumps ino he geomeric Brownian moion model (Secion 2), and show ha his price dynamics does no lead o a reasonable arbirage-free marke bu moivaes a dieren model. he new model is arbirage-free and leads o a pricing PDE ha we can use o price derivaives. Surprisingly, we obain he same pricing PDE hrough he local no-arbirage argumen and wih a very reasonable replicaing sraegy applied o he iniial price dynamics (Secion 3). hen, we simplify our PDE, derive is border condiions, presen some properies of he soluion and show how o solve i numerically (Secion 4). Finally, we presen numeric resuls (Secion 5) followed by some pracical advice for pricing applicaions (Secion 6) and he conclusion (Secion 7). 2 Sock price dynamics In he Black-Scholes framework for pricing derivaives, we have sandard geomeric Brownian moion as a model for sock price dynamics. We can also use i bu only o model sock price beween price correcions, and, as for he price correcions, we can have jumps a random imes dened by a sandard Poisson process. he jumps should be direced owards wha people hink is he fundamenal value of he sock, and we can model i as a sandard exponenial funcion. We can also consider he same model bu wih a compensaed Poisson 3

4 process. While i is no clear how i makes nancial sense, compensaed Poisson process is a maringale which is ofen desirable when pricing derivaives. here is one propery ha we would like our models o have. I comes from he noion ha price correcions are rear evens ha are managed and priced as risk. We wan o ensure ha we obain a pricing algorihm ha does no conradic risk managemen behaviors observed in he marke. Afer all, we wan an algorihm ha explicily prices and measures he risk of price correcions, no jus anoher algorihm for pricing derivaives. We are herefore going o inroduce wo models. One wih a sandard Poisson process and he oher wih is compensaed version. We will sar wih he rs model and show how, in ligh of he above requiremen, i moivaes he inroducion of he second. 2.1 Geomeric Brownian moion wih Poisson jumps Le us dene a lered probabiliy space {Ω, A, F = (F, 0 ), P } on which we have an independen pair of a sandard Brownian moion W and a Poisson process N wih consan inensiy λ. Here, we presen our rs aemp o model he sock price wih mean-revering jumps: (1) ds = µs d + σs dw + (S 0 e µ S )dn, S 0 L 1 (Ω, F 0, P ), 0 mauriy, where S 0, W, N are independen, and W, N are F-adaped. Here, we are using sandard noaions for he sock price S, is drif rae µ, volailiy σ, and ime. hese are inpus for he geomeric Brownian moion model. In addiion, we have he sandard Poisson process N as an exra source of randomness and he fundamenal value of he sock a ime : S 0 e µ = S wih consan S 0. A jump imes ( n, n = 1, 2,...), he las erm in he above equaion inroduces an incremen equal o he dierence beween he sock price and is fundamenal value, hus simulaing price correcions. Le us remark ha his SDE is solvable and admis he unique posiive soluion: (2) S = S 0 exp [(µ σ 2 /2) + σw ] > 0, N = 0 S = S 0 e µ N exp [(µ σ 2 /2)( N ) + σ(w W N )] > 0, N > 0, i.e. we have a sandard geomeric Brownian moion evolving from is iniial value a ime zero or from is fundamenal value a jump ime N. Proposiion 2.1 here exis F-adaped processes ψ, γ, (γ > 1) which dene a risk-neural probabiliy measure: Q = L P, where L = ρ W ρ N, where dρ W = ρ W ψ dw, dρ N = ρ N γ (dn λd). So his marke is viable, here is no arbirage opporuniy. Proof: Le us now look a how we can price derivaives and, following he sandard riskneural valuaion framework, we sar wih a change of probabiliy measure ha will make our process risk-neural, i.e. here exis maringales W and M such ha (3) ds = rs d + σs d W + (S 0 e µ S )d M, S 0, mauriy. 4

5 I means ha we are looking for a pair ψ, γ such ha he following processes are maringales: (4) dρ W = ρ W ψ dw dρ N = ρ N γ (dn λd), and from Girsanov's heorem (cf. Jeanblanc Yor page 69 and page 478 [10]) we know ha if he process L = ρ W ρ N is a maringale hen under Q = L.P : M = N W = W 0 0 ψ s ds λ(1 + γ s )ds, where W is a Q Brownian moion and N admis he inensiy λ(1+γ ) under Q. Subsiuing W and N in (1) yields: ds = µs d + σs (d W + ψ d) + (S 0 e µ S )(d M + λ(1 + γ )d), and we deduce (3) if he following equaion is saised: (5) r = µ + σψ + S 0e µ S S λ(1 + γ ). Since we do no have a unique soluion for ψ and γ (he marke is no complee), le us propose: (6) 1 + γ = inf(s, c), for some consan c > 0. his leads us o he following expression for ψ: ψ = r µ σ + (1 S 0e µ ) λ inf(s, c). S σ Such a ψ is bounded on he inerval [0, ] because 0 inf(s,c) S 1. From heorem II.2.a in [11] we know ha ρ W is a square inegrable maringale. Le us use M o denoe he maringale 0 γ s(dn s λd) and conrm from expression (6) ha he predicable compensaor of 0 γ sdn s, which is λ 0 γ sds, is bounded on [0, ]. I means (cf. heorem II.2.b in [11]) ha ρ N is a maringale wih inegrable variaion. We herefore have an equivalen probabiliy measure Q, under his measure he inensiy of price correcions λ is given by (7) λ = (1 + γ )λ, where γ is dened in (6). I is ineresing ha λ is a consan for S c, and his is wha people assume when pricing risks. Bu he drawback is ha λ depends on S when S is below c. Does i make any nancial sense, and if no why does i happen? We know from equaion (2) ha S > 0 P almos surely, so also Q almos surely because probabiliy measures P and Q are equivalen. Neverheless, wha acually sops us from posulaing ha, when pricing risk, one is using he physical probabiliy measure wih consan λ? I seems like a reasonable assumpion o make. ha would mean ha γ = 0, and he associaed process L could be only a local 5

