Stochastic Calculus and Option Pricing

Transcription

1 Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan , Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

2 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes Model 4 Mulivariae Iô Processes 5 SDEs 6 SDEs and PDEs 7 Risk-Neural Probabiliy 8 Risk-Neural Pricing c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

3 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes Model 4 Mulivariae Iô Processes 5 SDEs 6 SDEs and PDEs 7 Risk-Neural Probabiliy 8 Risk-Neural Pricing c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

4 Brownian Moion Consider a random walk x +nδ wih equally likely incremens of ± Δ. Le he ime sep of he random walk shrink o zero: Δ 0. The limi is a coninuous-ime process called Brownian moion, which we denoe Z, or Z (). We always se Z 0 = 0. Brownian moion is a basic building block of coninuous-ime models. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

5 Properies of Brownian Moion Brownian moion has independen incremens: if < <, hen Z Z is independen of Z Z. Incremens of he Brownian moion have normal disribuion wih zero mean, and variance equal o he ime inerval beween he observaion poins Thus, for example, Z Z N(0, ) E [Z Z ] = 0 Inuiion: Cenral Limi Theorem applied o he random walk. Trajecories of he Brownian moion are coninuous. Trajecories of he Brownian moion are nowhere differeniable, herefore sandard calculus rules do no apply. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

6 Io Inegral Io inegral, also called he sochasic inegral (wih respec o he Brownian moion) is an objec σ u dz u where σ u is a sochasic process. 0 Imporan: σ u can depend on he pas hisory of Z u, bu i canno depend on he fuure. σ u is called adaped o he hisory of he Brownian moion. Consider discree-ime approximaions N σ (i 1)Δ (Z iδ Z (i 1)Δ ), i=1 Δ = N and hen ake he limi of N (he limi mus be aken in he mean-squared-error sense). The limi is well defined, and is called he Io inegral. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

7 Properies of Iô Inegral Iô inegral is linear (a u + c b u ) dz u = a u dz u + c b u dz u X = σ 0 u dz u, is coninuous as a funcion of ime. Incremens of X have condiional mean of zero (under some echnical resricions on σ u ): E [X X ] = 0, > T T Noe: a sufficien condiion for E σ u dz u = 0 is E 0 0 σ2 u du <. Incremens of X are uncorrelaed over ime. If σ is a deerminisic funcion of ime and σ 2 du <, hen X 0 u is normally disribued wih mean zero, and variance E 0 [X 2 2 ] = σ u du 0 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

8 Iô Processes An Iô process is a coninuous-ime sochasic process X, or X(), of he form µ u du + σ u dz u 0 0 µ u is called he insananeous drif of X a ime u, and σ u is called he insananeous volailiy, or he diffusion coefficien. µ d capures he expeced change of X beween and + d. σ dz capures he unexpeced (sochasic) componen of he change of X beween and + d. Condiional mean and variance: E (X +d X ) = µ d + o(d), E (X +d X ) 2 2 = σ d + o(d) c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

9 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes Model 4 Mulivariae Iô Processes 5 SDEs 6 SDEs and PDEs 7 Risk-Neural Probabiliy 8 Risk-Neural Pricing c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

10 Quadraic Variaion Consider a ime discreizaion 0 = 1 < 2 <... < N = T, max n+1 n < Δ n=1,...,n 1 Quadraic variaion of an Iô process X () beween 0 and T is defined as N 1 [X] T = lim X( n+1 ) X ( n ) 2 Δ 0 n=1 For he Brownian moion, quadraic variaion is deerminisic: [Z ] T = T To see he inuiion, consider he random-walk approximaion o he Brownian moion: each incremen equals n+1 n in absolue value. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

11 Quadraic Variaion Quadraic variaion of an Iô process is given by X = µ u du + σ u dz u 0 0 T [X ] T = σ 2 d 0 Heurisically, he quadraic variaion formula saes ha Random walk inuiion: (dz ) 2 = d, d dz = o(d), (d) 2 = o(d) dz = d, d dz = (d) 3/2 = o(d), dz 2 = d Condiional variance of he Iô process can be esimaed by approximaing is quadraic variaion wih a discree sum. This is he basis for variance esimaion using high-frequency daa. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

12 Iô s Lemma Iô s Lemma saes ha if X is an Iô process, X = µ u du + σ u dz u 0 0 hen so is f (, X ), where f is a sufficienly smooh funcion, and f (, X ) f (, X ) df (, X ) = + µ f (, X ) σ 2 d + f (, X ) σ dz X 2 2 X X Iô s Lemma is, heurisically, a second-order Taylor expansion in and X, using he rule ha (dz ) 2 = d, d dz = o(d), (d) 2 = o(d) c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

