IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2013 c 2013 by Marin Haugh Black-Scholes and he Volailiy Surface When we sudied discree-ime models we used maringale pricing o derive he Black-Scholes formula for European opions. I was clear, however, ha we could also have used a replicaing sraegy argumen o derive he formula. In his par of he course, we will use he replicaing sraegy argumen in coninuous ime o derive he Black-Scholes parial differenial equaion. We will use his PDE and he Feynman-Kac equaion o demonsrae ha he price we obain from he replicaing sraegy argumen is consisen wih maringale pricing. We will also discuss he weaknesses of he Black-Scholes model, i.e. geomeric Brownian moion, and his leads us naurally o he concep of he volailiy surface which we will describe in some deail. We will also derive and sudy he Black-Scholes Greeks and discuss how hey are used in pracice o hedge opion porfolios. We will also derive Black s formula which emphasizes he role of he forward when pricing European opions. Finally, we will discuss he pricing of oher derivaive securiies and which securiies can be priced uniquely given he volailiy surface. Change of numeraire / measure mehods will also be demonsraed o price exchange opions. 1 The Black-Scholes PDE We now derive he Black-Scholes PDE for a call-opion on a non-dividend paying sock wih srike K and mauriy T. We assume ha he sock price follows a geomeric Brownian moion so ha ds = µs d + σs dw 1) where W is a sandard Brownian moion. We also assume ha ineres raes are consan so ha $1 invesed in he cash accoun a ime 0 will be worh B := $ expr) a ime. We will denoe by CS, ) he value of he call opion a ime. By Iô s lemma we know ha dcs, ) = C µs S + C + 1 ) 2 σ2 S 2 2 C S 2 d + σs C S dw 2) Le us now consider a self-financing rading sraegy where a each ime we hold x unis of he cash accoun and y unis of he sock. Then P, he ime value of his sraegy saisfies P = x B + y S. 3) We will choose x and y in such a way ha he sraegy replicaes he value of he opion. The self-financing assumpion implies ha dp = x db + y ds 4) = rx B d + y µs d + σs dw ) = rx B + y µs ) d + y σs dw. 5) Noe ha 4) is consisen wih our earlier definiion 1 of self-financing. In paricular, any gains or losses on he porfolio are due enirely o gains or losses in he underlying securiies, i.e. he cash-accoun and sock, and no due o changes in he holdings x and y. 1 I is also worh poining ou ha he mahemaical definiion of self-financing is obained by applying Iô s Lemma o 3) and seing he resul equal o he righ-hand-side of 4).
Black-Scholes and he Volailiy Surface 2 Reurning o our derivaion, we can equae erms in 2) wih he corresponding erms in 5) o obain y = C S rx B = C + 1 2 σ2 S 2 2 C S 2. 7) If we se C 0 = P 0, he iniial value of our self-financing sraegy, hen i mus be he case ha C = P for all since C and P have he same dynamics. This is rue by consrucion afer we equaed erms in 2) wih he corresponding erms in 5). Subsiuing 6) and 7) ino 3) we obain rs C S + C + 1 2 σ2 S 2 2 C rc = 0, 8) S2 he Black-Scholes PDE. In order o solve 8) boundary condiions mus also be provided. In he case of our call opion hose condiions are: CS, T ) = maxs K, 0), C0, ) = 0 for all and CS, ) S as S. The soluion o 8) in he case of a call opion is CS, ) = S Φd 1 ) e rt ) KΦd 2 ) 9) 6) where d 1 = log S K ) + r + σ 2 /2)T ) σ T and d 2 = d 1 σ T and Φ ) is he CDF of he sandard normal disribuion. One way o confirm 9) is o compue he various parial derivaives, hen subsiue hem ino 8) and check ha 8) holds. The price of a European pu-opion can also now be easily compued from pu-call pariy and 9). The mos ineresing feaure of he Black-Scholes PDE 8) is ha µ does no appear 2 anywhere. Noe ha he Black-Scholes PDE would also hold if we had assumed ha µ = r. However, if µ = r hen invesors would no demand a premium for holding he sock. Since his would generally only hold if invesors were risk-neural, his mehod of derivaives pricing came o be known as risk-neural pricing. 2 Maringale Pricing We can easily see ha he Black-Scholes PDE in 8) is consisen wih maringale pricing. Maringale pricing heory saes ha deflaed securiy prices are maringales. If we deflae by he cash accoun, hen he deflaed sock price process, Y say, saisfies Y := S /B. Then Iô s Lemma and Girsanov s Theorem imply dy = µ r)y d + σy dw = µ r)y d + σy dw Q η d) = µ r ση )Y d + σy dw Q. where Q denoes a new probabiliy measure and W Q is a Q-Brownian moion. Bu we know from maringale pricing ha if Q is an equivalen maringale measure hen i mus be he case ha Y is a maringale. This hen implies ha η = µ r)/σ for all. I also implies ha he dynamics of S saisfy ds = µs d + σs dw = rs d + σs dw Q. 10) Using 10), we can now derive 9) using maringale pricing. In paricular, we have [ ] CS, ) = E Q e rt ) maxs T K, 0) 11) 2 The discree-ime counerpar o his observaion was when we observed ha he rue probabiliies of up-moves and downmoves did no have an impac on opion prices.
Black-Scholes and he Volailiy Surface 3 where S T = S e r σ2 /2)T )+σw Q T W Q ) is log-normally disribued. While he calculaions are a lile edious, i is sraighforward o solve 11) and obain 9) as he soluion. Dividends If we assume ha he sock pays a coninuous dividend yield of q, i.e. he dividend paid over he inerval, + d] equals qs d, hen he dynamics of he sock price saisfy ds = r q)s d + σs dw Q. 12) In his case he oal gain process, i.e. he capial gain or loss from holding he securiy plus accumulaed dividends, is a Q-maringale. The call opion price is sill given by 11) bu now wih In his case he call opion price is given by S T = S e r q σ2 /2)T )+σw Q T. CS, ) = e qt ) S Φd 1 ) e rt ) KΦd 2 ) 13) where d 1 = log S K ) + r q + σ 2 /2)T ) σ T and d 2 = d 1 σ T. Exercise 1 Follow he argumen of he previous secion o derive he Black-Scholes PDE when he sock pays a coninuous dividend yield of q. Feynman-Kac We have already seen ha he Black-Scholes formula can be derived from eiher he maringale pricing approach or he replicaing sraegy / risk neural PDE approach. In fac we can go direcly from he Black-Scholes PDE o he maringale pricing equaion of 11) using he Feynman-Kac formula. Exercise 2 Derive he same PDE as in Exercise 1 bu his ime by using 12) and applying he Feynman-Kac formula o an analogous expression o 11). While he original derivaion of he Black-Scholes formula was based on he PDE approach, he maringale pricing approach is more general and ofen more amenable o compuaion. For example, numerical mehods for solving PDEs are usually oo slow if he number of dimensions are greaer 3 han hree. Mone-Carlo mehods can be used o evaluae 11) regardless of he number of sae variables, however, as long as we can simulae from he relevan probabiliy disribuions. The maringale pricing approach can also be used for problems ha are non-markovian. This is no he case for he PDE approach. The Black-Scholes Model is Complee I is worh menioning ha he Black-Scholes model is a complee model and so every derivaive securiy is aainable or replicable. In paricular, his means ha every securiy can be priced uniquely. Compleeness follows from he fac ha he EMM in 10) is unique: he only possible choice for η was η = µ r)/σ. 3 The Black-Scholes PDE is only wo-dimensional wih sae variable and S. The Black-Scholes PDE is herefore easy o solve numerically.
