Valuation Models of Inflation Derivatives using Pricing Kernel Workshop "Finance and Insurance", Jena (Germany) March 16-20, 2009 Koichi Miyazaki (with Sho Ito) University of Electro-Communications, Dept. of Systems Engineering 1 1. Introduction (Background, Basics and Purpose) 1-1. Origin of the corporate-debt valuation model: Merton (1974), the structural model. Time t Time T (Maturity ) S Case of F T F = K t T Default St WT = 0 Wt Equity : FT = K W = max()S − K ,0 Case of T T S Bond : T Non Default FT = min()ST , K WT The firm’s asset-value process St under physical probability measure P dS = µS dt +σ S dzS S 2 1-2. Extension of the Merton model and our model -Black and Cox (1976) and Geske (1977): incorporate the effects of the coupons and covenants of corporate bonds. Extension in terms of “Cash Flow” -Shimuko, Tejima, and Deventer (1993): incorporate the stochastic interest rate proposed by Vasicek (1977). R dR = κ R − R dt +σ dz Instantaneous nominal interest-rate f : f ( f f ) Rf Rf Extension in terms of “Modeling of nominal interest rate” -Longstaff and Schwarz (1995): incorporating the Vasicek (1977) model into the Black and Cox (1976) model. Extension in terms of “Cash Flow” and “Modeling of nominal interest rate” -Zhou (1997): applies the jump-diffusion process in Merton (1976) to the valuation of corporate debt. dS ω+ηε Nominal company-asset S : = µS dt +σ S dzS + (e −1)dNt . Extension in terms of “Firm Value Process” S z However, to best of our knowledge, no valuation model of inflation derivatives in the structural model. Our Extension is in terms of “Modeling of real interest rate” and “Modeling of inflation” 3 z Inflation-linked government bonds - The U.S.A. in mid 1990s, Japan since 2004, Germany and Italy at much the same time as in Japan. - In the U.K. (since 1982) and Canada, the history of such bonds is much longer. US TIPS (Anual Issuance and Ratio of UK Index-Linked Gilt (Ratio of Outstanding) Outstanding 35 800 10.00% 30 8.00% 600 25 6.00% 400 20 4.00% (%) 15 Oku-Dollar 200 2.00% 10 0 0.00% 5 8 2 6 0 997 9 001 0 005 0 1 19 1999 20002 20 2003 2004 2 20 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 Year Year Fig1: US TIPS Fig2: UK LINKER z From now on, the active issuance of inflation-linked financial product may become feasible. - Thus, it is very useful to introduce a model that evaluates inflation related products. 