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CLOSED FORM SOLUTIONS FOR TERM STRUCTURE DERIVATIVES WITH LOG-NORMAL INTEREST RATES KRISTIAN R. MILTERSEN, KLAUS SANDMANN, AND DIETER SONDERMANN Abstract. We derive a unified term structure of interest rates model which gives closed form solutions for caps and floors written on interest rates as well as puts and calls written on zero-coupon bonds. The crucial assumption is that the simple interest rate over a fixed finite period that matches the contract, which we want to price, is log-normally distributed. Moreover, this assumption is shown to be consistent with the Heath-Jarrow-Morton model for a specific choice of volatility. 1. Introduction Closed form solutions for interest rate derivatives, in particular caps, floors, and bond options, have been obtained by a number of authors for Markovian term structure models with normally distributed interest rates or alternatively log-normally distributed bond prices, see, e.g., Jamshidian (1989), Jamshidian (1991a), Heath, Jarrow, and Morton (1992), Brace and Musiela (1994), and Geman, El Karoui, and Rochet (1995). These models support Black-Scholes type formulas (cf. Black and Scholes (1973)) most frequently used by practitioners for pricing bond options, and swaptions. Unfortunately, these models imply negative interest rates with positive probabilities, and hence they are not arbitrage free in an economy with opportunities for riskless and costless storage of money. Briys, Crouhy, and Schöbel (1991) apply the Gaussian framework to derive closed form solutions for caps, floors and European zero-coupon bond options. To exclude the influence of negative forward rates on the pricing of zero-coupon bond options they introduce an additional boundary condition. As shown by Rady and Sandmann (1994) these pricing formulas are only supported by a term structure model with an absorbing boundary for the forward rate at zero, where the absorbing probability is not negligible at all, which for a term structure model is a problematic assumption. Alternatively, modeling log-normally distributed interest rates avoids these problems of negative interest rates. However, as shown by Morton (1988) and Hogan and Weintraub (1993) these rates explode with positive probability implying zero prices for bonds and hence also arbitrage opportunities. Furthermore, so far, no closed form solutions are known for these models. Date. March 1994. This version: April 28, 1999. Journal of Economic Literature Classification. G13. Key words and phrases. Heath-Jarrow-Morton model, interest rate derivatives, log-normal, non-negative interest rates, pricing of derivative securities, α-rates. Appears in The Journal of Finance, Vol. 52, No. 1, March 1997, pp. 409–430. An earlier version of this paper, Sandmann, Sondermann, and Miltersen (1995), was presented at the Seventh Annual European Futures Research Symposium in Bonn, Germany and at Norwegian School of Economics and Business Administration, Bergen, Norway. This version of the paper was presented at the Isaac Newton Institute at Cambridge University during the Financial Mathematics Programme, at the 12th AFFI Conference in Bordeaux, France, at the 22th EFA Annual Meeting in Milan, Italy, at the Symposium on Fixed Income Securities and Related Derivatives, Hindsgavl Castle, Middelfart, Denmark, at the Workshop on Quantitative Methods in Finance, Australian National University, Canberra, Australia, and at Stockholm School of Economics. We are grateful to an anonymous referee and to Simon Babbs, Peter Ove Christensen, Darrell Duffie, Jan Ericsson, Thomas Hey, Bjørn Nybo Jørgensen, Claus Munk, Marek Musiela, Johannes Raaballe, Sven Rady, Chris Rogers, Gunnar Stensland, and the two CBOT discussants Chris Veld and Frans de Roon for comments and assistance. The first author gratefully acknowledge financial support of the Danish Natural and Social Science Research Councils and all three authors gratefully acknowledge financial support of the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303 at Rheinische Friedrich-Wilhelms-Universität Bonn. Document typeset in LATEX. 1 2 KRISTIAN R. MILTERSEN, KLAUS SANDMANN, AND DIETER SONDERMANN As has been observed by Sandmann and Sondermann (1994) the problems of exploding interest rates result from an unfortunate choice of compounding of the interest rates modeled, namely the continuous compounding rate. Assuming that the continuously compounded interest rate is log-normally distributed results in “double exponential” expressions, i.e., the exponential function is itself an argument of an exponential function, thus giving rise to infinite expectations of the accumulation factor and of inverse bond prices under the martingale measure. The problem disappears, as shown in Sandmann and Sondermann (1994), if, instead of assuming that the continuously compounded interest rates are log-normally distributed, one assumes that simple interest rates over a fixed finite period are log-normally distributed. In practice, interest rates, both spot and forward, are quoted as simple rates per annum, that