Black-Scholes and he Volailiy Surface 4 3 The Volailiy Surface The Black-Scholes model is an elegan model bu i does no perform very well in pracice. For example, i is well known ha sock prices jump on occasions and do no always move in he smooh manner prediced by he GBM moion model. Sock prices also end o have faer ails han hose prediced by GBM. Finally, if he Black-Scholes model were correc hen we should have a fla implied volailiy surface. The volailiy surface is a funcion of srike, K, and ime-o-mauriy, T, and is defined implicily CS, K, T ) := BS S, T, r, q, K, σk, T )) 14) where CS, K, T ) denoes he curren marke price of a call opion wih ime-o-mauriy T and srike K, and BS ) is he Black-Scholes formula for pricing a call opion. In oher words, σk, T ) is he volailiy ha, when subsiued ino he Black-Scholes formula, gives he marke price, CS, K, T ). Because he Black-Scholes formula is coninuous and increasing in σ, here will always 4 be a unique soluion, σk, T ). If he Black-Scholes model were correc hen he volailiy surface would be fla wih σk, T ) = σ for all K and T. In pracice, however, no only is he volailiy surface no fla bu i acually varies, ofen significanly, wih ime. Figure 1: The Volailiy Surface In Figure 1 above we see a snapsho of he 5 volailiy surface for he Eurosoxx 50 index on November 28 h, 2007. The principal feaures of he volailiy surface is ha opions wih lower srikes end o have higher implied volailiies. For a given mauriy, T, his feaure is ypically referred o as he volailiy skew or smile. For a given srike, K, he implied volailiy can be eiher increasing or decreasing wih ime-o-mauriy. In general, however, σk, T ) ends o converge o a consan as T. For T small, however, we ofen observe an invered volailiy surface wih shor-erm opions having much higher volailiies han longer-erm opions. This is paricularly rue in imes of marke sress. I is worh poining ou ha differen implemenaions 6 of Black-Scholes will resul in differen implied volailiy surfaces. If he implemenaions are correc, however, hen we would expec he volailiy surfaces o be very 4 Assuming here is no arbirage in he marke-place. 5 Noe ha by pu-call pariy he implied volailiy σk, T ) for a given European call opion will be also be he implied volailiy for a European pu opion of he same srike and mauriy. Hence we can alk abou he implied volailiy surface. 6 For example differen mehods of handling dividends would resul in differen implemenaions.
Black-Scholes and he Volailiy Surface 5 similar in shape. Single-sock opions are generally American and in his case, pu and call opions will ypically give rise o differen surfaces. Noe ha pu-call pariy does no apply for American opions. Clearly hen he Black-Scholes model is far from accurae and marke paricipans are well aware of his. However, he language of Black-Scholes is pervasive. Every rading desk compues he Black-Scholes implied volailiy surface and he Greeks hey compue and use are Black-Scholes Greeks. Arbirage Consrains on he Volailiy Surface The shape of he implied volailiy surface is consrained by he absence of arbirage. In paricular: 1. We mus have σk, T ) 0 for all srikes K and expiraions T. 2. A any given mauriy, T, he skew canno be oo seep. Oherwise buerfly arbirages will exis. For example fix a mauriy, T and consider pu wo opions wih srikes K 1 < K 2. If here is no arbirage hen i mus be he case why?) ha P K 1 ) < P K 2 ) where P K i ) is he price of he pu opion wih srike K i. However, if he skew is oo seep hen we would obain why?) P K 1 ) > P K 2 ). 3. Likewise he erm srucure of implied volailiy canno be oo invered. Oherwise calendar spread arbirages will exis. This is mos easily seen in he case where r = q = 0. Then, fixing a srike K, we can le C T ) denoe he ime price of a call opion wih srike K and mauriy T. Maringale pricing implies ha C T ) = E [S T K) + ]. We have seen before ha S T K) + is a Q-submaringale and now sandard maringale resuls can be used o show ha C T ) mus be non-decreasing in T. This would be violaed Why?) if he erm srucure of implied volailiy was oo invered. In pracice he implied volailiy surface will no violae any of hese resricions as oherwise here would be an arbirage in he marke. These resricions can be difficul o enforce, however, when we are bumping or sressing he volailiy surface, a ask ha is commonly performed for risk managemen purposes. Why is here a Skew? For socks and sock indices he shape of he volailiy surface is always changing. There is generally a skew, however, so ha for any fixed mauriy, T, he implied volailiy decreases wih he srike, K. I is mos pronounced a shorer expiraions. There are wo principal explanaions for he skew. 1. Risk aversion which can appear as an explanaion in many guises: a) Socks do no follow GBM wih a fixed volailiy. Markes ofen jump and jumps o he downside end o be larger and more frequen han jumps o he upside. b) As markes go down, fear ses in and volailiy goes up. c) Supply and demand. Invesors like o proec heir porfolio by purchasing ou-of-he-money pus and so here is more demand for opions wih lower srikes. 2. The leverage effec which is due o he fac ha he oal value of company asses, i.e. deb + equiy, is a more naural candidae o follow GBM. If so, hen equiy volailiy should increase as he equiy value decreases. To see his consider he following: Le V, E and D denoe he oal value of a company, he company s equiy and he company s deb, respecively. Then he fundamenal accouning equaions saes ha V = D + E. 15) Equaion 15) is he basis for he classical srucural models ha are used o price risky deb and credi defaul swaps. Meron 1970 s) recognized ha he equiy value could be viewed as he value of a call opion on V wih srike equal o D. Le V, E and D be he change in values of V, E and D, respecively. Then V + V = E + E) + D + D) so ha V + V V = E + E V + D + D V
Black-Scholes and he Volailiy Surface 6 = E V E + E E ) + D V ) D + D If he equiy componen is subsanial so ha he deb is no oo risky, hen 16) implies D 16) σ V E V σ E where σ V and σ E are he firm value and equiy volailiies, respecively. We herefore have σ E V E σ V. 17) Example 1 The Leverage Effec) Suppose, for example, ha V = 1, E =.5 and σ V = 20%. Then 17) implies σ E 40%. Suppose σ V remains unchanged bu ha over ime he firm loses 20% of is value. Almos all of his loss is borne by equiy so ha now 17) implies σ E 53%. σ E has herefore increased despie he fac ha σ V has remained consan. I is ineresing o noe ha here was lile or no skew in he marke before he Wall sree crash of 1987. So i appears o be he case ha i ook he marke he bes par of wo decades before i undersood ha i was pricing opions incorrecly. Wha he Volailiy Surface Tells Us To be clear, we coninue o assume ha he volailiy surface has been consruced from European opion prices. Consider a buerfly sraegy cenered a K where you are: 1. long a call opion wih srike K K 2. long a call wih srike K + K 3. shor 2 call opions wih srike K The value of he buerfly, B 0, a ime = 0, saisfies B 0 = CK K, T ) 2CK, T ) + CK + K, T ) e rt ProbK K S T K + K) K/2 e rt fk, T ) 2 K K/2 = e rt fk, T ) K) 2 where fk, T ) is he probabiliy densiy funcion PDF) of S T evaluaed a K. We herefore have rt CK K, T ) 2CK, T ) + CK + K, T ) fk, T ) e K) 2. 18) Leing K 0 in 18), we obain fk, T ) = e rt 2 C K 2. The volailiy surface herefore gives he marginal risk-neural disribuion of he sock price, S T, for any ime, T. I ells us nohing abou he join disribuion of he sock price a muliple imes, T 1,..., T n. This should no be surprising since he volailiy surface is consruced from European opion prices and he laer only depend on he marginal disribuions of S T.
