Once we know he probabiliy densiy funcion (pdf) φ(s ) of S, a European call wih srike is priced a C() = E [e r d(s ) + ] = e r d { (S )φ(s ) ds } = e

Transcription

1 Opion Basics Conens ime-dependen Black-Scholes Formula Black-76 Model Local Volailiy Model Sochasic Volailiy Model Heson Model Example ime-dependen Black-Scholes Formula Le s begin wih re-discovering he BSM formula. Only his ime, we assume ime-dependen parameers. Assume under EMM measure, sock price follows SDE ds = r S d + σ S dw Here risk-free rae and volailiy are deerminisic (bu no consan) funcions of ime. he bank accoun follows an ODE db = r B d B = exp ( r d) Now suppose we are a ime. Use Io lemma on lns, dlns = 1 ds S 1 2 2S ds, ds = (r 1 2 σ 2 ) d + σ dw lns lns = (r 1 2 σ 2 ) d + σ dw In paricular, a ime, lns is normally disribued, or S is log-normally disribued, his is reached by using he fac lns ~N (lns + (r 1 2 σ 2 ) d Var ( σ dw ) = σ 2 d, σ 2 d)

2 Once we know he probabiliy densiy funcion (pdf) φ(s ) of S, a European call wih srike is priced a C() = E [e r d(s ) + ] = e r d { (S )φ(s ) ds } = e r d { e lns + r d N N ( ( lns + r d σ 2 d ln ) σ 2 d lns + r d 1 2 σ 2 d ln σ 2 d )} where o reach he second line, parial expecaion formula of lognormal disribuion (Wikipedia:Lognormal disribuion) is used. By defining he call opion price can be re-wrien as d 1 = ln (S ) + r d σ 2 d σ 2 d d 2 = ln (S ) + r d 1 2 σ 2 d σ 2 d C = S N(d 1 ) e r d(d 2 ) (1) I is he Black-Scholes formula where r and σ are replaced respecively by r = 1 r d and σ = 1 σ 2 d σ is referred o as he roo-mean-squared volailiy (Rebonao 24). Buying a call while simulaneously selling a pu wih he same srike is equivalen o buying a sock while borrowing e r d. his observaion leads o pu-call pariy C P = S e r d (2)

3 From equaions (1) and (2), he price of a pu is given by Back ou ime-dependen volailiy from implied volailiy P = N( d 2 )e r d N( d 1 )S (3) ha is o say, o calibrae ime-dependen Black-Scholes model o he marke observed implied volailiy erm srucure σ imp () observed a ime. Usually in his case σ imp () sands for a-hemoney (AM) volailiy erm srucure. In oher words, σ () = 1 σ 2 d = σ imp () σ 2 d = σ imp () 2 σ 2 = σ imp () 2 + 2σ imp () σ imp() σ = σ imp () 2 + 2σ imp () σ imp() In heory, his will give volailiy a any ime. In pracice, because we don have coninuous-ime (and differeniable) implied volailiy quoes, some addiional assumpions and inerpolaion mehods (such as piecewise consan) are needed in order o back ou he ime-dependen volailiy model parameer. Black-76 Model Le s coninue o assume ha under risk-neural measure, he sochasic process of sock price is ds = r S d + σ S dw From no-arbirage argumen, he forwards price mus saisfy hen, from Io lemma df(, ) = e r d = r S e r d = σ F(, )dw F(, ) = S e r d ds + S e r d d + σ S e r d ( r d) dw r S e r d d

4 his equaion implies wo poins. Firs is ha forward price is a maringale under risk-neural measure. In oher words, he -forward measure (defined in he Noaion Chaper) coincides wih he risk-neural measure. In fac, E[F(, )] = F(, ) E [ S() S() S() ] = E [ ] = E [ P(, ) P(, ) P(, ) ] ha is, S() is a maringale wih respec o zero coupon bond P(, ). Also noe ha here can be any enor on he forwards curve, e.g., one-year forwards, wo-year forwards, ec. his is useful o price a, say one-year opion on wo-year forwards. he second poin is ha forward s volailiy is exacly he sock volailiy. his conrass wih he ineres raes models (e.g., HJM model, see HJM chaper) where forward s volailiy is smaller han he spo rae volailiy by Now he call opion price is given by σ(, ) = σe a( ) Call = P(, )E[(F(, ) ) + ] Similarly o he BSM model, he forward price follows lognormal disribuion as lnf(, )~N (lnf(, ) 1 2 σ 2 d, σ 2 d) Follow he same logic in he BS model and use he fac F(, ) = S(), we have he Black-76 model where C() = e r d{f(, )N(d 1 ) N(d 2 )} (4) F(, ) ln ( ) + σ 2 2 F(, ) ln ( ) σ 2 2 d 1 =, d 2 = σ σ Buying a call while simulaneously selling a pu wih he same srike is equivalen o enering a sock forward conrac fixed a. his observaion leads o pu-call pariy C P = e r d E{F(, ) } = e r d(f(, ) ) (5) In he sense of sock forward conrac, he a-he-money (F = ) call opion and pu opion have he same price (C = P).

