MARKET MODEL OF STOCHASTIC IMPLIED VOLATILITY WITH APPLICATION TO THE BGM MODEL

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1 MARKET MODEL OF STOCHASTIC IMPLIED VOLATILITY WITH APPLICATION TO THE BGM MODEL ALAN BRACE, BEN GOLDYS, FIMA KLEBANER, AND ROB WOMERSLEY Absrac. Using a sochasic implied volailiy mehod we show how o inroduce smiles and skews ino he BGM ineres rae model. Conens. Inroducion.. Derivaive formulae 3. Dynamics of he implied volailiy surface 5 3. Differen formulaions 6 4. Some properies of soluions 9 5. Toy model 6. Applicaion o BGM 3 7. Marginal Disribuions Bachelier Model Derivaives Numerical resuls 7 Appendix A. Io-Vensel formula 7 References 7. Inroducion The aim of his paper is o presen a new model for implied volailiy. The main resuls and some properies of he model are announced wihou proofs. Those will be conained in he second par of his work, currenly under preparaion. Suppose, as in [], here are a full specrum of zero coupon bonds P, T )mauringa all imes T up o a finie horizon T,andleW T ) be Brownian moion under he forward measure P T locaed a mauriy T wih corresponding numeraire P, T )). Recall ha he forwards L, T ) over he inerval [T,T ], where T = T + δ, are relaed o zero coupons via he relaion L, T )= [ ] P, T ) δ P, T ). WORKING PAPER S-, DEPARTMENT OF STATISTICS, UNIVERSITY OF NEW SOUTH WALES

2 ALAN BRACE, BEN GOLDYS, FIMA KLEBANER, AND ROB WOMERSLEY From he arbirage free dynamics of he zero coupon bonds, L, T ) musbeaposiive maringale under he forward measure P T locaed a he end of [T,T ], so model i wih he SDE dl, T ) L, T ) = θ, T ) dw T ),.) where in general he volailiy funcion θ = θ, T ) is sochasic. The form of he SDE.) under P T is similar o ha of a sock under he spo measure P in he sandard Black-Scholes BS) model when ineres raes are zero. So o clarify ideas, firs consider modelling sochasic implied volailiy for a sock. Following he noaion and approach of Carr [4] bu assuming ineres raes are zero, under he spo arbirage free measure P, he numeraire will be uniy and he underlying sock S is a maringale which we may assume saisfies he SDE ds = S θ dw ),.) where θ is sochasic and dw is muli-dimensional Brownian moion under P. Noe ha, wih no loss of generaliy, we are aking all componens of he insananeous volailiy vecor θ o be zero excep he firs. The BS implied volailiy convenion says ha if he ime sochasic) implied volailiy of an opion exercising a ime T wih srike K is hen he ime price of a call opion will be where σ = σ T,K)=σ, T, K), C = C, T, K) =l S,σ T,K),T ; K),.3) l = l S, σ, τ; K) =SN h ) KN h ),.4) ln S h = K σ τ + σ τ, h = h σ τ. If he implied volailiy is also a diffusion saisfying an SDE like dσ = m T,K,S,θ,σ ) d + v T,K,S,θ,σ ) dw = m d + v dw,.5) hen because he calls C mus also be maringales under P, i follows ha he drif m and volvol v canno be arbirary, bu mus saisfy cerain exra condiions. Those condiions will lead naurally o a sysem of SDEs for he implied volailiy σ. The dependence of he volvol v T,K,S,θ,σ )onσ will be specified o ge rid of some roublesome singulariies. We also suppose here are a full specrum of call opions available for all srikes K and all mauriies T up o some horizon T. This assumpion leads o wo criical feedback condiions: The implied volailiy σ T,K)ofheT-mauring call mus remain finie a mauriy, ha is for T σ T,K)T ) and lim T σ T,K)T ) =..6)

