14Structural Credit Models

On 4 May 2005 General Motors’ share price rose 18% after Kirk Kerkorian offered to buy a large stake in the company: his company Tracinda then tendered for up to 28 million shares at 31 dollars each. The next day Standard & Poor’s cut the company’s debt two notches, to BB, below investment grade, and its bond and default-swap spreads widened. Trades that bet on the prices of the company’s debt and equity staying in line with each other lost on both legs, on consecutive days. The model behind many such trades is the subject of this chapter: a firm defaults when the value of its assets falls short of its debt, so that equity is an option on the firm and the prices of its shares and of its credit are two views of the same asset value. The reduced-form model of chapter 13 takes the hazard rate as given; the structural model explains it, and links it to the equity market, which is what makes it useful and what made May 2005 expensive.

14.1 Equity as a call on the firm

Equivalently, debt is a riskless bond minus a put on the assets struck at $D$: the creditors have sold the shareholders the right to hand over the firm instead of repaying. The credit spread of the debt is $s = -\ln(B_0/D)/T - r$, and the risk-neutral probability of default is $N(-d_2)$. Figure 14.1 shows the two claims as functions of the asset value.

The asset value and its volatility are not observed. Equity is: its market value $E_0$ and its volatility $\sigma_E$, implied from options or estimated from returns. By Itô’s formula on $E = C(V)$, $\sigma_EE_0 = N(d_1)\sigma_VV_0$, and with the pricing equation this gives two equations for the two unknowns.

def invert(E: float, sigma_E: float, D: float, T: float, r: float) -> tuple[float, float]:
    """Asset value and volatility matching equity value and equity volatility (Newton in two unknowns)."""
    V, s = E + D * math.exp(-r * T), sigma_E * E / (E + D * math.exp(-r * T))
    for _ in range(50):
        f1 = merton_equity(V, D, T, r, s) - E
        f2 = equity_vol(V, D, T, r, s) - sigma_E
        if abs(f1) < 1e-12 * E and abs(f2) < 1e-13:
            break
        hV, hs = 1e-6 * V, 1e-7
        a = (merton_equity(V + hV, D, T, r, s) - E - f1) / hV
        b = (merton_equity(V, D, T, r, s + hs) - E - f1) / hs
        c = (equity_vol(V + hV, D, T, r, s) - sigma_E - f2) / hV
        d = (equity_vol(V, D, T, r, s + hs) - sigma_E - f2) / hs
        det = a * d - b * c
        V -= (d * f1 - b * f2) / det
        s -= (a * f2 - c * f1) / det
    return V, s

14.2 Distance to default

The Merton model’s term structure of spreads depends on the quasi-leverage $d = De^{-rT}/V_0$ (Figure 14.2). A firm with low leverage has spreads that start at zero and rise with maturity, since it takes time for the assets to fall to the debt; a firm near the default point has a humped curve; an insolvent one ($d>1$) has spreads falling with maturity, since only time can rescue it.

14.3 First-passage models

def first_passage_survival(V: float, B: float, t: float, r: float, sigma: float, growth: float = 0.0) -> float:
    """P(min_{u<=t} V_u / (B e^{growth u}) > 1) for V with risk-neutral drift r (Black-Cox, reflection)."""
    if t <= 0.0:
        return 1.0
    nu = r - growth - 0.5 * sigma * sigma
    x = math.log(V / B)
    st = sigma * math.sqrt(t)
    return ncdf((x + nu * t) / st) - math.exp(-2.0 * nu * x / (sigma * sigma)) * ncdf((-x + nu * t) / st)

The survival function becomes a hazard curve on the premium dates, and chapter 13’s legs turn it into a model default-swap spread for each maturity, with a recovery set on the default swap’s convention (40%) rather than read from the asset value at the barrier.

14.4 Capital-structure signals and arbitrage

A structural model maps an equity price to a credit spread. Traders use the map in both directions: equity moves predict spread moves, and spreads far from the model’s value are candidate trades.

Yu (2006) ran such a convergence strategy with CreditGrades on 135 759 daily default-swap spreads of 261 North American names: single trades could lose heavily because spreads and equity prices were weakly correlated, while an equally weighted portfolio of all trades earned Sharpe ratios similar to other fixed-income arbitrage strategies.

14.5 Tutorial: from equity to spreads

14.6 Build: structural models

14.7 Exercises

14.8 Problem: The Capital-Structure Trade

14.9 Interview questions