6 maringale, because we do no have easy sucien condiion on ψ = r µ σ + (1 S 0e µ S ) λ σ (as Novikov's condiion) o be saised. And if L is no a maringale, we can no dene he equivalen probabiliy measure Q, and (under his consrain) he marke may allow an arbirage which is ne. echnical rading does mean arbirage, his is how echnical raders make money, and he inroducion of echnical rading is indeed our objecive here. On a posiive side, i leads o a much simpler pricing rule dened by (3) bu wih consan jump inensiy λ. Moreover, we can redene our lered probabiliy space and say ha his is our process under a measure Q (dieren from he Q above). hen we can inroduce a change of probabiliy measure beween P and Q by saing ha γ = 0, ψ = (r µ)/σ, and we are back o he sandard framework of nancial mahemaics. he only problem is ha our newly dened process allows negaive values bu here are many models ha do his, we jus need o posulae ha our case of ineres is when he probabiliy of such evens is low. 2.2 he Model We are operaing in a lered probabiliy space {Ω, A, (F, 0 ), P } endowed wih a lraion F generaed by a sandard Brownian moion W and a Poisson process N wih consan inensiy λ and independen of W. We can wrie our sock price dynamics under P as: ds = µs d + σs d W + (S 0 e µ S )(dn λd), S 0 R +, 0 mauriy. We now dene an equivalen probabiliy measures Q = L P wih he densiy of probabiliy µ r measure dl = L σ d W and under Q we have: (8) ds = rs d + σs dw + (S 0 e µ S )(dn λd), where dw = d W + µ r σ d. Here we only change he Brownian moion par, and under boh probabiliy measures P and Q he inensiy λ is he same. As in he previous model, here exiss a risk-neural probabiliy measure, namely Q, so once again his marke is viable, here is AOA (absence of opporuniy arbirage). Proposiion 2.2 he unique soluion o (8) under probabiliy measure Q, before he rs N jump ime 1 -meaning on he even { N = 0}, is ( σ2 ) (r+λ S = e 2 )+σw S 0 λ S 0 e µs σ2 (r+λ (9) e 2 )s σw s ds 0 and similarly, on he even { N > 0}, he soluion is: (10) ( σ2 (r+λ 2 S = e )( N )+σ(w W N ) S 0 e µ N λ S 0 e µs e (r+λ σ 2 2 )(s N ) σ(w s W N ) ds ). N Proof: We can check, via Iô's formula, ha, before he rs N jump ime 1, he unique soluion o (8) under probabiliy measure Q is S = Y A, where A = S 0 λ 0 S 0e µs Ys 1 ds and Y = exp[(r + λ σ 2 /2) + σw ]. Indeed, his is a produc of Y saisfying dy = Y [(r + λ)d + σdw ] 6

7 and he nie variaion process A saisfying da = λs 0 e µ Y 1 d. So he dierenial of he candidae soluion is Y da +A dy = Y λs 0 e µ Y 1 d+a Y [(r+λ)d+σdw ] = A Y [(r+λ)d+σdw ] λs 0 e µ d. hus, he explici soluion before he rs N jump ime 1 is ( σ2 ) (r+λ S = e 2 )+σw S 0 λ S 0 e µs σ2 (r+λ e 2 )s σw s ds on he even { N = 0}, and similarly, on he even { N > 0}, we can show ha he soluion is: ( σ2 (r+λ 2 S = e )( N )+σ(w W N ) S 0 e µ N λ S 0 e µs e (r+λ σ 2 2 )(s N ) σ(w s W N ) ds ). N 0 Noe ha, since he process λ 0 S 0e µs e (r+λ σ 2 2 )s σw s ds is coninuous and sricly increasing, S may ake negaive values, e.g. afer ime (11) τ := inf{ 0 : λ 0 S 0 e µs σ2 (r+λ e 2 )s σws ds S 0 } on he even {τ < 1 } and, more generally, beween wo jump imes when λ S 0 e µs e (r+λ σ 2 2 )(s N ) σ(w s W N ) ds S0 e µ N. N his is no desirable when pricing derivaives bu, on a posiive side, we believe ha having consan λ beer reecs pricing behaviors in he marke, and i also helps wih racabiliy of he soluion. Moreover, λ as a consan means ha here is a very reasonable replicaing sraegy ha we can apply o our iniial price dynamics (1) and, wih he help of he local no-arbirage argumen, derive he same pricing PDE as in his model. ha concludes he inroducion of he model for he sock price dynamics. We can now proceed o derive he pricing PDE. 3 Deriving he pricing PDE 3.1 Firs approach hroughou his secion we are operaing under he risk-neural probabiliy measure Q. Le us sar by noing ha he process (8) is a Markov process which leads us o conclude ha he opion's price C is a funcion of S. Indeed, we can invoke he fundamenal heorem of asse pricing and wrie: (12) C = e r( ) E Q [(S K) + /F ], and, since S is a Markov process, here exiss a regular funcion C such ha (13) C = C(, S ). 7