3 STOCHASTIC VOLATILITY 3 The insananeous volailiy θ of he underlying sock S mus equal he implied volailiy of he a-he-money opion mauring immediaely, ha is θ = σ, S )..7) Le us emphasise imporance of condiion.6). I imposes a very severe resricion on he volvol process v ). I becomes even more sriking if we rewrie he SDE for he process σ ) in erms of a new process ξ =T )σ see Secion 3 for deails). In ha case we end up wih a sochasic differenial equaion for he process ξ ) wih he iniial condiion ξ T,K)=fT,K) say) and he erminal condiion ξ T T,K) =. I is well known see for example [5], ha sochasic differenial equaions of his ype need no have adaped soluions, unless he coefficiens of his equaion saisfy cerain condiions. In our case, his fac is a source of mahemaical difficulies bu on he oher hand i allows o obain a closed sysem of equaions wih he coefficiens which are deermined inrinsically. Reurning o caps and caples, wo addiional problems ha mus be ackled o inegrae he above approach ino he ineres rae area are: How o approach a specrum of caples mauring a T and paying a T when he dynamics of each is specified under is own forward measure P T. How o use correlaion o ransfer feedback informaion from he immediaely mauring caple o laer caples... Derivaive formulae. Here are some formulae ha will be required laer, for he firs and second parial derivaives x sands for x ec) of he BS call wih respec o he underlying sock, srike and implied volailiy. Saring wih l = l S, σ, τ; K) =SN h ) KN h ), ln S h = K σ τ + σ τ, h = h σ τ, S h = S h = Sσ τ, Kh = K h = Kσ τ, σ h = σ h τ, σ h = h σ, σh = h σ, where N ) is he sandard normal cumulaive densiy funcion, and using K N h )=KN h )exp h σ τ ) σ τ = S N h ), he firs parial derivaives of l wih respec o S, σ and K are respecively S l = N h )+SN h ) S h KN h ) S h = N h ), σ l = SN h ) σ h KN h ) σ h = τkn h )= τsn h ), K l = SN h ) K h N h ) KN h ) K h = N h ).

4 4 ALAN BRACE, BEN GOLDYS, FIMA KLEBANER, AND ROB WOMERSLEY Then, recalling ha N x) = xn x), he second parial derivaives are Sl = N h ) h S = Sσ τ N h ), K l = N h ) h K = Kσ τ N h )= σ l = τsn h ) h σ = S τ σ h h N h ), S K l = N h ) h K = Kσ τ N h ), S K σ τ N h ), S σ l = N h ) h σ = h σ N h ), K σ l = N h ) h σ = h σ N h )= Sh Kσ N h ). In addiion we have he relaion τ l = σ S S l = σs τ N h ), which comes from he BS parial differenial equaion, and holds for any opion in he BS world. Exercise.. Show ha in he normal Bachelier model where ds = θ dw ), C = E { [S T K] + ) } S K F = σ T Φ σ, T wih Φ x) = x N u) du = N x)+xn x), and l = l S, σ, τ; K) =σ τφh), h = S K) σ, τ he equivalen expressions for he firs and second derivaives of l wih respec o S and σ are S l = K l = N h), σ l = τφh) τhn h) = τn h), S l = S K l = K l = σ τ N h), σ l = τn h) h σ = τ σ h N h), S σ l = K σ l = h σ N h), τ l = σ Sl = σ N h). τ

5 STOCHASTIC VOLATILITY 5. Dynamics of he implied volailiy surface Assuming ha he drif m ) and volvol v ) in.5) are well behaved funcions, applying Io o.3) produces he following SDE for a call opion: dc = τ C d + S C ds + S C d S + σ C dσ + σ C d σ + S σ C d S, σ, = σ S T N h ) d + N h ) ds + N h ) d S S σ T + T S N h ) dσ + S T h h N h ) d σ σ h N h ) d S, σ σ, = N h ) S θ dw ) + T S N h ) v dw.) + [ ] T S N θ h ) m + σ T ) σ T ) + h h v h θ v ) d. σ σ T For his o be a maringale under he arbirage free measure P, is drif mus be zero, and so m = = [ σ T ) σ T ) σ θ T ) h h v + T h θ v ) ln σ θ + K 4 σ T ) S v σ ln K σ S T )+ θ v ) σ, ],.) and he arbirage free dynamics of C becomes dc = N h ) S θ dw ) + T S N h ) v dw..3) The only reasonable choice for he dependence of he volvol v on he implied volailiy σ is now seen o be linear because ha removes he roublesome singulariies in.). Seing v = v T,K,S,θ,σ )=σ u T,K,S )=σ u, from.4),.5) and.) he dynamics of S,θ and σ are herefore deermined by he non-linear se of equaions ds = S θ dw ), θ = σ,, S ), lim σ T,K) <,.4) T σ + 4 σ4 T ) u σ T ) θ u ) dσ = σ T ) θ + u ln K d + σ u d dw, S and we now analyse his sysem furher.

