Why Default Rates Cannot Be Converted to Risk-Neutral Probabilities Alone
Summary
Historical default rates are probabilities under the real-world measure, but they cannot be converted directly into risk-neutral probabilities using Girsanov’s theorem alone. A model for the underlying process and its dynamics under the real-world measure must first be specified. The measure change is then represented by a Radon–Nikodym density, which weights real-world event probabilities to obtain risk-neutral probabilities.
Girsanov’s theorem relates process dynamics across equivalent measures within that modeling framework. A unique conversion requires additional assumptions, including a complete market and absence of arbitrage. The answer also notes that market practice often runs in the reverse direction: prices of traded instruments imply risk-neutral probabilities, which may then be related to real-world probabilities under a model. No specific credit model, calibration, or numerical example is supplied, so the explanation establishes conceptual requirements rather than a practical conversion recipe.
Key ideas
- Real-world default probabilities do not determine risk-neutral probabilities without a model.
- A measure change weights event probabilities using a Radon–Nikodym density.
- Girsanov’s theorem connects process dynamics across measures after the model is specified.
- A unique mapping requires market completeness and no arbitrage.
- Market prices typically imply risk-neutral probabilities, which can then be analyzed under a model.
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# Girsanov theorem and default rates in bond credit rating
# Girsanov theorem and default rates in bond credit rating
Default rates are kind of probabilities, right?
Is it possible to use the Girsanov theorem in that context? For example if we have a table of real world probabilities, could we use the Girsanov theorem to convert those real world probabilities into risk-neutral probabilities? Maybe even forward probabilities?
## Answer by Quantuple (score 2)
https://quant.stackexchange.com/a/29468
Suppose I give you objective probabilities $\mathbb{P}(S_T \geq K)$ of an equity finishing above a certain level $K$ at a future time $T$ (or in your case a survival probability in the form of default rates). Can you convert these to risk-neutral probabilities $\mathbb{Q}(S_T \geq K)$ ? Not immediately.
First, I need to give you a model for the behaviour of $S_T$, or equivalently to specify the dynamics of the process $(S_t)_{t\in[0,T]}$ under the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\in[0,T]}, \mathbb{P})$.
Then, simply noticing that $$ \mathbb{Q}(S_T \geq K) = \mathbb{E}^\mathbb{Q} [\mathbb{1}(S_T \geq K)] = \mathbb{E}^\mathbb{P} \left[\mathbb{1}(S_T \geq K) Z_T \right] $$ with $$ Z_T = \left. \frac{d\mathbb{Q}}{d\mathbb{P}} \right\vert_{\mathcal{F}_T} $$ the Radon-Nikodym derivative of the measure change should do the trick (with a tractability depending on your modelling assumptions of course). Obviously, you will appeal to Girsanov to translate the dynamics I provided you under $\mathbb{P}$ under the equivalent measure $\mathbb{Q}$.
So to answer your question: 'No, it is not possible', at least not without setting up a proper modelling framework. In addition, the conversion will be one to one if and only if you assume a complete market model + no arbitrage (see fundamental theorems of asset pricing).
Also notice that usually in the markets we do the opposite. We imply probabilities from quoted instruments and these are therefore obtained under $\mathbb{Q}$. The question is then, can we translate them under $\mathbb{P}$. The reasoning is the exact symmetric of the above.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.