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A game swaption, newly proposed in this paper, is a game version of usual interest-rate swaptions. It provides the both parties, fixed-rate payer and variable rate payer, with the right that they can choose an exercise time to enter a swap from a set of prespecified multiple exercise opportunities. We evaluate two types of game swaptions: game spot-start swaption and game forward-start swaption, under the generalized Ho-Lee model. The generalized Ho-Lee model is an arbitrage-free binomial-lattice interest-rate model. Using the generalized Ho-Lee model as a term structure model of interest rates, we propose an evaluation method of the arbitrage-free price for the game swaptions via a stochastic game formulation, and illustrate its effectiveness by some numerical results.

A game swaption, newly proposed in this paper, is a kind of exotic interest-rate derivatives whose payoff depends on interest rates or bond prices. After the early 1980’s, many financial institutions have served such exotic derivatives to respond various needs of clients. In general, it is commonly known that there is not any analytical solution to the pricing problems of many exotic derivatives due to their structural complexity, and the valuation of these derivatives has to rely on a method of a numerical computation. In order to evaluate such derivatives, tree methods and finite difference methods are widely used in many financial institutions.

In this paper, we propose an evaluation method of exotic interest-rate derivatives via a tree method based on the generalized Ho-Lee model [

In this paper, we consider valuation problems of swaptions with game features. Several papers have addressed such valuation problems of interest-rate derivatives with some game structures. Among them, Ben et al. [

A game swaption, newly proposed in this paper, is a game version of usual interest- rate swaptions. A usual swaption provides only one side of the two parties (fixed- rate payer and variable rate payer) with the right to enter a swap at a predetermined future time. In contrast, a game swaption provides the both parties with the right of choosing an exercise time to enter a swap from a set of prespecified multiple exercise opportunities. We evaluate two types of game swaptions: game spot-start swaption and game forward-start swaption. A game spot-start swaption allows us to enter the swap at the next setting time just after the exercise time, while a game forward-start swaption entitles us to enter the swap at a predetermined fixed calendar time regardless of the exercise time. In order to formulate the valuation problem of these two game swaptions, we apply a stochastic game formulation. The theory of stochastic games was originated by the seminal paper of Shapley [

This paper is organized as follows. We introduce and explain the generalized Ho-Lee model in the next Section 2. In Section 3, we first illustrate the game spot-start swaption and derive the optimality equation to evaluate the no-arbitrage values of the game spot-start swaption. Then, Section 4 provides the valuation for the game forward-start swaption as in the previous Section 3. Some numerical examples for these two game swaptions are shown in Section 5. Finally, we conclude the main contributions in this paper.

The generalized Ho-Lee model is an arbitrage-free binomial-lattice interest-rate model, where time is discrete. Let

In order to represent the degree of uncertainty of interest rates on the binomial lattice, we introduce the binomial volatilities

As the binomial volatilities become bigger, the uncertainty of interest rates also increases more. Let

where

Then, the one-period binomial volatility

where

The generalized Ho-Lee model is an arbitrage-free term-structure model of interest rates. Therefore, the bond prices for all different maturities at each node

under the risk-neutral probability

By using straightforwardly Equations (1) and (4), we can confirm Equation (5) to be an arbitrage-free condition. Thereby, as long as the T-period binomial volatility is defined by Equation (5), the generalized Ho-Lee model is no-arbitrage. Then, the one- period bond prices at node

Similarly, the T-period bond prices at node

Next, we explain the algorithm to derive the one-period bond prices based on the above arguments. The constructions of the generalized Ho-Lee model are decomposed into the following five steps:

Step 1. Derive one-period bond price at node

Step 2. Derive one-period bond price at node

Step 3. Derive one-period yields by one-period bond price:

Step 4. Derive one-period binomial volatilities:

Step 5. Derive T-period binomial volatilities:

Game swaptions can be classified into two types with respect to the timing of entering into the underlying swap. A game spot-start swaption allows us to enter the swap at the next setting time just after an exercise, while a game forward-start swaption allows us to enter the swap at a predetermined calender time regardless of the exercise time. First, in this section, we consider the game spot-start swaption.