6 6 ALAN BRACE, BEN GOLDYS, FIMA KLEBANER, AND ROB WOMERSLEY 3. Differen formulaions Our aim in his secion is o lis four differen formulaions of he sochasic implied volailiy problem, so as o have he flexibiliy of choosing one ha is mos convenien o he problem a hand. In.4) we derived SDEs for he implied volailiy σ T,K) expressed in erms of he absolue parameers T and K. Bu implied volailiy can also be expressed relaively as follows. For given consans x > andy, define η by he equaions η x, y) =σ + x, e y S ), σ T,K)=η T, ln K ). S The correc inerpreaion of his ransformaion is ha η, ) is he relaive volailiy surface a ime as seen by an observer moving wih he sock. The parameers x = T, y =ln K S, are, respecively, relaive mauriy x becomes zero as opions maure) and log-moneyness y is zero a-he-money and negaive for ou-of-he-money pus or in-he-money calls), and he relaive surface a ime is obained by ploing η x, y) agains x and y. Noe ha because θ = η, ), a sysem of SDEs for η will also include one for he spo volailiy θ. To ge SDEs for η = η x, y) we need o make he parameers T and K in σ = σ T,K) ino sochasic variables T and K like T = + x, dt = d K = e y S, dk = e y ds = e y S θ dw ), and hen rewrie he SDE.4) for σ using he Io-Vensel formula described in Appendix- A.As we shall see, ha will require he following parial derivaives: T σ T,K)= T η T, ln K ) = x η x, y); 3.) S K σ T,K)= K η T, ln K ) = S K yη x, y) = e y y η x, y); S Kσ T,K)= Kη T, ln K ) { = K S K yη T, ln K )}, S = { K y η x, y) y η x, y) } = { e y S y η x, y) y η x, y) }. We are now in a posiion o obain four sysems of SDEs describing he evoluion of he sochasic implied volailiy surface in erms of: The absolue implied volailiy σ = σ T,K) as in.4). The square of he absolue implied volailiy muliplied by ime o mauriy. ξ = ξ T,K)=σ T ) =σ T,K)T ). The relaive implied volailiy η = η x, y). The square of he relaive implied volailiy muliplied by relaive mauriy ζ = ζ x, y) =η x = η x, y) x.

7 Repeaing.4), he σ - formulaion is STOCHASTIC VOLATILITY 7 ds = S θ dw ) [spo SDE) dσ = σ σ T ) θ + u ln K ] d S [ ] + 8 σ3 T ) u σ θ u ) d + σ u dw, { σ T,K) specified iniial condiion), θ = σ,, S ) and lim T σ T,K) < feedback). 3.) Muliplying he SDE 3.) by σ T ) and using dξ = d [ σ T ) ] =T )σ dσ +T ) σ u d σ d, produces he ξ-formulaion ξ = ξ T,K)=σ T ) definiion) ds = S θ dw ) spo SDE) { [+ dξ = ξ 4 ξ ] } u θ u ) d θ + u ln K d +ξ u dw S ξ{ T,K) specified iniial condiion) ξ, K) = and θ feedback). = T ξ, K) 3.3) In he SDE 3.) for σ se T = + x, dt = d, K = e y S, dk = e y ds = e y S θ dw ), and apply he Io-Vensel TheoremA o ge dσ T,K )=dσ + x, e y S )=dη x, y) ds = S θ dw ), θ = σ, S ), lim σ T,K) <, T [ dσ = σ θ υ ln K θ u ) ln K ] d σ T ) S S [ + 8 σ3 T ) u ] σ θ u ) d + σ u dw [ + T σ T,K)+ K σ T,K) e y S θ + K σ υ ) ) e y S θ ] d + K σ T,K) e y S θ dw ). To express hese equaions solely in erms of η x, y) andx, y, assume u is expressed in erms of x and y raher han T and K, subsiue x = T, y =ln K S, dln S )=θ dw ) θ d,

