Valuation of Game Swaptions under the Generalized Ho-Lee Model

A game swaption, newly proposed in this paper, is a game version of usual interestrate 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 binomiallattice 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.

discount bond. Moreover, the zero-coupon bond prices for any remaining maturity $T$ at the initial time, $P(O, 0;T)$ , can be observed in the market, and these set determines the discount function at the initial time.

Valuation of game spot swaption
Game swaptions can be classified into two types with respect to the timing of entering into the swap. A game spot swaption allows us to enter the swap at the next coupon time just after an exercise, while a game forward swaption allows us to enter the swap at a predetermined calender time regardless of the exercise time. In this paper, we consider the game spot swaption.
A 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. We usually use LIBOR (London Interbank Offered Rate) as the variable rate. A swaption is an option on a swap. A payer swaption gives the holder the right to enter a particular swap agreement as the flxed rate payer. On the other hand, a receiver swaption gives the holder the right to enter a particular swap agreement as the fixed rate receiver. The holder of an European swaption is allowed to enter the swap only on the expiration time. In contrast, the holder of an American swaption is allowed to enter the swap on any time that falls within a range of two time instants. $A$ Bermudan swaption, which we consider in this paper, allows its holder to enter the swap on multiple prespecified times.
In this paper, we consider the 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 time sequence of coupon payment is where $N$ is an agreement time of the swap, $M_{0},$ $M_{1}$ , . . . , $M_{L}$ are the $L$ coupon payment times, and $N^{*}$ is a finite time horizon. Now, we suppose $N=M_{0}$ for a game spot swaption. The For the following discussions, we let $\kappa=1.$ Let $S(N,i)$ be a spot swap rate at the agreement time $N$ . The spot swap rate, $S(N, i)$ , specifies the flxed rate that makes the value of the interest rate swap equal zero at the agreement time $N$ and is given by Next, we define the exercise rate for a game swaption. If the fixed rate side exercises at an exercisable time, she will pay the fixed rate $K_{F}$ over the future period of swap. If the variable rate side exercises, the fixed rate side has to pay the fixed rate $K_{V}$ over the future period.
Given a two-person and zero-sum game specified by a payoff matrix $A\in \mathbb{R}^{mxn}(m, n\in N)$ , we define the value of the game as follows: where the second equality is due to the von Neumann Minimax Theorem, $p$ is an $m$ -dimensional vector representing a mixed strategy for the row player, and $q$ is an $n$ -dimensional vector representing a mixed strategy for the column player.
At the node $(n, i)$ where both players can exercise, we need to solve the following two-person and zero-sum game: 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 the saddle point in pure strategies. where $x$ and $y$ are $I$} $ure$ strategies of the fixed rate side player and the variable rate side player, respectively. Furthermore, if we denote $E$ and $N$ the pure strategies 'Exercise' and 'Not Exercise,' respectively, then the equilibrium strategy profile $(x, y)$ is as follows:  regions of the fixed rate player, while the lower surrounded area is the exercise regions of the variable rate player. When the swaption prices are positive, it has an advantage over the fixed rate player. In contrast, when the swaption prices are negative, it has an advantage over the variable rate player.
Secondly, the Bermudan-type game spot swaption is indicated in Table 2, It has properties that both players can exercise at any $n\in\{4$ , 8, 12, 16, 20 $\}$ , only a fixed rate player can also exercise at any $n\in\{6$ , 10, 14, 18 $\}$ . Similar to the first example, we suppose $K_{F}=5.3,$ $K_{B}=5.0$ , and $K_{V}=4.7.$ $0\alpha v\iota$ $0\alpha)0l4\alpha 1\alpha$ $10 11 12 \iota s 1 15 10 17 1S 19$ Table 1 The American-type game spot swaption prices and the exercise area 5 Concluding remarks This paper proposed game swaptions as game versions of a Bermudan swaption. Furthermore, we evaluated the no-arbitrage price of a game spot swaption under the generalized Ho-Lee model as the term structure model of interest rate. This game swaption is also considered to be a generalization of the payer swaption and receiver swaption.
Since, in this paper, we treated only the game spot swaption, we will consider the game forward swaption in the next research. Further, mathematical analysis of the structure of the exercise areas of both players in both game swaptions is also remained as our future research.  Table 2 The Bermudan-type game spot swaption prices and the exercise area