June 16th, 2003, 11:00 am
To incorporate the market price of risk in pricing instrument is essential when modelling quantities that are not directly tradable.In the "perfect" world of Black Scholes, in which you can always continuously and perfectly hedge your position, it is possible to build a portfolio that completely eliminates risk. In the Black Scholes world, you can always dynamically hedge your portfolio, and doing so you are eliminating risk totally, and the market price of risk is a quantity that does not need to appear in the model (or the differential equation that describes it).The situation changes radically if you are trading something that is not directly observable (e.g. volatility trading, interest rate products).Since volatilities (or interest rates) are not tradable, you will not be able to create a risk-free portfolio. In fact you will trade one or more assets that depends on that quantity, rather than the quantity itself (e.g. you want to take a position on an interest rate, you might want to trade bonds since their value depends on interest rates). The same happens for hedging, since you will need again to use another instrument similar to the original one to cover your position, but is not the underlying quantity.The result is that you will never have a portfolio that will completely eliminate risk, and as Sam said earlier, an agent will require a premium to balance his diminished utility function resulting from taking a risk,gc