8 8 ALAN BRACE, BEN GOLDYS, FIMA KLEBANER, AND ROB WOMERSLEY and apply he change of variable formulae 3.). The σ-formulaion hen becomes he η- formulaion ds = S θ dw ), spo SDE) dη = { η η x θ + yυ } { } d + 8 η3 x u η θ u ) d + η u dw )} + { x η + θ yη + θ y η u ) d + y η d ln S ) { η x, y) specified iniial condiion) θ = η, ) lim x η x, y) < feedback). 3.4) Similarly, applying Io-Vensel o he ξ-formulaion 3.3) produces he ζ-formulaion ζ = ζ x, y) =η x, y) x definiion) ds = S [ θ dw ) spo SDE) + dζ = ζ 4 ζ ) ] u θ u ) d θ + yu d )] + [ x ζ + θ y ζ +θ y ζ u ) d +ζ u dw + y ζ d ln S ) { ζ x, y) specified iniial condiion) ζ,,y)= and θ = feedback). xζ, ) 3.5) Exercise 3.. In he Bachelier model define moneyness by y = K S, and also ake he volvol v o be linear in he implied volailiy, ha is v = u σ. Show ha he equivalen formulaions for σ,ξ,η and ζ have dσ = σ T ) [ σ θ +K S ) u ] d + σ u dw, dξ = ξ u d θ +K S ) u d +ξ u dw, [ dη = η η x θ + yu ] d + η u dw + [ x η + θ yη ) ] + θ y η u ) d + y η ds, dζ = ζ u d θ + yu d +ζ u dw + [ x ζ + θ yζ ) ] +θ y ζ u ) d + y ζ ds. Remark 3.. Considering he ξ-formulaions in he Bachelier and Black-Scholes models dξ = ξ u d θ +K S ) u d +ξ u dw Bachelier) dξ = ξ {[+ ] } 4 ξ u θ u ) d θ + u ln K d +ξ u S dw BS),

9 STOCHASTIC VOLATILITY 9 one is sruck by he similariy beween he wo models. The main difference is he unpleasan non-linear erm 4 ξ u d in he drif of he Black-Scholes equaion.ha will cause us some problems. Remark 3.3. Le us commen on he feedback condiion which was inroduced in Secion 3. Noe ha on he boundary x =, he condiion ) ζ,,y)= y ζ,y)= y ζ,y)= y ζ υ ) = for all y, so ha he SDE for ζ,y) on he leading edge x = reduces o dζ,y)= θ + yu d + x ζ,y) d. Bu for ζ,y) o remain zero he incremen dζ,y) iself mus also be zero, which means x ζ,y)= θ + yu, y. In erms of η, he boundary condiions a x =are η,y)= θ + yu <, which we migh have obained direcly by asking ha he drif in 3.4) no have a singulariy a x =. Various skews and smiles of he familiar upward hook ype can be obained by changing he correlaion which shifs he verex of he underlying parabola. The iniial volailiy surface mus of course saisfy his equaion, which gives useful informaion abou how u varies wih y near he boundary x = remember also ha u,y) can depend on y). 4. Some properies of soluions Le Ω, F, F ), P) be a filered probabiliy space wih he filraion saisfying he usual condiions. ) We assume ha his space carries a wo-dimensional Wiener process W )= W ),W ). In his secion we will consider he equaion ) ) dζ = ζ u θ u ) d +ζ u dw θ + yu + θ d ) + ζ x + ζ ζ u ) θ y +θ ζ ) d + θ dw + y y 4 ζ d, 4.) where θ = x ζ, ), ζ x, y) =fx, y), and ζ,y)=. ) This equaion has a soluion only if he process u )= u ),u ) is chosen in a special way. We will sar wih a simpler problem. Namely, we will sudy equaion 4.) wihou he boundary condiion ζ,y) = and wih he process u ) given in advance and such ha Hypohesis 4.. The process u )=u x, y)) is adaped for each x, y), coninuous in, x, y)). Moreover, we assume ha he process y u ) x, y) is well defined and coninuous in, x, y).

