9Bermudans and Callables

In 2014 Taiwan’s legislature exempted foreign-currency bonds listed on the island from the cap on its life insurers’ overseas investments, and the insurers, with hundreds of billions of dollars to place, began buying thirty-year dollar zero-coupon bonds that the issuing banks could call every year after the first few. The investors earned a yield well above the Treasury curve for selling that right. The issuers did not keep it: they swapped each bond into floating funding with a dealer, and the swap carried the call, so that dealers ended up long a large stock of Bermudan options on long-dated rates. To hedge, they sold volatility in the swaption market, year after year. The instrument at the centre is the Bermudan swaption: an option to enter a swap on any one of several dates. It is worth more than the most valuable of the Europeans it can become, by an amount no European hedge captures. This chapter prices it on chapter 7’s tree and by regression Monte Carlo, measures that amount, and prices the callable zero-coupon bond the insurers bought.

9.1 Bermudan swaptions and their co-terminal Europeans

The lower bound is what a static hedge with Europeans secures; the switch option depends on how the co-terminal swap rates move together, which only a model of the whole curve can price.

9.2 Exercise by backward induction on a tree

def bermudan_tree(tree: HWTree, exercises: Sequence[int], final: int, K: float, receiver: bool = True) -> float:
    """Bermudan swaption by backward induction: exercise at year e into the swap from e to final."""
    per = int(round(1.0 / tree.dt))
    final_step = final * per
    ex_steps = {e * per: e for e in exercises}
    v = np.zeros(2 * tree.jmax + 1)
    for i in range(max(ex_steps), -1, -1):
        if i in ex_steps:
            sw = _swap_values_on_tree(tree, i, final_step, per, K)
            v = np.maximum(v, sw if receiver else -sw)
        if i > 0:
            v = tree.step_back(v, i - 1)
    return float(v[tree.jmax])

The exercise boundary of the receiver is a short rate below which exercising beats waiting (Figure 9.2). Early on the rate must fall far, to 0.82% in year one, since exercising gives up eight more chances; by year nine exercise is worth it below 2.30%, close to the strike.

9.3 Exercise by regression in Monte Carlo

Trees serve one or two factors; a market model (chapter 8) or any model with many state variables is simulated, and simulation runs forward while exercise is decided backward. The Longstaff–Schwartz method (One Quant Book 5, chapter 23) estimates the continuation value by regressing, at each exercise date, the discounted future cash flows of in-the-money paths on functions of the state.

    def run(seed_, coefs=None):
        xs = simulate_x(m, float(final), times, paths, seed=seed_)
        cf = np.zeros(paths)                                   # deflated cash flow of the policy
        fitted = {}
        for e in sorted(exercises, reverse=True):
            x = xs[float(e)]
            ex = _swap_value_x(m, e, final, K, x) / _numeraire_x(m, e, final, x)
            itm = ex > 0
            if e == max(exercises):
                cf = np.where(itm, ex, 0.0)
                continue
            basis = np.column_stack([np.ones(itm.sum()), ex[itm], ex[itm] ** 2])
            if coefs is None:
                beta, *_ = np.linalg.lstsq(basis, cf[itm], rcond=None)
                fitted[e] = beta
            else:
                beta = coefs[e]
            cont = basis @ beta
            exercise_now = np.zeros(paths, bool)
            exercise_now[np.flatnonzero(itm)[ex[itm] > cont]] = True
            cf = np.where(exercise_now, ex, cf)
        return cf, fitted

9.4 Upper bounds and the switch option

A regression price is a lower bound; how far below the true value it lies is measured by a dual upper bound (One Quant Book 5, chapter 23), which subtracts a martingale from the payoff and takes the maximum over dates along each path: the gap between the two bounds certifies the policy, and computing it (by nested simulation) belongs in the pricer’s validation. Once the gap is small, the practical risks lie elsewhere: in the model’s correlation and volatility structure, which set the switch option’s size.

9.5 The callable-bond issuance business

9.6 Tutorial: a Bermudan three ways

9.7 Build: Bermudan and callable pricers

9.8 Exercises

9.9 Problem: The Formosa Hedge

9.10 Interview questions