13 Iô s Lemma Using he Taylor expansion, Shor-hand noaion df (, X ) f (, X ) f (, X ) 1 2 f (, X ) d + dx + X 2 2 d f (, X ) + (dx ) f (, X ) d dx 2 X 2 X f (, X ) f (, X ) f (, X ) = d + µ d + σ dz X X 1 2 f (, X ) + σ 2 2 X 2 d + o(d) df (, X ) = f (, X ) f (, X ) 1 2 f (, X ) d + dx + (dx ) 2 X 2 X 2 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

14 Iô s Lemma Example Le X = exp(a + bz ). We can wrie X = f (, Z ), where f (, Z ) = exp(a + bz ) Using f (, Z ) f (, Z ) 2 f (, Z ) = af (, Z ), = bf (, Z ), = b 2 f (, Z ) Z 2 Z Iô s Lemma implies b 2 dx = a + X d + bx dz 2 Expeced growh rae of X is a + b 2 /2. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

15 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes Model 4 Mulivariae Iô Processes 5 SDEs 6 SDEs and PDEs 7 Risk-Neural Probabiliy 8 Risk-Neural Pricing c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

16 The Black-Scholes Model of he Marke Consider he marke wih a consan risk-free ineres rae r and a single risky asse, he sock. Assume he sock does no pay dividends and he price process of he sock is given by S = S 0 exp µ 1 σ 2 + σz 2 Because Brownian moion is normally disribued, using E 0 [exp(z )] = exp(/2), find Using Iô s Lemma (check) E 0 [S ] = S 0 exp(µ) ds = µ d + σ dz S µ is he expeced coninuously compounded sock reurn, σ is he volailiy of sock reurns. Sock reurns have consan volailiy. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

17 Dynamic Trading Consider a rading sraegy wih coninuous rebalancing. A each poin in ime, hold θ shares of socks in he porfolio. Le he porfolio value be W. Then W θ S dollars are invesed in he shor-erm risk-free bond. Porfolio is self-financing: no exogenous incoming or ougoing cash flows. Porfolio value changes according o dw = θ ds + (W θ S )r d Discree-ime analogy B +Δ W +Δ W = θ (S +Δ S ) + (W θ S ) 1 B c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

18 Opion Replicaion Consider a European opion wih he payoff H(S T ). We will consruc a self-financing porfolio replicaing he payoff of he opion. Look for he porfolio such ha for some funcion f (, S ). W = f (, S ) By Law of One Price, f (, S ) mus be he price of he opion a ime, being he cos of a rading sraegy wih an idenical payoff. Noe ha he self-financing condiion is imporan for he above argumen: we do no wan he porfolio o produce inermediae cash flows. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

19 Opion Replicaion Apply Iô s Lemma o porfolio value, W = f (, S ): f f 1 2 f dw = θ ds + (W θ S )r d = d + S ds + 2 S 2 (ds ) 2 where (ds ) 2 = σ 2 2 S d The above equaliy holds a all imes if θ = f (, S ) f (, S ) 1 2 f (, S ), + σ 2 2 f S 2 S 2 S r f (, S ) S = 0 S If we can find he soluion f (, S) o he PDE f (, S) f (, S) 1 σ 2 S 2 2 f (, S) r f (, S) + + rs + = 0 S 2 S 2 wih he boundary condiion f (T, S) = H(S), hen he porfolio wih replicaes he opion! f (, S ) W 0 = f (0, S 0 ), θ = S c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

20 Black-Scholes Opion Price We conclude ha he opion price can be compued as a soluion of he Black-Scholes PDE f f 1 2 f r f + + rs + σ 2 S 2 = 0 S 2 S 2 If he opion is a European call wih srike K, he PDE can be solved in closed form, yielding he Black-Scholes formula: C(, S ) = S N(z 1 ) exp( r(t ))KN(z 2 ), where N(.) is he cumulaive disribuion funcion of he sandard normal disribuion, log S K + r σ 2 (T ) z 1 =, σ T and z 2 = z 1 σ T. Noe ha µ does no ener he PDE or he B-S formula. This is inuiive from he perspecive of risk-neural pricing. Discuss laer. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