is on a yearly basis, even if the finite period is different from one year, e.g., three months. In order to compare simple rates over different finite periods effective annual rates1 are calculated and used as benchmark. Hence, simple interest rates over finite periods are directly observable at the market place and form a natural starting point for modeling the term structure. We are aware of two alternative approaches which are similar to our approach and which also avoid the problem of exploding rates: (i) Musiela (1994) models instantaneous forward rates with non-continuous compounding as log-normal and finds the corresponding dynamics of the continuously compounded rates. (ii) Stapleton and Subrahmanyam (1993) and Ho et al. (1994) model “bankers discount” rates as log-normal. However, this latter approach implies negative bond prices with positive probability. The main result of this paper is a unified model which provides closed form solutions for interest rate caps and floors as well as European puts and calls written on zero-coupon bonds within the context of a log-normal interest rate model. These solutions coincide with modifications of the Black-Scholes formula. In particular, for caps and floors we obtain the well-known standard Black formula (cf. Black (1976)) often used by market practitioners, however, without making the unrealistic assumption that the short interest rate used for discounting is deterministic and that the volatility of forward rates with different maturities are equal, cf. Hull (1993, p. 375–376). Thus, in this case our model supports market practice. For call and put options on zero-coupon bonds, our derived closed form solution matches the formula derived in Käsler (1991).2 In Käsler (1991) the formula is derived using no-arbitrage arguments on two bond prices only, in this paper we contribute with a supporting no-arbitrage term structure of interest rates model. Moreover, the log-normal assumption is shown to be consistent with the Heath-JarrowMorton model for a specific choice of volatility structure. Since the model implies non-negative interest rates with probability one, the model is arbitrage free in an economy with opportunities for riskless and costless storage of money. The paper is organized as follows. In Section 2 we present the model. Solutions for interest rate derivatives are then derived in Section 3. The relation to the Heath-Jarrow-Morton model is found in Section 4, and a discussion of the limitations of the model is found in the conclusion, Section 5. 2. A Model of the Term Structure of Interest Rates Let P (t, T ) denote the price, at date t, of a (default-free) zero-coupon bond that pays $1 at maturity date T . Let f (t, T, α) denote the simple forward rate prevailing at date t over a future time interval 1 By effective annual rates we mean the rate with annual compounding, which yields the same return as the original rate compounded appropriately. 2 This formula is published in Käsler’s Ph. D.-dissertation written in German. The formula appears in the English manuscript Rady and Sandmann (1994) which is a comparative study of different bond based no-arbitrage models. TERM STRUCTURE DERIVATIVES WITH LOG-NORMAL INTEREST RATES 3 f(t,T,α ) α t α α α α s + (n - 1) α s T Figure 1. Curve of the α-forward rates with marked points for the rates used to price the zero-coupon bond with maturity s + nα. [T, T + α]. That is, (1) P (t, T + α) = P (t, T ) 1 . 1 + αf (t, T, α) We term this simple forward rate the α-forward rate.3 The limit case α = 0 corresponds to the continuously compounded forward rate. This rate is given by the natural limit f (t, T, 0) := lim f (t, T, α), α→0 using l’Hospital’s rule from the standard definition of the continuously compounded forward rate ∂ P (t, T ) . f (t, T, 0) = − ∂T P (t, T ) Consider at date t an agreement between two parties to sell or buy the zero-coupon bond with maturity T + α at the future date T , which is known as a forward contract. The forward price F (t, T, α) of this contract is defined as the fixed price which the buyer agrees to pay at date T for the bond with maturity T + α such that the value of the forward contract at date t is zero. No arbitrage implies (2) F (t, T, α) = 1 P (t, T + α) = . P (t, T ) 1 + αf (t, T, α) Note that, at each date t, bond prices and α-forward rates are related by (3) P (t, s + nα) = P (t, s) n−1 Y i=0 3 For α= 1 4 n−1 Y P (t, s + (i + 1)α) 1 = P (t, s) , P (t, s + iα) 1 + αf (t, s + iα, α) i=0 the α-forward rate is simply the 3-month forward rate, hence, the name. 4 KRISTIAN R. MILTERSEN, KLAUS SANDMANN, AND DIETER SONDERMANN for n = 1, . . . and s ∈ [t, t + α), where, for pure simplicity, we have chosen the same fixed period length, α, for each interval. Figure 1 shows the points on the α-forward rate curve used to price the bond with maturity s + nα in the above situation. In our model the stochastic behavior of the term structure of interest rates is determined by the αforward rates. We assume that, at a given date t0 , we observe several simple forward rates with different maturities, T , and/or different