Bermudan Swaptions

Tutorial Contents

1. Fixed Income Notation
2. Fixed Income Probability Measures
3. Multi-Currency Markets
4. The HJM Framework
5. Basic Fixed Income Instruments
6. Caps, Floors and Swaptions
7. CMS Swaps, Caps and Floors
8. Bermudan Swaptions
9. Structured Notes and Exotic Swaps
10. Callable Libor Exotics and TARNs
11. Volatility Derivatives
12. Yield Curves and Curve Construction
13. Managing Yield Curve Risk
14. Vanilla Models With Local Volatility
15. Vanilla Models With Stochastic Volatility
16. Introduction to Term Structure Models
17. One-Factor Short Rate Models I
18. One-Factor Short Rate Models II
19. Multi-Factor Short Rate Models
20. The Libor Market Model I
21. The Libor Market Model II
22. Risk Management and Sensitivity Computations
23. P&L Attribution
24. Payoff Smoothing
25. Vegas in the Libor Market Model

Bermudan swaptions are a common form of option on swaps. They are similar to standard vanilla swaptions, except the holder has the right to enter into a fixed-floating swap on any of a list of specified fixing dates up to expiry. For a given tenor structure

the holder of a Bermudan swaption has the right to exercise on any of the dates $\{T_n\}_{n=0}^{N-1}$. Once exercised on date $T_n$, the option terminates, and the holder enters the swap with the first fixing date $T_n$ and the final payment date $T_N$. The period leading up to $T_0$ is known as the lockout or no-call period. In common jargon, a Bermudan swaption on a 10 year swap with a 2 year lockout period is known as a 10 no-call 2 or 10nc2 Bermudan swaption.

Demand for Bermudan swaptions is driven by various segments of the fixed income market. Mortgage companies use them to hedge their mortgage pools - the flexibility that the opportunity to exercise at various times gives them allows them to match the uncertain timings of mortgage prepayments. In contrast, investors seeking higher upfront income sell Bermudan swaptions - the additional optionality acts in favour of the holders of Bermudan-style options, leading to higher premiums versus an otherwise equivalent European swaption. Bermudan swaptions are also used in order to hedge callable Libor coupon bonds.

Pricing Bermudan Swaptions

Formally, at time $T_n$, the value of a payer Bermudan swaption, if exercised, is the swap value

Loosely speaking, a Bermudan swaption is an option to choose between $U_n(T_n)$ for different $n=0,\ldots,N-1$.

By no-arbitrage, the value $V(T_n)$ of the Bermudan swaption at time $T_n$ is then given by the maximum of $U_n(T_n)$ and the hold value $H_n(T_n)$, the value of the Bermudan swaption with only the exercise dates $\{T_i\}_{i=n+1}^{N-1}$ remaining. It is assumed that a rational investor at time $T_n$ will observe the exercise value $U_n(T_n)$ along with the hold value $H_n(T_n)$ and decide to exercise only if they stand to gain at that instant by doing so. Hence, we find that

Meanwhile, the hold value $H_n(T_n)$ simply represents the value $V(T_{n}^+)$ of the option at the instant after the choice not to exercise was made. For $t \in (T_n, T_{n+1})$, the investor has no ability to exercise, so we find that $H_n(t) = V(t)$ and hence, we have

Combining the above yields

This defines a useful iteration backwards in time from the terminal time $T_N$, to the initial time $0$. The idea behind this iterative scheme is known as dynamic programming. Loosely speaking, we work backwards to price Bermudan options. Hence, pricing Bermudan swaptions is a problem naturally suited to numerical schemes that proceed backwards through time, in particular, finite difference methods and tree-based methods fit this description well.

On the flipside of the above point, methods such as Monte Carlo pricing, which are inherently forward moving in time, are poorly suited to pricing Bermudan options, and more sophisticated numerical techniques must be considered to produce accurate pricing.