Black-Scholes and he Volailiy Surface 7 Example 2 Same marginals, differen join disribuions) Suppose here are wo ime periods, T 1 and T 2, of ineres and ha a non-dividend paying securiy has risk-neural disribuions given by S T1 = e r σ2 /2)T 1+σ T 1 Z 1 19) S T2 = e r σ2 /2)T ) 2+σ T 2 ρz 1+ 1 ρ 2 Z 2 20) where Z 1 and Z 2 are independen N0, 1) random variables. Noe ha a value of ρ > 0 can capure a momenum effec and a value of ρ < 0 can capure a mean-reversion effec. We are also implicily assuming ha S 0 = 1. Suppose now ha here are wo securiies, A and B say, wih prices S A) and S B) given by 19) and 20) a imes = T 1 and = T 2, and wih parameers ρ = ρ A and ρ = ρ B, respecively. Noe ha he marginal disribuion of S A) is idenical o he marginal disribuion of S B) for {T 1, T 2 }. I herefore follows ha opions on A and B wih he same srike and mauriy mus have he same price. A and B herefore have idenical volailiy surfaces. Bu now consider a knock-in pu opion wih srike 1 and expiraion T 2. In order o knock-in, he sock price a ime T 1 mus exceed he barrier price of 1.2. The payoff funcion is hen given by Payoff = max 1 S T2, 0) 1 {ST1 1.2}. Quesion: Would he knock-in pu opion on A have he same price as he knock-in pu opion on B? Quesion: How does your answer depend on ρ A and ρ B? Quesion: Wha does his say abou he abiliy of he volailiy surface o price barrier opions? 4 The Greeks We now urn o he sensiiviies of he opion prices o he various parameers. These sensiiviies, or he Greeks are usually compued using he Black-Scholes formula, despie he fac ha he Black-Scholes model is known o be a poor approximaion o realiy. Bu firs we reurn o pu-call pariy. Pu-Call Pariy Consider a European call opion and a European pu opion, respecively, each wih he same srike, K, and mauriy T. Assuming a coninuous dividend yield, q, hen pu-call pariy saes e rt K + Call Price = e qt S + Pu Price. 21) This of course follows from a simple arbirage argumen and he fac ha boh sides of 21) equal maxs T, K) a ime T. Pu-call pariy is useful for calculaing Greeks. For example 7, i implies ha VegaCall) = VegaPu) and ha GammaCall) = GammaPu). I is also exremely useful for calibraing dividends and consrucing he volailiy surface. The Greeks The principal Greeks for European call opions are described below. The Greeks for pu opions can be calculaed in he same manner or via pu-call pariy. Definiion: The dela of an opion is he sensiiviy of he opion price o a change in he price of he 7 See below for definiions of vega and gamma.
Black-Scholes and he Volailiy Surface 8 underlying securiy. The dela of a European call opion saisfies dela = C S = e qt Φd 1 ). This is he usual dela corresponding o a volailiy surface ha is sicky-by-srike. I assumes ha as he underlying securiy moves, he volailiy of he opion does no move. If he volailiy of he opion did move hen he dela would have an addiional erm of he form vega σk, T )/ S. In his case we would say ha he volailiy surface was sicky-by-dela. Equiy markes ypically use he sicky-by-srike approach when compuing delas. Foreign exchange markes, on he oher hand, end o use he sicky-by-dela approach. Similar commens apply o gamma as defined below. a) Dela for European Call and Pu Opions b) Dela for Call Opions as Time-To-Mauriy Varies Figure 2: Dela for European Opions By pu-call pariy, we have dela pu = dela call e qt. Figure 2a) shows he dela for a call and pu opion, respecively, as a funcion of he underlying sock price. In Figure 2b) we show he dela for a call opion as a funcion of he underlying sock price for hree differen imes-o-mauriy. I was assumed r = q = 0. Wha is he srike K? Noe ha he dela becomes seeper around K when ime-o-mauriy decreases. Noe also ha dela = Φd 1 ) = Probopion expires in he money). This is only approximaely rue when r and q are non-zero.) In Figure 3 we show he dela of a call opion as a funcion of ime-o-mauriy for hree opions of differen money-ness. Are here any surprises here? Wha would he corresponding plo for pu opions look like? Definiion: The gamma of an opion is he sensiiviy of he opion s dela o a change in he price of he underlying securiy. The gamma of a call opion saisfies where φ ) is he sandard normal PDF. gamma = 2 C S 2 = e qt φd 1) σs T In Figure 4a) we show he gamma of a European opion as a funcion of sock price for hree differen ime-o-mauriies. Noe ha by pu-call pariy, he gamma for European call and pu opions wih he same
Black-Scholes and he Volailiy Surface 9 Figure 3: Dela for European Call Opions as a Funcion of Time-To-Mauriy a) Gamma as a Funcion of Sock Price b) Gamma as a Funcion of Time-o-Mauriy Figure 4: Gamma for European Opions srike are equal. Gamma is always posiive due o opion convexiy. Traders who are long gamma can make money by gamma scalping. Gamma scalping is he process of regularly re-balancing your opions porfolio o be dela-neural. However, you mus pay for his long gamma posiion up fron wih he opion premium. In Figure 4b), we display gamma as a funcion of ime-o-mauriy. Can you explain he behavior of he hree curves in Figure 4b)? Definiion: The vega of an opion is he sensiiviy of he opion price o a change in volailiy. The vega of a call opion saisfies vega = C σ = e qt S T φd 1 ). In Figure 5b) we plo vega as a funcion of he underlying sock price. We assumed K = 100 and ha r = q = 0. Noe again ha by pu-call pariy, he vega of a call opion equals he vega of a pu opion wih he same srike. Why does vega increase wih ime-o-mauriy? For a given ime-o-mauriy, why is vega peaked near he srike? Turning o Figure 5b), noe ha he vega decreases o 0 as ime-o-mauriy goes o 0. This is
Black-Scholes and he Volailiy Surface 10 a) Vega as a Funcion of Sock Price b) Vega as a Funcion of Time-o-Mauriy Figure 5: Vega for European Opions consisen wih Figure 5a). I is also clear from he expression for vega. Quesion: Is here any inconsisency o alk abou vega when we use he Black-Scholes model? Definiion: The hea of an opion is he sensiiviy of he opion price o a negaive change in ime-o-mauriy. The hea of a call opion saisfies hea = C T = e qt σ Sφd 1 ) 2 T + qe qt SNd 1 ) rke rt Nd 2 ). a) Thea as a Funcion of Sock Price b) Thea as a Funcion of Time-o-Mauriy Figure 6: Thea for European Opions In Figure 6a) we plo hea for hree call opions of differen imes-o-mauriy as a funcion of he underlying