8 Proposiion 3.1 Assuming he funcion C dened on [0, ] R] is C 1,2, i is soluion o he parial derivaives equaion (14) C (, s)+1 2 σ2 s 2 2 C (, s) = r[c(, s) s C s2 s (, s)] λ[(c(, S 0e µ ) C(, s)) C s (, s)(s 0e µ s)], wih boundary condiions (15) C(, s) = (s K) +, s C(, s) = 0 if s 0. Proof: he process e r( ) C(, S ) is an (F, Q)-maringale and using Iô's formula we cancel is nie variaion par: C +rs C s σ2 S 2 2 C s 2 +λ[(c(, S 0e µ ) C) (S 0 e µ S ) C s ] = rc(, S ) d dq a.s.. he sochasic process S admis he suppor R, so we can replace S (ω) by s R and obain he PDE on R + R: C (, s)+1 2 σ2 s 2 2 C (, s) = r[c(, s) s C s2 s (, s)] λ[(c(, S 0e µ ) C(, s)) C s (, s)(s 0e µ s)], where we have [0, ), s (, + ), and he erminal condiion is: C(, s) = (s K) +. While border condiions (a s = 0 and s + ) are no necessary o dene he PDE, hey are useful for numeric mehods, and we will derive hem laer. 3.2 hrough he local no-arbirage argumen. I is ineresing ha we can derive he same PDE from he iniial price dynamics (1): (16) ds = µs d + σs dw + (S 0 e µ S )dn Noe ha, formally speaking, his is no our process process under Q or P because (12) prices he opion under Q, no P. his is he price dynamics we have sared from in order o moivae he inroducion of our model saed in he previous SDE (1). Le us sar wih looking for a self-nancing porfolio: (17) V = ϕ S + ψ B wih some predicable processes ϕ and ψ denoing our posiions in sock and he money marke accoun. We now ry o replicae he opion over a small ime inerval d. Recalling he assumpion ha he opion price process C is a regular funcion of (, S ) and wih he help of Iô's lemma we obain: dc(, S ) = C d + µs C s d + S C s σdw σ2 S 2 2 C (18) s 2 d + [C(, S 0e µ ) C(, S )]dn. On he oher hand, we know ha for he self-nancing porfolio V we can wrie: (19) dv = ϕ ds + ψ db = ϕ ds + r(v ϕ S )d. 8

9 hen, we assume ha C and V admi he same nie variaion par d dq almos surely on [0, ) Ω. hus, we dela hedge: ϕ = C s (, S ) beween jumps. Remark ha wih such a porfolio we equae Brownian moion coeciens on boh sides. However, we can no equae coeciens of Poisson process: jumps will inroduce a misalignmen beween he value of he replicaing porfolio and he call opion equal o: (20) C(, S 0 e µ ) C(, S ) ϕ (S 0 e µ S ) meaning ha jumps are managed as a risk, i.e. we are using he physical probabiliy measure and replicae on average": (21) E [dc dv ] = 0. ha is, if a jump happens, one needs o immediaely re-adjus his posiions in he sock price (as per he dela hedge) and he money marke accoun. his means ha we can no replicae he opion's price process exacly. Bes we can do is o ensure ha we replicae he payo on average": (22) (23) V = C(, S ) E[C(, S )/F ] = E[V /F ]. Since he above would hold rue for maringales, we jus need o idenify and equae nie variaion pars of semi-maringales (18) and (19) which yields: (24) C + µs C s σ2 S 2 2 C s 2 + λ[c(, S ) C(, S )] = µs C s + λ(s 0e µ S ) C s + r(c S C s ), and afer cancellaion we recover he PDE (14). We have herefore presened a replicaing sraegy for our iniial price dynamics (1) leading o he same PDE as in he no-arbirage marke model exhibied in Secion (2.2). he sraegy is very reasonable - markes do dela hedge and ignoring jumps means ha we are over or under inaing (vs. Black-Scholes formula) our posiion in he money marke accoun wih a view o aain he payo on average. Moreover, from (20) we can see ha, if C(, s) is a convex funcion of s, we are over inaing our posiion in he money marke accoun o ensure ha we can recover losses from jumps, i.e. we dela hedge and manage he possibiliy of jumps as a risk. Remark 1 Our iniial price dynamics (1) is sricly posiive bu he PDE is dened on s R. his hins a a possibiliy o resric i o s R + {0}. Noe, such resricion is no possible in general, and his resul is no rivial. 4 Pricing PDE Le us now move on o presen he pricing PDE and explore some of is properies. For simpliciy, we limi he scope o sandard vanilla call opions and posulae ha we operae in a risk-free environmen wih he pricing rule (12) and he sock price process (8). Wih 9

