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.


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.


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