21 Black-Scholes Opion Replicaion The replicaing sraegy requires holding f (, S ) θ = S sock shares in he porfolio. θ is called he opion s dela. I is possible o replicae any opion in he Black-Scholes seing because Price of he sock S is driven by a Brownian moion; Rebalancing of he replicaing porfolio is coninuous; There is a single Brownian moion affecing he payoff of he opion and he price of he sock. More on his laer, when we cover mulivariae Iô processes. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

22 Single-Facor Term Srucure Model Consider a model of he erm srucure of defaul-free bond yields. Assume ha he shor-erm ineres rae follows dr = α(r ) d + β(r ) dz Le P(, τ) denoe he ime- price of a discoun bond wih uni face value mauring a ime τ. We wan o consruc an arbirage-free model capuring, simulaneously, he dynamics of bond prices of many differen mauriies. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

23 Single-Facor Term Srucure Model Assume ha bond prices can be expressed as a funcion of he shor rae only (single-facor srucure) P(, τ) = f (, r, τ) Iô formula implies ha f f dp(, τ) = + α(r ) f β 2 (r ) d + f β(r ) dz r 2 r 2 r µ τ Consider a self-financing porfolio which invess P(, τ)/σ τ dollars in he bond mauring a τ, and P(, τ )/σ τ dollars in he bond mauring a τ. Self-financing requires ha he invesmen in he risk-free shor-erm bond is W P(, τ) P(, τ στ + στ ) σ τ c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

24 Single-Facor Term Srucure Model Porfolio value evolves according o dw = W P(, τ) P(, τ τ ) στ + στ r + µτ µ τ σ σ τ d+ σ τ σ τ dz σ τ σ τ = W P(, τ) P(, τ τ ) στ + στ r + µτ µ τ σ στ d Porfolio value changes are insananeously risk-free. To avoid arbirage, he porfolio value mus grow a he risk-free rae: W P(, τ) P(, τ στ + στ ) r + µ τ στ µ τ σ τ = W r c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

25 Single-Facor Term Srucure Model We conclude ha µ τ r P(, τ) = µ τ r P(, τ ), for any τ and τ σ τ Assume ha, for some τ, Then, for all bonds, mus have σ τ µ τ r P(, τ ) = η(, r ) σ τ µ τ r P(, τ) = η(, r )σ τ Recall he definiion of µ τ, σ τ o derive he pricing PDE on P(, τ) = f (, r, τ) f f 1 2 f f + α(r) + β 2 (r) rf = η(, r)β(r), f (τ, r, τ) = 1 r 2 r 2 r The soluion, indeed, has he form f (, r, τ). c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

26 Single-Facor Term Srucure Model Wha if µ τ r P(, τ) σ τ = η(, r ) for any funcion η, i.e., he LHS depends on somehing oher han r and? Then he erm srucure will no have a single-facor form. We have seen ha, for each choice of η(, r ), absence of arbirage implies ha bond prices mus saisfy he pricing PDE. The reverse is rue: if bond prices saisfy he pricing PDE (wih well-behaved η(, r )), here is no arbirage (show laer, using risk-neural pricing). The choice of η(, r ) deermines he join arbirage-free dynamics of bond prices (yields). η(, r ) is he price of ineres rae risk. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

27 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes Model 4 Mulivariae Iô Processes 5 SDEs 6 SDEs and PDEs 7 Risk-Neural Probabiliy 8 Risk-Neural Pricing c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

28 Muliple Brownian moions 1 Consider wo independen Brownian moions Z and Z 2. Consruc a hird process 1 X = ρz + 1 ρ 2 2 Z X is also a Brownian moion: X has IID normal incremens; X is coninuous. X and Z 1 are correlaed: E 0 X Z 1 = ρ Correlaed Brownian moions can be consruced from uncorrelaed ones, jus like wih normal random variables. Cross-variaion N 1 1 [Z, Z 2 ] T = lim (Z 1 ( n+1 ) Z 1 ( n )) (Z 2 ( n+1 ) Z 2 ( n )) = 0 Δ 0 n=1 Shor-hand rule dz 1 dz 2 = 0 dz 1 dx = ρ d c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

29 Mulivariae Iô Processes A mulivariae Iô process is a vecor process wih each coordinae driven by an Iô process. Consider a pair of processes dx =µ X d + σ X dz X, dy =µ Y d + σ Y dz Y, dz X dz Y =ρ d Iô s formula can be exended o muliple process as follows: f f f df (, X, Y ) = d + dx + dy + X Y 1 2 f 1 2 f 2 f (dx ) 2 + (dy ) 2 + dx dy 2 X 2 2 Y 2 X Y c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

