db t = r t B t dt (no Itô-correction term, as B has finite variation (FV), so ordinary Newton- Leibniz calculus applies). Now (again as B is FV)

Transcription

1 ullin4b.ex pm Wed The change-of-numeraire formula Here we follow [BM, 2.2]. For more deail, see he paper Brigo & Mercurio (2001c) cied here, and H. GEMAN, N. El KAROUI and J. C. ROCHET, Changes of numeraire, changes of probabiliy measure and pricing of opions. J. Applied Probabiliy 32 (1995), We begin our deailed analysis of he marke models by deriving he change-of-numeraire formula from (a mulivariae version of) Girsanov s heorem. This is worhwhile, as we will use he formula o derive boh he LMM and SMM dynamics. Also, his is a general approach ha can be used in many asse classes e.g. credi defaul swap (CDS) marke models; see e.g. [BM, ]. We shall also use i o prove he HJM drif condiion, which we saed wihou proof in IV.1. Recall (MATL480 and Ch. I) ha a numeraire is any non-dividendpaying radable asse whose value is always posiive. For a numeraire S, he measure Q S associaed wih i is a measure under which Y/S is a maringale for any Y which is he price of a non-dividend-paying radable asse. If he numeraire is he bank accoun dynamics db = r B d hen Q B is he classic risk-neural measure. Indeed, for any asse Y, wih dynamics dy = µ B (Y )d + σ (Y )dw B (), where W B is a Q B -BM, d(y /B ) = (1/B )dt + Y d(1/b ) (no Iô-correcion erm, as B has finie variaion (FV), so ordinary Newon- Leibniz calculus applies). Now (again as B is FV) Combining, d(1/b) = db/b 2 = rd/b. d(y /B ) = (µb (Y ) r Y ) d + σ (Y ) dw B (). B 1 B

2 Now by definiion of Q B, Y/B mus be a Q B -mg. To be a maringale for a regular diffusion process means zero drif. So µ B (Y ) r Y = 0 : µ B (Y ) = r Y. This says ha Q B is he measure under which all non-dividend-paying radable asses Y have he risk-free rae as growh-rae in he drif. Bu his is jus how he risk-neural measure Q is defined (MATL480; I.2). So (unsurprisingly) Q = Q B. Bu someimes i is useful o have numeraires oher han he bank accoun B we shall see examples shorly wih he LMM. So we now derive a general formula for changing numeraire, from B o S, say. Le X be an n-dimensional diffusion process whose dynamics under he measure Q S corresponding o numeraire S is given by dx = µ S (X )d + σ (X )CdW S (), W S BM under Q S. (S) Here σ is an n n square diagnal marix, wih µ S (x) and σ (x) deerminisic funcions of (, x) ha are smooh enough o allow he calculaions needed below (sufficien smoohness here is no a significan resricion in pracice, so we do no need o go ino deail here), and W S is sandard Q S -BM. The n n marix C is inroduced here o model correlaion in he resuling driving noise: CdW is equivalen o an n-dimensional BM wih insananeous correlaion marix ρ = CC T, where he superscrip T denoes ransposiion. (We use a square diffusion marix for simpliciy in using Girsanov s heorem laer. This is no in fac necessary, bu we do i here for convenience.) Now suppose we express he dynamics of X in a new numeraire U raher han he old one S. So changing S o U in (S) above, dx = µ U (X )d + σ (X )CdW U (), W U BM under Q S. (U) We can now use Girsanov s heorem o find he Radon-Nikodym (RN) derivaive beween Q S and Q U for he X-dynamics under he wo differen measures. We obain ζ T := dq S dq U F T 2

3 = exp{ 1 (σ (X )C) 1 [µ S (X ) µ U (X )] 2 d 2 0 ( T + (σ (X )C) 1 [µ S (X ) µ U (X )]) dwu ()}. 0 Then ζ is an exponenial maringale (MATL480 5a, VI.3): seing α := [µ S (X ) µ U (X )] T ((σ (X )C) 1 ) T gives exponenial maringale dynamics as he SDE for ζ: dζ = α ζ dw U (). (ζ : 1) On he oher hand, by definiion of Q S, for any radable asse price Z we have E Q S 0 [Z T /S T ] = E QU 0 [ U 0Z T ], S 0 U T boh being equal o Z 0 /S 0 (discouned asse prices are mgs, so have consan expecaion: on he lef, jus replace T by 0; on he righ, he U-erms are he discouning; he asse is Z/S 0 ). Bu by definiion of RN derivaive, we also have ha for all Z, E Q S 0 [Z T /S T ] = E QU 0 [ Z T S T. dq S dq U ]. Comparing he wo RHSs above, we have ha as Z is arbirary, and since ζ is a Q U -mg, So differeniaing his, ζ = E Q U ζ T := dq S dq U F T = U 0S T S 0 U T, [ζ T ] = E Q U [ U 0S T ] = U 0S. S 0 U T S 0 U dζ = U 0 S 0 d[ S U ]. Now S/U is boh a numeraire iself, and a Q U -mg. So i has mg dynamics: d(s /U ) = σ S/U CdW U () under Q U. 3 ( )