10 ALAN BRACE, BEN GOLDYS, FIMA KLEBANER, AND ROB WOMERSLEY By a local soluion o 4.) we mean a process {ζ x, y) :x,y R} and a sopping ime τ such ha he following holds.. The process ζ x, y)) is coninuous in, x, y) [,τ) R + R.. For each <τ he funcion ζ, ) C, R + R) andheprocesses ζ s ζs x, y), x y x, y) and ζ s x, y) y are coninuous in s<τ for each x, y) R + R. 3. For each f C, R + R) andx, y) R + R we have for <τ ) ζ = f + ζ s 4 ζ s + u s θ s u ) s ds + θ s y ζ s dw s ) + ζ s u sdw s + xζs + ) ) θs y ζ s +θ s y ζ s u ) s ds θ s + yu s + θ s where he dependence on x, y) is suppressed, wherever possible. Theorem 4.. Assume Hypohesis 4.. unique local soluion o 4.). Lemma 4.3. Assume Hypohesis 4.. Le and Then where M = ) ds, Then for each f C, R + R) here exiss a θ s dw ) s, Ns x, y) = u r x + r, y M r) dw r + X x, y) =e N x,y) f + x, y) s s s u r x + r, y S r ) dr ) 4 ζ r x + r, y M r ) θ r u ) r x + r, y M r ) dr. ζ x, y) =X x, y + M ), Exercise 4.4. In he Bachelier Model denoe N = e N s x,y) θ s +y S s ) u s x + s, y S s ) + θ s u sdw s u s ds. Show ha he process ξ if exiss, mus saisfy an inegral equaion T ξ = e N ft,k) e N e Ns θ s K S s ) u s ds, and herefore he feedback condiion akes he form ft,k)= T e Ns θ s K S s ) u s ds. ).

11 STOCHASTIC VOLATILITY Finally, show ha he process ξ ) exiss for all imes provided he process u ) is locally bounded in, T, K, Proposiion 4.5. Assume Hypohesis 4.. Moreover, assume ha he process u ) is chosen in such a way ha ζ,y)=for all <τ and y R. Then he unique local soluion ζ ) is sricly posiive: for each x, y R and <τ we have ζ x, y). Proof. Le ζ ) be a local soluion o 4.) and le ds = θ S dw ), < τ. I follows from Secion 3 ha for <τ ζ, x) =ξ + x, e y S ), and herefore i is enough o show ha ξ T,K) for<τ.le L = L T,K)= u sdw s ) 4 ξ s Then Since ξ T T,K) = we find ha Hence for all <τ. ft,k) u s + θ s u ) s T ξ T,K)=e L ft,k) e L e Ls θ s + u s ln K ds. S s e Ls θ s + u s ln K S s ds = T ) ds. e Ls θ s + u s ln K S s ds. T ξ T,K)=e L e Ls e Ls θ s + u s ln K ds >, S s 5. Toy model Sar wih he ζ-formulaion in he Bachelier model see Exercises. and 3.), and assume he volvol u is sochasic bu independen of boh x and y: dζ = ζ u d θ + yu d + y ζ θ dw ) +ζ u dw 5.) + [ x ζ + θ yζ ) ] +θ y ζ u ) d, ζ x, y) =f x, y), iniial condiion) x ζ,y)= θ + yu. feedback) If he iniial implied volailiy surface ζ x, y) =f x, y) is quadraic in y hen clearly ζ x, y) will remain quadraic in y because all erms in he SDE and boh iniial and feedback condiions appearing in 5.) are linear in ζ or quadraic in y. So suppose he iniial implied volailiy surface is given by e η = η λx ) x, y) = θ + yu, λx or ζ = ζ x, y) =f x, y) = e λx) θ + yu, λ

12 ALAN BRACE, BEN GOLDYS, FIMA KLEBANER, AND ROB WOMERSLEY For values of θ around %, u beween % and 4% and λ abou., such iniial surfaces are semi-believable. Seing M = θ dw ), N = u sdw s u s ds, a soluion o 5.) for all x andy R is ζ x, y) =e N f + x, y + M ) e N e Ns θ s +y + M M s ) u s ds. 5.) Differeniaing wih respec o x x ζ x, y) =e N x f + x, y + M )=e N e λ+x) θ +y + M ) u, and so he feedback condiion a x =gives x ζ,y)= θ + yu = e N λ θ +y + M ) u, or ) θ +yθ u ) + y u θ = e N λ +θ u ) M + u M ) +y θ u ) + u M + y u. Comparing coefficiens of powers of y yields u = e N λ u, 5.3) θ = e N λ θ + M u, ) θ u ) = e u ) N λ + u M, θ + M u ) e N = E u s dw s, M = θ dw ). Re-expressing he equaion for u as u = u E ) [u sdw s λd], d u = u [u sdw s λd], and inroducing he new Brownian moion W = u s dws u s,gives [ ] [ d u = u u d W λd = u d W λ ] u d, which in urn, afer applying Girsanov dŵ = d W λ u d), has form d u = u dŵ, which does no explode in finie ime. In his example i is relaively easy o simulae he equaions 5.3): ) simply incremen he processes M and N, calculae he values of θ and u = u ),u ) from he feedback equaions, and incremen again.