Black-Scholes and he Volailiy Surface 11 sock price. We have assumed ha r = q = 0%. Noe ha he call opion s hea is always negaive. Can you explain why his is he case? Why does hea become more negaively peaked as ime-o-mauriy decreases o 0? In Figure 6b) we again plo hea for hree call opions of differen money-ness, bu his ime as a funcion of ime-o-mauriy. Noe ha he ATM opion has he mos negaive hea and his ges more negaive as ime-o-mauriy goes o 0. Can you explain why? Opions Can Have Posiive Thea: In Figure 7 we plo hea for hree pu opions of differen money-ness as a funcion of ime-o-mauriy. We assume here ha q = 0 and r = 10%. Noe ha hea can be posiive for in-he-money pu opions. Why? We can also obain posiive hea for call opions when q is large. In ypical scenarios, however, hea for boh call and pu opions will be negaive. Figure 7: Posiive Thea is Possible The Relaionship beween Dela, Thea and Gamma Recall ha he Black-Scholes PDE saes ha any derivaive securiy wih price P mus saisfy P Wriing θ, δ and Γ for hea, dela and gamma, we obain P + r q)s S + 1 2 σ2 S 2 2 P = rp. 22) S2 θ + r q)sδ + 1 2 σ2 S 2 Γ = rp. 23) Equaion 23) holds in general for any porfolio of securiies. If he porfolio in quesion is dela-hedged so ha he porfolio δ = 0 hen we obain θ + 1 2 σ2 S 2 Γ = rp 24) I is clear from 24) ha any gain from gamma is offse by losses due o hea. This of course assumes ha he correc implied volailiy is assumed in he Black-Scholes model. Since we know ha he Black-Scholes model is wrong, his observaion should only be used o help your inuiion and no aken as a fac. Dela-Gamma-Vega Approximaions o Opion Prices A simple applicaion of Taylor s Theorem says CS + S, σ + σ) CS, σ) + S C S + 1 2 S)2 2 C C + σ S2 σ
Black-Scholes and he Volailiy Surface 12 = CS, σ) + S δ + 1 2 S)2 Γ + σ vega. where CS, σ) is he price of a derivaive securiy as a funcion 8 of he curren sock price, S, and he implied volailiy, σ. We herefore obain P&L = δ S + Γ 2 S)2 + vega σ = dela P&L + gamma P&L + vega P&L When σ = 0, we obain he well-known dela-gamma approximaion. This approximaion is ofen used, for example,in hisorical Value-a-Risk VaR) calculaions for porfolios ha include opions. We can also wrie P&L = δs ) S + ΓS2 S 2 ) 2 S + vega σ S = ESP Reurn + $ Gamma Reurn 2 + vega σ where ESP denoes he equivalen sock posiion or dollar dela. 5 Dela Hedging In he Black-Scholes model wih GBM, an opion can be replicaed exacly by dela-hedging he opion. In fac he Black-Scholes PDE we derived earlier was obained by a dela-hedging / replicaion argumen. The idea behind dela-hedging is o re-balance he porfolio of he opion and sock coninuously so ha you always have a new dela of zero. Of course i is no pracical in o hedge coninuously and so insead we hedge periodically. Periodic or discree hedging hen resuls in some replicaion error. Consider Figure 8 below which displays a screen-sho of an Excel spreadshee ha was used o simulae a dela-hedging sraegy. Figure 8: Dela-Hedging in Excel 8 The price may also depend on oher parameers, in paricular ime-o-mauriy, bu we suppress ha dependence here.
Black-Scholes and he Volailiy Surface 13 Mechanics of he Excel spreadshee In every period, he porfolio is re-balanced so ha i is dela-neural. This is done by using he dela of he opions porfolio o deermine he oal sock posiion. This sock posiion is funded hrough borrowing a he risk-free rae and i accrues dividends according o he dividend yield. The iming of he cash-flows is ignored when calculaing he hedging P&L. Le P denoe he ime value of he discree-ime self-financing sraegy ha aemps o replicae he opion payoff and le C 0 denoe he iniial value of he opion. The replicaing sraegy is hen given by P 0 := C 0 25) P i+1 = P i + P i δ i S i ) r + δ i Si+1 S i + qs i ) 26) where := i+1 i which we assume is consan for all i, r is he annual risk-free ineres rae assuming per-period compounding) and δ i is he Black-Scholes dela a ime i. This dela is a funcion of S i and some assumed implied volailiy, σ imp say. Noe ha 25) and 26) respec he self-financing condiion. Sock prices are simulaed assuming S GBMµ, σ) so ha S + = S e µ σ2 /2) +σ Z where Z N0, 1). Noe he opion implied volailiy, σ imp, need no equal σ which in urn need no equal he realized volailiy when we hedge periodically as opposed o coninuously). This has ineresing implicaions for he rading P&L which we may define as P&L := P T S T K) + in he case of a shor posiion in a call opion wih srike K and mauriy T. Noe ha P T is he erminal value of he replicaing sraegy in 26). Many ineresing quesions now arise: Quesion: If you sell opions, wha ypically happens he oal P&L if σ < σ imp? Quesion: If you sell opions, wha ypically happens he oal P&L if σ > σ imp? Quesion: If σ = σ imp wha ypically happens he oal P&L as he number of re-balances increases? Some Answers o Dela-Hedging Quesions Recall ha he price of an opion increases as he volailiy increases. Therefore if realized volailiy is higher han expeced, i.e. he level a which i was sold, we expec o lose money on average when we dela-hedge an opion ha we sold. Similarly, we expec o make money when we dela-hedge if he realized volailiy is lower han he level a which i was sold. In general, however, he payoff from dela-hedging an opion is pah-dependen, i.e. i depends on he price pah aken by he sock over he enire ime inerval. In fac, we can show ha he payoff from coninuously dela-hedging an opion saisfies P&L = T 0 S 2 2 2 V S 2 σ 2 imp σ 2 ) d 27) where V is he ime value of he opion and σ is he realized insananeous volailiy a ime. The erm S2 2 V 2 S is ofen called he dollar gamma, as discussed earlier. I is always posiive for a call or pu 2 opion, bu i goes o zero as he opion moves significanly ino or ou of he money. Reurning o self-financing rading sraegy of 25) and 26), noe ha we can choose any model we like for he securiy price dynamics. In paricular, we are no resriced o choosing geomeric Brownian moion and oher diffusion or jump-diffusion models could be used insead. I is ineresing o simulae hese alernaive models and o hen observe wha happens o he replicaion error in 27) where he δ i s are compued assuming incorrecly) a geomeric Brownian moion price dynamics.