4 So dζ() = U 0 S 0.σ S/U CdW U (). (ζ : 2) We now have wo expressions for dζ, (ζ : 1), (ζ : 2). Comparing hem, α ζ = U 0 S 0 σ S/U C. Subsiuing for ζ here from ( ), we obain S U α = σ S/U C. This and he definiion above of α give he following fundamenal resul: µ U (X ) = µ S (X ) U S σ (X )ρ(σ S/U ) T, ρ = CC T. This gives he change in he drif of a sochasic process when changing numeraire from S o U (or vice versa). I ofen happens ha, under he measure Q U, he S- and U-dynamics are given by SDEs of he form ds = ( )d + σ S CdW U (), du = ( )d + σ U CdW U () (he drifs can be anyhing here, bu if he diffusion erms are any furher apar han his, we canno draw a conclusion). Then (produc rule of Iô calculus) and by Iô s lemma, d( S U ) = 1 U ds + S d( 1 U ) + ds.d( 1 U ), d( 1 ) = 1 du U U (du U 3 ) 2. Combining, and reaining only dw U erms (so neglec erms in (d) 2, (dw U ) 2, as always in Iô calculus), ( σ S d(s/u) = (...)d + U S U 4 σ U U ) CdW u.

5 This idenifies he diffusion coefficien of he numeraire S/U: σ S/U = σs U S U σ U U. Subsiuing his in he resul above: Under he above circum- Theorem (Change-of-numeraire formula). sances, µ U (X ) = µ S (X ) U ( σ S σ (X )ρ S σ U ) T, ρ = CC T. S U U U Shocks. I is someimes helpful o consider wha happens in erms of shocks. Equaing he expressions for dx in (S) and (U) above, µ U (X )d + σ (X )CdW U () = µ U (X )d + σ (X )CdW U (). Subsiuing in he Theorem above gives ( σ S CdW S () = CdW U () ρ S σ U ) T d. U U U If we abbreviae he noaion by wriing he vecor diffusion coefficien of a diffusion X by DC(X), and we wrie he correlaed Brownian moion as dz = CdW, he above becomes ( DC(S) dz S () = dz U () ρ ((CBM) here sands for correlaed Brownian moion). S DC(U) U ) T d (CBM) Below, we will apply he change-of-numeraire echnique o hree hings: (i) Black s caple formula; (ii Black s swapion formula; (iii) he Heah-Jarrow-Moron drif condiion. 5

6 6. LMM (LIBOR Marke Model) dynamics We can use he resuls above o give a rigorous proof of Black s caple formula of 1976 (he echniques above came much laer). In III.5 above, ake (wih P (, T ) he bond prices as before, giving P (., T ) = ( P (, T )) as a funcion of ) U = P (., T i ), Q U = Q i. Since is a Q i -mg. Take F (; T i 1, T i ) = (1/τ i ) P (, T i 1) P (, T i ), P (, T i ) F i () := F (; T i 1, T i ) df i () = σ i ()F i ()dz i (), Q i, T i 1 (noaion as above). This is he LIBOR Marke Model (LMM) (his is he common name; Brigo and Mercurio [BM, 6.2, p.202] prefer he name Lognormal Forward-LIBOR Model (LMM). One ime-inerval. Consider he discouned T k 1 -caple (F k (T k 1 ) K) + B(0)/B(T k ). Wih E k [.] for Q k -expecaion, he ime-0 price of he caple is, by FACT 2 (V.1) B(0)E QB [(F k (T k 1 ) K) + /B(T k )] = P (0, T k )E k [(F k (T k 1 ) K) + /P (T k, T k )] = P (0, T k )B&S(F k (0), K, v k Tk 1 ), where we wrie B&S(.) for he Black-Scholes formula for calls, of which Black s caple formula is clearly a varian (argumens: iniial sock price, srike, volailiy), and v k := 1 k 1 σ k () 2 d. T k 1 0 The dynamics of F k is easy under Q k. Bu if we price a produc depending on several forward raes a he same ime, we need o fix a pricing measure, 6