30 Example of Iô s formula Consider wo asse price processes, X and Y, boh given by Iô processes dx = µ X d + σ X dz X dy = µ Y d + σ Y dz Y dz X dz Y = 0 Using Iô s formula, we can derive he process for he raio f = X /Y (use f (X, Y ) = X/Y ): 2 df dx dy dx dy dy = + f X Y X Y Y 2 X Y µ σ Y σ X σ Y = µ + X d + dz X Y 2 Y X Y dz Y c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

31 Example of Iô s formula We find ha he expeced growh rae of he raio X /Y is X Y σ Y 2 µ µ + X Y 2 Y Assume ha µ X = µ Y. Then, df σ Y 2 E = d f Y 2 Repeaing he same calculaion for he inverse raio, h = Y /X, we find X 2 dh σ E = d h 2 X I is possible for boh he raio X /Y and is inverse Y /X o be expeced o grow a he same ime. Applicaion o FX. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

32 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes Model 4 Mulivariae Iô Processes 5 SDEs 6 SDEs and PDEs 7 Risk-Neural Probabiliy 8 Risk-Neural Pricing c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

33 Sochasic Differenial Equaions Example: Heson s sochasic volailiy model ds =µ d + v dz S S dv = κ(v v) d + γ v v dz dz S dz v =ρ d Condiional variance v is described by a Sochasic Differenial Equaion. Definiion (SDE) The Iô process X saisfies a sochasic differenial equaion wih an iniial condiion X 0 if i saisfies dx = µ(, X ) d + σ(, X ) dz X = X 0 + µ(s, X s ) ds + σ(s, X s ) dz s. 0 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

34 Exisence of Soluions of SDEs Assume ha for some C, D > 0 µ(, X) + σ(, X ) C(1 + X ) and µ(, X ) µ(, Y ) + σ(, X) σ(, Y ) D X Y for any X and Y (Lipschiz propery). Then, he SDE dx = µ(, X ) d + σ(, X ) dz, X 0 = x, has a unique coninuous soluion X. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

35 Common SDEs Arihmeic Brownian Moion The soluion of he SDE dx = µ d + σ dz is given by X = X 0 + µ + σz. The process X is called an arihmeic Brownian moion, or Brownian moion wih a drif. Guess and verify. We ypically reduce an SDE o a few common cases wih explici soluions. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

36 Common SDEs Geomeric Brownian Moion Consider he SDE Define he process By Iô s Lemma, dx = µx d + σx dz Y = ln(x ). 1 1 dy = µx d + σx dz + 1 X X 2 ( 1 X 2 )σ2 2 X d = (µ σ 2 /2) d + σ dz. Y is an arihmeic Brownian moion, given in he previous example, and Y = Y 0 + (µ σ 2 /2) + σz. Then X = e Y = X 0 exp (µ σ 2 /2) + σz c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

37 Common SDEs Ornsein-Uhlenbeck process The mean-revering Ornsein-Uhlenbeck process is he soluion X o he sochasic differenial equaion dx = (X X ) d + σ dz We solve his equaion using e as an inegraing facor. Seing Y = e and using Iô s lemma for he funcion f (X, Y ) = X Y, we find d(e X ) = e (X d + σ dz ). Inegraing his beween 0 and, we find i.e., e X X 0 = e s Xds + e s σdz s, 0 0 X = e X 0 + (1 e )X + σ e s dz s. 0 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

38 Opion Replicaion in he Heson Model Assume he Heson sochasic-volailiy model for he sock. Aemp o replicae he opion payoff wih he sock and he risk-free bond. Can we find a rading sraegy ha would guaranee perfec replicaion? I is possible o replicae an opion using a bond, a sock, and anoher opion. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

39 Recap of APT Recall he logic of APT. Suppose we have N asses wih wo-facor srucure in heir reurns: R i = a i + b i 1 F 1 + b i 2 F 2 Ineres rae is r. While no saed explicily, all facor loadings may be sochasic. Unlike he general version of he APT, we assume ha reurns have no idiosyncraic componen. A ime, consider any porfolio wih fracion θ i in each asse i ha has zero exposure o boh facors: b 1 1 θ 1 + b 2 N 1 θ b 1 θ N = 0 b 1 2 θ 1 + b 2 N 2 θ b 2 θ N = 0 This porfolio mus have zero expeced excess reurn o avoid arbirage θ 1 E [R 1 ] r + θ 2 E [R 2 N ] r θ N E [R ] r = 0 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