Black-Scholes and he Volailiy Surface 14 6 Pricing Exoics In his secion we will discuss he pricing of hree exoic securiies: i) a digial opion ii) a range accrual and iii) an exchange opion. The firs wo can be priced using he implied volailiy surface and so heir prices are no model dependen. We will price he hird securiy using he Black-Scholes framework. While his is no how i would be priced in pracice, i does provide us wih an opporuniy o pracice change-of-measure mehods. Pricing a Digial Opion Suppose we wish o price a digial opion which pays $1 if he ime T sock price, S T, is greaer han K and 0 oherwise. Then i is easy 9 o see ha he digial price, DK, T ) is given by CK, T ) CK + K, T ) DK, T ) = lim K 0 K CK + K, T ) CK, T ) = lim K 0 K = CK, T ) K. In paricular his implies ha digial opions are uniquely priced from he volailiy surface. By definiion, CK, T ) = C BS K, T, σ BS K, T )) where we use C BS,, ) o denoe he Black-Scholes price of a call opion as a funcion of srike, ime-o-mauriy and volailiy. The chain rule now implies DK, T ) = C BSK, T, σ BS K, T )) K = C BS K C BS σ BS σ BS K = C BS K vega skew. Example 3 Pricing a digial) Suppose r = q = 0, T = 1 year, S 0 = 100 and K = 100 so he digial is a-he-money. Suppose also ha he skew is 2.5% per 10% change in srike and σ am = 25%. Then D100, 1) = Φ σ ) am σam ) S 0 φ.025 2 2.1S 0 = Φ σ ) am σam ) +.25 φ 2 2.45 +.25.4 =.55 Therefore he digial price = 55% of noional when priced correcly. If we ignored he skew and jus he Black-Scholes price using he ATM implied volailiy, he price would have been 45% of noional which is significanly less han he correc price. Exercise 3 Why does he skew make he digial more expensive in he example above? 9 This proof is an example of a saic replicaion argumen.
Black-Scholes and he Volailiy Surface 15 Example 4 Pricing a Range Accrual) Consider now a 3-monh range accrual on he Nikkei 225 index wih range 13, 000 o 14, 000. Afer 3 monhs he produc pays X% of noional where X = % of days over he 3 monhs ha index is inside he range e.g. If he noional is $10M and he index is inside he range 70% of he ime, hen he payoff is $7M. Quesion: Is i possible o calculae he price of his range accrual using he volailiy surface? Hin: Consider a porfolio consising of a pair of digial s for each dae beween now and he expiraion. Example 5 Pricing an Exchange Opion) Suppose now ha here are wo non-dividend-paying securiies wih dynamics given by dy dx = µ y Y d + σ y Y dw y) = µ x X d + σ x X dw x) so ha each securiy follows a GBM. We also assume dw x) have an insananeous correlaion of ρ. Le Z := Y /X. Then Iô s Lemma check!) implies dw y) = ρ d so ha he wo securiy reurns dz Z = µ y µ x ρσ x σ y + σx 2 ) d + σy dw y) σ x dw x). 28) The insananeous variance of dz/z is given by Now define a new process, W as ) 2 dz = Z dw = σ y σ σ y dw y) ) 2 σ x dw x) = σ 2 x + σ 2 y 2ρσ x σ y ) d dw y) σ x σ dw x) where σ 2 := σ 2 x + σ 2 y 2ρσ x σ y ). Then W is clearly a coninuous maringale. Moreover, dw ) 2 = = d. σ y dw y) ) σ x dw x) 2 σ Hence by Levy s Theorem, W is a Brownian moion and so Z is a GBM. Using 28) we can wrie is dynamics as dz = µ y µ x ρσ x σ y + σ 2 ) x d + σdw. 29) Z Consider now an exchange opion expiring a ime T where he payoff is given by Exchange Opion Payoff = max 0, Y T X T ). We could use maringale pricing o compue his direcly and explicily solve P 0 = E Q [ 0 e rt max 0, Y T X T ) ]
Black-Scholes and he Volailiy Surface 16 for he price of he opion. This involves solving a wo-dimensional inegral wih he bivariae normal disribuion which is possible bu somewha edious. Insead, however, we could price he opion by using asse X as our numeraire. Le Q x be he probabiliy measure associaed wih his new numeraire. Then maringale pricing implies [ ] P 0 max 0, YT X = E Qx T ) 0 X 0 X T = E Qx 0 [max 0, Z T 1)]. 30) Equaion 29) gives he dynamics of Z under our original probabiliy measure whichever one i was), bu we need o know is dynamics under he probabiliy measure Q x. Bu his is easy. We know from Girsanov s Theorem ha only he drif of Z will change so ha he volailiy will remain unchanged. We also know ha Z mus be a maringale and so under Q x his drif mus be zero. Bu hen he righ-hand side of 30) is simply he Black-Scholes opion price where we se he risk-free rae o zero, he volailiy o σ and he srike o 1. Pricing Oher Exoics Perhaps he wo mos commonly raded exoic derivaives are barrier opions and variance-swaps. In fac a his sage hese securiies are viewed as more semi-exoic han exoic. As suggesed by Example 2, he price of a barrier opion canno be priced using he volailiy surface as he laer only defines he marginal disribuions of he sock prices. While we could use Black-Scholes and GBM wih some consan volailiy o deermine a price, i is well known ha his leads o very inaccurae pricing. Moreover, a rule employed o deermine he consan volailiy migh well lead o arbirage opporuniies for oher marke paricipans. I is generally believed ha variance swaps can be priced uniquely from he volailiy surface. However, his is only rue for variance-swaps wih mauriies ha are less han wo or hree years. For mauriies beyond ha, i is probably necessary o include sochasic ineres raes and dividends in order o price variance swaps accuraely. Variance-swaps will be sudied in deail in he exercises. 7 Dividends, he Forward and Black s Model Le C = CS, K, r, q, σ, T ) be he price of a call opion on a sock. Then he Black-Scholes model says where C = Se qt ) Φd 1 ) Ke rt ) Φd 2 ) d 1 = logs/k) + r q + σ2 /2)T ) σ, T d 2 = d 1 σ T. Le F := Se r q)t ) so ha F is he ime forward price for delivery of he sock a ime T. Then we can wrie where d 2 = d 1 σ T. C = F e rt ) Nd 1 ) Ke rt ) Nd 2 ) 31) = e rt ) Expeced-Payoff-of-he-Opion d 1 = logf/k) + T )σ2 /2 σ, T Noe ha he opion price now only depends on F, K, r, σ and T. In fac we can wrie he call price as C = BlackF, K, r, σ, T ).