5 From pu-call pariy, he pu opion price is P() = e r d{n( d 2 ) F(, )N( d 1 )} (6) Remark: forward price is a maringale boh under risk-neural measure wih numeraire B(), and under forward measure wih numeraire P(, ), because his wo measures are essenially he same in he siuaion of deerminisic risk. In fac, using Proposiion of Brigo and Mercurio (26), he Radon- Nikodym derivaive beween hese wo measures is Local Volailiy Model dq B B()P(, ) = dqf B()P(, ) = 1 ime dependen Black-Scholes model can be calibraed o he AM implied volailiy erm srucure σ imp (), bu i can produce he implied volailiy surface σ imp (, ). Local volailiy model is described under risk-neural measure as ds = rsd + σ(s, )SdW he volailiy erm is no deerminisic any more. I depends on sae variable S and herefore is sochasic. his leads o a sochasic quadraic variaion which leads o implied volailiy surface (Rebonao 28). Local volailiy model is complee. Backing ou he local volailiy from call opion price his secion follows closely Wilmo (26). Given risk-neural pdf funcion φ(s, ), he call opion price is Apply Leibniz rule, Apply Leibniz rule again, C(, ) = e r (S ) φ(s, )ds C = e r φ(s, ) ds

6 2 C 2 = e r φ(s, ) ha is, he risk-neural disribuion of sock price a ime can be compleely backed ou from he marke quoes of European opions (long buerfly wih close srikes). Now differeniae C wih respec o and use he Fokker-Planck equaion, C = rc + e r φ (S ) ds = rc + e r (S ) [ S (σ2 S 2 φ) (rs S φ)] ds Inegraing his by pars wice, we ge Noice ha he equaion can be re-wrien as Invering his gives 2 C = rc e r σ 2 2 φ + re r e r e r σ 2 2 φ = σ C 2 S φds S φds = C + e r φds C = 1 2 σ2 2 2 C C r 2 C C + r σ(, ) = 2 2 C 2 In addiion, Gaheral (26) assers ha local variance (v L ) is he risk-neural expecaion of he insananeous (sochasic) variance v condiional on he final sock price S being equal o he srike price, or v L σ 2 (, ; S )=E[v S = ] o ge he implied volailiy from local volailiy, he auhor also found ha he Black-Scholes implied variance is he inegral of local variance along he pah ha maximizes he probabiliy densiy of he Brownian bridge beween S and S =, or,

7 Sochasic Volailiy Model σ BS (, ) 2 = 1 [σ2 (S, ; S ) mos prob pah of S = ]d his secion follows Lewis (2) and Gaheral (26). Consider a general SV model under measure P he process can be wrien as ds = μ S d + V S dw S V dv = α(s, V, )d + β(s, V, )dw dw S, dw V = ρd ds = μ S d + V S db 1 dv = α(s, V, )d + β(s, V, ) [ρdb ρ 2 db 2 ] db 1, db 2 = dw S = db 1 dw V = ρdb ρ 2 db 2 Le λ S and λ be he marke risk premium and hedging risk premium, respecively. hey correspond respecively o db 1 and db 2. In addiion, λ S is he original marke price of risk in BSM model. ha is, λ S = μ r V Because he marke is incomplee, λ is a free variable and depends on he marke uiliy funcion. Apply Girsanov heorem (Wikipedia:Girsanov heorem) and le (using he noaion on Wikipedia for a momen) dx = λ S db 1 λ db 2 db 1 = db 1 db 1, dx = db 1 + λ S d db 2 = db 2 db 2, dx = db 2 + λ d dq dp F = Ԑ(X ) = exp (X 1 2 [X ]) = exp { λ S d hen W V under risk-neural measure Q is λ d 1 2 [(λ S ) 2 + (λ ) 2 ]d} dw V = ρdb ρ 2 db 2

8 = ρ(db 1 + λ S d) + 1 ρ 2 (db 2 + λ d) = dw V + (ρλ S + 1 ρ 2 λ ) d = dw V + λ V d where λ V, defined by λ V = ρλ S + 1 ρ 2 λ is known as he marke price of volailiy risk. V he SV model under risk neural measure Q is usually wrien in erms of λ as ds = r S d + V S dw S dv = [α λ V β]d + βdw V dw S, dw V = ρd Le s drop he ils and re-wrie he SV model in risk-neural measure Q as S ds = r S d + V S dw V dv = α(s, V, )d + β(s, V, )dw dw S, dw V = ρd For any payoff F = F(S, V, ), i saisfies he following PDE F F 2 VS2 S 2 + ρβs V 2 F V S F F F 2 β2 + rs + α V2 S V = rf Heson Model he Heson model (Heson 1993) is a SV model ha has closed-form soluions for European opions. Under risk-neural measure Q, Heson model assumes, S ds = r S d + V S dw V dv = κ(θ V )d + σ V dw dw S, dw V = ρd

