Here is a short list of the most common ‘big-concept’ questions that I was asked throughout my years as a quant (whether coming from people on the trading floor, in control functions, or from newcomers to the team), in no particular order:

What is the risk-neutral measure?

What is arbitrage-free pricing?

What is a change of numeraire?

What is the ‘market price of risk’?

I don’t know of a single book on financial mathematics that attempts to give answers to these questions for a reader that is not familiar with stochastic calculus (which most traders are not, of course). Over a series of posts I will put down my own user-friendly answers to these questions, and we’ll end up with a grand *Guide to Financial Derivatives Pricing for a Non-Technical User*!

This post covers the first: what is the risk-neutral measure?

### A simple example with a coin-tossing game

Without even getting mixed up with stock and bond prices and suchlike, we can get a good sense of the *risk-premium* concept at work in a simple betting game.

The classic example, a game of coin tossing:

- a player hands over some money, say £X, to play,
- the host tosses an unbiased coin,
- if it comes up heads then the player is given £2,
- but if it comes up tails then nothing is given back.

A textbook on probability will tell you that the price of £1 per go is *fair* for this game because the concept of fair is defined in probability textbooks to mean that *the price paid should equal the value of the expected winnings*. Clearly it does for this example.

But let’s get savvy, step back from the theory, and ask how much would different players be prepared to pay for this game. Consider two different players:

- person A that has £1.50 in their pocket but is under pressure from a traffic warden to pay £2 for a parking ticket (and nothing less than £2 will do),
- person B that has £10 in their pocket and doesn’t really need anything more than that.

Don’t you think you could convince person A to pay up to their whole £1.50 for this game? Person B might be a harder sell, but perhaps they’d come around if we charged something like 50p a go and advertised the game as ‘potential 4 times returns on your investment’?

The important point is that the *theoretical* fair price may well be £1 for this game, but the *actual* price at which we sell the game may be something different since it will depend on the circumstances of the players we are selling it to.

The difference between the actual and theoretical price is called the *risk premium* for this game. Throwing in a bit of look-ahead market language, let’s write that:

the risk premium is the amount of premium (or discount) that needs to be added to the theoretical fair price in order to match the actual price of the trade in the market.

If you do a google search on *risk premium* you will see these concepts (amongst others):

- equity risk premium,
- inflation risk premium,

and these are just the theoretical musings of how much premium or discount there is in stock prices (to compensate for the volatility) or in bond prices (to compensate for the risk that inflation eats into your bond coupons and capital).

### The risk neutral measure — the flipside of the risk premium

The above examples showed that the price paid for a game is very likely to not be equal to the fair price for that game, ie. the value of the expected winnings.

In fact, it looks as though person A would buy the game for £1.50, which is a full 50p premium over the fair price of £1.

According to naive probability theory, person A would be paying the fair price only if the coin actually had a 75% probability of coming up heads, since the expected value would then be equal to the price paid:

expected winnings = 0.75 * £2 + 0.25 * £0 = £1.50.

This is the definition of the risk-neutral measure:

The risk neutral measure is the set of probabilities for which the given market prices of a collection of trades would be equal to the expectations of the winnings or losses of each trade.

**Remark:** It is *risk-neutral* because in this alternative reality *the price paid by player A for the game contains no risk premium* — the price is *exactly equal* to the value of the expected winnings of the game.

### Why is this so useful?

The risk-netural measure has a massively important property which is worth making very clear:

The price of any trade is equal to the expectation of the trade’s winnings and losses under the risk-neutral measure.

This property gives us a scheme for pricing derivatives:

- take a collection of prices of trades that exist in the market (eg swap rates, bond prices, swaption prices, cap/floor prices),
- back out the set of risk-neutral probabilities that these prices imply,
- calculate the expectation of the derivative trade’s payoff under these risk-neutral proabilities,
- that is the price of the derivative.

Wonderful! This is the reason why the risk-neutral measure is so important — it lies at the heart of the scheme for pricing derivatives (and therefore derivatives pricing is a sort of interpolation/extrapolation if you think about it).

### Am I telling you the *whole* truth?

This result as stated above in simple terms is not very far away from the *real* version that we use in derivatives pricing:

The Fundamental Theorem of Asset Pricing: There are no arbitrage opportunities in the market if, and only if, there is a unique equivalent martingale measure (read risk-neutral measure) under which all discounted asset prices are martingales.

So you see, even the simple coin tossing example above has been enough to take us quite far towards understanding this deep theorem, and without any substantially important lies too! (But don’t misunderstand me — the proof of the Fundamental Theorem requires some quite technical mathematics).

That’s it for now. In following posts I will give simple intuition to other ‘big concept’ questions, and we’ll make further progress towards understanding all the key elements in derivatives pricing.

### A footnote: how exactly do my quants derive the risk neutral probabilities from prices?

Step 2 in the scheme above might seem to be rather magical: *deduce the risk-neutral probabilities from the market prices*. Wow!