Black-Scholes and he Volailiy Surface 17 where he funcion Black ) is defined implicily by 31). When we wrie opion prices in erms of he forward and no he spo price, he resuling formula is ofen called Black s formula. I emphasizes he imporance of he forward price in esablishing he price of he opion. The spo price is only relevan in so far as i influences he forward price. Dividends and Opion Pricing As we have seen, he Black-Scholes formula easily accommodaes a coninuous dividend yield. In pracice, however, dividends are discree. In order o handle discree dividends we could conver hem ino dividend yields bu his can creae problems. For example, as an ex-dividend dae approaches, he dividend yield can grow arbirarily high. We would also need a differen dividend yield for each opion mauriy. A paricularly imporan problem is ha dela and he oher Greeks can become disored when we replace discree dividends wih a coninuous dividend yield. Example 6 Discree dividends) Consider a deep in-he-money call opion wih expiraion 1 week from now, a curren sock price = $100 and a $5 dividend going ex-dividend during he week. Then Bu wha do you hink he real dela is? Black-Scholes dela = e qt Φd 1 ) e qt = e.05 52)/52 = 95.12% Using a coninuous dividend yield can also creae major problems when pricing American opions. Consider, for example, an American call opion wih expiraion T on a sock ha goes ex-dividend on dae div < T. This is he only dividend ha he sock pays before he opion mauriy. We know he opion should only ever be exercised a eiher expiraion or immediaely before div. However, if we use a coninuous dividend yield, he pricing algorihm will never see his ex-dividend dae and so i will never exercise early, even when i is opimal o do so. There are many possible soluions o his problem of handling discree dividends. A common soluion is o ake X 0 = S 0 PVDividends) as he basic securiy where PVDividends) = presen value of dividends going ex-dividend beween now and opion expiraion. This works fine for European opions recall ha wha maers is he forward). For American opions, we could, for example, build a binomial laice for X. Then a each dae in he laice, we can deermine he sock price and accoun properly for he discree dividends, deermining correcly wheher i is opimal o early exercise or no. In fac his was he subjec of a quesion in an earlier assignmen. 8 Exensions of Black-Scholes The Black-Scholes model is easily applied o oher securiies. In addiion o opions on socks and indices, hese securiies include currency opions, opions on some commodiies and opions on index, sock and currency fuures. Of course, in all of hese cases i is well undersood ha he model has many weaknesses. As a resul, he model has been exended in many ways. These exensions include jump-diffusion models, sochasic volailiy models, local volailiy models, regime-swiching models, garch models and ohers. One of he principal uses of he Black-Scholes framework is ha is ofen used o quoe derivaives prices via implied volailiies. This is rue even for securiies where he GBM model is clearly inappropriae. Such securiies include, for example, caples and swapions in he fixed income markes, CDS opions in credi markes and opions on variance-swaps in equiy markes.
Black-Scholes and he Volailiy Surface 18 Exercises 1. Show ha a sock ha has a coninuous dividend yield of q has risk-neural dynamics ds = r q)s d + σs dw where W is a Q-Brownian moion corresponding o he cash accoun as he numeraire. 2. Consider a sock ha has a coninuous dividend yield of q wih risk-neural dynamics ds = r q)s d + σs dw Q where W Q is a Q-Brownian moion corresponding o he cash accoun as he numeraire. Show ha, as expeced, [ ] T S 0 = E Q 0 e r qs d + e rt S T. 32) You can assume ha exchanging he order of inegraions in 32) is jusified.) 3. Derive he Black-Scholes PDE when he underlying sock has a consan dividend yield of q. 0 4. Derive he same PDE as in Exercise 3 bu his ime by using 12) and applying he Feynman-Kac formula o an analogous expression o 11). 5. a) Use maringale pricing o derive he ime price, F T ), of a fuures conrac for delivery of a sock a ime T. You can assume ha he Black-Scholes model holds and ha he sock pays a dividend yield of q. You can use your knowledge of fuures prices from discree-ime models o jusify your answer.) b) Compue he fair price of an opion on a fuures conrac in he Black-Scholes model. You should assume ha he fuures conrac expires a ime T and ha he opion expires a ime τ < T. This is sraighforward using he original Black-Scholes formula and your answer from par a).) c) Confirm direcly ha he opion price you derived in par b) saisfies he Black-Scholes PDE of Exercise 3. 6. Variaion on Q3.11 in Back) Suppose an invesor invess in a porfolio wih price S and consan dividend yield q. Assume he invesor is charged a consan expense raio α which acs as a negaive dividend) and a dae T receives eiher his porfolio value or his iniial invesmen, whichever is higher. This is similar o a popular ype of variable annuiy. Leing D denoe he number of dollars invesed in he conrac, he conrac pays max D, a dae T. We can rearrange he expression 33) as max D, De q α)t S T S 0 ) De q α)t S T S 0 = D + max 0, ) = D + e αt D max De q α)t S T S 0 0, e qt S T S 0 ) D ) e αt. Thus he conrac payoff is equivalen o he amoun invesed plus a cerain number of call opions wrien on he gross holding period reurn e qt S T /S 0. Noe ha Z := e q S /S 0 is he dae- value of he porfolio ha sars wih 1/S 0 unis of he asse i.e., wih a $1 invesmen) and reinvess dividends. 33)
Black-Scholes and he Volailiy Surface 19 Thus, he call opions are call opions on a non-dividend paying porfolio wih he same volailiy as S and iniial price of $1. a) Compue he dae-0 value of he conrac o he invesor assuming Black-Scholes dynamics for he porfolio. In pracice such a conrac would be priced using he appropriae implied volailiy surface.) b) Creae a funcion in Malab, VBA or whaever you prefer) o compue he fair expense raio; i.e. find α such ha he dae-0 value of he conrac is equal o D. Hin: You can use α = 0 as a lower bound. Because he value of he conrac is decreasing as α increases, you can find an upper bound by ieraing unil he value of he conrac is less han D.) c) How does he fair expense raio vary wih he mauriy, T? 7. Consider a marke wih n risky securiies wih price processes, S 1),..., S n) ) := S, say. Suppose ha he P -dynamics of hese securiies are driven by m independen Brownian moions so ha ds = µ S d + Σ dw where W is an m 1 sandard Brownian moion ha generaes he filraion F, µ is an F -adaped n n diagonal marix and Σ is an F -adaped n m marix. Assume ha here is a cash-accoun ha earns ineres a a consan coninuously compounded risk-free rae of r. Use Girsanov s Theorem o deermine he condiions under which i) his marke is arbirage-free and ii) arbirage-free and complee and iii) arbirage-free and incomplee. Noe ha here is no loss of generaliy in assuming ha he m Brownian moions are independen. If hey were dependen, we could use he Cholesky Decomposiion o work wih independen Brownian moions. Finally, noe ha we could also easily allow r o be an adaped process driven by he same se of m Brownian moions.) 