Conversion between physical and risk-neutral default probabilities

Question

In the simple Merton structural credit risk model, the physical default probability is given by:

$$ DD_p = \frac{\ln(A / D) + (\mu -0.5\sigma^2)T}{\sigma \sqrt{T}} $$

$$ P=N(-DD_p) $$

Assuming that the physical distance-to-default(DD) and P are already provided, one needs to convert the physical default probability to risk-neutral probability in order to derive a fair value spread.

The "standard" way for this conversion in the literature seems to be the following: $$ Q=N[N^{-1}(P) + \lambda R \sqrt T] $$ where $\lambda$ is the market Sharpe ratio and R is the correlation of market and asset returns.

Essentially, the physical probability is transformed to a point in the CDF, a risk premium is added, then the sum is transformed back to arrive at the risk-neutral default probability.

My question is: why can we not simply plug in the risk-free rate(say the Treasury rate) into the distance-to-default formula since all assets have the same drift(r) under the risk-neutral measure?

$$ DD_q = \frac{\ln(A / D) + (r -0.5\sigma^2)T}{\sigma \sqrt{T}} $$

$$ Q=N(-DD_q) $$

Can someone point out what is wrong with my naive thinking? There must be a good reason why the literature is taking the more circuitous approach above.

Answer 1

I'm no expert on this topic but here's my two cents. Hopefully if I'm wrong someone will correct me.

From the 2 relations you wrote, we see that $$ DD_q = -N^{-1}(P) - \lambda R \sqrt{T} $$ or equivalently \begin{align} DD_q &= DD_p - \lambda R \sqrt{T} \\ &= \frac{\ln(A/D)+((\mu-\lambda \sigma R) - 0.5\sigma^2)T}{\sigma \sqrt{T}} \end{align} where the equivalent "risk-free" rate $r$ would be, $r := \mu - \lambda \sigma R $.

If the asset under scrutiny is tradable, you are right that $r$ should represent the cost of carrying that asset in the absence of arbitrage. However, if there is no way to straightforwardly "trade" it -- which could be the case here since $A$ represents the total assets of a firm -- assumptions have to be used. The assumptions used here are consistent with the CAPM.

Denoting asset-specific quantities using the index $a$ and market specific quantities with $m$, CAPM indeed writes \begin{align} \Bbb{E}[r_a] &= r + \beta_m (\Bbb{E}[r_m] - r) \\ &= r + \rho_{a,m} \frac{\sigma_a}{\sigma_m} (\Bbb{E}[r_m] - r) \\ &= r + \rho_{a,m} \sigma_a \lambda_m \end{align} whence $$ r = \Bbb{E}[r_a] - \rho_{a,m} \sigma_a \lambda_m $$ or using the notations above $$ r = \mu - \lambda \sigma R $$

Consequently, the method you refer to simply consists in specifying a particular form for the market price of risk by relying on CAPM assumptions.