# How to estimate real-world probabilities

In the world of finance, Risk-neutral pricing allow us to estimate the fair value of derivatives using the risk free rate as the expected return of the underlyings.

However, the behavior of financial assets in the real-world might be substantially different to the evolution used in a risk-neutral context.

For instance, if I want to estimate the real-world probability of an equity asset reaching certain thresholds, which models and calibration techniques could be used?

In particular, some questions that may arise in the estimation of real-world probabilities are:

• Calibration: Should real-world probabilities be calibrated to current market prices or, alternative, historical data should be used for this type of estimation?
• No-arbitrage conditions: Could they be relaxed or they still play a role in the assessment of real-world probabilities?
• Expected returns: Assuming that I have already estimated the expected return of an asset $\mu$, how accurate would be a real-world estimation that combines a widely used evolution model (e.g. Geometric Brownian motion), with the use of $\mu$ instead of the risk free rate $r$?

Per comments, I understand that in order to estimate real-world probabilities:

• I should use expected returns instead of the risk-free rate.
• The asset evolution should still respect the no-arbitrage conditions (i.e: the real-world dynamics should still reproduce the current prices of vanilla options).

However, if I just use $\mu$ instead of $r$, but leave unchanged all other inputs, the estimated asset behavior might not be consistent with the market prices of contingent claims. In particular, if I just change $r$ by $\mu$ (with $\mu>r$) the underlying asset dynamics will lead to call prices above its current market price, and put prices below its market price.

Therefore, in addition to use expected returns, which other adjustment might be needed in order to properly estimate real-world probabilities?

Any papers or references regarding real-world estimation will be greatly appreciated.

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If you liked one of the answer please consider accepting it - Thank you! – vonjd Nov 13 '13 at 15:04
@vonjd: There are several answers that I like and upvoted. However, I think current answers still lack some details. For instance, even if estimating $\mu$ is the main driver behind real-world evolutions, if you just substitute $\mu$ by $r$, but all other parameters are left unchanged, the proposed evolution will not be calibrated to current market prices. Therefore, if calibration (i.e.: no arbitrage conditions) is still necessary to estimate real-world evolutions, which other steps are need once you have already estimated $\mu$? – sets Nov 13 '13 at 15:50

This is indeed one of the most difficult tasks to do (if not next to impossible).

I would say the standard reference is the following:
Expected Returns: An Investor's Guide to Harvesting Market Rewards by Antti Ilmanen

An abridged (but still about 170 pages long), yet more current - and free (!) version in different formats (pdf, mobi for the Kindle and epub) can be found here:
Expected Returns on Major Asset Classes by Antti Ilmanen

Addendum: A 8-page long summary of the main points can be found here:
Understanding Expected Returns by Antti Ilmanen

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First of all, I must say that it's a very general question, and the answer can vary depending on type of assets you model.

In quant finance real world probabilities are generally used for risk management. It can be said, that in order to use real-world probabilities you have to calibrate your models to history. In order to obtain risk-neutral probabilities, you fit to market.

Simplest example - brownian motion for asset price. It is $\frac{dS}{S} = \mu dt + \sigma dW_t$ in real world and $\frac{dS}{S} = r dt + \sigma dW_t$ in risk-neutral world.

Where would you take $\mu$ from? The easiest way is to take history and estimate historical asset drift, or just calculate $\frac{1}{N}\sum_{i=1}^N\frac{S_{i+1}-S_i}{S_i}$.

Where would you take $r$ from? You just take current risk-free rate.

At the same time I must stress that quant finance models are IMHO unsuitable for long-term forecasting. In this case you have to seek for appropriate econometric model.

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You may want to consider splitting two important, yet very different concepts:

Pricing a derivative security with contingent payoff and forecasting an asset.

• Pricing a derivative can be achieved through setting up a hedge portfolio and track its evolution and "value" at any point in time before the derivative security pays off. Risk-neutral pricing is a handy tool to accomplish that. In most all cases do you need to possess knowledge of the underlying price dynamics which most likely depend on one or more random components, such as Brownian Motion.

• Estimating the probability of a non-contingent asset (such as a stock) reaching certain thresholds can be done entirely without the construct of any risk neutral probability measure. All you need is a pricing model and a parameter set (which you could estimate or derive from a fit to historical data) and run a simple Monte Carlo simulation. No need for risk neutral probabilities at all.

My point is that the concept of risk neutral pricing is not necessary if you want to estimate the probability of an asset with non contingent payoff to reach certain price levels. Your question was which models could be used to estimate the probability of reaching such thresholds: You can setup a pricing model, whose parameters you fit to past data, and throw it into a MC pricer. Check how many of the paths reach your thresholds and derive your probability. That is an example where you use real-world parameters to estimate a real-world probability.

EDIT (in response to edited question)

Calibration -> Calibrate real-world probabilities to historical data and models, incorporating risk-neutral probabilities, to current market prices.

No-arbitrage conditions: No they cannot be relaxed, and why would you want to do that? You look for a self-consistent model and if you calibrate to current market prices but throw overboard no-arbitrage conditions then you end up with incorrect probabilities because your model is distorted.

Historical fit: You can calibrate any model that incorporates real-world probabilities to historical data. Whether history repeats itself and whether your assumed risk premiums lead to the correct probabilities is an entirely different question.

Expected Returns: You do not have the choice when using real-probabilities; you cannot use the risk-free rate because in your world of real probabilities investors are risk averse and apply different utility curves, hence, you need to estimate risk premiums and expected returns instead of simply using a risk-free rate. As this is a pretty damning exercise it is the precise reason why risk-neutral probabilistic models are so attractive.

In short? A non risk-neutral probability model.

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I am afraid that this - while of course true - doesn't really address the question of the OP: "If I want to estimate the "real world" probability of an asset reaching certain thresholds, which models and alternatives could be used?" So I guess it would be helpful to expand your comment "All you need is a pricing model and a parameter set (which you could estimate or derive from a fit to historical data) and run a simple Monte Carlo simulation." – vonjd Jun 20 '13 at 17:08
Fair, though I wanted to stress that risk neutral probabilities and estimating the probability of an asset with non-contingent payoff reaching certain thresholds are two very different and unrelated exercises. But will try to elaborate. Thanks – Matt Wolf Jun 20 '13 at 17:14
+1: Thank you :-) – vonjd Jun 20 '13 at 17:20
@MattWolf, I have just updated the question. Will be very helpful if you could elaborate your answer. – sets Jun 27 '13 at 11:43
@sets, I edited my answer to reflect your edited question – Matt Wolf Jun 28 '13 at 4:42