40 Recap of APT To avoid arbirage, any porfolio saisfying b 1 1 θ 1 + b 2 N 1 θ b 1 θ N = 0 b 1 2 θ 1 + b 2 N 2 θ b 2 θ N = 0 mus saisfy θ 1 E [R 1 ] r + θ 2 E [R 2 N ] r θ N E [R ] r = 0 Resaing his in vecor form, any vecor orhogonal o 1 2 N 1 2 N (b 1, b 1,..., b 1 ) and (b 2, b 2,..., b 2 ) mus be orhogonal o (E [R 1 ] r, E [R 2 ] r,..., E [R N ] r). Conclude ha he hird vecor is spanned by he firs wo: here exis consans (prices of risk) (λ 1, λ 2 ) such ha i E [R ] r = λ 1 b i 1 + λ 2 i b 2, i = 1,..., N c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

41 Opion Pricing in he Heson Model Suppose here are N derivaives wih prices given by f i (, S, v ) where he firs opion is he sock iself: f 1 (, S, v ) = S. Using Io s lemma, heir prices saisfy df i (, S, v ) = a i d + f i (, S, v ) ds + f i (, S, v ) dv S v Compare he above o our APT argumen Conclude ha here exis λ S and λ v such ha E df i (, S, v ) rf i (, S, v ) d = f i (, S, v ) S λ d + f i (, S, v ) v λ d S v We work wih price changes insead of reurns, as we did in he APT, because some of he derivaives may have zero price. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

42 Opion Pricing in he Heson Model The APT pricing equaion, applied o he sock, implies ha λ v E [ds rs d] = (µ r)s d = λ S d is he price of volailiy risk, which deermines he risk premium on any invesmen wih exposure o dv. Wriing ou he pricing equaion explicily, wih he Io s lemma providing an expression for E df i (, S, v ), f i f i f i + µs + ( κ)(v v)+ S v 1 2 f i 1 2 f i 2 f i f i f i vs 2 + γ 2 v + ργsv rf i = (µ r)s + λ v 2 S 2 2 v 2 S v S v As long as we assume ha he price of volailiy risk is of he form λ v = λ v (, S, v ) he assumed funcional form for opion prices is jusified and we obain an arbirage-free opion pricing model. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

43 Numerical Soluion of SDEs Excep for a few special cases, SDEs do no have explici soluions. The mos basic and common mehod of approximaing soluions of SDEs numerically is using he firs-order Euler scheme. Use he grid i = iδ. X i+1 = X i + µ( i, X i ) Δ + σ( i, X i ) Δε i, where ε i are IID N(0, 1) random variables. Using a binomial disribuion for ε i, wih equal probabiliies of ±1, is also a valid procedure for approximaing he disribuion of X c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

44 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes Model 4 Mulivariae Iô Processes 5 SDEs 6 SDEs and PDEs 7 Risk-Neural Probabiliy 8 Risk-Neural Pricing c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

45 Momens of Diffusion Processes Ofen need o compue condiional momens of diffusion processes: Expeced reurns and variances of reurns on financial asses over finie ime inervals; Use he mehod of momens o esimae a diffusion process from discreely sampled daa; Compue prices of derivaives. One approach is o reduce he problem o a PDE, which can someimes be solved analyically. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

46 Kolmogorov Backward Equaion Diffusion process X wih coefficiens µ(, X) and σ(, X ). Objecive: compue a condiional expecaion f (, X) = E[g(X T ) X = X ] Suppose f (, X ) is a smooh funcion of and X. By he law of ieraed expecaions, f (, X ) = E [f ( + d, X +d )] E [df (, X )] = 0 Using Io s Lemma, f (, X ) f (, X ) 1 E [df (, X )] = + µ(, X ) + σ(, X) 2 2 f (, X ) d = 0 X 2 X 2 Kolmogorov backward equaion f (, X ) + µ(, X) f (, X) + 1 σ(, X ) 2 2 f (, X ) = 0, X 2 X 2 wih boundary condiion f (T, X) = g(x) c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

47 Example: Square-Roo Diffusion Consider a popular diffusion process used o model ineres raes and sochasic volailiy dx = κ(x X ) d + σ X dz We wan o compue he condiional momens of his process, o be used as a par of GMM esimaion. Compue he second non-cenral momen Using Kolmogorov backward equaion, f (, X) = E(X 2 T X = X) f (, X ) κ(x X ) f (, X ) + 1 σ 2 X 2 f (, X ) = 0, X 2 X 2 wih boundary condiion f (T, X) = X 2 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