period lengths, α. The observed simple forward rates are indexed by the set I.4 That is, we have observed {f (t0 , Ti , αi )}i∈I. Without loss of generality we assume Ti < Tj , for i, j ∈ I such that i < j and αi > 0, for i ∈ I. In practice, the set of observed simple forward rates is determined by maturities of existing bonds, Eurodollar futures contracts, swaps, etc. We model the processes of the simple forward rates indexed by the set I as log-normal diffusions,5 i.e., according to the following stochastic differential equation (SDE) t ∈ [t0 , Ti ], i ∈ I, df (·, Ti , αi )t = µ(t, Ti , αi )f (t, Ti , αi )dt + γ(t, Ti , αi )f (t, Ti , αi )dWt , where the stochastic processes, {f (t, Ti , αi )}t∈[t0 ,Ti ] i∈I, are initiated using the given term structure of (4) interest rates observable at date t0 f (t0 , T, α) = 1 P (t0 , T ) −1 . α P (t0 , T + α) No arbitrage implies that the set of forward rate processes satisfying the above log-normal assumption is restricted to those processes which cannot replicate each other. The mathematical reason is that the sum of log-normally distributed variables is itself not log-normally distributed. From the economic point of view, we have to select among all traded simple forward rates a non-redundant subset of simple forward rates which can be modeled as log-normal diffusions. There are lots of degrees of freedom in this selection process. E.g., we may select the forward rates given by all the 3-month Eurodollar futures contracts.6 In this case Ti would correspond to the settlement date of the i’th contract, and αi to the length of the period covered by this contract. The existence of a unique non-negative solution of the SDE (4) is proven (under suitable regularity conditions)7 in Brace, Gatarek, and Musiela (1995). To make a complete model, consider the situation where SDE (4) is satisfied for all T and one fixed α, say α0 . Hence, we have consistently determined the α0 -forward rates as log-normal for all maturities, T . This imply by the above argument that any α-forward rate with α 6= α0 cannot be log-normal. However, through the bond prices we have determined simultaneously the stochastic model for all α-forward rates for any α (including the continuously compounded rates). That is, bond prices are calculated using Equation (3). Given the bond prices, α-forward rates can be calculated using Equation (1) for any α. However, the domain of this stochastic description is only the time interval [t + α0 , ∞), cf. Figure 1. That is, for rates with α < α0 we have not determined the stochastic model of the α-forward rates (including the continuously compounded rates, i.e. α = 0) in the time interval [t, t + α0 − α). index set of observed simple forward rates, I, may be finite, infinite, or even non-countable infinite. the usual regularity conditions we can extend this to a multi-dimensional Wiener process. Similar closed form solutions can be derived in this situation. For simplicity of exposition we concentrate on the one-dimensional case. The Heath-Jarrow-Morton model uses the continuously compounded forward rates as starting point, whereas our modeling assumption is based on the simple forward rates. The relationship between these approaches is discussed in Section 4. 6 A 3-month Eurodollar futures quote of 94.47 corresponds to a 3-month forward LIBOR rate of 5.53 by common market practice, cf., e.g., Hull (1993, p. 98–99). That is, the market neglects the stochastic effect of margin payments and thus the difference between a futures contract and a forward rate agreement (FRA) based on the futures quote. With reference to the discussion in Section 3 it is an assumption of the model that the underlying interest rate is default free. 7 Taking up the ideas from our model, Brace, Gatarek, and Musiela (1995, Theorem 2.1) have shown existence of a solution to the SDE (4) in the case of a bounded and (piecewise) continuous volatility function γ(·, ·, α) : R2+ → R under the equivalent martingale measure. The existence of a solution to the SDE (4) under the original probability measure is well-known. 4 The 5 Under TERM STRUCTURE DERIVATIVES WITH LOG-NORMAL INTEREST RATES 5 Using Itô’s lemma on the forward price process from Equation (2) gives that vol dF (·, T, α)t = −F 2 (t, T, α)αγ(t, T, α)f (t, T, α)dWt (5) = −F (t, T, α) 1 − F (t, T, α) γ(t, T, α)dWt , where we are only calculating the diffusion part of the Itô processes in this paper, since we know from, e.g., Harrison and Kreps (1979) and Harrison and Pliska (1981) that the drift part will not play any role for the pricing of contingent claims. For that purpose, we have introduced the obvious notation “vol”. That is, for the Itô process dXt = ξ(Xt , t)dt + δ(Xt , t)dWt , we define vol(dXt ) := δ(Xt , t)dWt . 