Risk neutral Vs Real world

[Pulit Chhajer]

Risk neutral Vs Real world

Could you please break down the concept of "Risk neutral" and "Real World" using below narrative:

Risk neutral technique is a method to calculate the present value of cashflows by discounting risk-adjusted future cashflows with risk-free rates based. The risk-neutral method assumes no arbitrage and a complete market where there is no arbitrage opportunity and any derivative instruments can be perfectly available in the market. If these conditions hold the mathematical theory ensures that the expected value of the present value of the future cashflows based on a risk-free discount rates and a transformed probability distribution is equivalent to the present value of future cashflow based on adequate discount rates and a real-world probability distribution.

A real-world technique is a method to calculate the present value of cashflows by discounting projected cashflows with risk discount rates. Under this method, projected cashflows are not adjusted for uncertainty risk, which is the risk that the future cash flows can be different from those projected. To reflect the "price” of this uncertainty risk, it is common to set the risk-discount rates higher than risk-free rates. In risk-neutral technique the adjustment for the uncertainty can be consistent with observable market prices of securities

 

[Joe Hook]

Hi,

We can allow for the risk in cashflows in a number of different ways eg

1) Stress the cashflows
2) Stress the interest rate
3) Alter probabilities

In subject CT8/CM2 there's a fairly rigorous proof of a result called the risk-neutral pricing formulae which shows us how we can value derivatives in a risky world: by valuing them as the discounted expected payoff using risk-neutral probabilities. Under risk-neutral probabilities, all risky assets are expected to grow at the risk-free rate. Hence, the name risk-neutral. So we transform the probabilities and then we can use risk-free rates throughout our calculations.

Other assets may be priced or valued by using real-world cashflows and real-world probabilities. For example, you might value a share by coming up with possible future dividend values and associated probabilities and you can multiply payoff * probability * discount factor. As you pay something out now and receive income in the future, ie it's all income that's being discounted, we can allow for uncertainty by increasing the discount rate (known as the risk discount rate). This reduces the expected present value of income and so makes a asset or project appear less attractive. Hence, we'd need an asset/project that produces very high income to continue to appear attractive when we increase the discount rate.

Joe