Observed and inferred regimes

Every engine takes regime, a starting regime or a belief about it, and the engines that price a decision take information, "observed" or "inferred". This page says when the two differ and what is computed in each case.

A belief as the starting regime

regime=0 starts in a known regime. regime=[0.3, 0.7] starts from a belief: probability 0.3 that the regime is 0. For a payoff that depends only on what is observed, the price from a belief is the average over regimes of the price from each,

\[u(p) = \sum_ip_i\,u_i,\]

because pricing is linear when nobody has to act on the regime. This covers bonds, European and digital options on equity, barriers, Asian options and credit default swaps. Ratios are taken after averaging: the fair spread of a swap is the averaged protection leg over the averaged premium annuity.

chain = rl.RegimeChain.twoState(2.0, 3.0)
model = rl.SwitchingVasicek(chain, 0.03, 0.5, [0.07, 0.02], 0.01)
belief = [0.3, 0.7]
print(f"bond from regime 0        {library_bond(model, 4.0, regime=0):.8f}")
print(f"bond from regime 1        {library_bond(model, 4.0, regime=1):.8f}")
print(f"bond from the belief      {library_bond(model, 4.0, regime=belief):.8f}")
print(f"0.3 x first + 0.7 x second {0.3 * library_bond(model, 4.0, 0) + 0.7 * library_bond(model, 4.0, 1):.8f}")

bond from regime 0        0.84506184
bond from regime 1        0.85227814
bond from the belief      0.85011325
0.3 x first + 0.7 x second 0.85011325

Where information matters

The average stops being the price when someone acts on the regime. Two cases arise.

Options on bonds and swaps, including European ones. The payoff depends on the bond price at expiry, and that price depends on what the market knows about the regime then. If the regime is observed, the bond in regime \(j\) is \(P_j\) and the option pays \((P_j - K)^+\). If it is inferred, the bond is the belief-weighted \(\sum_jp_jP_j\) and the option pays \((\sum_jp_jP_j - K)^+\), which is smaller by Jensen’s inequality.
Early exercise. The holder of an American or Bermudan right exercises on what is known. With an observed regime there is one exercise boundary per regime; with an inferred one the boundary depends on the belief.

So the observed-regime price is an upper bound on the inferred-regime price.

When the path reveals the regime

In continuous time the quadratic variation of an observed path gives its diffusion coefficient at once. If that coefficient differs between every two regimes, the regime is known from the path and the two prices coincide. This is the case whenever a volatility switches with distinct values: the Black–Scholes or CEV volatility, the Vasicek or Hull–White volatility, the G2++ volatilities or correlation. The regime-by-regime constructions on Options on bonds, swaptions and caps and in the grid engine are then the price under either setting, and a starting belief is resolved at once into the average.

The two differ when only a drift level switches and the volatilities agree: the Vasicek mean level alone, or the CIR mean level, which is the only CIR parameter that switches.

regimelib.information.revealed(model) reports which case a model is in.

The belief as a state variable

For two regimes that differ only in the level the rate reverts to, write \(p_t\) for the probability of regime 0 given the path of the rate. Nobody sees the regime; everybody sees the rate, and the belief moves only when the rate surprises. With \(s(r) = \sigma\) for Vasicek and \(\sigma\sqrt r\) for CIR, and \(q_{01}\), \(q_{10}\) the switching rates,

\[dr_t = a\,\big(\bar b(p_t) - r_t\big)\,dt + s(r_t)\,d\nu_t, \qquad \bar b(p) = p\,b_0 + (1 - p)\,b_1,\]

\[dp_t = \big(q_{10}(1 - p_t) - q_{01}\,p_t\big)\,dt + p_t(1 - p_t)\,\frac{a\,(b_0 - b_1)}{s(r_t)}\,d\nu_t .\]

There is one Brownian motion \(\nu\), the innovation: the part of the rate’s move that the belief did not expect. The first term of \(dp\) is the chain pulling the belief towards its stationary value, and the second is learning, proportional to how different the two drifts are and inversely to the noise. The pair \((r, p)\) is Markov, so a claim is one function \(V(t, r, p)\) solving

\[\partial_tV + a(\bar b - r)\,V_r + \tfrac12s^2V_{rr} + \mu_pV_p + \tfrac12\gamma^2V_{pp} + s\gamma\,V_{rp} - rV = 0, \qquad \mu_p = q_{10}(1 - p) - q_{01}p, \quad \gamma = \frac{p(1 - p)\,a(b_0 - b_1)}{s(r)},\]

with no regime blocks. The bond at \((r, p)\) is \(p\,P_0(r) + (1 - p)\,P_1(r)\), exactly, so the exercise value of an option on bonds or swaps is a known function of \((r, p)\). SwitchingFDEngine solves this on a grid in \((r, p)\).

