The risky return can be positive or negative; it depends on the sign of P_x. If an asset rewards an investor for bearing the risk, then a hedge should produce a negative risky return (be “expensive”) so that a Delta-hedged portfolio generates nothing but the risk-free rate.

Using the risky return statement directly requires financial engineers to build pricing models so that they compute the risky return and, in particular, measure the derivative (P_x) of price P to risk factor x. Not only can this be a hassle, but taking such a measurement may not always be possible. For example, valuing path-dependent assets via the Monte-Carlo method is not sufficient to compute this derivative in a straightforward fashion.

2. Risk-neutral drift. There is a very simple and nice way to force a pricing system to produce the risky return by itself. Let us purposely add ps to the drift rate of our market factor x. With this intervention, each asset will get an additional return that is equal to the product of this rate by the asset’s exposure P_x/P (“duration”) to x, i.e. precisely ps P_x/P. The artificial drift is called the risk-neutral drift; a stochastic model that includes such a drift is called a risk-neutral model.

In short, instead of accounting for the risky return in each and every asset directly, we change the dynamics of our common risk factor x so that the required risky returns are generated automatically.

Suppose all bond are exposed to a single market factor, the overnight interest rate, which has an 80 bps annual volatility. Further, suppose that the market rewards 25 bps in return for bearing 1% of the asset’s volatility. Then, risk-neutral model for the short rate will feature an additional drift of 25 * 0.8 = 20 bp/yr.

Risk factor as a random parameter. In finance, uncertainty can come in various forms. We may be exposed to tomorrow’s economic release that will contain important indicators that look random from today’s point of view. Once released, these indicators won’t change for a while. Clearly, we deal here with random parameters (constants), but not with continuous randomness.

The arbitrage pricing works for multiple factors and for non-diffusive risks. With diffusive factors, risk-neutrality is achieved by adding an artificial drift rate. With random parameters, we do this by simply changing the value. To illustrate, let us return to the same short rate example now assuming that the factor is a random parameter rather than a random process (this is the actual assumption for the Black-Scholes equity option model). “Volatility” becomes an invalid term and has to be replaced by the usual standard deviation. Now, instead of adding 20 bps to the drift rate, we achieve risk-neutrality by adding 20 bps to the short rate itself.

Page 3 of 6 >>>