13 STOCHASTIC VOLATILITY 3 6. Applicaion o BGM To apply he above resuls o ineres raes, firs focus on wha will be maringales and under wha measures in a sochasic volailiy version of BGM. As menioned in he inroducion, from [] he Libor forward raes L, T ) mus be posiive maringales under P T and can be aken o saisfy SDEs like dl, T ) L, T ) = θ T ) dw T ), where he θ T ) are sochasic. We emphasise ha here he mauriy dependen volailiies θ T ) are vecors, unlike he spo volailiy θ used for socks above. The Black convenion for quoing cap prices in he presence of a volailiy smile or skew is similar o ha for socks. The volailiy in he Black cap formula, which adds componen caple values, is adjused o produce he correc price. To analyse furher, reurn o he sandard lognormal BGM model in which θ T ) is deerminisic. The presen value Cpl T ) of a caple sruck a κ, mauringat, and paying a T = T + δ is given by he Black caple formula Cpl T ) = P, T ) l {L, T ),σ T,κ),T ; κ}, 6.) T σ T,κ) = θ s T ) ds. T ) Suppose we can break he cap skew down ino a caple skew by disribuing he cap prices a differen srikes ino Black caple prices in such a way ha he corresponding caple volailiy profile σ T,κ) ploed agains T and κ, is reasonably smooh his sep could very well involve some heroic numerical work). Then afer inerpolaion, if neccessary) we can assume ha for all mauriies T we have an iniial implied volailiy surface σ = σ T,κ)whichcan be inpu as a sar parameer. Now suppose, following on from our work on socks above, ha he implied volailiies σ = σ T,κ) saisfy SDEs of he form dσ = m T,κ,L, T ),θ,σ ) d + σ u T,κ,L, T ),θ ) dw T ), = m d + σ u dw T ) under he forward measures P T, and ha caple values are given by 6.) wih σ = σ T,κ) now sochasic. Because he caples Cpl T ) are asses, heir presen values divided by he numeraire P, T ) mus be maringales under he forward measure P T. Tha is, for all posiive T and κ, he expression l {L, T ),σ T,κ),T ; κ} mus be a maringale under he P T forward measure. Similarly o 3.3), ha leads o he sysem ξ = ξ T,κ)=σ T ), dl, T )=L, T ) θ, T ) dw T ), 6.) dξ = ξ {[+ ] } 4 ξ u θ u ) d θ κ + u ln L, T ) d +ξ u dw T ), ξ T,κ)=Tσ T,κ)=fT,κ) specified iniial condiion), ξ T T,κ)= T feedback).

14 4 ALAN BRACE, BEN GOLDYS, FIMA KLEBANER, AND ROB WOMERSLEY The Libor forward rae volailiy θ T )andheimpliedvolvolu T,κ) mus now be linked ino a feedback loop, oherwise he sysem 6.) for L, T ) and is derivaive caple volailiies will be under-specified. Assuming θ T )andu T,κ) are well defined, a formal soluion o 6.) is ξ T,κ) = e NT,κ) f T,κ) e NT,κ) e NsT,κ) κ θ s + u s T,κ)ln ds, N T,κ) = u s T,κ) dw T s) {[ 4 ξ s T,κ) ] L s, T ) u s T,κ) + θ s u ) s T,κ) } ds. The feedback condiion ξ T T,κ) = for all mauriies T implies T f T,κ)= e NsT,κ) κ θ s T )+ u s T,κ)ln L s, T ) ds, and T f T,κ)=e N T T,κ) θ κ T + u T T,κ)ln L T,T) [ + T e NsT,κ) κ ] θ s + u s T,κ)ln L s, T ). =T which means T ξ T,κ)=ξ T,κ) T N T,κ) } +e NT,κ) T {f T,κ) e NsT,κ) κ θ s + u s T,κ)ln L s, T ) ds, { and [ T ξ T,κ)] =T = e N T T,κ) e N T T,κ) θ κ } T + u T T,κ)ln, L T,T) = θ κ T + u T T,κ)ln L T,T). In oher words ξ T T,κ)= [ T ξ T,κ)] =T = θ κ T T )+u T T,κ)ln L T,T). 6.3) Puing κ = L T,T) in 6.3) yields he Libor volailiy link θ T T ) =[ T ξ T,LT,T))] =T or θ ) = σ, L, )), 6.4) which can be exended o include θ T ).a laer mauriies using correlaion informaion. Suppose ha from hisorical daa analysis wih a sandard BGM model, we have consruced a deerminisic vecor volailiy funcion γ T ).which reflecs he correlaion srucure we would like our model o exhibi. Namely, ha he insananeous correlaion a ime beween he T i and T j Libor forward raes is ρ T i,t j )= γ T i) γ T j ) γ T i ) γ T j ). For he forward rae volailiy vecor ry θ T )=γ T ) ψ, 6.5)