4 1.3 Risk-neutral valuation and Pricing kernel valuation under the Merton model - Risk-neutral valuation (Valuation under Risk-neutral measure) dS Q ⎛⎛ 1 2 ⎞ Q ⎞ = R f dt + σ S dzS , ST = St exp⎜⎜ R f − σ S ⎟()T − t +σ S zS ()T − t ⎟ S ⎝⎝ 2 ⎠ ⎠ −Rf ()T −t Q + Equity value: W (t,T ) = e E [(ST − K ) ], Bond value: F()t,T = St −W (t,T ) - Pricing kernel valuation (Valuation under Physical measure) dM = −R dt +φ dz +φ dz dz dz Pricing kernel (Stochastic Discount Factor): M f S S u u , where u ⊥ S dS ⎛ dM ⎞ Nominal firm-asset S : = (R f + σ S λS )dt + σ S dzS , where λS dt = −Cov⎜ ,dzS ⎟ = −φS dt S ⎝ M ⎠ ⎡M T ⎤ Risk-free Bond value: P()t,T = Et ⎢ ⋅1⎥ ⎣ M t ⎦ ⎡M ⎤ W ()t,T = E T ()S − K + Equity value: t ⎢ T ⎥ , Bond value: F(t,T ) = St −W (t,T ) ⎣ M t ⎦ 5 1.4 Computational Method of Expectation in the Pricing kernel valuation Utilize moment generating function (m.g.f.) 2 2 When stochastic variable ()X ,Y ~ N 2 (µ1 , µ 2 ,σ 1 ,σ 2 , ρ ), its m.g.f. mXY (u,v)is given by ⎛ σ 2u 2 + 2ρσ σ uv +σ 2v2 ⎞ m u,v = exp ux + vy f x, y dxdy = exp⎜µ u + µ v + 1 1 2 2 ⎟ XY ()∫∫ ( )() ⎜ 1 2 ⎟ . ⎝ 2 ⎠ ⎡M T ⎤ −R f ()T −t MT ⎛⎛ 1 2 2 ⎞ ⎞ Risk-free Bond value: P()t,T = Et ⎢ ⋅1⎥ = e , = exp⎜⎜− R f − ()φS +φu ⎟()T − t +φS zS ()T − t +φu zu ()T − t ⎟ ⎣ M t ⎦ M t ⎝⎝ 2 ⎠ ⎠ ⎡M + ⎤ x ⎛ M ⎞ ⎛ S ⎞ + W ()t,T = E T ()S − K = e g ()()y f x, y dxdy X = ln⎜ T ⎟ Y = ln⎜ T ⎟ y Equity value: t ⎢ T ⎥ ∫∫ , ⎜ ⎟ , ⎜ ⎟ and g()y = (St e − K ) . ⎣ M t ⎦ ⎝ M t ⎠ ⎝ St ⎠ ⎛ 1 2 2 ⎞ 2 2 ⎛ 1 2 ⎞ 2 E[]X = ⎜− R f − ()φS + φu ⎟()T − t , V []X = (φS + φu )(T − t), E[]Y = ⎜ R f + σ S λS − σ S ⎟()T − t , V [Y ] = σ S (T − t), Cov(X ,Y ) = φSσ S (T − t). ⎝ 2 ⎠ ⎝ 2 ⎠ ∞ x Define f ()y ≡ e f (x, y )dx ， ∫−∞ ∞ ∞ ∞ ∞ ∞ evy f ()y dy = evy e x f ()x, y dxdy = e x+vy f ()x, y dxdy = m ()1, v ∫−∞ ∫−∞ ∫−∞ ∫∫−∞ −∞ 2 ⎧ ⎛ 1 ⎞ ⎫ 2 2 ⎛ 2 ⎞ ⎛ σ ⎞ ⎛ σ ⎞ ⎪ ⎜ y − ⎜ R f − σ s ⎟()T − t ⎟ ⎪ = exp⎜ µ + 1 ⎟ exp⎜()µ + ρσ σ v + 2 v2 ⎟ 1 ⎪ ⎝ ⎝ 2 ⎠ ⎠ ⎪ ⎜ 1 ⎟ ⎜ 2 1 2 ⎟ f ()y = P (t,T ) exp⎨− ⎬ ⎝ 2 ⎠ ⎝ 2 ⎠ 2 2πσ S T − t ⎪ 2σ S ()T − t ⎪ ⎛ 1 σ 2 ()T − t ⎞ ⎪ ⎪ ⎜⎛ 2 ⎞ S 2 ⎟ ⎩ ⎭ = P()t,T exp⎜⎜ R f − σ S + ()λS + φS σ S ⎟v + v ⎟ ⎝⎝ 2 ⎠ 2 ⎠ 6 1.5 Equity value formula in the Merton model Equity value formula: ∞ ⎡M T + ⎤ x W ()t,T = Et (ST − K )= e g ()(y f x, y )dxdy = g ()()y f y dy We have only to evaluate ⎢ ⎥ ∫∫ ∫−∞ using the probability ⎣ M t ⎦ ⎧ 2 ⎫ ⎛ ⎛ 1 2 ⎞ ⎞ ⎪ ⎜ y − ⎜ R f − σ s ⎟()T − t ⎟ ⎪ 1 ⎪ ⎝ 2 ⎠ ⎪ ⎝ ⎠ . density function f ()y = P (t,T ) exp⎨− 2 ⎬ 2πσ S T − t ⎪ 2σ S ()T − t ⎪ ⎪ ⎪ ⎩ ⎭ ⎧ 2 ⎫ ⎛ ⎛ 1 2 ⎞ ⎞ ⎪ ⎜ y − ⎜ R f − σ s ⎟()T − t ⎟ ⎪ ∞ 1 ⎪ ⎝ ⎝ 2 ⎠ ⎠ ⎪ W ()t,T = g ()()y P t,T exp − dy ∫−∞ ⎨ 2 ⎬ 2πσ S T − t ⎪ 2σ S ()T − t ⎪ ⎪ ⎪ ⎩ ⎭ ⎧ 2 ⎫ ⎛ ⎛ 1 2 ⎞ ⎞ ⎪ ⎜ y − ⎜ R f − σ s ⎟()T − t ⎟ ⎪ ∞ ⎜ ⎟ y + 1 ⎪ ⎝ ⎝ 2 ⎠ ⎠ ⎪ = P()t,T ()St e − K exp − dy ∫−∞ ⎨ 2 ⎬ 2πσ S T − t ⎪ 2σ S ()T − t ⎪ ⎪ ⎪ ⎩ ⎭ ⎛ ⎛ 1 2 ⎞ ⎞ ⎛ ⎛ 1 2 ⎞ ⎞ ⎜ ln()St K + ⎜ R f + σ S ⎟()T − t ⎟ ⎜ ln()St K + ⎜ R f − σ S ⎟()T − t ⎟ ⎜ ⎝ 2 ⎠ ⎟ ⎜ ⎝ 2 ⎠ ⎟ = St Φ − KP()t,T Φ ⎜ σ T − t ⎟ ⎜ σ T − t ⎟ ⎜ S ⎟ ⎜ S ⎟ ⎝ ⎠ ⎝ ⎠ 7 1.6 Purpose of Our Research Using Pricing Kernel, (1-A) we derive closed-form valuation formulas of nominal and real (inflation-linked) corporate bonds. (1-B) we