9 he square roo process of variance says posiive if 2κθ > σ 2. he variance erm follows a CIR process in order o say posiive. Apply he general SV model o his special (square roo) form, he Heson PDE is F F 2 VS2 S 2 + ρσsv 2 F V S σ2 V 2 F F F + rs + κ(θ V) V2 S V = rf Heson (1993) found he soluion by using Fourier ransformaion. Given curren saes (S, V ) and τ =, he European call and pu opion is priced as where C(, τ; S, V ) = S [P 1 (, τ) ] e rτ [P 2 (, τ) ] (7) P(, τ; S, V ) = S [P 1 (, τ) 1 2 ] e rτ [P 2 (, τ) 1 2 ] (8) P j (, τ) = 1 π and he characerisic funcions φ j are given by and where Re φ j (τ, φ) [e iφln ] dφ j = 1,2 iφ φ j (τ, φ) = exp{c j (τ, φ) + D j (τ, φ)v + iφlns } j = 1,2 C j (τ, φ) = rφiτ + κθ σ 2 {(b j ρξφi + d j )τ 2ln [ 1 g je djτ ]} 1 g j D j (τ, φ) = b j ρσφi + d j σ 2 [ 1 edjτ 1 g j e d jτ ] b 1 = κ ρσ b 2 = κ u 1 =.5, u 2 =.5 d j = (ρσφi b j ) 2 σ 2 (2u j φi φ 2 ) g j = b j ρσφi + d j b j ρσφi d j As poined ou by Mikhailov and Nogel (23), C++ sandard library provides complex<> emplae o handle complex numbers; and Quanlib has Gaussian Quadraure inegraion in is mah library.

10 Albrecher e al. (27) claimed ha he following formulaion is idenical o Heson s original formulaion, bu offers beer numerical performance. his is also documened in Gaheral (26). he pu-call pariy sill holds: Example g j = 1 = b j ρσφi d j g j b j ρσφi + d j e d jτ C j (τ, φ) = rφiτ + κθ σ 2 {(b j ρσφi d j )τ 2ln [ 1 g j ]} 1 g j D j = b j ρσφi d j σ 2 [ 1 e djτ 1 g j e d jτ ] C P = S e rτ he accompanying C++ example illusraes Heson model calibraion on SPX index opions. Unlike SPY opions, SPX opions are rade solely in CBOE. hey are European syle, may be exercised only on he las business day before expiraion. he Expiraion dae is se on he Saurday following he hird Friday of he expiraion monh. hey are quoed in decimals, seled in cash. One conrac conains 1 shares of index (muliplier is 1). On Monday, , he call opion ha maures in May is quoed as ID Mauriy Srike Bid Ask Las SPX 5/21/11 C On ha day, S&P 5 index closed a herefore he call opion has inrinsic value of $1.25 and ime value of $12.4. he resuls show ha calibraed model volailiy surface fis he marke implied volailiy surface and is smooher.

11 Marke Implied Vol S/ Model Implied Vol S/ Relaed Bloomberg Commands: SPX US <Eqy> hen OMON ime (years) ime (years) Appendix Fokker-Planck equaion Consider a random variable ha follows SDE dx = μ(x, )d + σ(x, )dw

12 and is value a ime is x. Denoe probabiliy of reaching x a ime > by is ransiion probabiliy funcion f(x, ; x, ). hen he evoluion of his ransiion probabiliy funcion follows he 1-D Fokker- Planck (olmogorov forward) equaion (Wikipedia: Fokker-Planck equaion) Paricularly in he case of BSM, f(x, ) = 1 2 x 2 [σ(x, )2 f(x, )] [μ(x, )f(x, )] x 2 ds = μsd + σsdw Denoe φ(s, ) he probabiliy of reaching S a ime from S a ime. hen is evoluion follows Leibniz inegral rule (wikipedia:leibniz inegral rule) Reference d dy b(y) a(y) φ = S (σ2 S 2 φ) (μs S φ) 2 f(x, y)dx = db(y) da(y) f(b(y), y) dy dy f(a(y), y) + f(x, y) dx y [1] Albrecher, H. and Mayer, P. and Schouens, W. and isaer, J. (27). he lile Heson rap. pp Wilmo Magazine. [2] Bingham, N.H. and iesel, R. (24). Risk-neural valuaion: Pricing and hedging of financial derivaives. Springer Verlag. [3] Brigo, D. and Mercurio, F (26). Ineres rae models: heory and pracice: wih smile, inflaion, and credi. Springer Verlag. [4] Gaheral, J. (26). he volailiy surface: a praciioner's guide. Wiley. [5] Hull, J. (29). Opions, fuures and oher derivaives. Pearson Prenice Hall. [6] Lewis, A.L. (2). Opion Valuaion under Sochasic Volailiy. Finance press. [7] Mikhailov, S. and Nogel, U. (23). Heson s sochasic volailiy model implemenaion, calibraion and some exensions. pp Wilmo. [8] Rebonao, R. (24). Volailiy and correlaion: he perfec hedger and he fox. Wiley. [9] Shreve, S.E. (24). Sochasic calculus for finance: Coninuous-ime models. Springer Verlag. b(y) a(y)

13 [1] Wilmo, P. (26). Paul Wilmo on quaniaive finance, 2 nd. John Wiley & Sons.