4Vanilla Rates Options

In June 2014 the ECB set its deposit rate at $-0.10\%$, and within two years short euro rates and many forward rates were below zero. Black’s model, in which a rate is lognormal, cannot price an option on a negative forward, or struck at a negative rate: its formula takes the logarithm of both. For more than twenty years caps and swaptions had been quoted as Black volatilities; the quotes that could not exist began to appear as normal volatilities, in basis points a year, or as volatilities of a shifted rate. The prices did not care which model was used to quote them; the risk numbers did. This chapter prices the two vanilla rates options, caplets and swaptions, in both models, strips the volatility of each caplet out of the quoted caps, and shows what the settlement terms of a swaption do to its price. The products and their market are those of One Quant Book 2, chapter 13; the curves are chapter 2’s euro curves.

4.1 Black and Bachelier for rates

A caplet on six-month Euribor fixing at $T_{k-1}$ and paid at $T_k$ pays $N\delta_k\max(F_k(T_{k-1})-K,0)$, where $F_k(t)$ is the forward of that fixing. Under the $T_k$-forward measure $\mathbb Q^{T_k}$ of the discount curve (One Quant Book 4, chapter 5), $F_k$ is a martingale, and so is the par rate $S_{a,b}$ of a swap under the annuity measure $\mathbb Q^{a,b}$. Both options are therefore options on a martingale, and the model only has to say how that martingale is distributed.

A normal model priced at one volatility for all strikes is a steep negative skew in lognormal terms (Figure 4.1): for a five-year option on a forward of 2.50% at 80 basis points of normal volatility, the Black volatility is 32.7% at the money (the approximation gives $80/250 = 32\%$), 71.1% at a strike of 0.50% and 23.8% at 4.50%. A market whose smile is flat in normal terms therefore looks strongly skewed to anyone quoting in Black terms, and a model choice is also a choice of how the smile moves when rates move.

4.2 Caps, floors and caplet stripping

A cap is quoted as one number, its price or a single volatility for all its caplets. The caplets of a five-year cap are the same caplets as the first four years of a ten-year cap, and cannot have two prices.

def strip_caplet_vols(start: dt.date, maturities: list[int], flat_vols: list[float], strike: float, proj, disc,
                      model: str = "normal", months: int = 6) -> list[tuple[float, float]]:
    """Piecewise-constant caplet volatilities, one level per interval between cap maturities, such
    that every cap reprices at its flat volatility. Returns (last fixing time of the interval, vol)."""
    knots: list[tuple[float, float]] = []

    def stripped(t: float) -> float:
        for tk, v in knots:
            if t <= tk + 1e-9:
                return v
        return knots[-1][1]
    for m, fv in zip(maturities, flat_vols, strict=True):
        cap = Cap(start, m, strike, months)
        target = cap_price(cap, proj, disc, fv, model)
        last_fix = caplets(cap, proj, disc)[-1]["t"]
        lo, hi = 1e-6, 5.0 if model == "lognormal" else 0.05
        for _ in range(100):
            mid = 0.5 * (lo + hi)
            knots.append((last_fix, mid))
            p = cap_price(cap, proj, disc, stripped, model)
            knots.pop()
            lo, hi = (mid, hi) if p < target else (lo, mid)
        knots.append((last_fix, 0.5 * (lo + hi)))
    return knots

4.3 Swaptions: physical and cash settlement

A physically settled swaption delivers the swap; its value is the annuity times an option on the forward swap rate (Proposition 4.1). In euros and sterling most swaptions are settled in cash, and the settlement amount needs an annuity that both parties can compute on the exercise date from one number.

def cash_annuity(rate: float, tenor: int, freq: int = 1) -> float:
    """Annuity of a swap discounted at its own rate (par-yield, or IRR, method)."""
    return sum((1.0 / freq) / (1.0 + rate / freq) ** i for i in range(1, tenor * freq + 1))


def swaption_cash_irr(expiry: dt.date, tenor: int, strike: float, proj, disc, vol: float, model: str = "normal",
                      payer: bool = True, notional: float = 1.0) -> float:
    """The traditional market formula for a par-yield cash-settled swaption: discount to expiry,
    times the cash annuity at the forward rate, times the option on the rate."""
    s, _ = forward_swap(proj, disc, expiry, tenor)
    t = disc.t(expiry)
    return notional * disc.df(expiry) * cash_annuity(s, tenor) * PRICERS[model](s, strike, t, vol, 1.0, payer)

4.4 Vega, the smile, and the choice of model

The normal vega of a caplet, $\partial\mathrm{Caplet}/\partial\sigma_N = N\delta P\sqrt T\varphi(d)$ with $d=(F-K)/(\sigma_N\sqrt T)$, is largest at the money and does not depend on the level of rates; the Black vega is proportional to $F$. A desk that hedges in one model and is marked in the other sees vega move when rates move with no change in the market. Two rules follow. Hedge in the model in which the market’s smile is most stable (for euro and dollar rates, normal or shifted models since the negative-rate years); and report vega per basis point of normal volatility, which is additive across caplets and swaptions of different forwards.

4.5 Tutorial: strip, price, compare

4.6 Build: caps, floors and swaptions

4.7 Exercises

4.8 Problem: The Treasurer’s Cap

4.9 Interview questions