48 Example: Square-Roo Diffusion Look for he soluion in he he form Subsiue f (, X ) ino he PDE f (, X ) = a 0 () + a 1 ()X + a 2() X 2 2 a 0 () + a 1 ()X + a 2 () X 2 κ(x X )(a 1 () + a 2 ()X ) + σ2 Xa 2 () = Collec erms wih differen powers of X, zero, one and wo, o ge wih iniial condiions a 0 () + κxa 1 () = 0 σ 2 a 1 () κa 1 () + + κx a 2 () = 0 2 a 2 () 2κa 2 () = 0 a 0 (T ) = a 1 (T ) = 0, a 2 (T ) = 2 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

49 Example: Square-Roo Diffusion We solve he sysem of equaions saring from he hird one and working up o he firs: 1 a 0 () = XA 2 e2κ( T ) e κ( T ) a 1 () = A e κ( T ) e 2κ( T ) a 2 () = 2e 2κ( T ) A = σ2 + 2κX κ Compare he exac expression above o an approximae expression, obained by assuming ha T = Δ is small. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

50 Example: Square-Roo Diffusion Assume T = Δ is small. Using Taylor expansion, a 0 () = o(δ), a 1 () = AκΔ + o(δ) a 2 () = 2(1 2κΔ) + o(δ) Then E (X 2 +Δ X = X ) X 2 + Δ (σ 2 + 2κX )X 2κX 2 Alernaively, X +Δ X κ(x X ) Δ + σ X Δε, ε N(0, 1) Therefore 2 E (X +Δ X = X ) X 2 + Δ σ 2 X 2κX (X X ) c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

51 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes Model 4 Mulivariae Iô Processes 5 SDEs 6 SDEs and PDEs 7 Risk-Neural Probabiliy 8 Risk-Neural Pricing c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

52 Risk-Neural Probabiliy Measure Under he risk-neural probabiliy measure Q, expeced condiional asse reurns mus equal he risk-free rae. Alernaively, using he discouned cash flow formula, he price P of an asse wih payoff H T a ime T is given by T P = E Q exp r s ds H T where r s is he insananeous risk-free ineres rae a ime s. The DCF formula under he risk-neural probabiliy can be used o compue asse prices by Mone Carlo simulaion (e.g., using an Euler scheme o approximae he soluions of SDEs). Alernaively, one can derive a PDE characerizing asse prices using he connecion beween PDEs and SDEs. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

53 Change of Measure We ofen wan o consider a probabiliy measure Q differen from P, bu he one ha agrees wih P on which evens have zero probabiliy (equivalen o P) (e.g., Q could be a risk-neural measure). Differen probabiliy measures assign differen relaive likelihoods o he rajecories of he Brownian moion. I is easy o express a new probabiliy measure Q using is densiy dq ξ T = dp T For any random variable X T, E Q 0 [X T ] = E P 0 [ξ T X T ], E Q [X T ] = E P ξ T XT, ξ E P [ξ T ] Q is equivalen o P if ξ T is posiive (wih probabiliy one). ξ c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

54 Change of Measure If we consider a probabiliy measure Q differen from P, bu he one ha agrees wih P on which evens have zero probabiliy, hen he P-Brownian moion Z P becomes an Io process under Q: dz P = dz Q η d for some η. Z Q is a Brownian moion under Q. When we change probabiliy measures his way, only he drif of he Brownian moion changes, no he variance. Inuiion: a probabiliy measure assigns relaive likelihood o differen rajecories of he Brownian moion. Variance of he Io process can be recovered from he shape of a single rajecory (quadraic variaion), so i does no depend on he relaive likelihood of he rajecories, hence, does no depend on he choice of he probabiliy measure. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

55 Risk-Neural Probabiliy Measure Under he risk-neural probabiliy measure, expeced condiional asse reurns mus equal he risk-free rae. Sar wih he sock price process under P: Under he risk-neural measure Q, Thus, if dz P = η d + dz Q, ds = µ S d + σ S dz P ds = r S d + σ S dz Q µ σ η = r η is he price of risk. The risk-neural measure Q is such ha he process Z Q defined by is a Brownian moion under Q. dz Q = µ r d + dz P σ c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