3. Closed Form Solutions for Interest Rate Derivatives In this section, we focus on the no-arbitrage price of interest rate derivatives. More precisely, we consider two special interest rate derivatives: European options with a zero-coupon bond as underlying security and interest rate caps and floors. Since we assume log-normality of the underlying interest rate model, we should expect pricing formulas similar to the Black-Scholes model for derivatives written directly on the interest rates such as caps and floors. Caps and floors are special types of options where a nominal interest rate is the underlying security. The underlying interest rate could be, for example, the 3-month or 6-month LIBOR. A cap is an insurance against upward movements in the interest rate and a floor is an insurance against downward movements in the interest rate. Let {rt } be the nominal α-interest rate process, e.g., for α = 14 the process {rt } is the quoted 3-month LIBOR. In our model the α-interest rate is included in the α-forward rate, i.e., rt = f (t, t, α). It is an assumption of the model that the underlying interest rate, f , is default-free since it is used to price default-free bonds. In practice, the LIBOR is based on a “replenished” AA rate and, hence, not default-free. However, (i) assuming that the short position of the cap or floor contract has the same credit quality as the one on which LIBOR is based and (ii) modeling the default risk as in Duffie and Singleton (1994) and Duffie, Schroder, and Skiadas (1994) the same formulas apply with the volatility process adjusted to include the default spread on LIBOR. As it is shown in Duffie (1994), the volatility of the credit spread and of the default-free rate simply adds together to give the volatility of the defaultable rate. This result also applies to our model, SDE (4), with an appropriate dynamics of the default risk. Duffie and Singleton (1994) then show that options etc. written on defaultable interest rates can be priced using standard option pricing techniques, such as valuing expectations under an equivalent martingale measure or solving PDEs, by (i) simply substituting the default-free volatility with the volatility of the defaultable rate and (ii) using the defaultable rate as the short rate in the option pricing model.8 Let t1 < · · · < tN be a set of dates and {rti }N i=1 a corresponding set of αi -interest rates. That is, the simple interest rate over the time period [ti , ti + αi ] is rti , to be paid at date ti + αi . A cap contract with N cap rate L, face value V , underlying nominal αi -interest rates, {rti }N i=1 , and payment dates {ti + αi }i=1 is defined by the payoff at all dates ti + αi + V αi rti − L = V αi max rti − L, 0 , 8 We are indebted to Darrell Duffie for pointing this out to us. 6 KRISTIAN R. MILTERSEN, KLAUS SANDMANN, AND DIETER SONDERMANN if payments are made in arrear. A cap with one payment date is called a caplet. Since all caps are portfolios of caplets, we concentrate on pricing a caplet. Since this rate is known at date ti , the payoff of a caplet at date ti + αi is also known at date ti , hence the present value of this payoff, at date ti , is equal to h i+ + V 1 + αi f (ti , ti , αi ) − (1 + αi L) P (ti , ti + αi )V αi f (ti , ti , αi ) − L = 1 + αi f (ti , ti , αi ) i+ h 1 + αi L (6) =V 1− 1 + αi f (ti , ti , αi ) h i+ 1 − P (ti , ti + αi ) . = V (1 + αi L) 1 + αi L The floor is just the opposite contract, and the present value at date ti is given by i+ h 1 (7) . V (1 + αi L) P (ti , ti + αi ) − 1 + αi L Hence, the value of the payoff of a cap or a floor, at each date ti , is equivalent to V (1 + αi L) times the payoff of a European put option or a European call option, respectively, with exercise date ti , exercise 1 , and a zero-coupon bond with maturity ti + αi as the underlying security. Thus, the price K = 1+α iL price of a cap or a floor is equal to the price of a portfolio of European put options or European call options, respectively. Note that in this model the relation between caps and floors and zero-coupon bond options is an exact relation in opposition to, e.g., Briys, Crouhy, and Schöbel (1991) where the relation between caps and floors and zero-coupon bond options is an approximation. Because of this exact relation, consider the pricing of European options on zero-coupon bonds. Proposition 3.1. Consider a European call option with exercise price K and exercise date T written on a zero-coupon bond with maturity date T + α. If the simple forward rate process {f (t, T, α)}t≤T is log-normally distributed, i.e. satisfies SDE (4), then the price of the call option, at date t ≤ T , is Call = P (t, T + α)N (e+ ) − KP (t, T )N (e−) − KP (t, T + α) N (e+ ) − N (e− ) (8) = (1 − K)P (t, T + α)N (e+ ) − K P (t, T ) − P (t, T + α) N (e− ), where P (t, T + α)(1 − K) 1 σ 2 (t, T, α) ± ln , σ(t, T, α) 2 P (t, T ) − P (t, T + α) K Z T 2 γ 2 (s, T, α)ds, σ (t, T, α) = e± = t where N (·) denotes the standard normal distribution. Proof. The proof consists of two steps. First, we derive a partial differential equation (PDE) which the option price function must fulfill by a change of measure technique. This PDE is solved in the second step. Under the T -forward risk adjusted measure, QT , we have by no arbitrage that h P (T, T + α) i P (t, T + α) = E QT (9) Ft = E QT P (T, T + α) Ft , P (t, T ) P (T, T ) where E QT [·|Ft ] denotes the conditional expectation given the information at date t under the