APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY Appl. Stochastic Models Bus. Ind. 2008; 24:359–368 Published online 30 April 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/asmb.718 Factors’ correlation in the Heath–Jarrow–Morton interest rate model Leonard Tchuindjo1, 2, ∗, † 1 George 2 Fannie Washington University, SEAS, 1776 G Street NW, Washington, DC 20052, U.S.A. Mae, Capital Market Pricing Group, 4000 Wisconsin Avenue NW, Washington, DC 20016, U.S.A. SUMMARY We propose a new derivation of the Heath–Jarrow–Morton risk-neutral drift restriction that takes into account nonzero instantaneous correlations between factors. The result allows avoiding the orthogonalization of factors and provides an approach by which interest rate derivatives can be priced by preserving the economic meaning of each underlying factor. An application is given for the term structure of creditrisky bonds, driven by two correlated factors—the risk-free forward rate and the forward credit spreads. Copyright q 2008 John Wiley & Sons, Ltd. Received 22 June 2007; Revised 4 February 2008; Accepted 25 February 2008 KEY WORDS: Hilbert space; HJM model; equivalent measure; risk-neutral drift; correlation 1. INTRODUCTION The Heath–Jarrow–Morton (HJM) [1] approach to the term structure model provides a framework for pricing interest rate derivative securities. This approach assumes that all factors are orthogonal, uses the equivalent martingale measure in the pricing procedure, and calibrates the model to the current yield curve without considering the market price of risk. The market no-arbitrage condition imposes a relationship, called the risk-neutral drift restriction, between the drift and the volatility terms. This article contributes to the literature by deriving a risk-neutral drift restriction for the multi-factor HJM model when there are nonzero instantaneous correlations between factors. A reformulation of the HJM model in a finite-dimensional Hilbert space prepares a framework for the integration of an instantaneous correlation matrix between factors. ∗ Correspondence to: Leonard Tchuindjo, Fannie Mae, Capital Market Pricing Group, 4000 Wisconsin Avenue NW, Washington, DC 20016, U.S.A. † E-mail: [email protected] Copyright q 2008 John Wiley & Sons, Ltd. 360 L. TCHUINDJO Keeping the correlations between factors in the HJM model has the appealing feature of preserving the economic meaning of each factor. Although it is well known that one can always transform correlated factors into an orthogonal set of factors before deriving the HJM risk-neutral drift restriction, such orthogonalized factors are mathematical constructs that lack financial meanings. Therefore, in a multiple factor environment, the connection between interest rate derivative prices and some market observable variables is unclear. For example, following the empirical studies of Longstaff and Schwartz [2], Duffee [3], Collin-Dufresne et al. [4], and Papageorgiou and Skinner [5] that find significant negative correlations between the two main factors driving the prices of corporate bonds, the risk-free interest rate and the credit spread, it is more intuitive to model the term structure of corporate bonds by these two observable correlated factors than to use two independent factors, which are the result of orthogonalization and do not have economic meaning. In an attempt to address this issue, the risk-neutral drift restriction of a multi-correlated-factor HJM derived in this paper enables the pricing of interest rate derivatives using easy-to-interpret factors that can be directly observed on, or implied from, the market. Thereby, the results of this paper enable one to avoid the use of economically nonintuitive factors, which is what often results when correlated factors are orthogonalized. These results are applied to the forward rate dynamics of the term structure of credit-risky bonds driven by two observable correlated factors—the risk-free forward rate and the forward default intensity—to obtain an extra risk-neutral drift term. The remainder of this paper is organized as follows. Section 2 reformulates the HJM model in a finite-dimensional Hilbert space. In Section 3, a more general HJM risk-neutral drift restriction, which takes correlation between factors, is derived. Section 4 is an application that derives an extra drift term for the term structure of credit-risky bonds, using an intensity based reduced-form approach. Section 5 is a numerical illustration. 