56 Risk-Neural Probabiliy Measure Under he risk-neural probabiliy measure, expeced condiional asse reurns mus equal he risk-free rae. We conclude ha for any asse (paying no dividends), he condiional risk premium is given by E P ds r d = E P ds E Q ds S S S Thus, mahemaically, he risk premium is he difference beween expeced reurns under he P and Q probabiliies. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

57 Risk-Neural Probabiliy Measure If we wan o connec Q o P explicily, how can we compue he densiy, dq/dp? In discree ime, he densiy was condiionally lognormal. The densiy ξ T = (dq/dp) T is given by P ξ = exp η u dz u 1 η 2 u 2 du, 0 T 0 0 The sae-price densiy is given by π = exp r u du ξ 0 The reverse is rue: if we define ξ T as above, for any process η saisfying cerain regulariy condiions (e.g., η is bounded, or saisfies he Novikov s condiion as in Back 2005, Appendix B.1), hen measure Q is equivalen o P and is a Brownian moion under Q. Z Q = Z P + η u du 0 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

58 Risk-Neural Probabiliy and Arbirage If here exiss a risk-neural probabiliy measure, hen he model is arbirage-free. If here exiss a unique risk-neural probabiliy measure in a model, hen all opions are redundan and can be replicaed by rading in he underlying asses and he risk-free bond. A convenien way o build arbirage-free models is o describe hem direcly under he risk-neural probabiliy. One does no need o describe he P measure explicily o specify an arbirage-free model. However, o esimae models using hisorical daa, paricularly, o esimae risk premia, one mus specify he price of risk, i.e., he link beween Q and P. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

59 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes Model 4 Mulivariae Iô Processes 5 SDEs 6 SDEs and PDEs 7 Risk-Neural Probabiliy 8 Risk-Neural Pricing c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

60 Black-Scholes Model Assume ha he sock pays no dividends and he sock price follows ds = µ d + σ dz P S Assume ha he ineres rae is consan, r. Under he risk-neural probabiliy Q, he sock price process is ds = r d + σ dz Q S Terminal sock price S T is lognormally disribued: ln S T = ln S 0 + r σ2 T + σ T ε Q, ε Q N(0, 1) 2 Price of any European opion wih payoff H(S T ) can be compued as = E Q r(t P e ) H(S T ) c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

61 Term Srucure of Ineres Raes Consider he Vasicek model of bond prices. A single-facor arbirage-free model. To guaranee ha he model is arbirage-free, build i under he risk-neural probabiliy measure. Assume he shor-erm risk-free rae process under Q dr = κ(r r) d + σ dz Q Price of a pure discoun bond mauring a T is given by T P(, T ) = E Q exp r s ds Characerize P(, T ) as a soluion of a PDE. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

62 Term Srucure of Ineres Raes Look for P(, T ) = f (, r ). Using Io s lemma, E Q f (, r ) [df (, r )] = κ(r r) f (, r ) 1 + σ 2 2 f (, r ) d r 2 2 r Risk-neural pricing requires ha E Q [df (, r )] = r f (, r ) d and herefore f (, r ) mus saisfy he PDE f (, r) κ(r r) f (, r) + 1 σ 2 2 f (, r) = rf (, r) r 2 r 2 wih he boundary condiion f (T, r) = 1 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

63 Term Srucure of Ineres Raes Look for he soluion in he form f (, r ) = exp ( a(t ) b(t )r ) Derive a sysem of ODEs on a() and b() o find a(t ) = r(t ) r 1 e κ(t ) κ σ 2 2κ(T ) e 2κ(T ) + 4e κ(t ) 3 4κ 3 1 κ(t ) b(t ) = 1 e κ c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

64 Term Srucure of Ineres Raes Assume a consan price of risk η. Wha does his imply for he ineres rae process under he physical measure P and for he bond risk premia? Use he relaion dz P = η d + dz Q o derive P dr = κ(r r) d + ση d + σ dz = κ r r + ση P d + σ dz κ Expeced bond reurns saisfy E P dp(, T ) = (r + σ P η) d P(, T ) where σ P = 1 f (, r ) σ = b(t )σ f (, r ) r b(t )σ is he volailiy of bond reurns. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

