Puttable Bond


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Puttable Bonds Valuation


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A puttable bond is a bond in which the investor has the right to sell the bond back to the issuer at specified times for a specified price.


1. Puttable Bond Introduction

At each puttable date prior to the bond maturity, the investor may get the investment money back by selling the bond back to the issuer. The underlying bonds can be fixed rate bonds or floating rate bonds. A puttable bond can therefore be considered a vanilla underlying bond with an embedded Bermudan style option. Puttable bonds protect investors. Therefore, a puttable bond normally pays investors a lower coupon than a non-puttable bond.

Although a puttable bond is a higher cost to the investor and an uncertainty to the issuer comparing to a regular bond, it is actually quite attractive to both issuers and investors. For investors, puttable bonds allow them to reduce interest costs at a future date should rate increase. For issuers, puttable bonds allow them to pay a lower interest rate of return until the bonds are sold back. If interest rates have increased since the issuer first issues the bond, the investor is like to call its current bond and reinvest it at a higher coupon. This presentation gives an overview of puttable bond and valuation model.


2. Puttable Bond Payoffs

Assume that the maturity of a fixed rate puttable bond is T. there are n put dates denoted as .

The payoff of a puttable bond at maturity can be expressed as

Puttable bond payoff in FinPricing

The payoff of the puttable bond at any call date T_i can be represented as

Puttable bond valuation in FinPricing

3. Valuation Model Selection Criteria

Given the complexity of puttable bond valuation, there is no closed form solution. Therefore, we need to select an interest rate term structure model and a numeric solution to price Bermudan swaptions.

Popular IR term structure models in the market are Hull-White, Linear Gaussian Model (LGM), Quadratic Gaussian Model (QGM), Heath Jarrow Morton (HJM), Libor Market Model (LMM). HJM and LMM are too complex while Hull-White is inaccurate for computing sensitivities. Therefore, we choose either LGM or QGM.

After selecting a term structure model, we need to choose a numeric approach to approximate the underlying stochastic process of the model.Commonly used numeric approaches are tree, partial differential equation (PDE), lattice, and Monte Carlo simulation. Tree and Monte Carlo are notorious for inaccuracy in sensitivity calculation. Therefore, we choose either PDE or lattice. Our final decision is to use LGM plus lattice.


4. LGM Model
Linear Gaussian Model (LGM) in FinPricing

5. LGM Assumptions and Calibration

The LGM model is mathematically equivalent to the Hull-White model but offers significant improvements in calibration stability and accuracy. It is also more accurate and stable in sensitivity calculation. The state variable is normally distributed under the appropriate measure. The LGM model has only one stochastic driver (one-factor), thus changes in rates are perfected correlated.

At time t, X(0)=0 and H(0)=0. Thus Z(0,0;T)=D(T). In other words, the LGM automatically fits today’s discount curve or yield curve. To calibrate swaption implied volatilities, first select a group of market swaptions and then solve parameters by minimizing the relative error between the market swaption prices and the LGM model swaption prices.


6. Valuation Practical Guide
  • Calibrate the LGM model first.
  • Create the lattice based on the LGM: the grid range should cover at least 3 standard deviations.
  • Find the underlying bond value at each final note.
  • Conduct backward induction process iteratively rolling back from final dates until reaching the valuation date.
  • Compare exercise values with intrinsic values at each exercise date.
  • The value at the valuation date is the price of the puttable bond.

7. Related Topics