A (plain vanilla) swap is an agreement to exchange a fixed rate and a variable rate (or floating rate) for a common notional principal over a prespecified period (e.g., Hull [

In this paper, we consider a game swaption which is an extension of Bermudan swaption. The game swaption entitles both of the fixed-rate side and variable-rate side to enter into the swap at multiple prespecified times. The sequence of setting/payment times is

where N is an agreement time of the swap,

For the following discussions, we further let

Let

Next, we define the exercise rate for a game swaption. If the fixed-rate side exercises at an exercisable time, he will pay the fixed rate

Moreover, the sets

respectively. We let

Now, we apply the theory of two-person and zero-sum stopping game to the valuation of the game swaption. The players of the game are the fixed-rate-payer side and the variable-rate-payer side. We shortly call them fixed-rate player and variable-rate player, respectively. At a jointly exercisable node (n, i) (

Definition 3.1. When the game spot-start swaption is exercised at an exercisable node

If the both players do not exercise at an admissible time, the stochastic game moves to the following node

at the next time, where

Then, the both players face a two-person and zero-sum stage game whose payoff is dependent on a state of interest rates and their strategies at every exercisable nodes

Given a two-person and zero-sum game specified by a payoff matrix

where

Let

Step 0. (Terminal condition) for

Step 1. (Recursion) from

Case 1-1. (When both players can exercise) for

Case 1-2. (When only fixed-rate player can exercise) for

Case 1-3. (When only variable-rate player can exercise) for

Case 1-4. (When neither player can exercise) for

where

In the terminal condition,

At a node

where the fixed-rate side chooses the row as a maximizer and the variable-rate side chooses the column as a minimizer. In general, a saddle point equilibrium of two-person and zero-sum game is known to exist in mixed strategies including pure strategies. However, the following theorem shows that the above game has a saddle point in pure strategies.

Theorem 1. Suppose

where x and y are pure strategies of the fixed-rate player and the variable-rate player, respectively. Furthermore, if we denote E and N the pure strategies “Exercise” and “Not Exercise”, respectively, then the equilibrium-strategy profile

Proof. We suppose the former case

1) If

2) If

3) If

A game forward-start swaption entitles the both parties to enter into the swap at a predetermined calender time regardless of the exercise time. It is more practical than the game spot-start swaptions. The solution method is similar to the game spot-start swaptions, thus we discuss only the different points.

In previous section, we suppose that the sequence of setting/payment times of the game spot-start swaption starts

Then, a forward-start swap rate at an agreement time N is given by

Note that

As in the previous section, we apply a stochastic game formulation to the valuation of the game forward-start swaption. At a jointly exercisable node

Definition 4.1. When the game forward-start swaption is exercised at an exercisable node

Let

Step 0. (Terminal condition) for

Step 1. (Recursion) from

Case 1-1. (When both players can exercise) for

Case 1-2. (When only fixed-rate player can exercise) for

Case 1-3. (When only variable-rate player can exercise) for

Case 1-4. (When neither player can exercise) for

where we define a continuation value at node

The following theorem shows that the above game for the forward-start swaption has a saddle point in pure strategies.

Theorem 2. Suppose

Furthermore, the equilibrium strategy profiles

Proof. Since the proof is almost the same as Theorem 1, it is omitted.

In this section, we show some numerical examples for a game spot-start swaption and a game forward-start swaption. Firstly, we consider a game spot-start swaption with the swaption maturity of 5 years, the protection period of 1 year, and the swap period of 5 years. The both players can choose to exercise at any time in prescribed exercisable time-intervals after the protection period. If either or both players exercise the option, they enter the 5-years swap at the next setting/payment time. We set

The upper surrounded area stands for the exercise area of the fixed-rate player, while the lower surrounded area stands for the one of the variable-rate player. According to the numerical results, we can confirm that the spot-start swap rate

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