65 Equiy Opions wih Sochasic Volailiy Consider again he Heson s model. Assume ha under he risk-neural probabiliy Q, sock price is given by d ln S = (r 1 v ) d + v dz Q,S 2 dv = κ(v v) d + γρ v dz Q,S + γ 1 ρ 2 v dz Q,v dz Q,S dz Q,v = 0 Z Q,v models volailiy shocks uncorrelaed wih sock reurns. Consan ineres rae r. The price of a European opion wih a payoff H(S T ) can be compued as P = E Q [exp ( r(t )) H(S T )] We haven said anyhing abou he physical process for sochasic volailiy. In paricular, how is volailiy risk priced? c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

66 Equiy Opions wih Sochasic Volailiy In he Heson s model, assume ha he price of volailiy risk is consan, η v, and he price of sock price risk is consan, η S. Then, under P, sock reurns follow d ln S = r + η S 1 v v d + P,S v dz 2 dz P,S dz P,v = 0 We have used dv = κ(v v) + γ v ρη S + 1 ρ 2 η v d+ γρ v dz P,S + γ 1 ρ 2 v dz P,v dz P,S = η S d + dz Q,S dz P,v = η v d + dz Q,v Condiional expeced excess sock reurn is E P ds r d = E P d ln S + v d r d = η S v d S 2 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

67 Equiy Opions wih Sochasic Volailiy Our assumpions regarding he marke prices of risk ranslae direcly ino implicaions for reurn predicabiliy. For sock reurns, our assumpion of consan price of risk predics a nonlinear paern in excess reurns: expeced excess sock reurns proporional o condiional volailiy. Suppose we consruc a posiion in opions wih he exposure λ o sochasic volailiy shocks and no exposure o he sock price: dw = [...] d + λ dz P,v Then he condiional expeced gain on such a posiion is (W r + λ η v ) d c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

68 Heson s Model of Sochasic Volailiy Assume ha under he risk-neural probabiliy Q, sock price is given by d ln S = (r 1 v ) d + v dz Q,S 2 dv = κ(v v) d + γρ v dz Q,S + γ 1 ρ 2 v dz Q,v dz Q,S dz Q,v = 0 Z Q,v models volailiy shocks uncorrelaed wih sock reurns. Consan ineres rae r. The price of a European opion wih a payoff H(S T ) can be compued as P = E Q [exp ( r(t )) H(S T )] Assume ha he price of volailiy risk is consan, η v, and he price of sock price risk is consan, η S. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

69 Variance Swap in Heson s Model Consider a variance swap, paying T (d ln S u ) 2 K 2 a ime T. Wha should be he srike price of he swap, K, o make sure ha he marke value of he swap a ime is zero? Using he resul on quadraic variaion, (d ln S ) 2 = v d he srike price mus be such ha T e r(t ) E Q 2 v u du K Need o compue K 2 = E Q T v u du = 0 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

70 Variance Swap in Heson s Model Since u u Q,S v u = v κ(v s v) ds + γρ vs dz s + γ u 1 ρ 2 Q,v vs dz s we find ha u u E Q [v u ] = v E Q κ(v s v) ds = v κ(e Q [v s] v) ds Solving he above equaion for E Q [v u ], we find We obain he srike price E Q [v u ] = v + e κ(u ) (v v) T 1 2 K = E Q κ(t ) v u du = v(t ) + (v v) 1 e κ c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

71 Expeced Profi/Loss on a Variance Swap To compue expeced profi/loss on a variance swap, we need o evaluae T E P (d ln S u ) 2 K 2 Insananeous expeced gain on a long posiion in he swap is easy o compue in closed form. The marke value of a swap sars a 0 a ime, and a s > becomes s T P s E Q e r (T s) v u du + v u du K 2 s s = e r(t s) 2 2 v u du + K s K We conclude ha he insananeous gain on he long swap posiion a ime s equals [...] ds + 1 e r(t s) 1 e κ(t s) dv s κ c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74 s

72 Expeced Profi/Loss on a Variance Swap We conclude ha he insananeous gain on he long swap posiion a ime s equals [...] ds + 1 e r(t s) 1 e κ(t s) dv s κ Given our assumed marke prices of risk, η S and η v, he ime-s expeced insananeous gain on he swap opened a ime is r (T s) P s r ds + γ 1 v s e 1 e κ κ(t s) ρη S + 1 ρ 2 η v ds c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

73 Summary Risk-neural pricing is a convenien framework for developing arbirage-free pricing models. Connecion o classical resuls: risk-neural expecaion can be characerized by a PDE. Risk premium on an asse is he difference beween expeced reurn under P and under Q probabiliy measures. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

74 Readings Back 2005, Secions , , 2.11, 13.2, 13.3, Appendix B.1. c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus , Fall / 74

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