A zero-strike call on a bond is the bond, whose value from a belief is known exactly, so it checks the belief’s dynamics with a switching level:

cir = rl.SwitchingCoxIngersollRoss(chain, 0.03, [0.07, 0.02], 0.5, 0.08)
for name, m in (("Vasicek", model), ("CIR", cir)):
    option = rl.CouponBondOption("call", 0.0, 1.0, [(3.0, 1.0)])
    option.setPricingEngine(rl.SwitchingFDEngine(m, regime=belief, n=(201, 41), steps=200))
    print(f"{name:8s} belief grid {option.NPV():.7f}   exact bond from the belief {library_bond(m, 3.0, regime=belief):.7f}")

Vasicek  belief grid 0.8901135   exact bond from the belief 0.8901135
CIR      belief grid 0.8903052   exact bond from the belief 0.8903048

Observed against inferred

A payer swaption and its Bermudan version under a Vasicek rate whose level alone switches, on a slow chain, from an even belief:

slow = rl.RegimeChain.twoState(0.5, 0.5)
level = rl.SwitchingVasicek(slow, 0.04, 0.5, [0.07, 0.02], 0.01)
fixed = [2.0, 3.0, 4.0, 5.0, 6.0]

def price(information, exerciseTimes=None):
    swaption = rl.Swaption("payer", 1.0, fixed, 0.045, notional=100.0, exerciseTimes=exerciseTimes)
    n = (301, 61) if information == "inferred" else 401
    swaption.setPricingEngine(rl.SwitchingFDEngine(level, regime=[0.5, 0.5], n=n, steps=200, information=information))
    return swaption.NPV()

print("                     observed   inferred")
print(f"European swaption    {price('observed'):.4f}     {price('inferred'):.4f}")
print(f"Bermudan swaption    {price('observed', [1.0, 2.0, 3.0, 4.0]):.4f}     {price('inferred', [1.0, 2.0, 3.0, 4.0]):.4f}")

                     observed   inferred
European swaption    1.3834     1.1690
Bermudan swaption    2.0221     1.9078

When the volatility also switches the regime is revealed, and the characteristic-function engines price the same swaption under either setting:

both = rl.SwitchingVasicek(slow, 0.04, 0.5, [0.07, 0.02], [0.012, 0.008])
swaption = rl.Swaption("payer", 1.0, fixed, 0.045, notional=100.0)
for information in ("observed", "inferred"):
    swaption.setPricingEngine(rl.NumericalSwitchingEngine(both, regime=[0.5, 0.5], information=information))
    print(f"{information:9s} {swaption.NPV():.6f}")
swaption.setPricingEngine(rl.NumericalSwitchingEngine(level, regime=[0.5, 0.5]))
try:
    swaption.NPV()
except NotImplementedError as error:
    print("level only, inferred: refused by this engine; use SwitchingFDEngine")

observed  1.382383
inferred  1.382383
level only, inferred: refused by this engine; use SwitchingFDEngine

What each engine does

Case	`information="observed"`	`information="inferred"` (the default)
Payoff of the observed state	average over the belief	average over the belief
Rate option or early exercise, regime revealed by a volatility	regime by regime, averaged over the belief	the same
Rate option, level only, two regimes, Vasicek or CIR	regime by regime	`SwitchingFDEngine` on the \((r, p)\) grid; the characteristic-function engines refuse
Anything else that depends on information	regime by regime	refused, with the observed price named as the upper bound

With three or more regimes the belief has two or more dimensions, and with a switching level inside a two-factor model the grid would need a third; neither is implemented.