probability measure QT , cf. Jamshidian (1991b) or Geman, El Karoui, and Rochet (1995). Given the definition of TERM STRUCTURE DERIVATIVES WITH LOG-NORMAL INTEREST RATES 7 F (t, T, α), Equation (9) can equivalently be written F (t, T, α) = E QT F (T, T, α) Ft . (10) Hence, {F (t, T, α)}t≤T is a QT -martingale. Let ĉ(t, F ) denote the forward price of the European call option as function of calendar time, t, and the forward price, F , of the underlying security and let Ĉ denote the forward price of the European call option as a stochastic process, i.e., t ≤ T. Ĉt = ĉ t, F (t, T, α) , By Itô’s lemma 1 dĈt = ĉt (t, F )dt + ĉF F (t, F )dhF (·, T, α)it + ĉF (t, F )dF (·, T, α)t , 2 where subscripts of ĉ denotes partial derivatives. Again by no arbitrage, {Ĉt }t≤T is a QT -martingale. Thus the drift of Ĉ is zero which imply that (11) 1 ĉt (t, F )dt + ĉF F (t, F )dhF (·, T, α)it = 0, 2 since the drift of {F (t, T, α)}t≤T is also zero as observed above in Equation (10). Combining Equation (11) with the quadratic variation process of the forward price from Equation (5) yields the following PDE which ĉ has to fulfill (12) 1 ĉt (t, F ) + ĉF F (t, F )γ 2 (t, T, α)F 2 (1 − F )2 = 0. 2 Moreover, for a European call option the boundary condition is that ĉ(T, F ) = [F − K]+ . The solution of the PDE (12) with the stated boundary condition follows the presentation in Rady and Sandmann (1994). For completeness we give the outline of the proof in Appendix A.1. Finally the spot price of the European call option is given by Call = P (t, T )ĉ t, F (t, T, α) . The corresponding self-financing duplicating strategy can easily be derived as ! ! (1 − K)N (e+ ) + KN (e− ) φ1t , = φ2t −KN (e− ) where φ1t (resp. φ2t ) is the number of bonds with maturity T + α (resp. T ) to hold at date t ≤ T . Using the put-call parity the value for the put option is (13) Put = K P (t, T ) − P (t, T + α) N (−e− ) − (1 − K)P (t, T + α)N (−e+ ), where N (·), e+ , e− , and σ are as defined in Proposition 3.1. For the put option the self-financing duplicating strategy on the bond market is given by φ1t − 1 and φ2t + K, respectively. We have written two versions of the closed form solution of the call option price in Equation (8). The first version has three terms, where the first two terms look similar to the Black-Scholes formula (but note that e+ and e− are not the usual arguments of N (·)), and then there is a third correction term. The second version is in structure a Black-Scholes formula, where (1 − K)P (t, T + α) should be interpreted as the price of the underlying security and K P (t, T ) − P (t, T + α) as the present value of the exercise price. A closed form solution of the type of Equation (8) was first derived by Käsler (1991) under specific assumptions for the two underlying zero-coupon bonds. A discussion of this model relative to other 8 KRISTIAN R. MILTERSEN, KLAUS SANDMANN, AND DIETER SONDERMANN bond-price based models9 can be found in Rady and Sandmann (1994). A very recent bond-price based model is Rady (1995). Rady derives the same pricing formulas for zero-coupon bond options as we do in Equations (8) and (13) using the change of measure technique in a purely probabilistic framework. The initial step of this approach is also to observe that the forward price of the call option is a martingale under the T -forward risk adjusted measure, QT . The main difficulty with his approach is to determine the transition density q(F ; t, F0 ). By solving the PDE (12) with proper boundary conditions the transition density for F ∈ (0, 1) is F0 1 − e q(F ; t, F0 ) := p − 2πσ 2 (t) F 2 (1 − F ) (14) (ln 1−F0 1−F −ln + 1 σ2 (t))2 F F0 2 2σ2 (t) , RT R1 where σ 2 (t) = t γ 2 (u, T, α)du. Note that the PDE (12) is satisfied by the function ĉ(t, F ) = 0 [v − 1 the substitution ρ K]+ q(v; t, F )dv. Since F is equal to the forward price at date t, i.e., F = 1+αf (t,T,α) = ln 1−F αF yields ĉ(t, F ) = p 1 2πσ 2 (t) Z +∞ h −∞ i+ 1 + α exp (ρ) (ρ−ln f + 2 σ 1 − 2σ2 (t) −K e 1 + α exp (ρ) 1 + αf 1 2 (t))2 dρ. The solution can be calculated in the same way as the one in Appendix A.1.10 We can now apply Proposition 3.1 and Equation (13) combined with Equations (6) and (7) to the pricing of interest rate caps and floors. Proposition 3.2. Consider a cap with cap rate L, face value V , underlying nominal αi -interest rates N N {f (ti , ti , αi )}N i=1 and (arrear) payment dates {ti +αi }i=1 . If the simple forward rate processes {f (t, ti , αi )}t≤ti i=1 satisfy the SDE (4), then the price of the cap at date t ≤ t1 is (15) Cap = V N X αi P (t, ti + αi ) f (t, ti , αi )N d+ (t, ti , αi ) − LN d− (t, ti , αi ) , i=1 where f (t, s, α) σ 2 (t, s, α) 1 ln ± , σ(t, s, α) L 2 Z s γ 2 (u, s, α)du, σ 2 (t, s, α) = d± (t, s, α) = t where N (·) denotes the standard normal distribution. Proof. By no arbitrage we can reduce the problem to the pricing of a caplet with arrear payment date ti + αi . From Equation (6), the payoff of the caplet is equivalent to V (1 + αi L) times the payoff of a European put option with exercise price 1 1+αi L , where the underlying security is a zero-coupon bond with 9 By the term bond-price based model we mean a model in which the bond prices are the exogenously given basic underlying variables. This is in contrast to the model of this paper—and many other models—where the exogenously given basic underlying variable is the term structure of interest rates from which the bond prices are then derived endogenously. 