یک روش برنامه ریزی پویا برای گزینه های قیمت گذاری تعبیه شده در اوراق قرضه

We propose a dynamic programming (DP) approach for pricing options embedded in bonds, the focus being on call and put options with advance notice. An efficient procedure is developed for the cases where the interest-rate process follows the Vasicek, Cox–Ingersoll–Ross (CIR), or generalized Vasicek models. Our DP methodology uses the exact joint distribution of the interest rate and integrated interest rate at a future date, conditional on the current value of the interest rate. We provide numerical illustrations, for the Vasicek and CIR models, comparing our DP method with finite-difference methods. Our procedure compares quite favorably in terms of both efficiency and accuracy. An important advantage of the our DP approach is that it can be applied to more general models calibrated to capture the term structure of interest rates (e.g., the generalized Vasicek model).

A bond is a contract that pays its holder a known amount, the principal, at a known future date, called the maturity. The bond may also pay periodically to its holder fixed cash dividends, called the coupons. When it gives no dividend, it is known as a zero-coupon bond. A bond can be interpreted as a loan with a known principal and interest payments equal to the coupons (if any). The borrower is the issuer of the bond and the lender, i.e., the holder of the bond, is the investor. Several bonds contain one or several options coming in various flavors. The call option gives the issuer the right to purchase back its debt for a known amount, the call price, during a specified period within the bond's life. Several government bonds contain a call feature (see Bliss and Ronn, 1995 for the history of callable US Treasury bonds from 1917). The put option gives the investor the right to return the bond to the issuer for a known amount, the put price, during a specified period within the bond's life. These options are an integral part of a bond, and cannot be traded alone, as is the case for call and put options on stocks (for example). They are said to be embedded in the bond. In general, they are of the American-type and, thus, allow for early exercising, so that the bond with its embedded options can be interpreted as an American-style interest-rate derivative. This paper focuses on call and put options embedded in bonds with advance notice, that is, options with exercise decisions prior to exercise benefits. As it is often the case in practice, we assume that exercise of the call and put options is limited to the coupon dates posterior to a known protection period and that there is a notice period of fixed duration ΔtΔt. Thus, consider a coupon date tmtm where the exercise is possible, and let CmCm and PmPm be the call and put prices at tmtm. The decision to exercise or not by the issuer and the investor must be taken at tm-Δttm-Δt. If the issuer calls back the bond at tm-Δttm-Δt, he pays CmCm to the investor at tmtm, and, similarly, if the investor puts the bond at tm-Δttm-Δt, he receives PmPm from the issuer at tmtm. If no option is exercised at tmtm, by the no-arbitrage principle of asset-pricing ( Elliott and Kopp, 1999), the value of the bond is equal to the expected value of the bond at the next decision date, discounted at the interest rate. This expectation is taken under the so-called risk-neutral probability measure, where the uncertainty lies in the future trajectory of the (risk-free short-term) interest rate. There are no analytical formulas for valuing American derivatives, even under very simplified assumptions. Numerical methods, essentially trees and finite differences, are usually used for pricing. Recall that trees are numerical representations of discrete-time and finite-space models and finite differences are numerical solution methods for partial differential equations. The pricing of American financial derivatives can also be formulated as a Markov decision process, that is, a stochastic dynamic programming (DP) problem, as pointed out by Barraquand and Martineau (1995). Here the DP value function, that is, the value of the bond with its embedded options, is a function of the current time and of the current interest rate, namely the state variable. This value function verifies a DP recurrence via the no-arbitrage principle of asset pricing, the solution of which yields both the bond value and the optimal exercise strategies of its embedded options at all time during the bond's life. For an extensive coverage of stochastic DP, see Bertsekas (1995). The pricing of options embedded in bonds can be traced back to Brennan and Schwartz (1977). They used a standard finite-difference approach, which is however known to be instable in this context. Much later, Hull and White (1990a) proposed trinomial trees in the context of generalized versions with time-dependent parameters for the state process. Trees crudely approximate the dynamics of the underlying asset and convergence is not easy to attain. Büttler and Waldvogel (1996) (indicated here by BW) suggested that finite differences could not be used to price callable bonds with advance notice because of discontinuities at notice dates. They proposed an approach based on Green's functions, which turns out to be a DP procedure combined with finite elements, where the transition parameters are obtained by numerical integration. Recently, d’Halluin et al. (2001) (which we indicate by DFVL) were able to stabilize the finite-difference approach via flux limiters and timestepping. Using a Crank–Nicholson time weighing, they compared their results with those of BW, concluding that their method showed better accuracy than other methods. Here, we propose a DP approach where the transition parameters are derived in closed-form. As BW and DFVL, we use the Vasicek and CIR specification of the interest rate dynamics for our numerical experiments. Results are compared to those of BW and DFVL and show the efficiency and robustness of our DP method, with an accuracy comparable to DFVL and superior to BW. An interesting advantage of our approach is that it can be applied to generalized models calibrated to capture the term structure of interest rates, such as the ones proposed by Hull and White (1990a). The rest of the paper is organized as follows. Section 2 presents a short review of the interest rate dynamics that have been proposed in the literature. Section 3 presents the model, the general DP formulation and the approximation procedure. Section 4 derives the exact joint distribution of the interest rate and integrated interest rate at a future date, conditional on the current value of the interest rate, for the Vasicek, CIR, and generalized Vasicek models. These joint distributions are used to obtain closed-form formulas for some constants used in our DP procedure. This turns out to be a key ingredient for its efficiency. Section 5 reports our numerical experimentations for the Vasicek and CIR models. Section 6 contains a conclusion. Appendix A contains some auxiliary results and proofs.

In this paper, we model the valuation of options embedded in bonds as a stochastic dynamic programming problem. We propose a simple and efficient approximation for the case of three affine models of the term structure. Our method is based on a piecewise linear approximation of the value function and is quite easy to implement. It differs from the usual lattice-based and finite-difference methods in that the exact conditional probability distribution of the interest rate is used in order to compute the transition probabilities. We provide numerical results for the Vasicek and CIR models. These results demonstrate convergence and an efficiency that compares favorably with that of the other available methods. In addition, our methodology readily extends to handle the generalized Vasicek model with time-dependent long-run mean, provided that a function can be fitted to the observed zero-coupon bond prices. Extensive empirical studies of the term structure generated by callable and puttable bonds have been lacking in the literature, mainly due to the lack of a robust and efficient numerical procedure for pricing these bonds. Our approach can provide this type of numerical procedure for many situations, where it can be used to investigate empirically the cross-sectional and time series properties of bonds with embedded options. Our solution technique is flexible and can be extended to other models involving different continuous-time as well as discrete-time processes, whenever the transition parameters can be obtained efficiently. In particular, it can be adapted to models with multifactor affine term structure (Dai and Singleton, 2000) and more generally to the valuation of other Bermudan or American-style derivatives in the setting of affine diffusion state processes (Duffie et al., 2000 and Duffie et al., 2003).