8. The curren index price is $100 and he erm srucure of ineres raes is consan a 3%. European call and pu opion prices of various srikes and mauriies are presened below. T.25.5 1 1.5 Srike 60 40.2844 42.4249 50.8521 59.1664 Call Prices 70 30.5281 33.5355 42.6656 51.2181 80 21.0415 24.9642 34.4358 42.9436 90 12.2459 16.9652 26.4453 34.7890 100 5.2025 10.1717 19.4706 27.8938 110 1.3448 5.4318 14.4225 23.3305 120 0.2052 2.7647 11.2103 20.7206 130 0.0216 1.4204 9.1497 19.1828 140 0.0019 0.7542 7.7410 18.1858 T.25.5 1 1.5 Srike 60 0.0858 2.1546 10.6907 19.3603 Pu Prices 70 0.2548 3.1164 12.2087 20.9720 80 0.6934 4.3962 13.6833 22.2575 90 1.8232 6.2483 15.3972 23.6629 100 4.7050 9.3060 18.1270 26.3276 110 10.7725 14.4171 22.7834 31.3243 120 19.5582 21.6012 29.2757 38.2744 130 29.2999 30.1080 36.9195 46.2965 140 39.2055 39.2929 45.2152 54.8595
Black-Scholes and he Volailiy Surface 20 a) Use pu-call pariy o deermine a piece-wise consan dividend yield implied by he opion prices. Does your dividend yield depend on he srikes you choose? b) Wrie a piece of code o deermine he Black-Scholes implied volailiy for each opion and plo he volailiy surface. In pracice, only he bid and ask prices of opions are available in he marke place and some pre-processing will be necessary o build he volailiy surface and calibrae he implied dividends. For example, some opions will have very wide bid-offers and are herefore less informaive. Moreover, because hese opions are less liquid i is also he case ha hese bid-offers may no have been updaed as recenly as he more liquid opions. I is ofen preferable hen o ignore hem when building he volailiy surface.) 9. Wrie a compuer program ha simulaes he dela-hedging of a long posiion in a European opion in he Black-Scholes model. Your code should ake as inpus he iniial sock price S 0, opion expiraion T, implied volailiy σ imp, risk-free rae r, dividend yield q and srike K as well as wheher he opion is a call or pu. Your code should also ake as inpus: i) he number of re-balancing periods N and ii) he drif and volailiy, µ and σ respecively, of he geomeric Brownian moion used o simulae a pah of he underlying sock price. Noe ha σ imp and σ need no be he same. A he very leas your code should oupu he opion payoff and he oal P&L from holding he opion and execuing he dela-hedging sraegy. See Figure 8 in he Black-Scholes and he Volailiy Surface lecure noes for an example where he code was wrien in VBA wih he oupu in Excel.) Once you have esed your code answer he following quesions: a) When σ imp = σ how does he oal P&L behave as a funcion of N? Wha happens on average if σ imp < σ? If σ imp > σ? b) For a fixed N, how does he oal P&L behave as σ imp = σ increases? c) How does he drif, µ, affec he oal P&L? d) Run your code repeaedly for σ imp = 20% and σ = 40% wih S 0 = K = $50. Why does he oal P&L move abou so much? How does he variance of he oal P&L depend on he money-ness of he opion? 10. Referring o Example 5, suppose asse X pays a coninuous dividend yield of q x. Show ha only he srike in 30) needs o be changed in order o obain he correc opion price. 11. Referring again o Example 5, suppose asse Y pays a coninuous dividend yield of q y. Show ha 30) is sill valid bu ha we mus now assume Z pays he same dividend yield. 12. Call on he Maximum of 2 Asses) Suppose here are wo asses wih price processes S 1) and S 2), respecively, ha saisfy ds 1) = r q 1 )S 1) d + σ 1 S 1) dw 1) ds 2) = r q 2 )S 2) d + σ 2 S 2) dw 2) where W 1) and W 2) are Q-Brownian moions wih correlaion coefficien, ρ. Q is he EMM corresponding o aking he cash accoun as numeraire and q 1 and q 2 are he respecive dividend yields of he socks. A call-on-he-max opion wih srike K and expiraion T has a payoff given by max 0, max S 1) T, S2) T ) ) K = max 0, S 1) T K, S2) T K ).
Black-Scholes and he Volailiy Surface 21 a) Show ha he value of he opion a mauriy T may be wrien as xs 1) T + ys2) T zk where x, y and z are binary random variables aking he values 0 or 1. b) By considering numeraires V 1) := e q1 S 1), V 2) := e q2 S 2) and R := e r, show ha he ime 0 value of he opion, C say, is given by C = e q1t S 1) 0 Prob V1 x = 1) + e q2t S 2) 0 Prob V2 y = 1) e rt K Prob R z = 1) 34) where Prob Z ) denoes a risk-neural probabiliy corresponding o he numeraire Z. c) Compue he probabiliies in b) and herefore deermine he price of he opion. 13. Variaion on Back Q7.2) In he Black-Scholes framework, deermine he dela-hedging sraegy for a call opion on a fuures conrac where he he fuures conrac is wrien on a sock ha has a coninuous dividend yield of q. Wihou doing any calculaions you should be able o ell ha as par of he dela hedge you always inves C in he cash accoun a ime where C is he ime value of he opion. Why is his he case? This assumes ha you use he fuures conrac and he cash accoun as your hedging securiies raher han he underlying sock and he cash accoun. The answer o Exercise 5 should be useful in deermining he ime posiion in he underlying fuures conrac.) Remark: Here s he way o hink abou he self-financing condiion wih fuures as he underlying: Le P denoe he value of he self-financing replicaing sraegy a ime and assume you hold x unis of he cash accoun, B, and y unis of he fuures conrac a ime. Then P = x B + y A 35) where A is he ime value of he fuures conrac. Yes, A is in fac idenically 0 as we know bu le s leave i as A for now.) Also le F be he ime fuures price. If you hink abou i for jus a couple of seconds you ll see ha he correc way o view a fuures conrac is ha i is a securiy ha is always worh 0 bu ha pays ou a coninuous dividend yield of df. Therefore he self-financing condiion applied o 35) yields: dp = x db + y da + df ) 36) which is jus he self-financing condiion for a dividend paying securiy. Of course A = 0 for all so da = 0 so 36) becomes dp = x db + y df which from an economic perspecive is clearly correc! 14. Idealized Variance-Swap) A variance-swap wih mauriy T is a derivaive securiy whose ime T payoff is a funcion of annualized realized variance beween = 0 and = T. In paricular, he purchaser of he variance-swap will receive N var σrealized 2 σsrike 2 ) 37) upon expiraion a ime T where σrealized 2 is he annualized realized variance, σ2 Srike is he srike and N var is he variance noional or number of variance unis. The vega noional of he variance-swap is defined by N vega = 2 N var σ Srike.) The srike, σsrike 2, is chosen a ime 0 so ha he iniial value of he variance-swap is zero. Suppose now ha he risk-neural dynamics of a sock price, S, saisfy ds = r q)s d + σ S dw 38) where W is a Q-Brownian moion. Then he annualized coninuous realized variance is given by σ 2 Realized = 1 T T 0 σ 2 d.