10 he help of Iô's lemma, we have derived PDE (14), and we can now simplify i furher by inroducing an auxiliary funcion u on [0, ] R + saisfying: (25) C(, s) = u(, s)e (r+λ)( ) + λe r( ) e λ( v) u(v, S 0 e µv )dv. Proposiion 4.1 Assuming he exisence of a funcion u saisfying (25), he PDE (14) is equivalen o he following PDE wih u being a soluion of i: (26) u σ2 s 2 2 u = s(r + λ) u s2 s + λs 0e µ u s, u(, s) = (s K)+. Proof: We can use he expression for C(, s) (25) o calculae is derivaives C, C s, 2 C s 2 and C(, S 0 e µ ) wih respec o he derivaives of he fncion u. hen, plugging all of hese derivaives ino he PDE (14) and is erminal condiion (15) leads o u σ2 s 2 2 u = s(r + λ) u s2 s + λs 0e µ u and obviously u(, s) = C(, s) = (s K) +. Conversely, if u is a soluion of (26), he funcion C dened by (25) saises he original PDE (24). When working wih jumps, i is normal o end up wih equaions of an inegro-dierenial ype. he inegro" par comes from changes in he derivaive's price following jumps, and i could signicanly complicae he soluion. I is ineresing ha here is no inegro" par in he above PDE (26) - we have removed i by subsiuing C(, s) wih u(, s). his is acually a special case, i is no possible o remove he inegro" componen in general. As for he soluion o he PDE iself, we have a sandard Black-Scholes Hea equaion bu wih a convecion erm arising from he drif λ(s 0 e µ S )d in he price dynamics (8). he drif has a non-proporional componen wih respec o he sock price S which signicanly undermines analyical racabiliy of he PDE soluion. A similar siuaion would arise, for example, if we have non-proporional dividends paid on a sock. While his is how dividends are normally paid, i is ofen assumed ha hey are proporional o he sock price, and analyical formulae can be derived and used as good enough esimaes. Unforunaely, if we are o assume ha he fundamenal value of he sock is proporional o is spo price, we are going o remove he mean reversion. If we are o solve he PDE (26) as i is, we need o hink of a case where similar equaions may appear. Using sandard Feynman-Kac's formula, e.g. [10] , we ge he corollary Corollary 4.2 he unique soluion o (26) is he following (27) u(, s) = E[(X,s K)+ ], where (28) X,s = e(r+λ σ 2 2 )( )+σ(w W ) (s λs 0 e µ Remark 2 Such a funcion u belongs o C 1,2 s ) σ2 (r+λ µ e 2 )(u ) σ(w u W ) du. because we have a linear second-order PDE of he parabolic ype, and arguing from he PDE heory, our payo funcion is of he righ ype o ensure exisence and uniqueness of he PDE soluion which is C 1,2. Counable number of disconinuiies in he payo funcion do no maer. 10

11 Proof: he PDE (26) is associaed o he operaor Lf = f σ2 s 2 2 f + s(r + λ) f s2 s λs 0e which generaes he sochasic dierenial equaion he soluion of which being: X,s µ f dx = X ((r + λ)d + σdw ) λs 0 e µ d, X = s, = e(r+λ σ 2 2 )( )+σ(w W ) (s λs 0 e µ For ha, i helps o re-wrie u(, s) as an expecaion: where we obain X,s X,s u(, s) = E[(X,s K)+ ], s ) σ2 (r+λ µ e 2 )(u ) σ(wu W) du. hrough a sraighforward applicaion of he Feynman-Kac's formula: = e(r+λ σ 2 2 )( )+σ(w W ) (s λs 0 e µ ) σ2 (r+λ µ e 2 )(u ) σ(w u W ) du. In oher words, acually i is S before he rs N jump ime 1 (9) bu evolving from sae s a ime. he removal of he inegro" componen in PDE (14) is equivalen o he removal of jumps from he underlying SDE (8). he inegral in he above equaion is proporional o he average of a geomeric Brownian moion over ime, i.e. when we manage he risk of price correcions, he risk premium depends on he full pah of he Brownian moion on [, ], no jus on is value a mauriy. From he Finance poin of view, i means ha, when esimaing he risk premium, one would no jus care abou where he sock price is going o end up a mauriy (which is he case in he Black-Scholes world) bu would also hink abou he average posiion of is price wih respec o he fundamenal value during he life ime of he opion. his reasoning suggess ha, since he pah-dependence is an imporan par of he risk managemen behaviour, i could be dicul o improve analyical racabiliy of he soluion because i is he inegral over ime ha inroduces λs 0 e µ s u erm ino our PDE. he mos prominen case of pah-dependen opions are Asian opions. Here, averaging over ime is used o reduce he impac of he las minue price jumps on he nal payou a mauriy which acually is an example of a risk managemen behaviour bu on he opion buyer's side. In conras, we are inroducing a risk premium for he opion sellers. Neverheless, he mahemaical formulaion of he problems appears o be very similar: in boh cases we have pah-dependence and averaging over ime. 4.1 Properies of he soluion he problem of solving PDE (26) is mahemaically equivalen o he problem of pricing Asian opions, and here are no known analyical ime-domain soluions o i. Only he Laplace ransform wih respec o ime is available in erms of hypergeomeric funcions [15]. Neverheless, here are sill some insighs we can ge from he equaion iself. For example, 11

