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We propose a valuation for the bond in which an issuer and a holder are simultaneously granted the right to exercise a call and put options. As the term structure model of interest rate, we use the Generalized Ho-Lee model that is an arbitrage-free binomial lattice interest rate model. The issuer and the holder play a series of stage games in each exercisable node on the lattice whose payoff structure is dependent on the nodes. We formulate the valuation problem as a stochastic game or a Markov game. Our stochastic games possess saddle points in pure strategies for each stage game. We derive the optimality equation to solve backwardly the bond values and the exercise strategies from the maturity to the initial time. Our numerical results are useful to intuitively understand the risk to a change of interest rates for options embedded in bond.

Callable and putable bonds are one of the major interest rate derivatives due to its ability to respond the needs of various markets participants. Callable bonds give an issuer the right which can call a bond before the maturity date. The issuer is allowed to exercise the right at the several prescribed exercise times. If the issuer exercises the option, she buys back the bond at a call price from a holder. Then, the issuer is a buyer of the call option and a holder has an obligation to respond to the demand of the issuer. Thereby, a holder of a callable bond generally receives a higher yield than a usual coupon bond in exchange for taking prepayment risk. On the other hand, putable bonds give a holder the right which can sell back the bond before the maturity date. If a holder exercises the option, the issuer must buy back the bond for a put price. Then, the holder is a buyer of the put option, while the issuer is a seller. Thus, the putable bond has a lower yield than a straight bond because of the put option. In this paper, we consider the bond that the issuer and the holder are simultaneously granted the right to call and put a bond, respectively. We call this bond a game option bond.

To represent an interest rate dynamics in markets, we make use of the Generalized Ho-Lee model [

The theory of stochastic games was originated by the seminal paper of Shapley [

Furthermore, we report key rate duration which measures the price sensitivity of the bond to the key rate shift in a yield curve [

The organization of this paper is as follows. In section 2, we illustrate the Generalized Ho-Lee model. In section 3, we formulate the game option bond as the stochastic game and derive the optimality equation to evaluate the non-arbitrage values of the game option bonds. In section 4, we show numerical examples for the initial values of bonds and the exercise strategies for the players, and report the key rate durations. Finally, in section 5, we conclude.

The Generalized Ho-Lee model is an arbitrage-free term structure model of interest rates [

The Ho-Lee model, which is the first arbitrage-free term structure model of interest rates, assumes that the binomial volatilities are independent of the time and state [

where

where the parameters

for

Similarly, the T-period zero-coupon bond prices are as follows:

We use an equal risk?neutral probability of 0.5 for the up-state and down-state. To be an arbitrage-free term structure model of interest rates, the bond prices have to satisfy the following arbitrage-free conditions:

for each

Solving recursively Equations (1)-(5), we derive the arbitrage?free term structure model of interest rates on every node.

Callable bonds give an issuer the right to purchase back the bond for a pre-specified price (a call price) before the maturity time. If the issuer exercises the right, she pays the call price to a holder and calls back the bond. In contrast, putable bonds give a holder the right to sell back the bond at a pre-specified price (a put price) before the maturity time. If the holder exercises the right, the issuer must buy back the bond for the put price from the holder. In this paper, we consider the bond that both the issuer and the holder have the right to exercise the call and the put options, respectively. We call its bond a game option bond.

Let

not to exercise the options at the coupon times

exercise at an exercisable time

Let

We now apply a stochastic game or a Markov game approach to the valuation of the game option bond. Let the issuer and the holder be the players of the game. Each player has two pure strategies, Exercise or Not Exercise. The holder chooses a strategy x and the issuer chooses a strategy y from the strategy set

at the next coupon time

where

Given a two-person zero sum game (a matrix game) defined by a payoff matrix

where

In Equation (7), the second equality is due to the von Neumann minimax theorem.

Let

Optimality Equation:

where F denotes the principal of the bond,

in order to find the exercise strategies for the players, where the holder is the row player because of a maximizer, and the issuer is the column player because of a minimizer. In the Matrix (11), for example, if the holder chooses to exercise and the issuer chooses not to exercise, then the value of the payoff matrix is

Next, we show the equilibrium strategies in the stage-game at each exercisable node. In general, it is well known that a two-person zero sum game has a saddle point in mixed strategies including pure strategies. However, the next result shows that a saddle point of our stochastic game exists in pure strategies.

Theorem 1. Suppose

Furthermore, the equilibrium strategy profiles are

where E denotes the pure strategy “Exercise” and N “Not Exercise”.

