Risk-neutral probabilities (FRM T5-07)

One of the harder ideas in fixed income is risk-neutral probabilities. In this video, I'd like to specifically illustrate, and define, what we mean by risk-neutral probabilities. I will do this in three steps. The first one is just a simple example of a coin toss, where my objective is to illustrate what we mean by risk-neutral probabilities. These are the probabilities that equate the expected discounted value to the market price of the instrument. Then, having defined the concept, in the second sheet I've replicated Bruce Tuckman's example in Chapter 7 where he retrieves the risk-neutral probabilities that are implied by a 1 year zero coupon bond. We'll take those risk-neutral probabilities and go to the third sheet and use them to price or value an option on that same bond so that's a contingent claim. Then we'll see why there is some magic to these risk-neutral probabilities because we're going to be able to use them to price the option and we'll get a price that's necessarily equal to the price if we were to value the option with a replicating portfolio, which is something of an unambiguous value that would be indifferent to our risk preferences. So, I look forward to that illustration of risk-neutral probabilities.

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[shree]

The best explanation I've encountered for risk neutral probabilities definition. To summarize in my own words :- Risk neutral probabilities are the implied probabilities, when the market price of your instrument, equals the expected payoff

[mohammedsaeed]

The risk-neutral probs are derived from the market price, and the discount rate is used to derive them is the RISK-FREE rate. Thank you this was very informative.

[michalcertik]

Working for over 10 years in quantitative finance, this is so far the best explanation on real-world vs risk-neutral pricing.

[johnsonwayne]

I wish I could understand your language better. It is not your fault - it is mine.
I am mathematically trained but not economically or financially. Some of the
words and phrases you use to describe a term or an equation throw me but I
will try to learn more about this economic/financial language. Great video
though since I can understand a lot of this which explains debt market activity
and demystifies what "expectations" means. Also, as I consume this kind of material
it becomes foundationaly clear that human emotion and mom and pop are not
market drivers but suckers in the debt markets. And another thing. Although
many pundits swear that the central banks do not have the power to set interest
rates, this kind of information shows that they absolutely do and are doing it every second.
Their close competition would be the hedge fund people who are using the same
formulations to pursue CB policy to "front run" the market, then followed by the
pension funds and the others.
This was a great video for me.

[excelisfun]

You make such great videos! Thanks, Bionic Turtle!!!

[SzTz]

Hay the Bionic man, great to see you are still around.

[jiqi]

Thank you! It is an intuitive way to understand risk neutral probability finally

[AugustNocturne]

Thank you for your explanation for a hard concept!

[d]

Knew how to calculate and work with risk neutral probabilities but now I actually finally understand what it means :D thank you

[kathirkamanathan]

In the bond pricing example, the 1 year risk free rate is available (=5.15%) from the curve. So why should we try to estimate it from the 6-month risk free rate (=5%)?

[investwithvincent]

20:17

I see two different discounted prices 1.46 and .58 .... From my understanding these two prices create a no arbritrage opportunity. How would an investor explore this opportunity?

[investwithvincent]

7:15

It is at this point do i say that reminds me of the blackscholes model because of the concept of implied discount rate being incorporated in this example

[sakuranooka]

Why bother with RN probabilities if we could instead use a risk-adapted discount rate (eg 16.3%) together with the real-world probs?

[syphiliticpangloss]

Please always explain risk neutral properly in terms of transformations of probability distributions. It is not possible to determing the Q from a single observation.

The Q is determined from P *and* a transformation function (utility). So under some conditions you can write E_P(f(X)) as E_Q(X) where Q is the f-transformed probability distribution.

If f is linear it is called risk-neutral.

Please correct me if anything I am saying is wrong.

All of the examples with binary events are nice but they are just confusing people or tricking people into thinking they understand something.