12 (25) yields s C(, s) = e (r+λ)( ) s u(, s) and 2 ssc(, s) = e (r+λ)( ) 2 ssu(, s). hen, we can diereniae he expecaion" form of u(, s) (27) wih respec o s and obain (29) 2 s C(, s) = E[e σ2 2 ( )+σ(w W ) θ(x s, K)] ssc(, s) = E[e (r+λ σ2 )( )+2σ(W W ) δ(x s, K)], where diereniaing he call opion payo gives us Heaviside θ and Dela δ funcions. For he rs resul, we can commue expecaion wih diereniaion because we can bound he non-negaive expression inside expecaion's brackes by he coecien e σ2 2 ( )+σ(w W ) which does no depend on s and has a nie expecaion which is equal o 1. he resul iself follows from he mean value and dominaed convergence heorems. Furhermore, he coecien e σ2 2 ( )+σ(w W ) can operae as a probabiliy measure change (Girsanov's heorem) meaning ha s C(, s) is in fac a cumulaive disribuion funcion of he following random variable: (30) Y,K = Ke (r+λ σ 2 2 )( ) σ(w W ) + λs 0 e µ σ2 (r+λ µ+ e 2 )(u ) σ(w u W ) du, and 2 ssc(, s) is is probabiliy densiy funcion: (31) s C(, s) = E[θ(s Y,K )] = P (Y,K s) 2 ssc(, s) = s P (Y,K s) = f Y,K (s) where acually P F = e σ2 2 ( )+σ(w W ) Q F. Boh derivaives are sricly posiive suggesing ha he call opion's price is a sricly increasing and convex funcion wih respec o he sock price. We can also see ha s C(, s) = P (Y,K s) goes o 1 as s goes o inniy because i is a disribuion funcion. he rs derivaive wih respec o s denoes he posiion of he dela hedge, and i herefore makes sense for he value of he hedge o be an increasing funcion wih respec o he sock price and no o exceed he sock price iself. hese are essenially he same properies of he soluion ha we have in he Black-Scholes world. he convexiy also means ha, afer a jump, he value of he replicaing porfolio will always be below he call opion's price because is value evolves along he angen line which would be below he call opion's price if i is a convex funcion of s. his means ha jumps are always bad for opion sellers, hey presen hemselves as a risk, and one should herefore manage hem as such. I is also ineresing o see wha happens if we vary he payo funcion which is equivalen o changing he erminal condiion in he PDE for u (26). For example, we can make i equal o he sock price iself, and a search for a soluion in he form of se (r+λ)( ) + f(), f(), f(0) = 0 will give us: (32) u(, s) = se (r+λ)( ) λs 0e µ [e µ( ) e (r+λ)( ) ]. µ r λ hen, we can plug he above expression for u(, s) ino (25) and obain he following rivial ideniy: (33) S = s. 12

13 he resul is no as sraighforward as i may seem, one needs o ake several inegrals o derive i bu he equaliy neverheless does hold which means ha we can use he same pricing rule o price he sock iself. I is also clear ha we can apply our pricing rule o anoher rivial asse - a bond ha pays one uni of currency" a mauriy. Seing he erminal condiion in he PDE (26) o 1 gives us u(, s) = 1 and from (25) (34) P = e (r+λ)( ) + λe r( ) e λ( v) dv = e r( ). hese wo ideniies were cerainly expeced. I is obvious ha we can use he risk-neural measure Q o price he sock and a bond no jus call opions. hey also imply ha he call-pu pariy relaionship beween call and pu opion's prices (C, P ) will be rue as well: (35) C P = S Ke r( ). his is because he dierence of he call and pu payous is a linear funcion of he sock price iself, and we can easily derive he above equaion by aking expecaions in our original pricing rule (12). 4.2 Boundary condiions In order o solve PDE (26) numerically, i would be useful o have boundary condiions for large values of s and for s = 0. Proposiion 4.3 he boundary condiions for he funcions u and C are: u(, s) = s C(, s) = s u(, s) = 0 for s 0 and as a special case: u(, 0) = s C(, 0) = s u(, 0) = 0. hus (36) lim su(, s) = 0 ; s lim s C(, s) = 0. s u(, + ) se (r+λ)( ) K λs 0e µ [e µ( ) e (r+λ)( ) ] µ r λ u(, 0) = s u(, 0) = 0, and he expression for C(, s) (25) gives us: C(, + ) s Ke (r+λ)( ) λs 0e µ [e (µ r λ)( ) 1] + λe r( ) µ r λ (37) C(, 0) = λe r( ) e λ( v) u(v, S 0 e µv )dv ; s C(, 0) = 0. e λ( v) u(v, S 0 e µv )dv Proof: Recall ha s u(, s) = e (r+λ)( ) s C(, s) which is proporional o he hedge posiion ϕ. We do expec i o be zero because a s = 0 he only risk we have is he risk of jumps, and we are no hedging for i. Indeed, le us recall he soluion exhibied in (28): X,s = e(r+λ σ 2 2 )( )+σ(w W ) (s λs 0 e µ ) σ2 (r+λ µ e 2 )(u ) σ(w u W ) du, 13

14 and noe ha, since he process λs 0 e µ sricly increasing, X,s (38) τ := inf{ : shall ake negaive values afer ime λs 0 e µ e (r+λ µ σ 2 2 )(u ) σ(w u W ) du is coninuous and σ2 (r+λ µ e 2 )(u ) σ(w u W ) du = s} <. Moreover, he process will ake sricly negaive values a mauriy if i sars from zero or below zero. ha however means ha expecaions in he expressions for u(, s) (27) and s C(, s) (29) are zeros because he payo and Heaviside funcions are zeros for negaive values of heir argumens, and we can immediaely see ha u(, s) = s C(, s) = s u(, s) = 0 for s 0 and as a special case: u(, 0) = s C(, 0) = s u(, 0) = 0. As for he boundary condiion for large values of s, we can derive i from PDE (26) by replacing is erminal condiion wih u(, s) = s K. In fac, we can use he soluion ha we had obained for he erminal condiion u(, s) = s: (32). Since PDE (26) does no have u(, s) bu only is derivaives, we only need o reduce i by K: (39) u(, + ) se (r+λ)( ) K λs 0e µ [e µ( ) e (r+λ)( ) ]. µ r λ We also wan o check ha he following asympoic expression holds: (40) se (r+λ)( ) K λs 0e µ [e µ( ) e (r+λ)( ) ] µ r λ as s +. I helps o re-wrie i as u(, s) 0 (41) E[(X,s K)] E[(X,s K)+ ] = E[(K X,s )+ ] which ends o zero as s + because i is posiive and a sricly decreasing funcion of s for s > 0. he laer is because he above expecaion solves for he pu opion, and he pu-call pariy relaionship (35) suggess ha he pu opion's derivaive wih respec o s is in (0, 1] and ends o zero when s + which is wha wha we would expec in he Black-Scholes world. his concludes he proof. I is ineresing ha, according o Remark 1, we can resric PDE (26) o s R + {0} and forge abou negaive values of he sock price. Noe ha his is a very special case which comes from he fac ha he process X never goes back o posiive values afer i ouches he line s = Solving he PDE numerically Since our problem (26) is mahemaically equivalen o he problem of pricing Asian opions, one may refer o an abundance of numerical mehods available in his domain. Unforunaely, here are no analyical soluions which signicanly undermines pracical applicaions of our model. However, i is helpful o noe ha, since we are rying o explain he implied volailiy phenomena, our objecive is o adjus he Black-Scholes formula raher han o come up wih fundamenally dieren soluions. I is herefore naural o consider asympoic mehods, and, in paricular, hose ha have somehing like he Black-Scholes formula as heir rs approximaion. he β expansion asympoic mehod [14] developed for he purpose of pricing a oaing-srike nicely s his crieria. I is an example of he mached asympoic expansions mehod which has been rs inroduced ino he eld of nancial mahemaics by [9], and i 14