Black-Scholes and he Volailiy Surface 22 a) Show ha 1 T T 0 σ 2 d = 2 T T 0 ds S ) ) ST ln. 39) S 0 Equaion 39) implies ha he realized leg of a var-swap can be replicaed by aking a shor posiion in he log conrac and by following a dynamic rading sraegy ha a each ime holds 1/S shares of he underlying sock. b) Use equaions 37), 38) and 39) o show ha he fair srike of a variance swap saisfies σsrike 2 = 2 [ ) [ )]] x r q)t ln E Q ST 0 ln T x where x > 0 is any consan. c) Use he mahemaical ideniy y ln x) y x) = x + x 0 S 0 1 max 0, k y) dk + k2 x 1 max 0, y k) dk k2 o express he fair srike of he variance-swap in erms of call and pu opions wih expiraion T. Simplify your answer by aking x equal o he ime T forward value of he sock. You have now shown ha he fair srike of a variance swap only depends on he prices of vanilla European opions wih expiraion T. Our only model assumpions were i) ha he sock price follow a diffusion as in 38) and ha ii) ineres raes were consan. Noe ha we did no assume σ in 38) was consan or even deerminisic. Of course we did assume ha he realized variance was observed coninuously. In pracice realized variance is accumulaed a discree ime inervals, usually via daily or weekly observaions. We will address his in he nex quesion.) 15. A Variance-Swap in Pracice) In Exercise 14 we considered he coninuous-ime version of a variance swap. In pracice, variance-swap reurns are based on discree ypically daily or weekly) observaions. As before, he payoff of a shor posiion in a variance swap saisfies Payoff = N K 2 σrealized 2 ) bu now σrealized 2 is calculaed as A ) 2 M i=1 ln Si S i 1 σ realized = 100 M where M is he number of observaion periods and A is he annualizaion facor. For example, if daily reurns are used hen we ypically have A = 252. There are approximaely 252 business days in a year. See he sample erm shee ha is posed on he course web-sie.) a) Use i) he approximaion ln1 + x) x for small x and ii) Taylor s Theorem o show ha M i=1 ln S i S i 1 ) 2 2 M i=1 ) 1 ST S i S i 1 ) 2 ln. S i 1 S 0 Show ha risk-neural pricing herefore implies ha he fair srike saisfies K 2 10, 000 2 [ )]) rt T E Q ST 0 ln S 0 40) where r is he risk-free ineres rae and T = M/A is he ime-o-mauriy. In pracice variance-swaps on indices are no dividend-adjused whereas variance-swaps on single socks are. If a variance-swap is
Black-Scholes and he Volailiy Surface 23 dividend-adjused hen, assuming a sock goes ex-dividend on dae i 1 wih dividend d i 1, he variance ) 2 ) S 2. is calculaed wih a conribuion of ln i S i 1+d i 1 insead of ln Si S i 1 Recall also from Quesion 4 ha he log conrac in 40) can be replicaed using call and pu opions wih mauriy T. In paricular a variance-swap can be priced using he volailiy smile for he mauriy of he variance-swap.) b) Suppose we wan o mark-o-marke a variance swap ha expires a ime T and ha was iniiaed a ime 0 wih a oal of M observaions and srike K. Assume oday is dae and ha exacly m observaions have already occurred. Le σexpeced, 2 := expeced realized variance given he reurns up-o ime. Show ha σexpeced, 2 = m M σ2 0, + M m M K2,T where K,T is he fair srike a ime for a new variance swap expiring a ime T and σ0, 2 is he realized variance o dae. Hence show ha he ime value of he variance-swap, V, saisfies V = e rt ) N K 2 σexpeced, 2 ) [ = e rt ) N m K 2 σ 2 M m) M 0,) + N K 2 K 2 ) ],T M = Realized P&L + Implied P&L. This resul is no surprising since i is easily seen ha a variance-swap is simply a sum of 1-period variance swaps. Noe ha his is no rue of volailiy swaps which are herefore much harder o price. Noe also ha in pracice many-variance-swaps are raded wih caps on he realized variance. These caps are deep ou-of-he-money opions on variance.) c) Greeks for variance-swaps can be calculaed eiher analyically or by bumping, i.e. shifing he parameer by a small amoun, recompuing he value of he variance-swap and compuing he derivaive numerically. i) In a Black-Scholes world wih a fla volailiy surface, does a variance-swap have any dela exposure a he beginning of an observaion? Does i have a dela exposure in he middle of an observaion? ii) Wha is he daily hea of a variance swap, i.e. he amoun you will lose or earn over he nex day if he sock price does no move? You can assume he variance-swap is based on daily observaions.) iii) Describe a leas 2 differen mehods by which he vega of a variance-swap could be calculaed. iv) Assuming a consan volailiy surface, compue he gamma of a variance-swap a he beginning of an observaion. Wha is he dollar-gamma of he variance-swap. 16. Download he Excel spreadshee VarSwapUnwind.xls from he course websie where he deails of a long variance-swap posiion can be found. The posiion was iniiaed on May 10 h 2010 a a srike of 27.2% and wih a vega-noional of $200k, i.e. 2NK = 200. The mauriy of he variance-swap is May 12 h 2011. Today s dae is July 2 nd 2010 and he closing price of he underlying index oday is 3772.59. The fair value oday for a variance-swap ha expires on May 12 h 2011 is 35.3%. We wish o unwind he variance-swap immediaely. Wha is he realized P&L of he variance swap afer we unwind i? 17. Use he volailiy surface you compued in Exercise 8 o esimae he price of a digial opion ha pays $1 in he even ha ha he sock price in exacly 1 year from now is greaer han or equal o $120. In order o do his you will need some way of esimaing he volailiy a non-raded srikes. A convenien way o do his is by fiing a spline. This is easy o do in Malab using jus one or wo lines of code.)