10 This is not just an alternative way of proving Proposition 3.1. The price of any European contingent claim with Borel measurable payoff function h(·), at date T , on the zero-coupon bond with maturity T + α is determined by Z 1 , h(v)q(v; t, F0 )dv. P (t, T )E QT h F (T, T, α) F (t, T, α) = F0 = P (t, T ) 0 Hence, the pricing of such claims is reduced to (numerical) calculation of a one-dimensional integral equation. TERM STRUCTURE DERIVATIVES WITH LOG-NORMAL INTEREST RATES 9 maturity date ti + αi . Then, according to Equation (13), the price of the caplet is 1 P (t, ti ) − P (t, ti + αi ) N (−e− ) Caplet = V (1 + αi L) 1 + αi L 1 P (t, ti + αi )N (−e+ ) − 1− 1 + αi L (16) P (t, ti ) − 1 N (−e− ) − αi LN (−e+ ) = V P (t, ti + αi ) P (t, ti + αi ) = V P (t, ti + αi ) αi f (t, ti , αi )N (−e− ) − αi LN (−e+ ) , where 1 P (t, ti + αi ) 1 − 1+α −1 σ 2 (t, ti , αi ) iL 1 − ln −e− = σ(t, ti , αi ) 2 P (t, ti ) − P (t, ti + αi ) 1+αi L 2 αi L σ (t, ti , αi ) −1 ln − = d+ (t, ti , αi ). = σ(t, ti , αi ) αi f (t, ti , αi ) 2 By summing the respective caplets, this yields the pricing formula for a cap. The hedge strategy for each caplet is determined by a long and a short position on the bond market. For the caplet with payment date ti + αi the hedge strategy for t ≤ ti is equal to a position, φ1t , in the bond with maturity ti + αi and a position, φ2t , in the bond with maturity ti , where ! ! −V N d+ (t, ti , αi ) + αi LN d− (t, ti , αi ) φ1t . = φ2t V N d+ (t, ti , αi ) A similar proof gives, under the same conditions as in Proposition 3.2, that the price of a floor is (17) Floor = V N X αi P (t, ti + αi ) LN −d− (t, ti , αi ) − f (t, ti , αi )N −d+ (t, ti , αi ) , i=1 where d± and σ are defined as in Proposition 3.2. The Formulas (15) and (17) coincide with the formulas frequently used by market practitioners for the pricing of caps and floors. Up until now there has been no convincing theoretical argument justifying this practice. In particular, it has been an open question whether a consistent term structure of interest rate model existed in which these formulas were valid, ruling out arbitrage across different maturities. The proof of Proposition 3.2 uses the relationship between a caplet and a European put option on a zero-coupon bond. Alternatively one can calculate the price of a caplet directly under the T -forward risk adjusted measure QT . The transition probability q(F ; t, F0 ), where F0 is the date t forward price is given 1 and the substitution ρ = ln 1−F by Equation (14). With F0 = 1+αf αF this implies h i+ 1 QT − F (T, T, α) F (t, T, α) = F0 Caplet = P (t, ti )V (1 + αi L)E 1 + αL Z +∞ h 1 + αL i+ 1 + α exp (ρ) − (ρ−ln f +212 σ2 )2 2σ e dρ 1− = P (t, ti )V 1 + α exp (ρ) 1 + αf −∞ Z (ρ−ln f + 1 σ2 )2 2 P (t, ti )V α +∞ ρ 2σ2 (e − L)e− dρ = 1 + αf ln L = P (t, ti + αi )V α f N (d+ ) − LN (d− ) , where for simplicity we have set f := f (t, T, α), 2 Z σ := t T γ 2 (u, T, α)du, and d± := σ2 1 f ln ± . σ L 2 10 KRISTIAN R. MILTERSEN, KLAUS SANDMANN, AND DIETER SONDERMANN However, under the T -forward risk adjusted measure, QT , the simple forward rate {f (t, T, α)}t≤T is not a martingale since 1 + αf (t, T, α) exp σ 2 (t, T, α) QT . f (T, T, α) F (t, T, α) = F0 = f (t, T, α) E 1 + αf (t, T, α) In order to clarify the relation between the different forward rates and to strengthen the intuition behind these formulas we adapt a suggestion made by a referee of this journal. It was observed by the referee (and by Brace, Gatarek, and Musiela (1995)) that a probabilistic proof of the cap pricing formula—without recurrence to the PDE (12) and the put option price from Equation (13)—may be obtained as follows:11 . . . show that under an appropriate change of measure the caplet formula (16) is the expected value of its payoff at maturity under this measure. This involves identifying the change of measure; explicitly identifying the distribution under the change of measure; and explicitly identifying how to evaluate the integral. Given expression (4), show the form of (4) under the change of measure. Show how to compute the expression for the cap. This can be done as follows: According to the following Lemma {f (t, T, α)}t≤T is a log-normal martingale under QT +α . Hence Black’s formula gives immediately c(t) = αP (t, T + α) f (t, T, α)N (d+ ) − LN (d− ) . Lemma 3.3. The forward rate process of SDE (4) satisfies the stochastic differential equation (18) df (·, T, α)t = γ(t, T, α)f (t, T, α)dWtT +α , where W T +α denotes a Wiener process under the (T + α)-forward measure QT +α . Proof. Consider the following trading strategy: At date t, buy 1 α 1 − F (t, T, α) zero-coupon bonds with maturity T and short α1 forward contracts giving the holder the right to buy one zero-coupon bond with maturity date T + α at date