examine the sensitivity of nominal and real credit spreads on company parameters related to inflation. (2-A) we derive closed-form valuation formula of the non-defaultable inflation derivatives. (2-B) we examine the sensitivity of the non-defaultable inflation derivative prices. (3-A) we derive semi-closed-form valuation formula of the defaultable inflation derivatives. (3-B) we examine the sensitivity of the default premium in the inflation derivative prices. to attain some useful implication for corporate finance. 8 2 The setting and the valuation of the nominal government bond 2.1 Setting (Replacing the equity process with company value process in Brennan and Xia (2002)) dΠ z Price level Π : = πdt + σ Π dz Π (1) Π z Instantaneous expected inflation π : dπ = α(π − π )dt + σ π dzπ (2) dM M = −rdt + φS dz S + φr dzr + φπ dzπ + φu dzu = −rdt + φ′dz + φu dzu z Real-pricing kernel : M , (3) where φi (i = S,r,π ,u ) are constants and determine the corresponding market price of risks λS , λr , λπ and λu . z Instantaneous real interest-rate r : dr = κ(r − r)dt +σ rdzr (4) dS z Nominal firm-asset S : = (R f + σ S λS )dt + σ S dzS (5) S where λS is the constant unit-risk premium associated with the innovation, dzS , and Rf is the nominal interest rate. ′ z Price level Π process can be written by the innovations dz = []dzS ,dzr ,dzπ and the projection residualξudzu . dΠ = πdt + σ Π dzΠ = πdt + ξ S dz S + ξ r dz r + ξπ dzπ + ξ u dzu ≡ πdt + ξ ′dz + ξ u dzu (6) Π 9 2.2 Nominal government-bond price (Brennan and Xia (2002)) z The nominal-pricing kernel: M Π z The price of the nominal government bond that pays $1 at the maturity T , as follows: ⎡M T M t ⎤ P()t,T = Et ⎢ ⋅1⎥ (7) ⎣ ΠT Π t ⎦ T ⎛ 1 ⎞ T T ln()M T M t = ⎜− r ()s − VM ⎟ds + φ′dz + φu dzu (8) ∫t ⎝ 2 ⎠ ∫t ∫t T T T ⎛ 1 2 ⎞ ln()()Π T Π t = ⎜π s − VΠ ⎟ds + ξ ′dz + ξ u dzu (9) ∫t ⎝ 2 ⎠ ∫t ∫t ⎡ ⎤ ⎧ ⎫ ⎡M T M t ⎤ ⎧ ⎛ M T ⎞ ⎛ Π T ⎞⎫ ⎪ ⎡ ⎛ M T ⎞ ⎛ Π T ⎞⎤ 1 ⎡ ⎛ M T ⎞ ⎛ ΠT ⎞⎤⎪ (Proof) P()t,T = Et ⎢ ⋅1⎥ = Et ⎢exp⎨ln⎜ ⎟ − ln⎜ ⎟⎬⎥ = exp⎨Et ⎢ln⎜ ⎟ − ln⎜ ⎟⎥ + Vart ⎢ln⎜ ⎟ − ln⎜ ⎟⎥⎬ ⎜ M ⎟ ⎜ ⎟ ⎜ M ⎟ ⎜ ⎟ 2 ⎜ M ⎟ ⎜ ⎟ ⎣ ΠT Π t ⎦ ⎣⎢ ⎩ ⎝ t ⎠ ⎝ Π t ⎠⎭⎦⎥ ⎩⎪ ⎣ ⎝ t ⎠ ⎝ Π t ⎠⎦ ⎣ ⎝ t ⎠ ⎝ Π t ⎠⎦⎭⎪ ⎛ M ⎞ ⎛ Π ⎞ X = ln⎜ T ⎟ − ln⎜ T ⎟ E [exp X ] m ()1 m (t) Putting T ⎜ ⎟ ⎜ ⎟ , t T is equivalent to X T , where X T is moment generation function ⎝ M t ⎠ ⎝ Π t ⎠ ⎧ 1 2 ⎫ of X T and is given by exp⎨Et ()X T t + Vart ()X T t ⎬ . Therefore, we have only to evaluate Et [X T ] and Vart []X T . ⎩ 2 ⎭ ■ 10 3 Valuation of corporate bond and inflation-linked corporate bond (1-A) 3.1 Corporate bond (The discount bond that has maturity T and face amount K ) value F(t,T ) - First, find out the equity value by evaluating equation (10), then subtract it from the firm value to derive F()t,T .