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I am wondering how traders come up with prices that are not perfectly explained by the models common for the respective products they are trading. If they use a pricing library, this library should present them a price which is consistent with the model. This would mean market prices should be matched by the most common model quite well.

For example, I am studying the SABR model, which to my understanding is a common model for interest rate swaptions. But the best fit of the model to the market data is not extremely good. Where are these prices coming from; in other words: how did the traders come up with these implied volatilities?

Example from the paper Managing Smile Risk by Hagan et al.: enter image description here The implied volatilities in the market don't look like there could be a common model that explains (interpolates) them all perfectly. How did those prices come to be?

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    $\begingroup$ Market frictions and other effects not accounted for un the models. Also remember that each market participant is not exposed to the same funding costs and tax rules! $\endgroup$ – Quantuple Aug 26 '16 at 16:54
  • $\begingroup$ Expanding on @Quantuple market participants also don't have the same financial motivations. A trader trying to maximize profits is willing to pay less than a hedge fund using options as a "Risk Control". The hedge fund gains additional economic utility from the transaction by marketing their lower risks and gaining assets. $\endgroup$ – drobertson Aug 26 '16 at 17:36
  • $\begingroup$ Economics theoreticians taught us that goods (like apples) are priced based on supply and demand while derivatives are priced entirely on replication arguments; but experience shows that to some extent demand/supply conditions affect the price of derivatives also. But is is a second order effect. $\endgroup$ – noob2 Aug 26 '16 at 18:02
  • $\begingroup$ @noob2 The last time they let economists trade, Long Term Capital Management almost took down the banking system. Theories are great for defending your PhD, but reality can mess you up if you trade off them. I am not saying they are completely wrong, just don't ever mistake a formula for the market. :-) $\endgroup$ – drobertson Aug 27 '16 at 5:46
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The simplest answer is that prices are not derived from the models. Prices are a result of trading in the market and express the sum of the information and opinions of all market participants at the time.

Despite what many academics would like you to believe, markets participants are not always perfectly rational and not always 100% motivated by the maximum utility of their money. There are many human decisions that drive market choices.

Sometimes a trader has to pay the bills (or go on vacation, or put Jimmy through school, or quickly max out your IRA contribution, whatever..) In aggregate, we can assume market efficiency is somewhat true, but even then there is a big gap between the models and reality.

The better way to think about any model is that it is a way to describe what the market price represents by wrapping it in a mathematical framework. The nice thing is that if the model is any good it allows you to understand some specific things about what the market believes (in aggregate) based on the price of the asset.

A good example is interpreting the future price volatility implied by an asset's option price. By using an option pricing model we are able to calculate the Implied Volatility based on market prices created by people trading in the market. This is literally using a model to infer the range of probable price movement of an asset based on the price of the option contract.

Once you are comfortable that the model is a good representation of the pricing mechanism, you can also use the model to do what-if analysis of the pricing based on your assumptions of what will happen. This allows you to engineer trades to perform specific functions in your portfolio, such as hedges or balancing risks.

It is a serious mistake to believe that market prices are a result of the model. Check out Long Term Capital Management for an example of what can happen when you make that mistake.

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The prices come from the market. One common source is Reuters. For items not traded in open markets you might assemble a set of bids from various counter parties.

Volatilities are derived from the market prices.

Many places use the market prices as inputs to calibrate shocks to their models.

Traders might get "correct" prices from their models which they use to decide when to buy and sell and what. However, just because their model says the correct price is $X does not mean they can get that price in the market.

The models are an attempt to estimate what the correct price is but come with a lot of assumptions that aren't necessarily true such as (and varying depending on the model):

  1. Prices are independent of each other.
  2. Prices form a random distribution.
  3. Prices form a normal random distribution.
  4. Everyone is a rational actor.
  5. Prices adjust as continuous function with no discontinuities.
  6. Markets clear instantly.

As drobertson points out a study of LTCM (a good book is When Genius Failled) will introduce you to a lot of potential issues mapping the model to the real world.

For the prices to exactly match the model requires magical thinking, specifically of the "the map is the terrain" sort.

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  • $\begingroup$ Understood, but how do they actually come up with prices then? $\endgroup$ – Beginner Aug 26 '16 at 18:37
  • $\begingroup$ The prices used to calibrate are drawn from the actual market close. For example, at my work the calibration team starts gathering various price quotes from various sources as soon as markets close. The are used as inputs to various processes. $\endgroup$ – HerbN Aug 26 '16 at 18:58
  • $\begingroup$ I think my question wasn't phrased well. I wanted to ask how traders come up with non-model prices which don't expose them to arbitrage opportunities? $\endgroup$ – Beginner Aug 26 '16 at 19:09
  • $\begingroup$ Because of the bid ask spread and various market frictions there is a little bit of wiggle room around the price before arbitrage can kick in. $\endgroup$ – noob2 Aug 26 '16 at 20:38
  • $\begingroup$ @Beginner That question is very different from the question you posted above. It requires a lot more space to answer than a comment provides. If you want to make a new question along the lines of "How did markets determine prices before (or without) financial models?" I would be happy to write up my "Parable of the efficient market" as an answer. I will give you the 50 cent answer in the next comment. $\endgroup$ – drobertson Aug 26 '16 at 20:41

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