T at the price F (t, T, α). At date T , short the α1 zero-coupon bonds with maturity T + α yielding a cash inflow of α1 F (t, T, α). Added together with the initial investment in the zero coupon bond with maturity T we now have $ α1 in cash. Invest this amount in the zero-coupon bond with maturity T + α. Hence, at date T + α we have 1 1 1 + αf (T, T, α) − = f (T, T, α). α α That is, we have found a trading strategy yielding the cash flow f (T, T, α) at date T + α. The initial price of this strategy is 1 Xt := P (t, T ) 1 − F (t, T, α) . α Xt By no arbitrage, P (t,T +α) t≤T is a QT +α -martingale, where QT +α denotes the (T + α)-forward risk adjusted measure. Now observe, P (t, T ) 1 − F (t, T, α) 1 − F (t, T, α) Xt = = = f (t, T, α). P (t, T + α) αP (t, T + α) αF (t, T, α) Thus, we have shown that {f (t, T, α)}t≤T is a QT +α -martingale. That the diffusion term is as written in Equation (18) follows from the SDE (4) since according to Girsanov’s theorem the quadratic variation of the process is unaffected by a change of measure operation. 11 Quoting from a referee report on an earlier version of this paper. TERM STRUCTURE DERIVATIVES WITH LOG-NORMAL INTEREST RATES 11 4. The Corresponding Term Structure Model of the Continuously Compounded Interest Rates Let us consider the complete model described in Section 2 where we have specified the α-forward rates according to the SDE (4) for one specific α and for all T . We want to show how to specify the volatility of the Heath-Jarrow-Morton model such that this model will give the crucially needed log-normal α-forward rates. In the Heath-Jarrow-Morton model the continuously compounded forward rate, {f (t, T, 0)}t≤T , is the basic modeling element. This process is modeled as an Itô process in the following way df (·, T, 0)t = µ t, T, f (t, T, 0) dt + η t, T, f (t, T, 0) dWt . The relation between the continuously compounded forward rates and the α-forward rates is given by R T +α 1 = F (t, T, α) = e− T f (t,s,0)ds , 1 + αf (t, T, α) t ≤ T. On the first hand, define Y (t, T, α) = − ln F (t, T, α) then ∂ Y (t, T, α) = f (t, T + α, 0) − f (t, T, 0). ∂T On the other hand, Y (t, T, α) = ln 1 + αf (t, T, α) , therefore, (19) (20) 1 ∂ Y (t, T, α) = αfT (t, T, α) = F (t, T, α)αfT (t, T, α), ∂T 1 + αf (t, T, α) where fT (t, T, α) denotes ∂ ∂T f (t, T, α). Combining Equation (19) and (20) yields f (t, T + α, 0) − f (t, T, 0) = αF (t, T, α)fT (t, T, α). (21) Solving the simple difference equation (21) gives (22) f (t, s + nα, 0) = f (t, s, 0) + n−1 X αF (t, s + iα, α)fT (t, s + iα, α), s ∈ [t, t + α), i=0 with initial condition Z (23) t+α f (t, s, 0)ds = ln 1 + αf (t, t, α) . t This is compatible with our earlier findings in Section 2, that is, when specifying the Itô process of the αforward rates, we do not specify the continuously compounded interest rates in the time interval [t, t + α). So any (non-negative) value of the continuously compounded forward rate in that interval, fulfilling the initial condition (23), is valid, because the continuously compounded rates specified in the time interval [t + α, ∞) by Equation (22) is defined in such a way that the forward rates integrate to the right bond prices. Again, the reader is referred to Figure 1 to get the intuition. To find the volatility of the corresponding continuously compounded interest rates we just have to find vol dX(·, T, α)t , where X(t, T, α) = αfT (t, T, α)F (t, T, α) and then use Equation (22). We already know vol dF (·, T, α)t from Equation (5). Moreover, using the Itô process description from SDE (4) and a result from Fernique et al. (1983, Chapter 2), vol dfT (·, T, α)t can be calculated as (24) ∂ f (t, T, α)γ(t, T, α) dWt vol dfT (·, T, α)t = ∂T = fT (t, T, α)γ(t, T, α) + f (t, T, α)γT (t, T, α) dWt . 12 KRISTIAN R. MILTERSEN, KLAUS SANDMANN, AND DIETER SONDERMANN Finally, using Itô’s lemma and that αf (t, T, α)F (t, T, α) = 1 − F (t, T, α) implies vol dX(·, T, α)t = α −fT (t, T, α)F (t, T, α) 1 − F (t, T, α) γ(t, T, α) (25) + F (t, T, α) fT (t, T, α)γ(t, T, α) + f (t, T, α)γT (t, T, α) dWt = αfT (t, T, α)F 2 (t, T, α)γ(t, T, α) + 1 − F (t, T, α) γT (t, T, α) dWt = −FT (t, T, α)γ(t, T, α) + 1 − F (t, T, α) γT (t, T, α) dWt ∂ 1 − F (t, T, α) γ(t, T, α) dWt . = ∂T Using Itô’s lemma on Equation (22) and the result of Equation (25) yields the volatility of the corresponding continuously compounded forward rate model. However, it should be emphasized that this volatility process of the Heath-Jarrow-Morton model is state dependent. 