Proof. We suppose

holder is a pure strategy E, and hence

the holder chooses a pure strategy N which is the dominant strategy for her(a maximizer). Then, the best response strategy for the issuer is choosing a pure strategy E, and hence

Note that if

This section proposes the value of the game option bond using the Generalized Ho-Lee model. We consider the bond with a 10-year maturity, the protection period of 5 years,

We first show the performance surfaces of the bond to a parallel shift in the yield curve and a change in the volatility. The yield curve is shifted in parallel from 4.5% to 5.5% every a 10 bp, and the volatility from 0% to 20% every a 2% increment.

We next consider the effect on the exercise regions of the issuer and the holder to the parallel shift in the parameters of interest rate. Figures 2-4 present the exercise regions of the game option bond in yield curve 4%, 5%, and 6%, respectively. In Figures 2-4, the leftmost node represents the initial time, the rightmost nodes the maturity time, the upper nodes the higher state of the interest rate, and a number in each node the game option bond value. In addition, the upper gray-areas are the holder’s exercise region, while the lower enclosed areas are the issuer’s one. Figures 2-4 show that the exercise region of the issuer expands with the downward shift of interest rates. Generally, if an interest rate declines after debt issues, an issuer is likely to want to refinance the loan at a lower interest rate, and hence she will exercise the call option early. In contrast, the exercise region of the holder expands with the upward shift of interest rates. This is because that the holder then can obtain higher yields by switching to other bonds, and the holder of put option will finally sell back for the issuer.

rise a value of an option on a contingent claim. As seen from the previous

Key rate duration is a measure of the interest rate risk of a bond. We follow Ho [

Bond type | Key year | Effective duration | ||||||
---|---|---|---|---|---|---|---|---|

0.25 | 1 | 2 | 3 | 5 | 7 | 10 | ||

Coupon bond | 0.011 | 0.050 | 0.100 | 0.235 | 0.431 | 0.703 | 6.130 | 7.660 |

Callable bond | 0.012 | 0.051 | 0.103 | 0.240 | 3.240 | 0.893 | 1.039 | 5.577 |

Putable bond | 0.011 | 0.050 | 0.100 | 0.235 | 0.498 | 0.763 | 5.627 | 7.284 |

Game option bond | 0.012 | 0.051 | 0.103 | 0.240 | 3.317 | 0.938 | 0.549 | 5.209 |

the coupon and the putable bonds are sensitive to the key rate shift in ten years and are alike in the key rate duration profiles. In comparing the effective durations,

To consider the effect on game option bond value of the call option, we finally propose the key rate duration to different call price.

In this paper, we propose a valuation of the game option bond under the Generalized Ho-Lee model. We for-

Call price | Key year | Effective duration | ||||||
---|---|---|---|---|---|---|---|---|

0.25 | 1 | 2 | 3 | 5 | 7 | 10 | ||

1.01 | 0.012 | 0.051 | 0.103 | 0.240 | 3.317 | 0.938 | 0.549 | 5.209 |

1.02 | 0.012 | 0.051 | 0.102 | 0.238 | 3.000 | 0.857 | 1.144 | 5.403 |

1.03 | 0.011 | 0.050 | 0.102 | 0.237 | 2.616 | 0.861 | 1.971 | 5.849 |

1.04 | 0.011 | 0.050 | 0.101 | 0.237 | 2.188 | 0.833 | 2.963 | 6.384 |

mulate its valuation as a stochastic game on the binomial lattice. We show that our stochastic games always possess saddle points in pure strategies at exercisable nodes. Thereby, we can efficiently solve the optimality equation to value the game option bond. Moreover, then we can obtain simultaneously both the non-arbitrage price of the game option bond and the optimal exercise strategies for the players. We provide the numerical results for the values of the bonds with a variety of contingent claims and the exercise regions for the players of the game option bond. In addition, we report the price sensitivity of the bond to a small parallel shift in some key rates by using key rate duration. The results in key rate duration show that the option embedded in a bond, particularly the call option, has the effect of reducing the exposure with respect to the interest rate change. These numerical results are useful to intuitively understand the interest rate risk in the bonds with an embedded option. However, further studies will be needed in order to value the non-arbitrage price for various interest rate derivatives related to both the credit risk and the interest rate risk [

NatsumiOchiai,MasamitsuOhnishi, (2015) Valuation of Game Option Bonds under the Generalized Ho-Lee Model: A Stochastic Game Approach. Journal of Mathematical Finance,05,412-422. doi: 10.4236/jmf.2015.54035