7 say Q i, and model all raes F k under his same measure Q i. This is handled as above for k = i, bu no when i < k or i > k (below). Black volailiy. The v k above is a volailiy as in he Black-Scholes formula, and he caple price above is an opion price (on an ineres rae, raher han a sock as in Black-Scholes). Recall (MATL480) ha (wih vega he parial derivaive of he opion price wr volailiy) vega is posiive ( opions like volailiy ). So (as a coninuous sricly increasing funcion has a well-defined inverse funcion) here is a one-one correspondence beween opion prices and volailiies, and one can go back and forh beween he wo, i.e. obain eiher from he oher. We can see he prices a which opions are raded in he marke; he corresponding volailiy is he implied volailiy. The same applies here. Traders in caples speak of he v k above, obained as an implied volailiy in his way, as he Black volailiy, or Black vol for shor. They have a very well-developed inuiion for i (as sock-marke raders do for implied vol here): his is he way raders hink. See e.g. [BM, p.197, p ]. Several ime-inervals. We are now going o handle he i < k and i > k cases lef open above by he change-of-numeraire oolki of V.5 above. i < k. We use (CBM) from V.5 above: ( DC(S) dz S () = dz U () ρ DC(U) ) T d. (CBM) S U Here DC is a linear operaor on diffusions: DC(X ) is he row-vecor v in dx = ( )d + vdz, for diffusion processes X describable in erms of a common column-vecor of driving noise, a vecor BM Z. So if df 1 = σ 1 F 1 dz 1, hen DC(F 1 ) = (σ 1 F 1, 0,, 0) = σ 1 F 1 e 1, 7

8 say. The correlaion marix ρ is he insananeous correlaion beween he shocks (he same under any measure), dz i dz j = ρ ij d. The oolki (CBM) above can also be wrien For, dz S () = dz U () ρ(dc(log(s/u))) T d. (CBM ) DC(S) S DC(U) U = DC(log S) DC(log U) = DC(log S log U) = DC(log(S/U)). We now apply he oolki: aking S = P (., T k ) and U = P (., T i ), (CBM ) gives dz k () = dz i () ρ(dc(log(p (., T k )/P (., T i ))) T d. Now by (F j ) (V.1, W4a), ( P (, Tk ) ) log P (, T i ) So lineariy of DC gives ( P (, Tk ) P (, T k 1 ) = log P (, T k 1 ) P (, T k 2 ) P (, T ) i+1) P (, T i ) ( 1 = log 1 + τ k F k () τ k 1 F k 1 (). 1 ) 1 + τ i+1 F i+1 () = log (1/[ k ) (1 + F j ())] = log(1 + F j ()). ( P (, Tk ) ) DC log P (, T i ) = = DC log(1 + F j ()) DC(1 + F j ()) 1 + F j () 8

9 (as in he calculaion for DC above) = = DC(F j ()) 1 + F j () σ j ()F j ()e j 1 + F j (), wih e j he row-vecor (δ ij ) (Kronecker dela 1 in he jh posiion, 0 elsewhere). Combining, dz k () = dz i () + ρ Pre-muliply boh sides by e k. We obain dz k k = dz k i + [ρ k1, ρ k2,, ρ kn ] σ j ()F j ()e j 1 + F j () d. σ j ()F j ()e j 1 + F j () d, in an obvious noaion (he superscrips k here denoe evaluaion a ime T k ) = dz k i + σ j ()F j ()ρ kj 1 + F j () d. Subsiue his in he dynamics wrien in he above noaion, o obain df k = σ k F k (dz k i + df k = σ k F k dz k k σ j ()F j ()ρ ) kj 1 + F j () d. This finally gives he dynamics of a forward rae wih mauriy k under he forward measure wih mauriy i < k. The case i > k is handled similarly. Wrie he drifs here as µ m i := m σ j ()F j ()ρ mj 1 + F j (). 9