15 is an exension of he work by [5] which derives Black-Scholes-like asympoic soluions for he Asian" PDE. From now on, is xed, so is S = s, and s = S. Along wih µ, r, σ, λ, K and, hese now become inpus ino our pricing algorihms. he following resul is proved in [14]. Proposiion 4.4 he ime being xed and τ =, he soluion of a oaing-srike pu Asian PDE can be reduced o he soluion of he following PDE: (42) τ ψ(ξ, τ, ρ,, λ) = 1 2 (ξ 1 ρ (1 e ρτ )) 2 ξξ ψ(ξ, τ, ρ,, λ) ; ψ(ξ, 0, ρ,, λ) = (ξ λ) + wih some real consans ξ, τ 0, ρ, > 0, and λ > 0. hus we can reduce he above PDE (42) for pricing PDE (26) hrough a sraighforward change of variables. he soluion saises: [ (43) u(, s) = ψ ζ (s, s, µ, λ, τ, σ, K, r), σ 2 τ, µ r λ ] σ 2, σ2 λse µτ, K, where he funcion ψ is he same as in [14], ζ is dened as a naural Log funcion similar o he one in he Black-Scholes formula: [ ( s (44) ζ (s, s, µ, λ, τ, σ, K, r) = Log K e(r+λ)τ λseµτ 1 e (µ r λ)τ ) ] + 1 K(µ r λ) 2 σ2 τ, and he expression for ψ also looks like he Black-Scholes model: (45) [ ] [ ] ψ(ζ, τ, ρ, L, K) = e ζ τ ζ ζ τ 2 KΦG τ KΦ G + ϵ(ζ, τ, ρ, L, K), τ where Φ G denoes he cumulaive normal disribuion funcion, and ϵ is an asympoic expansion of he remainder: (46) ϵ(ζ, τ, ρ, L, K) = 1 ( τ ) e ζ2 2τ 2πτ L n β n ξ j 1 n τ n=1 i=0 j=0 k=0 b n,i,j,k ζ i τ j ρ k n!(kl) n 1. he real-valued consans {b n,i,j,k } can be obained from he algorihm described in [14]. In he spiri of he mached asympoic expansions mehod, he soluion is obained as a closed form expression suiable for all pracical purposes. We should noe ha here is no proof for he approximaion o converge, and we should no deviae oo much from he Black-Scholes formula. he mehod does however make he model usable for all pracical purposes because sandard mehods for solving he problem of Asian opions may no be fas enough. As we can see, he algorihm for u(, s) is a relaively simple analyical funcion including a cumulaive normal disribuion funcion and nie sums bu, in order o obain he funcion C (25), we need o ake a nie one-dimensional inegral over ime. he inegrand however is a coninuous, sricly increasing funcion wih no singulariies on [, ], and we can herefore easily approximae i wih a sum. While he asympoic mehod gives us an essenially analyical soluion o our problem, crieria for is convergence are no known and i is herefore necessary o benchmark is 15

16 performance agains a sandard algorihm used o solve his paricular problem. For his purpose, we used he nie dierence mehod wih an implici nie dierence scheme [3]. In summary, we found ha, if we are o consider high (λ 3) Poisson inensiies for price correcions, he asympoic algorihm does no perform well. Large deviaions from he Black-Scholes world are no desirable. Anoher criical parameer is he ime o mauriy τ = which works ogeher wih he volailiy σ (he criical inpu is σ 2 τ), he algorihm gives beer accuracy for lower volailiies or/and when closer o mauriy. I is ineresing o see wha happens when we ge close o he borders of he allowable ranges and see how errors and numerical insabiliies arise. Figure 1 demonsraes wha goes wrong wih he asympoic expansion algorihm. When going far enough from mauriy we see a seady increase in he absolue error slowly oscillaing along srikes. he error will sar developing closer o mauriy if we increase λ or/and σ bu he shape will be similar o wha we see in he gure. As for he nie dierence scheme iself, i does conribue o some errors bu hey will reduce if we are o reduce he size of seps in he mesh. In paricular, he ridge" along he srike of around 100, for boh u(, s) and opion's Dela, is because he mesh seps are no small enough. he asympoic expansion algorihm is herefore good for all pracical purposes. his is for calculaing boh opion's price and Dela, one jus needs o rea resuls as esimaes if we have unusually high Poisson inensiies (more han 3) or/and long mauriies (more han 6 monhs). Figure 1: Finie dierence vs. Bea expansion algorihm: dierence for u(τ, s) on lef hand and opion's Dela on righ hand as funcions of τ = and K; σ = 0.2, µ = 4.125%, r = 0.15%, λ = 0.25, s = 100, s = Calibraion algorihms Having algorihms for he opion's price may no be sucien in iself. I is ofen ha we wan o solve a reverse problem - wha are he parameers implied by he model given hisoric prices? his is paricularly imporan for our problem because one of our objecives is o have a mechanism o measure he probabiliy of price correcions λ implied by he marke. he process for solving his problem is usually referred o as calibraion. We need o calibrae some of our inpu parameers (e.g. σ, s, λ and may be µ) so ha o achieve he bes possible 16

