Can I use Real Probabilities to Price Derivatives?

Short answer

Yes. But you may need to move away from classical quantitative finance.

Example

Some modern derivatives models use ideas from utility theory to price derivatives. Such models may find a use in pricing derivatives that cannot be dynamically hedged.

Long answer

Yes and no. There are lots of reasons why risk-neutral pricing doesn't work perfectly in practice, because markets are incomplete and dynamic hedging is impossible. If you can't continuously dynamically hedge then you cannot eliminate risk and so risk neutrality is not so relevant. You might be tempted to try to price using real probabilities instead. This is fine, and there are plenty of theories on this topic, usually with some element of utility theory about them. For example, some theories use ideas from Modern Portfolio Theory and look at real averages and real standard deviations.

For example you could value options as the certainty equivalent value under the real random walk, or maybe as the real expectation of the present value of the option's payoff plus or minus some multiple of the standard deviation. (Plus if you are selling, minus if buying.) The 'multiple' represents a measure of your risk aversion.

But there are two main problems with this.

1. You need to be able to measure real probabilities. In classical stochastic differential equation models this means knowing the real drift rate, often denoted by x for equities. This can be very hard, much harder than measuring volatility. Is it even possible to say whether we are in a bull or bear market? Often not! And you need to project forward, again even harder, and harder than forecasting volatility. 2. You need to decide on a utility function or a measure of risk aversion. This is not impossible, a bank could tell all its employees 'From this day forward the bank's utility function is Or tests can be used to estimate an individual's utility function by asking questions about his attitude to various trades, this can all be quantified. But at the moment this subject is still seen as too academic.

Although the assumptions that lead to risk neutrality are clearly invalid the results that follow, and the avoidance of the above two problems, means that more people than not are swayed by its advantages.

References and Further Reading

Ahn, H & Wilmott, P 2003 Stochastic volatility and mean-variance analysis. Wilmott magazine No. 03 84-90

Ahn, H & Wilmott, P 2007 Jump diffusion, mean and variance: how to dynamically hedge, statically hedge and to price. Wilmott magazine May 96-109

Ahn, H & Wilmott, P 2008 Dynamic hedging is dead! Long live static hedging! Wilmott magazine January 80-87

What is Volatility?

Short answer

Volatility is annualized standard deviation of returns. Or is it? Because that is a statistical measure, necessarily backward looking, and because volatility seems to vary, and we want to know what it will be in the future, and because people have different views on what volatility will be in the future, things are not that simple.

Example

Actual volatility is the a that goes into the Black-Scholes partial differential equation. Implied volatility is the number in the Black-Scholes formula that makes a theoretical price match a market price.

Long answer

Actual volatility is a measure of the amount of randomness in a financial quantity at any point in time. It's what Desmond Fitzgerald calls the 'bouncy, bouncy.' It's difficult to measure, and even harder to forecast but it's one of the main inputs into option-pricing models.

It's difficult to measure since it is defined mathematically via standard deviations, which requires historical data to calculate. Yet actual volatility is not a historical quantity but an instantaneous one.

Realized/historical volatilities are associated with a period of time, actually two periods of time. We might say that the daily volatility over the last sixty days has been 27%. This means that we take the last sixty days' worth of daily asset prices and calculate the volatility. Let me stress that this has two associated timescales, whereas actual volatility has none. This tends to be the default estimate of future volatility in the absence of any more sophisticated model. For example, we might assume that the volatility of the next sixty days is the same as over the previous sixty days. This will give us an idea of what a sixty-day option might be worth.

Implied volatility is the number you have to put into the Black-Scholes option-pricing equation to get the theoretical price to match the market price. Often said to be the market's estimate of volatility.

Let's recap. We have actual volatility, which is the instantaneous amount of noise in a stock price return. It is sometimes modelled as a simple constant, sometimes as time dependent, sometimes as stock and time dependent, sometimes as stochastic, sometimes as a jump process, and sometimes as uncertain, that is, lying within a range. It is impossible to measure exactly; the best you can do is to get a statistical estimate based on past data. But this is the parameter we would dearly love to know because of its importance in pricing derivatives. Some hedge funds believe that their edge is in forecasting this parameter better than other people, and so profit from options that are mispriced in the market.

Since you can't see actual volatility people often rely on measuring historical or realized volatility. This is a backward looking statistical measure of what volatility has been. And then one assumes that there is some information in this data that will tell us what volatility will be in the future. There are several models for measuring and forecasting volatility and we will come back to them shortly.

Implied volatility is the number you have to put into the Black-Scholes option-pricing formula to get the theoretical price to match the market price. This is often said to be the market's estimate of volatility. More correctly, option prices are governed by supply and demand. Is that the same as the market taking a view on future volatility? Not necessarily because most people buying options are taking a directional view on the market and so supply and demand reflects direction rather than volatility. But because people who hedge options are not exposed to direction only volatility it looks to them as if people are taking a view on volatility when they are more probably taking a view on direction, or simply buying out-of-the-money puts as insurance against a crash. For example, the market falls, people panic, they buy puts, the price of puts and hence implied volatility goes up. Where the price stops depends on supply and demand, not on anyone's estimate of future volatility, within reason.

Implied volatility levels the playing field so you can compare and contrast option prices across strikes and expirations.

There is also forward volatility. The adjective 'forward' is added to anything financial to mean values in the future. So forward volatility would usually mean volatility, either actual or implied, over some time period in the future. Finally hedging volatility means the parameter that you plug into a delta calculation to tell you how many of the underlying to sell short for hedging purposes.

Since volatility is so difficult to pin down it is a natural quantity for some interesting modelling. Here are some of the approaches used to model or forecast volatility.

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