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Total Posts: 415
Joined: Mar 2018
Posted: 2020-06-27 19:24
Risk neutral pricing seems to be very hard to understand at depth (emphasis on depth).

I have read standard texts and lectures on this subject but I am still struggling with a number of issues.

1) How can we find the risk neutral probabilities in practice? In all texts they descend into beautiful maths (girsanov, radon nikodym) but it is lost on me as to how to find the risk neutral measure in practice (emphasis on in practice)

2) Why does this always become a martingale?


Total Posts: 117
Joined: Apr 2018
Posted: 2020-06-27 20:46
2) is literally by definition. A risk neutral measure is defined as a probability measure such that the discounted price process is martingale.

1) Risk neutral probability is ultimately a change of variable. It's just a symbolic manipulation. It's probably best not to think of it as actual "probability". A lot of textbooks say that the price of a derivative is the expected value of the payoff under risk neutral measure. While this is not untrue, it used to confuse me a lot. Why do we use some made up probability measure? Why use expectation, does this mean derivative pricing is only correct "on average"?

A better way to understand this is to forget about probabilities. Derivatives are priced by the cost of exact replication. The price is exact if you can replicate the payoff, it has nothing to do with probabilities or expectation. This cost of replication turns out to be equal to the expectation under some probability measure. But calling this measure "risk neutral" confuses a lot of people, including me.


Total Posts: 415
Joined: Mar 2018
Posted: 2020-06-27 21:10
3) So if I fit a model to traded products and I back out the parameters, are you saying that this is the risk neutral parameters I can use for a new, related products? It seems to have nothing to do with any of the math in the textbooks

4) If given a general SDE

How can you "spot" the transformation to make this risk neutral? and how would this help you price a new product in practice?

5) Does the transformation differ if it is not a SDE?


Total Posts: 255
Joined: Dec 2010
Posted: 2020-06-28 03:53
Back in the days when I looked into understanding this a bit better, I found that the precise question you are asking was covered well in Bjork's book. In general this is a reasonably nice book that tries to connect the math underlying arbitrage theory and some market aspects, although this dude is clearly an academic.

Have a look. THere may be much better more modern books out there now, but I don't know them.

From a practical perspective, honestly once you adhere to the central result that numeraire normalized payouts are martingales under a certain corresponding measure, and that you can change the measure, you are good to go. That's a lot of what people in the industry do. I don't want to dumb shit down here and I am not saying that there aren't people out there who master the theory, but in the industry there is more to it than just that.


Total Posts: 415
Joined: Mar 2018
Posted: 2020-06-28 06:29
Ive got Bjork's book, its just the same as the other books, full of math and giving me personally no idea how the risk neutral measure is used or evaluated in practice. What part is it that you recommend?

Sorry but I dont really understand what your last paragraph means. What does you can just change the measure mean in practice?


Total Posts: 9
Joined: Jun 2020
Posted: 2020-06-28 10:38
Try Financial Calculus from Baxter and Rennie.

The first 3 chapters are a pretty good intro to risk neutral pricing.

Back in 96 Rennie was a quant in the old UBS teaching FI derivatives for the training programme.


Total Posts: 1352
Joined: Jun 2005
Posted: 2020-06-28 11:35

> full of math


Are you Permian (followed by Triassic and Jurassic) trying to turn NP into another stackexchange?

... What is a man
If his chief good and market of his time
Be but to sleep and feed? (c)


Total Posts: 130
Joined: Dec 2007
Posted: 2020-06-28 14:53

I echo mtsm's suggestion to go through Bjork's book (again), which I personally find one of the best introductory books available. I went out of my way to buy the first edition (at around 300 pages nicely compact and more than sufficient for an introduction, I mean seriously can anyone read a technical book more than 300p long?), as later editions often suffer from authors adding 100 pages for each new edition (e.g. Hull).

It took me 10 yrs to somewhat understand risk-neutral pricing, and still I regularly fail to understand it when confronted with a new situation / paradigm.

The phrench superior ecole minded like to dive right away into measure theory, functional analysis, you name it. Bigly unreadable and useless if you are just starting out in the subject.

The point is the hedging portfolio: make the claim riskless using hedge instruments, which leads you to a particular PDE, and through Feynman-Kac you see that you can act as if the world is risk-neutral, i.e. regardless of the volatility (risk) the drift of a tradable asset is $r$ (the 'risk-free rate'), hence 'risk-neutral'.

Just understanding this fundamental principle / idea (for which BMS rightly got the Nobel Prize) should be more than enough for many purposes. All other things are just variations on a theme.

You don't need a Ph.D to understand this stuff, and if there is something you don't understand it probably wasn't explained clearly.

No vanna, no cry


Total Posts: 415
Joined: Mar 2018
Posted: 2020-06-28 20:36
Ive never seen the Feynman Kac formula used much in finance textbooks on this. I cant remember it being prominent

In here it says to "back out the set of risk-neutral probabilities that these prices imply" How is this done?

Then, "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)." What is the probability model here of?