5. Conclusion Under the assumption of log-normality of simple rates we have derived simple and intuitive closed form solutions for pricing and hedging caps and floors in a consistent no-arbitrage term structure model of the Heath-Jarrow-Morton type. These formulas support common market practice to price caplets by a naive application of Black’s formula on interest rates. However, there is another common market practice which is inconsistent with the caplet formula: namely, to apply Black’s formula also to the pricing of bond options and swaptions. A necessary (but in a term structure model by no means sufficient) condition for the application of Black’s formula is that the underlying security is log-normal. But bond prices and rates cannot be log-normal at the same time. E.g., if bond prices are log-normal, continuously compounded interest rates must be normal. Thus, there is an obvious inconsistency in the market practice of pricing interest rate derivatives. We have also shown in this paper how the formula for call and puts on zerocoupon bonds has to be modified in order to be consistent with the caplet formula. However, our model does not strictly support the application of the Black type formulas to all interest rate derivatives in large portfolios. Recall the crucial assumption of log-normality of the αi -forward rates, where we are free to choose the interval lengths αi . In practice one would choose these intervals according to the most liquid markets for forward rates; e.g., first model the 3-month forward rates using either the Eurodollar futures—or the FRA—contracts for the first two or three years, then space the αi according to market information for longer forward rates. But if, e.g., two consecutive 3-month rates are log-normal, the 6-month rate for the same period cannot be log-normal at the same time. Thus if we have priced the two 3-month caplets consistently, the caplet formula for the 6-month caplet must be considered with caution, since Black’s formula will not give the exact no-arbitrage price. However, the Heath-Jarrow-Morton solution to our term structure model is valid for all forward rates and provides the exact no-arbitrage price also for any composed or interpolated rate. However, this solution may require numerical techniques. In general, if some of the needed forward rate processes are replicable, there are two ways to proceed: either (i) one can solve the corresponding PDE by numerical procedures or (ii) one can inconsistently assume that the true underlying simple interest rate process is log-normally distributed. Surely, at first glance (ii) is questionable. However, the question which arises is: How big is the mispricing if, in spite of the inconsistency, one uses the closed form solutions to price different options? Further research is needed to measure the size of this problem. This mispricing should be counterbalanced with the extra calculations needed to do numerical procedures. This second problem is analogous to the problem of using TERM STRUCTURE DERIVATIVES WITH LOG-NORMAL INTEREST RATES 13 the Black-Scholes formula on individual assets simultaneously with using the Black-Scholes formula on an arithmetic index of the same assets. An inconsistency problem which practitioners do not care much about, because the magnitude of this problem is smaller than many other theoretically inconsistency problems of using the Black-Scholes formula, cf. Hull (1993, Chapter 11, Footnote 2, p. 253). Appendix A. Proofs A.1. Solution of the PDE (12). The proof follows the arguments given in Rady and Sandmann (1994). Proof. Given the assumptions of Proposition 3.1, we have to solve the PDE (12) on [0, T ] × (0, 1), i.e., 1 ĉt (t, x) + γ 2 (t, T, α)x2 (1 − x)2 ĉxx (t, x) = 0, 2 with boundary condition ĉ(T, x) = [x− K]+ , for all x ∈ [0, 1]. This problem is transformed by introducing the new time variable, s, as Z T s := s(t, T ) := γ 2 (r, T, α)dr, t and the new space variable, z, as z := ln x 1−x ⇐⇒ x= 1 , 1 + exp (−z) and finally setting ĉ(t, x) = a(z)b(s)h(s, z). The idea is now to choose differentiable functions a(·) and b(·) in such a way that any solution h(·, ·) of the heat equation yields a solution ĉ(·, ·) of the original PDE. As shown in Rady and Sandmann (1994) this can be done by setting a(z) := z 2 exp s 1 , + exp − z2 b(s) := e− 8 . That is, ĉ(t, x) = exp z 2 s 1 e− 8 h(s, z). + exp − z2 The transformed problem on [0, T ] × R is 12 hzz − hs = 0 with boundary condition i+ h z z 1 −K . h(0, z) = e 2 + e− 2 1 + exp (−z) The well-known solution to this problem is Z ∞ ρ2 √ √ 1 1 1 1 √ − K e− 2 dρ e 2 (z+ρ s) + e− 2 (z+ρ s) h(s, z) = √ K 1 + exp (−(z + ρ s)) 2π √1s (ln 1−K −z) = (1 − K)I1 − KI2 , where 1 1−K s + e e dρ = e e N √ z + ln , 1 K s K 2 √ (ln 1−K −z) s Z ∞ 2 √ 1 1−K s 1 − 12 (z+ρ s) − ρ2 − z2 8s − e e dρ = e e N √ z + ln . I2 = √ K s K 2 2π √1s (ln 1−K −z) 1 I1 = √ 2π Z ∞ √ 1 2 (z+ρ s) 2 − ρ2 z 2 s 8 14 KRISTIAN R. MILTERSEN, KLAUS SANDMANN, AND DIETER SONDERMANN Hence, exp − 8s h(s, z) ĉ(t, x) = exp z2 + exp − z2 exp − z2 exp z2 N (e+ ) − K N (e− ). = (1 − K) exp z2 + exp − z2 exp z2 + exp − z2 {z } {z } | | =x =1−x References Black, F. (1976): “The Pricing of Commodity Contracts,” Journal of Financial Economics, 3:167–179. Black, F. and M. 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Department of Management, Odense Universitet, Campusvej 55, DK-5230 Odense M, Denmark E-mail address: [email protected] Department of Statistics, Rheinische Friedrich-Wilhelms-Universität Bonn, Adenauerallee 24–42, D-53113 Bonn, Germany E-mail address: [email protected] Department of Statistics, Rheinische Friedrich-Wilhelms-Universität Bonn, Adenauerallee 24–42, D-53113 Bonn, Germany E-mail address: [email protected]