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# What are the Bastard Greeks?

The greeks are sensitivities of values, such as option prices, to other financial quantities, such as price. Bastard means 'illegitimate, here in the sense that sometimes such a concept is not mathematically justified and can give misleading information.

Example

Suppose you value a barrier option assuming constant volatility, a, of 20% but are then worried whether that volatility is correct. You might measure |^ so that you know how sensitive the option s value is to volatility and whether or not it matters that you have used 20%. Because you are assuming volatility to be constant and then are effectively varying that constant you are measuring a strange sort of hybrid sensitivity which is not the true sensitivity. This could be very dangerous.

Bastard greeks are sensitivities to parameters that have been assumed constant. The classic example is the measurement of vega, Let's work with the above example in a bit more detail, and draw some graphs.

Suppose we had an up-and-out call option and we naively priced it using first a 17% volatility and then a 23% volatility, just to get a feel for the possible range of option values. We would get two curves looking like those in Figure 2.16.

This figure suggests that there is a point at which the value is insensitive to the volatility. Vega is zero. So you might think that at this point the option value is insensitive to your choice of volatility. Not so. Figure 2.16: Barrier option valued using two different constant volatilities.

Actually, the value is very sensitive to volatility as we shall now see (and in the process also see why vega can be a poor measure of sensitivity).

Figure 2.17 shows the values for the same option, but using best- and worst-case volatilities. The volatility is still in the range 17-23% but, crucially, it is not a constant. Observe the very wide range of values at the point where the vega is zero. The wide range of values is a genuine measure of the sensitivity of the option value to a range of, non-constant, volatilities. Vega is not. Figure 2.17: Barrier option valued using best and worst cases.

It is illegitimate to measure sensitivity to a parameter that has been assumed constant. (Of course, one way around this is to treat volatility as another variable in a stochastic volatility model for example.)

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

# What are the Stupidest Things People have Said about Risk Neutrality?

Where do I start? Probably the stupidest and most dangerous thing is to implicitly (or sometimes even explicitly) assume that one can use ideas and results from risk neutrality in situations where it is not valid.

Example

Naming no names, I have seen the following written in research papers: 'Using risk-neutral pricing we replace x with the risk-free rate r.' Naughty! As explained below you can only do this under certain very restrictive assumptions.

Risk-neutral pricing means that you price as if traded contracts grow at the risk-free interest rate, and that non-traded financial quantities have a growth that is the real growth adjusted for risk. Sticking with the case of traded underlyings, it means that instead of using the 15.9% (or whatever) growth rate we have estimated from data we 'pretend' that the growth rate is actually the 4.5% (or whatever) risk-free rate.

For risk-neutral pricing to work you need complete markets. And that means enough traded quantities with which to hedge risk. Also you need to be able to hedge that risk continuously (and usually costlessly). And you need to be sure of parameters. And you usually can't have jumps. And so on. All highly unlikely.

Here are some things people say that are wrong because of the confusion of real and risk neutral:

• The forward price of some traded quantity is the market's expected value of it in the future. Wrong. Under various assumptions there is a simple arbitrage between spot price and forward price that links them via the interest rate. So there is no information in the forward price about anyone s expectations.

• The forward curve (of interest rates) is the market's expected value of the spot interest rate at future times. Wrong. If any expectation is involved then it is a risk-neutral expectation. The forward curve contains information about the expected future spot interest rate, yes, but it also contains information about the market s risk aversion. There are different risks in rolling over your money in an instant-access bank account and tying it up for years in a bond. That s why a risk premium is built into the forward curve.

• Using risk-neutral pricing we replace ¡1 with the risk-free rate r. Only if your assumptions allow you to do this. (And, of course, reality never does!) Otherwise it's an assumption in its own right.

• The delta of an option is the probability of it ending up in the money. Wrong for two reasons. One is that the probability of ending up in the money depends on the real probabilities, and the real growth rate, and that s disappeared from the option value so it can t be true. The second reason is that there s a sign wrong! (If we have a call option then the probability you want is N(d^, where the prime () means use x instead of r, and not

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

# What is the Best-Kept Secret in Quantitative Finance?

That inventors/discoverers/creators of models usually don't use them. They often use simpler models instead.

Example

Yeah, right, as if I'm going to give names!

Named models are not necessarily used by their authors. Ok, perhaps not the best-kept secret but it is something that newbie's ought to be aware of, so that they don't have unwarranted respect for models just because they've got a famous name attached to them.

In the early 1990s I was chatting to a famous quant who'd generously given his name to a fixed-income model. Let's call this person Dr X, and his model the X model. I asked him what fixed-income model his bank used. I was expecting an answer along the lines of 'We use the three-factor X model, of course.' No, his answer was 'We use the Vasicek model.' Dr X's model was at that time pretty sophisticated and so it was rather surprising to hear him admit to using what is essentially the 'starter' model.

A decade later I asked another inventor of a then state-of-the-art fixed-income model, let's call him Dr Y and his model the Y model, whether he used the Y model himself. Dr Y had just moved from a bank, and his reply was the very illuminating 'No! I work for a hedge fund now, and I need to make money!' You can figure out the implications.

I then asked another inventor of a popular ... Dr Z ... His answer: 'No, we don't use our model. Have you ever tried to calibrate it? It's terrible! We only published it to mislead other banks!' and he then named the model that he used in practice. Again, it was a much simpler model than his own.

The moral of this story is the same moral that we get from many quantitative finance experiences: Do not believe what it says on the tin, do your own modelling and think for yourself. Found a mistake? Please highlight the word and press Shift + Enter 