2. THE HJM MODEL IN A FINITE-DIMENSIONAL HILBERT SPACE Hereafter, all random variables and stochastic processes are defined on a filtered probability space (, FT , (Ft )(t 0) , P), where 1. is the set of all possible states of nature. 2. T <∞ is a fixed time horizon. 3. (Ft )(t 0) is the filtration representing the information structure, and it satisfies the usual conditions of Jacod and Shiryaev [6]. 4. P is the physical measure associated with objective probabilities. Further, we define Hn to be the vector space of real column vectors of nonzero dimension n ∈ N. We equip Hn with a real inner product ·, · : Hn ×Hn → R, defined by A, B = AT B, for all A, B ∈ Hn , such that Hn becomes a pre-Hilbert space. More specifically, Hn is isometric to Rn . Hence, Hn is an n-dimensional Hilbert space. Let { f (t, T, t )}(0t T ) be the stochastic process of the instantaneous forward rate at time t for a maturity date T t. For a given nonnegative finite T , let the stochastic differential equation of f be given by d f (t, T, t ) = (t, T, t ) dt +V(t, T, t ), dZ(t, t ) Copyright q 2008 John Wiley & Sons, Ltd. (1) Appl. Stochastic Models Bus. Ind. 2008; 24:359–368 DOI: 10.1002/asmb HEATH–JARROW–MORTON INTEREST RATE MODEL 361 where 1. {Z(t, t )}(t 0) = [{z 1 (t, t )}(t 0) , {z 2 (t, t )}(t 0) , . . . , {z n (t, t )}(t 0) ]T is a vector of n independent P-standard Brownian motions and represents the path of these processes. 2. (t, T, t ) : {(t, T ) : 0tT <∞}× → R is the drift, which is P-almost everywhere absolutely integrable on any finite time horizon. 3. V(·, T, ·) is such that V(t, T, t ) = [v1 (t, T, t ), v2 (t, T, t ), . . . , vn (t, T, t )]T belongs to S, the set of all functions defined from {(t, T ) : 0tT <∞}× onto Rn , that are P-almost everywhere nonnegative, bounded, square integrable on any finite time horizon, and Lipschitz continuous with respect to the second variable. According to Heath et al. [1], the absence of arbitrage implies that there exists a unique measure Q, equivalent to P, as in the standard pricing theory of Harrison and Kreps [7], and Harrison and Pliska [8], under which the stochastic differential equation of the forward rate is given by T d f (t, T, t ) = V(t, T, t ), V(t, s, t ) ds dt +V(t, T, t ), dW(t, t ) (2) t where {W(t, t )}(t 0) = [{w1 (t, t )}(t 0) , {w2 (t, t )}(t 0) , . . . , {wn (t, t )}(t 0) ]T is a vector of n independent Q-standard Brownian motions. The following equation T (t, T, t ) = V(t, T, t ), V(t, s, t ) ds , (Q—a.e.) (3) t is called the risk-neutral drift restriction. It shows that the drift term is well defined whenever the volatility term is given. As a result, the HJM model is completely specified if the volatility process is specified. Many studies have focused on the different classes of models that arise when different assumptions about the form of the volatility process are made (see, e.g. Ritchken and Sankarasubramanian [9] and Chiarella and Kwon [10]); but studies on the correlation between factors that can drive the forward rate process have been limited. In the following section, a more general form of the drift term is presented for the case where the Brownian motions driving the forward rate dynamics are correlated. Hereafter, for simplicity of notation, the argument t representing the path dependence of the process will be omitted. 