Total Posts: 255
Joined: Dec 2010
Posted: 2020-06-29 00:02
Agreed, it is not used much in practice, but it is a way to see the connection between the evolution equation for the distribution of a process and the equivalent pathwise approach of measuring the distribution through expectations, moments and stuff. I am getting sketchy here, I should not even talk about this, ouyt of my depth.

Anyway, in one edition of Bjork's book there is a chapter that talks about what it is that sets the martingale measure. I thought that is the question you were asking.

Well there are various ways of doing this. It depends on some details. There are some connections between certain contingent claims and the distribution governing the process of the underlying. Breeden and Litzenberger 1978 is one way of donig that for european contingent claims I suppose. But there are other ways. If you have picked a stochastic model of some sort and fit it to discoverable prices, let's say, you may probably recover the distribution (under certain conditions).

Hey, but from a practical perspective you are overthniking things a bit. Who says that you need to recover the actual risk-neutral distribution? Many of the times there is no need for this, in practice. It depends. That's where you need some practical experience. The important thing here is that there is an underlying framework that give you results that you can rely on to milk results on top of that. Do you see?

As to your last statement, that is a bit sarcastic I think. Personally I find the value of classical quantitative finance very relative. The models coming out of Bachelier, Black, Scholes, etc... are relatively complicated on a mathematical level, but kind of rock bottom stupid on an economical level. And these models aren't used in practice more as a yardstick. That's probably why quants aren't that big in the industry, but that's another discussion to be had.

The probability model is a normal model for the 5y5y swap rate. Literally.

Where do you work? There are some good working documents in banks that would help. But the questions you are asking are relevant and it's good to ask them, not sure this is the right place.


Total Posts: 3604
Joined: Jun 2004
Posted: 2020-06-29 08:00
My two cents.

There is a fundamental theorem (look for the article
J. Michael Harrison Stanley R. Pliska Martingales and stochastic integrals in the theory of continuous trading )
(works also for the discrete time as well)
which tells that if you do not allow for the arbitrage (i.e. money for free) your equivalent pricing measure
needs to be such that the underlying tradeable assets are martingales.

(Thus answering to your second question)

Now, in some models, mainly when the assets are modeled by the diffusion processes

we have the only such measure (the so called complete markets).
And for the diffusion it is

(lots of technical details swept under the rug)
Hopefully, that answers your first question.

As for the book, I would heartily recommend the Joshi's book

The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken


Total Posts: 679
Joined: May 2006
Posted: 2020-06-29 09:04
I think the best intuitive explanation you got is from @gaj. It's a change of coordinates, nothing more, nothing less. Girsanov tells you how to change coordinates. Radon Nikodym is the contribution you see from the change of coordinates - is basically the Coriolis effect of quant finance.

As it happens, there is one particular set of coordinates in which there is a simple pricing formula.

Why this one? Because when you hedge out the deltas, you are left with a cash position. So the coordinates that follow your cash position make things simple.

You back out implied probabilities by constructing point payoffs from call or put spreads. The formula is really simple, but you really should try to derive it yourself.

"There is a SIX am?" -- Arthur


Total Posts: 58
Joined: Jun 2017
Posted: 2020-06-30 22:22
Perhaps this is too elementary, but find that it helps to take the simplest case.

First of all you are risk-neutral if you dont care whether you get 1$ With certainty or 2$/0$ with 50/50 chance. You just want to make EV.

1: Price a future (replication)
Borrow what ever money X cost now, which is X(0), then B) you go and buy X.
This is going to cost you e(r x T) x X(0).
So you charge the client F(T) = e(r x T) x X(0) to buy the future.

2: Price a Stock (Q measure):
Should be quite clear that prices are always expectations (because by trading you are betting on an event). Given this fact. Assume you Stock can only do 2 things:

Up 10% with p probability
Down 50% with 1-p probability

Then e(-r x T) x [ p x1.1 X(0)+ (1-p)x 0.5 X(0) ]

Should be what you want to pay right now to buy Stock X. So you can make EV if the following is not true: X(0) = e(-r x T) x [ p x 1.1 X(0)+ (1-p) x 0.5 X(0) ]

3: Put things together

F(T) = e(r x T) x X(0) = [ p x 1.1 X(0)+ (1-p) x 0.5 X(0) ] =: E[ X(T)]

So the price of the future is an expectation. In reality we do not know p. You can observe X(0) in a newspaper. And given X(0) you can solve for a p such that the equation X (0) = e(- r x T) E[X(T)] holds ... this “calibration” of p is just implied probability. Just like implied odds from a poker or from a football game.

This yield a new method:

A) You can solve for that implied probability precisely when you know X(0) and the discount rate r.
B) Given that market implied probability p, to calculate the futures price you just calculate that expectation instead.


Total Posts: 415
Joined: Mar 2018
Posted: 2020-07-01 08:47
@day1pnl great answer. one thing i dont get is "So you can make EV if the following is not true: X(0) = e(-r x T) x [ p x 1.1 X(0)+ (1-p) x 0.5 X(0) ]". why if this is not true?
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