17 (in he leas squares sense) of he model's oupu o hisorical daa, and here are many guides for praciioners on how o do i, e.g. [4]. We can eiher calibrae o prices or implied volailiies. he laer approach is preferable because he volailiy smile and skews are he manifesaion of he markes aversion o risk, and we can guess some parameers from he shape of he volailiy surface iself, e.g. i will ell us where we are wih respec o he fundamenal value of he sock - above, below or approximaely a i. here is only one hing ha we will improve on o ake in accoun he liquidiy - we will inroduce weighs o implied volailiy errors by dividing hem wih he dierence in opion's bid and ask prices. his approach is ofen used in he indusry because he weighs amplify errors for more liquid opions hus skewing he resul owards opions ha are more acively raded. As for he opimizaion procedure iself, one of he popular algorihms arising in his conex is he Levenberg-Marquard algorihm [12] which is an example of an ieraive algorihm for nding a local minimum in a muli-dimensional seing. 5 Numerical resuls he objecive of his secion is o es our model wih he real markes daa. If he volailiy surface is formed in response o he risk of price correcions, we should expec he model o produce similar shapes of he volailiy surface (like skews and smiles) as observed in realiy. Moreover, i should acual daa poins wih reasonable values for he hree new (vs. Black- Scholes) parameers: s, λ and µ. he laer is o make sure ha we can inerpre hem as marke's percepions abou he fundamenal value of he sock s, inensiy of price correcions λ, and expeced rae of reurn on he sock µ. We sar by showing how well we can he model o daa by comparing is performance wih he Heson model and hen coninue o explore ranges of parameers obained hrough calibraion. 5.1 Calibraion o marke Socks from he NASDAQ 100 index are highly liquid, and sandard American call and pu conracs are raded a many mauriies and srikes. We look a one monh worh of opion daa for randomly seleced en socks from he index. Our primary objecive is o calibrae he model's oupus o implied volailiies. Afer all, we assume ha he shape of he implied volailiy surface comes from he markes aversion o risk, and i is his risk ha we wan o price. In all insances, we used he nie dierence mehod because i is guaraneed o converge wih no numeric insabiliies, and we have o allow for a more general case of American opions wih discree dividends. When calibraing o daa, we learned several hings ha dened our calibraion sraegy. Firsly, when calibraing o conracs wih dieren daes, we could ge relaively sable values for σ bu no for s. hey reec marke's percepions abou he fundamenal value of he sock and Poisson inensiy of price correcions owards his value, and hese percepions are jus no sable over ime. he bullish" marke in he underlying may become bearish" in an insan afer a new informaion is released o he marke. We do expec hese parameers o be piecewise consan (over quie, no new informaion periods in he marke aciviy) which will sill make he analyics hold. However, since i is hard o know if he ime period was really quie" and, jus for he sake of simpliciy, we decided o calibrae o only one volailiy shape, i.e. o conracs wih he same sock symbol, dae, ime o mauriy and he opion ype (call/pu). One monh worh of daa over en randomly seleced socks gave us

18 calibraion resuls across 6,238 conracs. Secondly, we found ha he calibraion could be very slow and may resul in unreasonable values for µ, especially when he sock is no priced on value (s is far from s). We hypohesized ha i mus be because he model's oupus have low sensiiviy o µ, and i makes sense o se i o a consan (le us say 4.125% per year). As well as making he calibraion process faser, i has allowed o remove one degree of freedom and use he following expression for he Sandard Esimaion Error: (47) SEE = N ([impl. vol. model] i [impl. vol. daa] i ) 2, N 3 i=1 where [impl. vol. daa] is obained from he marke's daa (i is volailiy implied by he Black-Scholes formula and marke prices), [impl. vol. model] is a resul of he calibraion algorihm (i is volailiy implied by our model and marke prices), and he sum is over N daa poins. his is a sandard expression for he SEE wih hree degrees of freedom (σ, s, λ), and i is he rs hing ha diereniaes our model from he Heson model. he laer has ve degrees of freedom, and, when measuring is goodness of, 5 has o be used insead of 3. We also found ha calibraion resuls could be sensiive o he iniial value of s, he value from which he calibraion process sars. here are presumingly muliple minima because large correcions wih low probabiliies demand similar price as small correcions wih high probabiliies. We have herefore calibraed wih hree dieren saring poins for s (50%, 100% and 150% of he sock price) and seleced bes opimizaion resuls wih respec o SEEs. 5.2 Performance vs. he Heson model We found ha, from he goodness of poin of view, he comparison wih he Heson model is no very ineresing. As he box and scaer plos from he Figure 2 show, he models perform very similarly, and he errors are srongly correlaed. We can obain a very good linear regression model (R 2 = 0.50), and he correlaion coecien is close o one (0.78). Figure 2: Sandard Esimaion Errors (SEEs) for my model (SS) vs. Heson model, box and scaer plos for SEEs. 18