10 Then he above becomes: df k () = µ k i (, F ())σ k ()F k ()d + σ k ()F k ()dz i k() df k () = σ k ()F k ()dz k k () (i = k), df k () = µ i k(, F ())σ k ()F k ()d + σ k ()F k ()dz i k() (i < k), (i > k). These SDEs may be shown o have a unique soluion. We omi he deails; see [BM, 6.3.2]. Black s swapion formula; he swapion marke model (SMM). See e.g. [BM, 6.7, ], and for hedging, especially [BM 6.7.1]. Incompaibiliy beween LMM and SMM We refer o [BM, 6.8] for deails. We have already menioned he resul, and will feel free o use i. This incompaibiliy is no a serious problem in pracice, as he wo give resuls in good agreemen. Noe. The analogy wih Physics may be useful. The wo grea advances in Physics in he 20h cenury were Quanum Theory (dealing wih he very small subaomic paricles, ec.), and Einsein s General Theory of Relaiviy (dealing wih he very large cosmology, galaxies ec.). We know ha each is righ raher han wrong. We also know ha he wo are incompaible. The search for a Grand Unified Theory (o unify he four fundamenal forces of Naure graviy [in relaiviy] wih elecromagneism and he weak and srong nuclear forces [in quanum heory]) is moivaed by his. We do no know wheher his search will ever succeed; meanwhile Physics goes on, using differen mehods in differen conexs. Similarly here. 7. The Heah-Jarrow-Moron (HJM) drif condiion We discussed earlier (IV.1) he HJM framework for he forward raes f(, T ). While we ake he view ha mos useful models are for r (Ch. II) or F i, S ij (Ch. V), HJM is sill imporan, in a number of areas (commodiies, ec.) and hisorically. We saed he HJM drif condiion earlier wihou proof; we now have he ools o prove i, so we do so. Recall ha under he risk-neural measure Q wih bank accoun B as numeraire, df(, T ) = σ(, T )( σ T (, s)ds)d + σ(, T )dw B () 10 (HJM)

11 as needed o give no arbirage (NA) which we need. Tha is, o avoid arbirage, he drif is compleely deermined by he volailiies. We work in n dimensions, wih σ a row n-vecor and W a column n-vecor BM. Correlaions will be presen, bu we pu hem in he inner produc σσ T raher han in he BM W (recall he correlaed BMs involving C in V.6 above). We use he change-of-numeraire echnique. Recall ha f(, T ) = 1 P (, T ) P (, T ) T P (, T ) P (, T + T ) P (, T ) T for small T. So his is a radable asse (difference of wo bonds) divided by a second asse (he bond P (, T )), and by FACT 1 of V.1 i is a mg under he P (., T ) numeraire measure Q T, which we call T -forward measure. Since a mg has zero drif, df(, T ) = σ(, T )dw T () under he T -forward measure. Now use he change-of-numeraire oolki formula (CBM ) of V.6 above. As Z here is W, which has independen componens (above), he Brownian covariance marix here is he ideniy: ρ = I. We choose numeraires S = B (bank accoun) and U = P (., T ). Then As before, dw B () = dw T () (DC(log(B/P (., T )))) T d. DC(log(B/P (., T )) = DC(log B) DC(log P (., T )) = DC(log P (., T )). So we now need o find DC(log P (., T )). Inegraing he definiion f(, T ) = P (, T ) = exp{ f(, u)du} : Differeniae wr : d log P (, T ) = f(, )d log P (, T ), T d f(, u)du = 11 log P (, T ) = f(, u)du. [( )d + σ(, u)dw ]du,

12 whichever measure we are in, provided σ(, u) is he vecor volailiy for df(, u). This SDE gives he diffusion coefficien of d log P (, T ) as DC(log P (., T )) = σ(, u)du. As he bank-accoun numeraire B has no diffusion coefficien, his gives So (CBM ) (V.6) gives dw B () = dw T () DC(log(B/P (., T )) = Subsiuing his ino our iniial SDE gives σ(, u)du. σ(, u)du : dw T () = dw B () + df(, T ) = σ(, T )dw T () df(, T ) = σ(, T )[dw B () + ( σ T (, u)du)d] = σ(, T )( σ T (, u)du)d + σ(, T )dw B (), giving he HJM drif condiion, as required. // σ(, u)du. Noe. Models developed according o he general HJM framework are ofen non-markovian, and can even be infinie-dimensional. Bu if he volailiy srucure of he forward raes saisfy cerain condiions, hen an HJM model can be expressed enirely by a finie-sae Markov chain, making i compuaionally feasible. Examples include a one-facor, wo sae model: O. Cheyee, Term Srucure Dynamics and Morgage Valuaion, J. Fixed Income 1, 1992; P. Richken and L. Sankarasubramanian, Volailiy Srucures of Forward Raes and he Dynamics of Term Srucure, Mah. Finance 5, 1995), and laer muli-facor versions. 12