Well the truth is that it sounds more mystical than it actually is in practice, just because this way of describing it is really putting the cart before the horse. Here is how it works:

- Start with a collection of prices of market traded products, from which we will deduce the risk-neutral probabilities.
- Do your best to think up a realistic probabilistic mathematical model for the key elements that determine the payoffs of these traded products (e.g. let’s hypothesise a normal distribution for the 5y5y swap rate).
- Use your model to calculate the expectations of the winnings/losses for each trade.
- If this expectation is exactly equal to the market-traded prices then you’ve done it.
- Otherwise, fiddle with the parameters of your model (e.g. mean & standard deviations) until the calculated expectations of each trade’s winnings/losses in your model are equal to the prices in the market.

Once (if) you can do this you can then say that you have built a consistent model to explain the market prices, and can then ‘ask’ this model to calculate any probabilities you like: e.g. what is the proability that the 5y5y rate is above 10%?

Note that the process of adjusting your model’s parameters until you hit the market prices is called *calibration*.

Quants love to come up with clever mathematical routines that make calibration automatic and very quick. That’s a large part of their *raison d’être*.

**Remark**: if you only have a small collection of market prices to calibrate to then you may actually have a few different models that can be well calibrated to the prices. In fact, you might be surprised to find that there are some quite different models which can be well calibrated to lots of different market prices.

This is not really a problem until you find that you get quite quite different prices for the derivative you are pricing. More on this later.

God explanation Robert! You are absolutely right about the lack of pedagogical skill most textbooks have on the subject. I’ve spent endless hour’s trying to figure out what these things is all about. Sadly, I’ve learnt more from the Web and people like you than from professors in the subject at University. Regardless if you’re a mathematician or not, a solid intuition about what’s going on is essential. I think this is what separates the sheep from the goats. I’ll gladly continue following your posts and try to giving feedback. (:

Thanks Pontus.

I am writing more on the risk-neutral measure, looking into the concept of the risk premium in interest-rate models.

Glad you liked this one.

Thank you for an inspiring explanation. Very intuitive and easy to understand

Many thanks.

Fantastic explanation. Everything makes much more sense now. I’m having to learn stochastic processes for an internship at an IB and whilst the maths isn’t bad (being a math masters graduate new to this area) the applied “real-world” view of it is (not anymore thanks to this post).

Hi Robert,

It is an excellent explanation. For my understanding from your post, risk-neutral probability is basically from the people’s preference (i.e., the existing price of the asset), not from what the real probability of the asset itself. However, in your example, you only mentioned the price of 1.5 (the price of Person A), how about the price of Person B? From his price preference, we can deduce another probability? Is this a risk-neutral probability?

In addition, in your step 1:

1. take a collection of prices of trades that exist in the market (eg swap rates, bond prices, swaption prices, cap/floor prices).

These prices may deduce to different probability? How to handle this problem?

Thanks.

Pantelis, you are quite right — the risk-neural probabilities implied by the price that B would trade are going to be different.

You will actually find some maths texts that make it clear how the risk-neutral measure can actually be different for different people. I recall an article by Chris Rogers that says something just like that (see this very interesting article: http://www.skokholm.co.uk/wp-content/uploads/2013/02/origins.pdf): ‘In general, there will be no uniqueness of EMMs; any two agents in the same market would generate an EMM, and these will in general be different’. Note that ‘EMM’ is ‘Equivalent Martingale Measure’, the same idea as a risk-neutral measure.

In the case of the markets, the fact that we all agree a single price (open/closing/live/whatever) means that we only ever talk about a single risk-neutral measure.

brilliant example! can u do the same on Radon Nikodym theorem and derivative please? thanks!

Thank you! And thanks for the suggestions.

Hopefully I will soon get back to the writing, but at the moment I am so busy at PrismFP that I have not really found the time to look after this blog as much as I would like 😦

Very good explanation of seemingly abstract concepts. Learnt quite a bit from this. Please keep it up. Thanks.

Reblogged this on Human Mathematics and commented:

Just like Aaron Brown’s argument in Red Blooded Risk.

Thank You very much, this concept came across my actuarial studies and i didn’t get it until i read your post.But one question, how do we deduce risk neutral probabilities in real world.

I mention this briefly in the post: you think up a mathematical model which and use it to obtain the theoretical prices of the assets. If those theoretical prices match the ones you actually see then you have got a ‘good’ model and you can use it to deduce probabilities.

Dear Robert, Thank you for very insightful post (I know it was a few years ago). I notice you have been very busy with lot of other things. But if you could find time for the remaining three articles in this series, it will be immensely appreciated. I just wanted to let you know that your ability to simplify the things and still convey the essence is astounding.

Many thanks for the kind words.

Yes, life has been rather busy (work, family and children :-).

I also now run the Full Stack Quants community in London UK which takes time.

I will try my best to get back to this series soon.

My question is if one calculates the risk-neutral price of an asset. What is the relationship between the observed risk-free rates (e.g., US T-bill rates) and the observed price of the asset in the real market?

Rob, You have a great talent to explain such a complicated topic in a crystal clear way. I have been puzzled by these concepts for very long time. I hope I found your article earlier. I understood you may have changed your interest. But I want you know how much help you have given to people, somebody like me, by updating this Blog. I really hope I could read more about this topic, especially “What is a change of numeraire?”

Many thanks again.

Lee

Thanks Lee for your kind words. I do intend to get back to these things at some point (family life has been a bit too busy recently, that’s my excuse).