Fear and Greed: The Market Price of Risk

FearAndGreed

The Market Price of Risk is a much-neglected quantity. It is a concept that you’ll find in models of incomplete markets. In a nutshell, if a market is incomplete and you can’t hedge away some risk then you have to say how that risk is valued. The Market Price of Risk (MPR) quantifies this, and allows you to price all derivatives on the same underlying(s) consistently with each other. (If they have the same source(s) of risk then those risks ought to be treated alike.)

You don’t see it discussed much because we tend to talk about the risk-neutral world alone, whereas the MPR defines the difference between the real and risk-neutral worlds. (You’ll find the MPR in the drift of the stochastic variables.) And once you calibrate to the market prices of derivatives you won’t see it anymore. (That’s why it’s hard to spot in the HJM and BGM models which calibrate right from the start.)

Nevertheless this quantity is very important since it levels the playing field for all investments, no matter how complex. You see it in Markowitz’s Modern Portfolio Theory, and the MPR for each source of randomness and the correlations between them can be used to choose optimal portfolios.

But what does the MPR look like? Is it a time-stable constant, representing the compensation for taking risk of rational investors. Is it slowly varying representing the changing attitude towards risk of different generations? No, neither of these. The figure above shows what the MPR looks like for US interest rates. (Technically, this is the Market Price of Spot Interest Rate Risk.) It appears random. The spikes can be interpreted as over or undercompensation for taking risk, fear versus greed.

Details of how to find this MPR and how to use it for modelling as a second stochastic factor (incidentally, introducing the Market Price of Market Price of Risk Risk!) and also for trading, as in stat arb, can be found in Ahmad, R & Wilmott, P 2007 The Market Price of Interest-rate Risk: Measuring and Modelling Fear and Greed in the Fixed-income Markets. Wilmott magazine, January 64-70.

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