15 STOCHASTIC VOLATILITY 5 where ψ is a scalar sochasic variable free o be deermined by feedback he deerminisic vecor γ T ) is of course already fully specified from he hisorical daa analysis). From 6.4) ψ = θ ) γ ) = σ, L, )), γ ) and so Moreover θ T )= σ, L, )) γ ) γ T ). d L,T) L, T ) d L,T i ),L,T j ) L, T i ) L, T j ) = θ T ) d = ψ γ T ) d, = θ T i) θ T j ) d = ψ γ T i) γ T j ) d, which reurns he required insananeous correlaion θ T i) θ T j ) θ T i ) θ T j ) = ρ T i,t j ). The volvol link for u T,κ) can be specified in a similar, bu somewha looser, fashion.. Thespovolvolu, κ) is largely deermined by he feedback condiion 6.3), alhough as in he sock case) here is sill considerable freedom o engineer is dependence on he srike κ, and is disribuion ino componens. For laer mauriies, u T,κ) can be specified in erm of is spo value u, κ) so as o exhibi he same sor of decay wih respec o mauriy T ha is seen in hisorical daa. Hence he BGM ξ-formulaion ξ = ξ T,K)=σ T ) definiion ξ) θ T )=ψ γ T ) volailiy of forwards) dl, T )=L, T ) θ T ) dw T ) forward dynamics) { [+ dξ = ξ 4 ξ ] } u θ u ) d θ κ + u ln L, T ) d +ξ u dw T ) ξ T,κ)=Tσ T,κ)=fT,κ) specified iniial condiion) ξ T T,κ)= [ T ξ T,κ)] =T = θ κ T T )+u T T,κ)ln feedback). L T,T) 6.6) 7. Marginal Disribuions Le p S, T ) denoe he marginal disribuion, so he price C of a European Call wih srike K and expiry T can be wrien as CK, T )= maxs K)p S, T )ds.

16 6 ALAN BRACE, BEN GOLDYS, FIMA KLEBANER, AND ROB WOMERSLEY Following Breeden and Lizenberger [3] he marginal disribuion can be recovered by CK, T ) = CK, T ) K = K S K)p S, T )ds K p S, T )ds CK, T ) K = p K, T ). 7.. Bachelier Model. In he Bachelier model Cσ, K,, T )=σ T )Φh) 7.) where Φu) = x h = S K σ T, Nu)du = xnx)+n x), and N is he sandard normal cumulaive densiy funcion. If σ does no depend on he srike K hen C K = N h) σ T. In our model σ = σk, T ) depends on he srike K and oher variables). Noing ha ξ = σ T ), he iniial form for σ is implied by he iniial volailiy surface ξx, y) = fx, y)wherex = T is he mauriy and y = K S is he moneyness. 7.. Derivaives. To use he Breeden and Lizenberger resul we need CσK),K,,T) K. As K = S + y i does no maer if we differeniae w.r. y or K. Usingξ = σ x and Also σ y = σ y = σ f x y 4σ 3 x f y ) + f σx y. h y = σ x + y σ σ x y. Wriing C for CσK),y,x,T) he Bachelier formula 7.) evenually gives ) C C + ynh) σ K = Nh)+ σ y C K = N h) σ x yn h) σ x σ y + y N h) σ 3 x ) σ + y C + ynh) σ ) σ y.