19 I is sill rue ha models may perform dierenly for a cerain range of parameers, and we indeed see ha he model does seem o perform slighly beer when he conracs mauriy is less han hree weeks. However, he p-value for he hypohesis ha here is no dierence in he means of SEEs (-es) is 0.18 which is oo high o claim he resul as saisically signican. We also nd no saisically signican and/or ineresing dierences in he means of SEEs when looking across oher parameers including sock symbol and he ype of he volailiy surface involved (smile vs. skews, where we dene smile as 0.9 s/s 1.1). 5.3 Volailiy σ he meaning of he parameer σ in our model is no dieren o ha in he Black-Scholes model. I is a muliplier in fron of he Brownian moion, a scaling parameer ha denes he ampliude of noise driving he sock price. We obain i hrough calibraion bu i can also be esimaed hrough he saisical properies of log reurns - we expec hem o be normally disribued wih he sandard deviaion dened by he ime scaled σ. ha sill holds rue in our model excep ha we need o exclude jumps from he ime series. he jumps are no par of he noise, hey are correcions in response o a new informaion released o he marke. We herefore expec he range of σ o be similar o he range of implied volailiies, and his is indeed is he case. Looking a Figure 3 we can see ha for all socks he median is slighly below he median of implied volailiies, and for all socks we see a reducion in he daa range for σ. Figure 3: Box plos for implied volailiies (σ BS ) lef hand vs. volailiies from our model (σ) righ hand. he resul was almos cerainly expeced. In he Black-Scholes model, σ BS is slighly 19

20 inaed because i needs o accoun for he risk premium of price correcions and, by ing he volailiy surface wih our model, we signicanly reduce daily variaions in he parameer resuling in he daa range being smaller over ime. We can also see ha, for he purpose of calibraion, i is very easy and inuiive o guess σ. When we look he shape of an implied volailiy curve (e.g. as a funcion of srikes), we know ha σ should be slighly less han all implied volailiies. his is very much unlike he Heson model where σ is a volailiy of he square of he volailiy. I has a very dieren meaning, and one can no guess i as easily as σ in our model. 5.4 Fundamenal value of he sock s and λ he remaining wo parameers are he mos ineresing. hey are new (vs. Black-Scholes model), and hey are all abou he risk of price correcions. Le us sar by looking a he ranges of opimum values ha we have managed o achieve hrough calibraion. Figure 4 shows hisograms for calibraed s/s and λ. Figure 4: Hisograms for s/s and λ. Firs hing we noice is ha we have hree disinc clusers for he fundamenal value of he sock. We have a spike below 10% ( close o zero, no immediaely visible), in he range of % and somehing could be going on above 150% of he sock price. he rs cluser is when markes are expecing he sock o defaul, second is he implied volailiy smile region where he sock is perceived o be priced on value, and he righ cluser is when an exreme upward correcion is expeced. For he remaining values, we have a skew ha is no well pronounced and his is a grey region where markes are undecided if here should be an exreme movemen or if he sock is priced on value afer all. hese are marke behaviours implied by he model. We also have he Poisson inensiy of price correcions λ which is concenraed around smaller values, in fac, is median is around 0.5 meaning ha in approximaely half of he cases markes expec less han one correcion in wo years on average. his is a very desirable resul because high values for λ will ake us ouside of he sandard risk-neural valuaion framework and may also resul in numeric insabiliies for he asympoic algorihm. he low values of λ do undermine he moivaion behind he Heson model which relies on local changes in volailiy o explain he implied volailiy surface. Fundamenally, jumps wih he Poisson inensiy of less han 0.5 can no be localized o any shor erm saisic of 20

21 able 1: Calibraion resuls for he ranges of s/s and λ low inensiy high inensiy λ 0.5 λ > 0.5 exreme upward correcion is expeced, s/s he sock is priced on value, 0.9 s/s defaul is expeced, s/s undecided 0.1 s/s he sock price. hey are abou he fear of price correcions, no abou he price correcions hemselves. I is also ineresing o see how he ranges of s/s and λ are relaed. As able 1 shows, when he sock is priced on value (i.e. when we have he implied volailiy smile, second line), he more ofen cases are in case of a high inensiy for Poisson jumps (λ > 0.5). While as he defaul and exreme upward correcion cases are concerned (lines 1 and 3) - all resuls have calibraed o some λ less han 0.5. here are no many resuls relaed o he exreme upward correcion case (only six), markes seem o have rarely expeced his o happen. However, i is reasonable o assume his region o be a symmerical reecion of he defaul case and lead o similar ndings as he defaul case. Figure 5: λ versus s/s. We can see he dependency beween s/s more clearly on gure 5. When sock is perceived as priced on value, we ge a spike in he inensiy of jumps. his is reasonable, when λ is high, jumps are more likely, and having large s/s 1 would lead o larger perceived pros for echnical raders holding posiions in sock. his will creae higher demand or supply and reduce s/s 1. On he oher hand, if λ is small, here is no oo much jump risk, and markes can aord higher misalignmen beween he sock price and i's fundamenal value. he dependency may also undermine our assumpion ha λ and s are consans - hey are no, hey change over ime. In order for he analyics o hold, we have o assume ha hese parameers are piecewise consan. his is a reasonable assumpion because we expec λ and s o change only when here is a new informaion released o he marke. hese parameers 21