Gisanov theoremGisanov theorem represents measure change techique. This is similar to change variable x on g(x) in an integral in the case when x is from a functional space. When the BSE was developed and all people suddenly felt that "no free lunch" in option pricing someone probably noted that underlying iin BS pricing actually differs from one which actually should be underlying. The difference is the drift. We calculate price for the option on ( mu , signa ) while BS option formula has underlying on ( r , sigma ) security. Because BS formula is thinking as magical no arbitrage someone decided to apply Girsanov theorem as a tool that can replace real mu on r.In probability Girsanov theorem is used differently than in Finance. In probability we have original probability space with a measure P. In Finance it is refferred as to 'real'world. On the real world we introduce given stochastic process=real stock price equation with mu drift. Then one can make the measure change and state that there exists risk-neutral measure Q such that mu-GBM would get r drift. This is standard math way. This way actually fails to present underlying with r drift because integrals with respect to measure Q along the risk-neutral process with r drift are equal to these integrals with respect to measure P along GBM with mu drift. In other words Kolmogorov equation will have mu coefficients in front of the derivative of the first order. Then mathematical experts improved risk neutral concept. The settle original mu-stock equation on risk neutral probability space with measure Q. The Q is chosen such that its image on real space with measure P has drift r. This is the same to consider that the given GBM process has drift r and state that there exists probability space with measer Q that given r-GBM will have mu drift. This is nonsense because given mu GBM which govern the stock prices is defined and existed on the real world (original probability space) regardless whether or not options exist along with the risk free bond.The correct initial question to professors who understand mathematics before we beging to talk about option pricing is that they need to defiene original probability space with pr. measure P and the stock equation. If the define stock with respect to P we could not arrive at r in BSE. It is first case in above. If they try to settle it risk-neutral world then we can ask what is the expected return for our stock? They should answr'r' but you can argue that our stock should have expected return mu not r and therefore this is not our stock. We do not step here to derivatives. This risk-neutral confuse does not eliminate our lovely BS pricing. It is a confusion in undrstanding mathematics.

What is the risk-neutral measure?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:

1. a player hands over some money, say £X, to play,2. the host tosses an unbiased coin,3. if it comes up heads then the player is given £2,4. 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:

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

2. back out the set of risk-neutral probabilities that these prices imply,3. calculate the expectation of the derivative trade’s payoff under these risk-neutral

proabilities,4. 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:

1. Start with a collection of prices of market traded products, from which we will deduce the risk-neutral probabilities.

2. 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).

3. Use your model to calculate the expectations of the winnings/losses for each trade.4. If this expectation is exactly equal to the market-traded prices then you’ve done it.5. 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.

The true probabilities underlying the B-S equation are actually postulated. The pricing process is assumed to follow the stochastic process dSt=μStdt+σStdWt, where Wt is the Wiener process.

It means that (for simplicity, let's talk about European call) lnST is distributed as N(ln(S0)+(μ−12σ2)T,σ2T)Correct me if I'm wrong, you'd like to find EP(C)=e−rTEP[max(ST−K,0)], where P is a "physical" probability measure. Just to make sure, this expected value won't represent the fair price of the option.

If my calculations are correct, this expected value is equal to S0N(d1(μ))e(μ−r)T−KN(d2(μ))e−rTthe terms d1 d2 are from the B-S formula, with the adjustment to replace risk-free rate r there with "risky" μ

Now, I write down some derivation steps, please check them.

Let's rewrite expectation as follows, EP[...]=EP[I(ST≥K)(ST−K)], where I(.) is the indicator function.

Notice that the inequality ST≥K is equivalent to lnST≥lnKThen, ...=EP[STI(lnST≥lnK)]−EP[KI(lnST≥lnK)]=EP[elnSTI(lnST≥)lnK)]−KN(d2(μ))To calculate the first term, use the following lemma: if X distributed as N(a,s2) then E(eXI(l<X))=es+12s2N(μ+s2−ls)Take lnST as X and l as lnK, obtain EP[STI(lnST≥lnK)]=elnS0+(μ−12σ2)T+12σ2TN(lnS0+(μ−12σ2)T+σ2T−lnKσT√)=S0eμN(d1(μ))Finally, discount it with the risk-free rate r and we get the result.

You cannot get "true probabilities" (empirical distribution) from the BS model. Option price is required initial investment, which is risk neutral expectation of payout. “True probabilities” are irrelevant in Black

ou cannot deduce the real-world probabilities from the option prices.

It may seem strange, but here is a simple example which might help you to understand.

Suppose that everyone in the market agrees on the real-world probabilities, and that they are not changing for any external reason.

Then suppose that the investment board of a large pension fund decides that they need to increase the amount of options they have bought because they get a feeling that they would like to hold more protection against an adverse move (and since most pension funds are net long equities, this is likely to mean that they want to buy out-of-the-money equity put options to protect against a sell off in the equity market).

The pension fund will come to the dealers (investment banks probably) and will buy a whole load of put options, say. Naturally the price in the market will go up (simple law of supply/demand, and demand has increased), which implies that the implied vols will go up.

In summary: no change in the real-world probabilities, but a big change in the implied volatilities which will in turn lead to a change in the implied underlying probability distribution.