Past exam of the martingales and financial mathematics course of the University of Oxford / 2009 / B / 4

Apply Itô's formula to $\log X$ for positive initial value $x_0$. The drift correction is $-{\sigma }^2/2$, soThis is geometric Brownian motion. Under the physical probability measure, $W_T/\sqrt{T}$ has the standard normal distribution. For positive volatility,The drift here is $\mu$, because this is a physical bankruptcy probability. The risk-neutral measure will instead give drift $r$ for pricing. If $\sigma =0$, the value is deterministic and the default probability is the indicator of $x_0{e}^{\mu T}<m$; the usual positive-volatility convention is used in the remaining formulas.

In the Merton model, the total equity payoff is $(X_T-m{)}^{+}$, exactly a European call option on the company value with strike $m$. Total debt pays $\min (X_T,m)=X_T-(X_T-m{)}^{+}$. The first payoff has slope zero below $m$ and slope one above it; the second has slope one below $m$ and is constant above it.

LetThe Black-Scholes formula gives total equity value $C=x_0\Phi (d_1)-m{e}^{-rT}\Phi (d_2)$. Because the total company value is a traded asset, replication gives total debt value $x_0-C$. Divide by the numbers of securities to obtainThese are prices per security; the payoff diagram represents the two aggregate claims.

As insurance against credit default, the payment is the positive shortfall of recovery below face value:Thus the credit default swap in this question is one $m$th of a European put option on the firm value. Its price isEquivalently, adding this insurance to one corporate bond produces a sure unit payment, so its price is $e^{-rT}$ minus the bond price from part (b). This also fixes the positive orientation of the printed “difference”: insurance compensates face value minus recovery.

At maturity, one stock share is worth $(X_T-m{)}^{+}/n$. For a nonnegative strike $K$, the digital call option therefore pays one exactly onUnder the risk-neutral measure, $\log (X_T/x_0)$ is normal with mean $(r-{\sigma }^2/2)T$ and variance ${\sigma }^{2}T$. The risk-neutral pricing consequently givesThe effective company-value strike is $m+nK$, because both the outstanding debt and the number of shares matter. At $K=0$, the strict positive-share event remains $X_T>m$.