17 STOCHASTIC VOLATILITY 7 The marginal disribuion implied by he iniial volailiy surface is hen obained by seing =,sox = T,andy = K S Numerical resuls. In he simulaions he iniial volailiy surface is given by fx, y) = e λx qy) λ where qy) =θ +ρθ νy + ν y, where ν is he volvol, λ conrols he flaening ou as mauriy x increases, and ρ is he correlaion in he quadraic dependence of he surface on moneyness y. Thus iniially a = σ K) = [ e λt ] θ +ρθ νk S )+ν K S ). T λ The Bachelier model was simulaed over 5 years using 4 ime seps. The parameers in he iniial volailiy surface were ν = %, λ =.5, θ = % and differen values for he correlaion ρ. Figures, and 3 show he iniial volailiy surface and he empirical using, simulaions) and analyic marginal disribuions a.5 and 5 years for ρ =,.7,.5 respecively. The agreemen in all cases is remarkable. Appendix A. Io-Vensel formula To wrie down an SDE for η = η, x, y) we will need a generalizaion of he Io-Vensel formula as derived in [6] and [8]. Theorem A.. Le W be muli-dimensional Brownian moion. Suppose F, u) is wice differeniable wih respec o he parameer u and saisfies he SDE df, u) =A, u) d + B, u) dw. If u saisfies he SDE du = C, u ) d + D, u ) dw, hen an SDE for F, u ) is df, u )=A, u ) d + B, u ) dw + u F, u ) du + u F, u ) D, u ) d + u B, u ) D, u ) d. References [] Brace A, Gaarek D & Musiela M 997) The marke model of ineres rae dynamics Mah Finance 7, p7-54. [] Balland P & Hughson LP ) Pricing and hedging wih a sicky-dela smile April Risk Conference, Paris [3] Breeden and Lizenberger 978) Prices of Coningen claims implied in opion prices, Journal of Business 5, [4] Carr P ) A survey of preference free opion valuaion wih sochasic volailiy April Risk Conference, Paris.

18 8 ALAN BRACE, BEN GOLDYS, FIMA KLEBANER, AND ROB WOMERSLEY [5] Nualar D and Pardoux E 99) Boundary value problems for sochasic differenial equaions, Ann. Probab. 9, 8 44 [6] Rozovskii BL 973) On he Io-Vensel formula Vesnik Moskovskogo Universiea, Maemaika Vol 8 No p6-3. [7] Schonbucher PJ 998) A marke model for sochasic implied volailiy Working paper, Universiy Bonn. [8] Vensel AD 965) On he equaions of he heory of condiional Markov processes Teoriya veroyan i ee primenen, X No p Naional Ausralia Bank and Financial Mahemaical Modelling and Analysis address: abrace@oz .com.au School of Mahemaics, The Universiy of New Souh Wales, Sydney, NSW 5, Ausralia address: B.Goldys@unsw.edu.au Deparmen of Mahemaics and Saisics, Universiy of Melbourne, Parkeville, VIC 35, Ausralia address: F.Klebaner@ms.unimelb.edu.au School of Mahemaics, The Universiy of New Souh Wales, Sydney 5, Ausralia address: R.Womersley@unsw.edu.au

19 STOCHASTIC VOLATILITY 9 Iniial volailiy surface, ν = %, ρ =., λ = Mauriy x Moneyness y Marginal disribuion a.5 years Simulaion Analyic Marginal disribuion a 5 years Simulaion Analyic Moneyness Figure. Iniial volailiy surface and marginal disribuions, ρ =

20 ALAN BRACE, BEN GOLDYS, FIMA KLEBANER, AND ROB WOMERSLEY Iniial volailiy surface, ν = %, ρ =.7, λ = Mauriy x Moneyness y.5 Marginal disribuion a.5 years Simulaion Analyic Marginal disribuion a 5 years Simulaion Analyic Moneyness Figure. Iniial volailiy surface and marginal disribuions, ρ =.7

21 STOCHASTIC VOLATILITY Iniial volailiy surface, ν = %, ρ =.5, λ = Mauriy x Moneyness y Marginal disribuion a.5 years Simulaion Analyic Marginal disribuion a 5 years Simulaion Analyic Moneyness Figure 3. Iniial volailiy surface and marginal disribuions, ρ =.5