3. THE HJM MODEL WITH CORRELATED FACTORS The main assumption of this section is that all factors can be correlated. Hence, the aim is to derive a more general drift restriction under the equivalent measure Q, such that the original HJM result becomes a particular case. Now let {Z̃(t)}(t 0) = [{z̃ 1 (t)}(t 0) , {z̃ 2 (t)}(t 0) , . . . , {z̃ n (t)}(t 0) ]T (4) be a vector of n correlated (Ft )(t 0) -adapted standard Brownian motions on (, FT , P) such that, almost everywhere with respect to the physical probability measure P and the sigma algebra Ft , we have dZ̃(t) dZ̃T (t) = K(t) dt Copyright q 2008 John Wiley & Sons, Ltd. (P—a.e.) (5) Appl. Stochastic Models Bus. Ind. 2008; 24:359–368 DOI: 10.1002/asmb 362 L. TCHUINDJO where K(t) is the n ×n correlation coefficient matrix of {Z̃(t)}(t 0) at time t0. It is defined as K(t) = [ki, j (t)](i, j=1,2,...,n) , such that −1ki, j (t) = k j,i (t)1 and ki,i (t) = 1. Let {C(t)}(t 0) = [{1 (t)}(t 0) , {2 (t)}(t 0) , . . . , {n (t)}(t 0) ]T be a vector of n processes that are (Ft )(t 0) -predictable processes and satisfy Novikov’s condition. Then the process {m(t)}(t 0) , t t defined by m(t) = exp( 0 C(s), dZ(s)−( 12 ) 0 C(s), C(s) ds) is a P-martingale, and hence, for 0t<∞ we have E P [m(t)] = E P [m(0)] = 1, where E P (·) is the expectation with respect to the physical probability measure P. A new probability measure Q, equivalent to P, can be defined on FT , such that ∀A ∈ FT we have Q(A) = E Q [1{A} ] = E P [1{A} m(T )], where 1{A} is the indicator function of the even A. m(T ) = dQ/dP |FT is the Radon–Nikodym derivative, at time T , of the measure Q with respect to the measure P. In what follows, we use the fact that this change of measure to an equivalent one preserves the correlation structure between standard Brownian motions (see, e.g. Brigo and Mercurio [11, p. 33]) to generalize the HJM risk-neutral drift restriction. Theorem 3.1 If the stochastic differential equation of the forward rate dynamics is given by d f (t, T ) = (t, T ) dt +V(t, T ), dZ̃(t) (6) where (t, T ) and V(t, T ) are defined similar to those in Equation (1), and {Z̃(t)}(t 0) is defined T by Equations (4) and (5), then d f (t, T ) = V(t, T ), K(t) t V(t, s) ds dt +V(t, T ), dW̃(t), where {W̃(t)}(t 0) is a n-dimensional vector of correlated Q-standard Brownian motions with correlation matrix K(t). Proof See Appendix. Theorem 3.1 leads to an adjusted risk-neutral drift restriction given by T (t, T ) = V(t, T ), K(t) V(t, s) ds (7) t Clearly Equation (7) shows that in addition to the components of a regular HJM risk-neutral drift, there are components coming from each pair of correlated factors. The following section illustrates these additional terms in the case of the term structure of credit-risky bonds. 4. AN APPLICATION TO THE TERM STRUCTURE OF DEFAULTABLE BONDS Using the intensity-based reduced-form approach, we derive in this section an extra risk-neutral drift term of the forward rate dynamics of the term structure of credit-risky bonds driven by two correlated factors—the risk-free forward rate and the forward default intensity. Theorem 4.1 The instantaneous credit-risky forward rate, f˜(t, T ), can be modeled as the sum of the corresponding instantaneous default-free forward rate and forward credit spread. i.e. for any finite time t such that 0tT , almost everywhere (w.r.t. Q) we have f˜(t, T ) = f (t, T )+h(t, T ), where h(t, T ) is the instantaneous forward credit spread for the maturity date T , as seen at time t. Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2008; 24:359–368 DOI: 10.1002/asmb 363 HEATH–JARROW–MORTON INTEREST RATE MODEL Proof See, e.g. Schonbucher [12, pp. 206–211]. Theorem 4.1 specifies the stochastic dynamics of the credit-risky forward rate in such a way that the correlation between default-free forward rate and the forward credit spread can be explicitly shown in a closed-form solution. We can define the dynamics of the default-free forward rate and the forward credit spread as d f (t, T ) = f (t, T ) dt + f (t, T ) dw̃ f (t) (8) where f (t, T ) is a time-dependent drift, w̃ f (t) is a Q-standard Brownian motions, and f (t, T ) is the volatility of the forward rate; and dh(t, T ) = h (t, T ) dt +h (t, T ) dw̃h (t) (9) where h (t, T ) is a time-dependent drift, w̃h (t) is a Q-standard Brownian motion, and h (t, T ) is the volatility of the forward credit spread. Using Theorem 4.1, Equations (8) and (9), the credit-risky forward rate dynamics is given by d f˜(t, T ) = (t, T ) dt + f (t, T ) dw̃ f (t)+h (t, T ) dw̃h (t) (10) where (t, T ) = f (t, T )+h (t, T ). According to the standard HJM risk-neutral drift restriction, if w̃ f (t) and w̃h (t) are independent, the absence of arbitrage will impose the drift term of Equation (10) to be given by (t, T ) = f (t, T ) T f (t, s) ds +h (t, T ) t T h (t, s) ds (Q—a.e.) (11) t It should be noticed that the standard HJM risk-neutral drift restriction does not take into account the correlations between factors. As the change of measure to an equivalent one preserves the correlation structure between standard Brownian motions, if there is a nonzero correlation, between the forward interest rate process and the forward credit spread process under the real-world probability measure, the two processes will have the same correlation under the equivalent martingale measure. Further, if k f,h (t) is the correlation coefficient at time t between the forward risk-free rate and forward credit spread, by Equation (7) the no-arbitrage condition implies that under the equivalent martingale measure (t, T ) = f (t, T ) T T f (t, s) ds +h (t, T ) t h (t, s) ds t + k f,h (t) f (t, T ) t T h (t, s) ds +h (t, T ) T f (t, s) ds (Q—a.e.) (12) t As a result, an extra term is added to the risk-neutral drift, and that term becomes bigger as the correlation coefficient is closer to 1 in absolute value. As both the volatilities of the risk-free rate and the credit spread are positive numbers, a positive (negative) correlation increases (decreases) the risk-neutral drift term. Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2008; 24:359–368 DOI: 10.1002/asmb 364 L. TCHUINDJO 5. NUMERICAL APPLICATIONS In this subsection, we use Monte Carlo simulation to forecast the evolution (from January 1, 2007 to January 31, 2007) of the 2-month tenor forward U.S. Dollar Libor rate maturing on January 31, 2008 (i.e. effective from January 31 to March 31, 2008). Only U.S. business days are considered in this application, and 1-month and 1-year periods have 21 and 252 business days, respectively. 5.1. Description of data The U.S. Dollar Libor rate is higher than risk-free rate, as it takes into account the credit risk of the banking system. In this analysis, we first consider the yield of the U.S. Treasury Bill to be a good proxy for the risk-free rate. Then we use Theorem 4.1 to represent the 2-month tenor forward Libor as the sum of the 2-month tenor forward risk-free rate and the 2-month tenor forward credit spread. All data points are obtained from BloombergTM where the tickers of the U.S. Dollar Libor rate and the U.S. Treasury Bill rate are US00 and USGG, respectively: • We derived the 2-month tenor forward Libor from the daily values of the 1-month and 3-month U.S. Dollar Libor rates for the last half of 2007 (i.e. from July 2, 2007 through December 28, 2007). • We derived the 2-month tenor forward risk-free rate from the daily values of the 1-month and 3-month U.S. Treasury bill rates for same period. • Then the 2-month tenor forward credit spread is the difference in the two preceding rates. 