Home Business & Finance Frequently Asked Questions in Quantitative Finance

# Why Does Risk-Neutral Valuation Work?

Risk-neutral valuation means that you can value options in terms of their expected payoffs, discounted from expiration to the present, assuming that they grow on average at the risk-free rate.

Option value = Expected present value of payoff (under a risk-neutral random walk).

Therefore the real rate at which the underlying grows on average doesn't affect the value. Of course, the volatility, related to the standard deviation of the underlying s return, does matter. In practice, it s usually much, much harder to estimate this average growth than the volatility, so we are rather spoiled in derivatives, that we only need to estimate the relatively stable parameter, volatility.2 The reason that this is true is that by hedging an option with the underlying we remove any exposure to the direction of the stock, whether it goes up or down ceases to matter. By eliminating risk in this way we also remove any dependence on the value of risk. End result is that we may as well imagine we are in a world in which no one values risk at all, and all tradable assets grow at the risk-free rate on average.

For any derivative product, as long as we can hedge it dynamically and perfectly (supposing we can as in the case of known, deterministic volatility and no defaults) the hedged portfolio loses its randomness and behaves like a bond.

2I should emphasize the word 'relatively.' Volatility does vary in reality, but probably not as much as the growth rate.

Example

A stock whose value is currently \$44.75 is growing on average by 15% per annum. Its volatility is 22%. The interest rate is 4%. You want to value a call option with a strike of \$45, expiring in two months' time. What can you do?

First of all, the 15% average growth is totally irrelevant. The stock's growth and therefore its real direction does not affect the value of derivatives. What you can do is simulate many, many future paths of a stock with an average growth of 4% per annum, since that is the risk-free interest rate, and a 22% volatility, to find out where it may be in two months' time. Then calculate the call payoff for each of these paths. Present value each of these back to today, and calculate the average over all paths. That's your option value. (For this simple example of the call option there is a formula for its value, so you don't need to do all these simulations. And in that formula you'll see an r for the risk-free interest rate, and no mention of the real drift rate.)

Risk-neutral valuation of derivatives exploits the perfect correlation between the changes in the value of an option and its underlying asset. As long as the underlying is the only random factor then this correlation should be perfect. So if an option goes up in value with a rise in the stock then a long option and sufficiently short stock position shouldn't have any random fluctuations, therefore the stock hedges the option. The resulting portfolio is risk free.

Of course, you need to know the correct number of the stock to sell short. That's called the 'delta' and usually comes from a model. Because we usually need a mathematical model to calculate the delta, and because quantitative finance models are necessarily less than perfect, the theoretical elimination of risk by delta hedging is also less than perfect in practice. There are several such imperfections with risk-neutral valuation. First, it requires continuous rebalancing of the hedge.

Delta is constantly changing so you must always be buying or selling stock to maintain a risk-free position. Obviously, this is not possible in practice. Second, it hinges on the accuracy of the model. The underlying has to be consistent with certain assumptions, such as being Brownian motion without any jumps, and with known volatility.

One of the most important side effects of risk-neutral pricing is that we can value derivatives by doing simulations of the risk-neutral path of underlyings, to calculate payoffs for the derivatives. These payoffs are then discounted to the present, and finally averaged. This average that we find is the contract's fair value.

Here are some further explanations of risk-neutral pricing.

Explanation 1: If you hedge correctly in a Black-Scholes world then all risk is eliminated. If there is no risk then we should not expect any compensation for risk. We can therefore work under a measure in which everything grows at the risk-free interest rate.

Explanation 2: If the model for the asset is dS = ixSdt + oSdX then the ixs cancel in the derivation of the Black-Scholes equation.

Explanation 3: Two measures are equivalent if they have the same sets of zero probability. Because zero probability sets don't change, a portfolio is an arbitrage under one measure if and only if it is one under all equivalent measures. Therefore a price is non-arbitrageable in the real world if and only if it is non-arbitrageable in the risk-neutral world. The risk-neutral price is always non-arbitrageable. If everything has a discounted asset price process which is a martingale then there can be no arbitrage. So if we change to a measure in which all the fundamental assets, for example the stock and bond, are martingales after discounting, and then define the option price to be the discounted expectation making it into a martingale too, we have that everything is a martingale in the risk-neutral world. Therefore there is no arbitrage in the real world.

Explanation 4: If we have calls with a continuous distribution of strikes from zero to infinity then we can synthesize arbitrarily well any payoff with the same expiration. But these calls define the risk-neutral probability density function for that expiration, and so we can interpret the synthesized option in terms of risk-neutral random walks. When such a static replication is possible then it is model independent, we can price complex derivatives in terms of vanillas. (Of course, the continuous distribution requirement does spoil this argument to some extent.)

It should be noted that risk-neutral pricing only works under assumptions of continuous hedging, zero transaction costs, continuous asset paths, etc. Once we move away from this simplifying world we may find that it doesn't work. Found a mistake? Please highlight the word and press Shift + Enter 