How Girsanov Changes Can Make Financial Models Less Tractable
Summary
The document asks why changing from the physical probability measure to a risk-neutral measure can make a stochastic asset model harder to use, even when the stochastic differential equation appears to retain its form. It also describes a related modeling question: whether a traded asset can be represented as a transformation of non-traded variables, such as macroeconomic data or a firm’s asset value, so that the discounted transformed process is a local martingale.
The text presents these as questions raised while reading a paper, rather than providing an example or a worked answer. It therefore offers no specific model, calibration result, or evidence that demonstrates a loss of tractability. The practical issue is left unresolved: changing drift may preserve a familiar equation in some cases, while the resulting parameters, dependencies, or pricing calculations can still be harder to handle. The discussion flags non-traded inputs and structural credit modeling as contexts where the distinction matters.
Key ideas
- A change of measure can alter stochastic process dynamics used for derivative pricing.
- A model may become less tractable under the risk-neutral measure even if its equation retains a similar form.
- The document asks whether traded assets can be modeled as transformations of non-traded variables.
- The text raises these issues but gives no concrete example or resolution.
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Full text
# 77062 # Why is it said that Girsanov’s theorem destroys the tractability of the process which is undesirable for quantitative finance applications？ I am reading the paper "Risk-neutral pricing techniques and examples" by Robert A. Jarrow et al., and it is said that Girsanov’s theorem destroys the tractability of the process which is undesirable for quantitative finance applications. There are two paragraphs in the paper that involve this view: "The traditional approach to pricing derivatives using risk-neutral valuation is to do a change of measure using Girsanov’s theorem. Girsanov’s theorem describes how the dynamics of stochastic processes change when the original measure is changed to an equivalent probability measure. In mathematical finance this theorem tells how to convert from the physical measure to the risk neutral measure. However, even when one starts with a stochastic process which has various properties that can capture the behavior of a financial asset, and is easily tractable, in many cases after changing measure it loses its tractability." "On the other hand, there are some additional issues in risk-neutral pricing. For example, we may need to model some assets that are not directly traded in the market, which need not to follow the local-martingale requirement due to their non-tradability. Then, the question naturally arises if it is possible to represent some other traded asset as a function or transformation of these non-traded assets, such that the discounted transformed process is a local martingale under the same measure? For example, we may want to represent a stock index in terms of macro-economic data, or represent a firm’s stock price in terms of its (non-traded) asset value, etc. This is particularly useful when we consider Merton’s structural model of credit risk, see Merton (2012), which has long been criticized for being unrealistic because a firm’s value is not tradable." I can't quite understand the meaning of "loses its tractability" when changing measure by Girsanov theorem. Because I think though the parameters of SDE may be different in two measure, their form is identical, and what we need to do is to calibrate parameters. Can someone give me a example of losing tractability? Thanks a lot!
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