5.2. Computational approaches The correlated HJM approach: Using the above data we compute the annualized historical volatilities of the series of both the 2-month tenor forward risk-free rate and the 2-month tenor forward credit spread. Their values are 11.4041 and 9.5781%, respectively. The historical correlation coefficient of these two series is negative 95.7741% for this period. Then we use the volatilities, the correlation, and the initial value (4.7775% as of December 28, 2007) of the 2-month tenor forward U.S. Dollar Libor rate maturing on January 31, 2008 to simulate its daily path till maturity, as explained in Equations (10) and (12). The standard HJM approach: In conjunction with the above two series, we use the principal component analysis to create two orthogonal series. These two new series have the annualized volatilities of 14.794 and 2.1316%, respectively. Then we use these volatilities, Equations (10) and (11), and the initial value of the 2-month tenor forward U.S. Dollar Libor rate maturing on January 31, 2008 to simulate its daily path till its maturity. 5.3. Results and discussion Table I presents the results, in percentage, of the forecast of the evolution of the 2-month tenor forward U.S. Dollar Libor rate. The first column shows the business day for which the forecasts are done. The second and third columns are the forecast results by the correlated and the standard HJM approaches, respectively. The last column is the difference in the two forecast results. The forecast results by the two different approaches are very close. The average and the maximum difference over the forecasting period are 0.36 and 2.43 basis points, respectively (a basis point is 10−4 ). The mean square difference of the two forecasts is 2×10−8 . However, the annualized Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2008; 24:359–368 DOI: 10.1002/asmb 365 HEATH–JARROW–MORTON INTEREST RATE MODEL Table I. Forecast results in percentage. U.S. Business day January January January January January January January January January January January January January January January January January January January January January 02, 03, 04, 07, 08, 09, 10, 11, 14, 15, 16, 17, 18, 22, 23, 24, 25, 28, 29, 30, 31, Correlated HJM Standard HJM 4.7724 4.7712 4.7721 4.7723 4.7670 4.7774 4.7741 4.7724 4.7799 4.7831 4.7812 4.7889 4.7901 4.7930 4.8022 4.8029 4.8048 4.8031 4.7936 4.7925 4.7931 4.7945 4.7688 4.7733 4.7931 4.7452 4.7904 4.7721 4.7687 4.7688 4.7591 4.7772 4.8132 4.8130 4.7961 4.8232 4.8258 4.8203 4.8083 4.7867 4.7703 4.7951 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 2008 Difference −0.0221 0.0024 −0.0012 −0.0209 0.0217 −0.0131 0.0020 0.0038 0.0111 0.0240 0.0040 −0.0243 −0.0229 −0.0031 −0.0210 −0.0229 −0.0155 −0.0053 0.0069 0.0223 −0.0020 volatility of the forecasted result by the correlated HJM is less than the one of the forecast resulted by the standard HJM: 19.51 versus 35.92%. In regard to this application, this feature allows the correlated HJM approach to be more appropriate for the pricing purpose. APPENDIX A: PROOF OF THEOREM 3.1 Equation (6) can be rewritten in stochastic integral form as f (t, T ) = f (0, T )+ t (s, T ) ds + 0 t V(s, T ), dZ̃(s) (A1) 0 Let B(t, T ) be the price at time t of a zero-coupon bond that matures at time T . We have B(t, T ) = exp − T f (t, s) ds (A2) t By Leibnitz’s rule of differentiation, the previous equation becomes d ln B(t, T ) = f (t, t)− T t Copyright q 2008 John Wiley & Sons, Ltd. T * d f (t, s) ds f (t, s) ds dt = f (t, t) dt − *t t (A3) Appl. Stochastic Models Bus. Ind. 2008; 24:359–368 DOI: 10.1002/asmb 366 L. TCHUINDJO Substituting f (t, t) from Equation (A1) and d f (t, s) from Equation (6) into Equation (A3), we obtain t t d ln B(t, T ) = f (0, t) dt + (s, t) ds dt + V(s, t), dZ̃(s) dt 0 − T 0 (t, s) dt ds − t T t = f (0, t)+ (s, t) ds + 0 T − V(t, s), dZ̃(t) ds t t V(s, t), dZ̃(s)− 0 T (t, s) ds dt t V(t, s) ds, dZ̃(t) (A4) t Applying Ito’s lemma to the preceding equation, the stochastic differential equation of the zero-coupon bond price can be expressed as t t T dB(t, T )/B(t, T ) = f (0, t)+ (s, t) ds + V(s, t), dZ̃(s)− (t, s) ds dt 0 t T 2 1 V(t, s) ds, dZ̃(t) + V(t, s) ds, dZ̃(t) 2 t T − 0 t (A5) T As t V(t, s) ds, dZ̃(t) is a scalar and ·, · is a symmetric form on Hn , we have 2 T T = V(t, s) ds, dZ̃(t) t T V (t, s) ds dZ̃(t) dZ̃ (t) T t = V(t, s) ds t T T T V (t, s) ds [dZ̃(t)dZ̃ (t)] T T t V(t, s) ds t Using Equation (5), the preceding equation becomes 2 T = V(t, s) ds, dZ̃(t) t T V(t, s) ds, K(t) t T V(t, s) ds dt (A6) t Hence, Equation (A5) can be rewritten as t t dB(t, T )/B(t, T ) = f (0, t)+ (s, t) ds + V(t, s), dZ̃(s) 0 T − t − 1 (t, s) ds + 2 0 T V(t, s) ds, K(t) t T V(t, s) ds, dZ̃(t) T V(t, s) ds dt t (A7) t Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2008; 24:359–368 DOI: 10.1002/asmb 367 HEATH–JARROW–MORTON INTEREST RATE MODEL We can define a nonlinear operator, D, on S such that ∀V(·, ·) ∈ S, T t t (s, t) ds − (t, s) ds + V(s, t), dZ̃(s) D[V(t, T )] = f (0, t)+ + 1 2 0 t T V(t, s) ds, K(t) t T 0 V(t, s) ds (A8) t By the Hilbert space representation theorem, we can find an n-dimensional bounded (Ft )(t 0) predictable process, {U(t)}(t 0) , such that T V(t, s) ds, U(t) (A9) D[V(t, T )] = t Equations (A8) and (A9) imply that T t V(t, s) ds, U(t) = f (0, t)+ (s, t) ds − t 1 + 2 0 T (t, s) ds + t T V(t, s) ds, K(t) t T t V(s, t), dZ̃(s) 0 V(t, s) ds (A10) t For every time t0, the vector U(t) = [1 (t), 2 (t), . . . , n (t)]T will provide a proportional relationship between the drift rate of change of prices and the amount of risk in price changes stemming from each Brownian motion. U(t) can be viewed as the n-dimensional market price of risk at time t. Provided that U(t) satisfies Novikov’s condition, the process {W̃(t)}(t 0) defined by dW̃(t) = dZ̃(t)−U(t) dt (A11) is an n-dimensional vector of correlated Q-standard Brownian motions, having K(t) as their correlation matrix. By differentiating both sides of Equation (A10), with respect to T , we obtain T (t, T ) = V(t, T ), K(t) V(t, s) ds −V(t, T ), U(t) (A12) t Substituting Equation (A12) into Equation (6), the forward rate dynamics under the new equivalent probability measure Q is given by the following equation: T d f (t, T ) = V(t, T ), K(t) V(t, s) ds +V(t, T ), U(t) dt −V(t, T ), dZ̃(t) t As ·, · is a bilinear form on Hn , the preceding equation becomes T d f (t, T ) = V(t, T ), K(t) V(t, s) ds dt +V(t, T ), [dZ̃(t)−U(t) dt] t Using Equation (A11), the preceding equation becomes T d f (t, T ) = V(t, T ), K(t) V(t, s) ds dt +V(t, T ), dW̃(t) t Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2008; 24:359–368 DOI: 10.1002/asmb 368 L. TCHUINDJO ACKNOWLEDGEMENTS The author would like to thank the editor and an anonymous referee for their suggestions which helped in improving the original version of this paper. REFERENCES 1. Heath D, Jarrow R, Morton A. Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica 1992; 60:77–105. 2. Longstaff FA, Schwartz ES. A simple approach to valuing risky fixed and floating rate debt. Journal of Finance 1995; 50:789–819. 3. Duffee GR. The relation between treasury yields and corporate yield spreads. Journal of Finance 1998; 53: 2225–2241. 4. Collin-Dufresne P, Goldstein RS, Martin JS. The determinants of credit spread changes. Journal of Finance 2001; 56:2177–2207. 5. Papageorgiou N, Skinner FS. Credit spreads and the zero-coupon treasury spot curve. Journal of Financial Research 2006; 29:421–439. 6. Jacod J, Shiryaev AN. Limit Theorems for Stochastic Processes. Springer: New York, 1988. 7. Harrison JM, Kreps DM. Martingales and arbitrage in miltiperiod securities markets. Journal of Economic Theory 1979; 20:381–408. 8. Harrison JM, Pliska SR. Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 1981; 11:215–260. 9. Ritchken P, Sankarasubramanian L. Volatility structure of forward rates and the dynamics of the term structure. Mathematical Finance 1995; 5:55–72. 10. Chiarella C, Kwon OK. Classes of interest rate models under the HJM framework. Asia—Pacific Financial Markets 2001; 8:1–22. 11. Brigo D, Mercurio F. Interest Rate Models—Theory and Practice (2nd edn). Springer: Berlin, 2006. 12. Schonbucher PJ. Credit Derivatives Pricing Models. Wiley: New York, 2004. Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2008; 24:359–368 DOI: 10.1002/asmb

1/--страниц