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Taleb's Proof

Nassim Taleb has recently written a very interesting paper where he gives a theoretic proof of Risk Neutral Pricing of options without relying on dynamic delta hedging.

Risk Neutral Option Pricing Without Dynamic Hedging, A Measure-Theoretic Proof, by Nassim Taleb

‘”There have been a couple of predecessors to the present thesis that Put-Call parity enforces risk-neutrality, such as Derman and Taleb (2005), Haug and Taleb (2010), which were based on heuristic methods, robust though "hand- waving". This paper uses a completely distribution-free, expectation-based and proves the risk-neutral argument with- out dynamic hedging.”

The dynamic delta hedging argument used by Black, Scholes and Merton to argue for risk neutral valuation of options is unnecessary and also breaks down in practice. Dynamic delta hedging can in no way remove enough risk to argue for risk neutral valuation if we have jumps, and in practice we have jumps in every market.

Nassim's proof is the final blow to the Black, Scholes, Merton way of deriving the formula, at least if you are interested in methods that also work well in practice. The Bachelier-Thorp formula is robust and is the one used by veteran option traders.

Option Traders Use (very) Sophisticated Heuristics, Never the Black–Scholes–Merton Formula

Demand and Supply

The demand for my Option Pricing Formuals Collection is now higher than supply? or may be not. At least the price has gone up: New copies from $ 899.98 (Amazon May 23). Time will tell if a bubble or not! The price of my book is clearly following a jump process.

$899.98 is still cheap, like less than $10 per formula..

There are rumours of a a short squeeze or buy back program, others will call it sector inflation !

Option traders use (very) sophisticated heuristics, never the Black–Scholes–Merton formula

The paper draws on historical trading methods and 19th and early 20th century references ignored by the finance literature.

It is time to stop using the wrong designation for option pricing.

Schrödinger Banks are they dead or alive?

Personally I do not agree on the philosophy behind the Schrödinger’s cat in physics, the so called copenhagen interpretation of quantum mechanics (will get back to this at some point I think), but I think a somewhat similar thought experiment much better fits many financial institutions, what we can name Schrödinger Banks:

That is if the bank "is" dead or alive will depend on the observation of it. Banks knows they funding cost and their life or death could depend on if the external observers could truly observe their operations in depth. If things looks bad their credit spread will widen, they will get less credit lines, this will worsen the situation and if on the edge of life or death this could be crucial. So yes the bank managers are very good at giving out the information they want the observer to observe and polish the information they not are so happy for the market to get.

Several Banks that are alive are not truly observed, lots of their activities are off balance sheet, only notional volumes are reported for many derivatives etc, that is close to meaningless information in many cases, the risk is typically given out is in terms of flawed Value at Risk measures based on Gaussian type models etc.. If external observers (the market) could take a detailed look at their books (I mean their uncooked books) in the depth some of them would probably collapse very quickly, others would naturally stay alive and some even increase in value.

The same could be generalized to other corporations and even to Schrödinger countries.

The Duality Bird

Look into the past and into the future, what do you see?

The past is more uncertain than most people think, but it is more deterministic than some people think. The future is more stochastic than most people think, but it is less stochastic than some people think.

Only the surface of duality is partly understood, the depth of duality holds a lot of secrets.

Stochastic Volatility Models

Every stochastic volatility model assumes yes stochastic volatility. All the stochastic volatility models I have looked into however assume constant volatility of volatility. Empirical research (mostly unpublished) shows the volatility of volatility is highly stochastic (make sure you not get tricked by sampling errors). Okay so what? Every model needs to make some assumptions. Models are only models. The problem is every stochastic volatility model I have looked at is highly sensitive to changes in the volatility of volatility. This combined with stochastic volatility of volatility is a rather bad combination. You do not know what the value of the option should be, and you do not know what the delta hedge should be. In particular out-of-the money options are extremely sensitive to the volatility of volatility (both the price and the hedge).

Stochastic volatility models are in general non-robust. They have moved the problem from one parameter to another parameter. And in addition they have added even more parameters to estimate (vol of vol, correlation between vol and underlying, bla bla bla.)

Well yes stochastic volatility models are better than the Black, Scholes and Merton model. But then knowledgeable traders do no use the Black, Scholes, Merton model. Yes they use the wording “Black-Scholes” or “Black-Scholes-Merton” but this do not mean that this is what they actually use. Trading is not about giving proper references to who did what when. GOOD Trading is about trying to make money, or at least making sure you not can blow up, traders use wording only for communication, few of them are too interested in getting their names in academic journals or about who published what. They don't even care if they call something A that actually not is A but B.

Quants and academics working with options think they have understood fat-tails for a very long time, because they have stochastic volatility models, jump-diffusion models, kurotsis adjusted models, local vol models etc etc.

Stochastic vol models and jump-diffusion models was however a nice attempt to move in the right direction, but I am afraid the approach "failed" compared to what many of us (including me) had hoped for.

So is the solution to extend stochastic volatility models to stochastic volatility of volatility models with constant volatility of volatility of volatility or to combine stochastic volatility with jumps and stochastic volatility surfaces. NO PLEASE STOP IT!. Yes you will probably get published something like that in a prestigious academic journal, but this is not giving us any improvements in practical option trading and hedging (the real problem to solve I personally think is in a completely different direction).

You will do far better than any of these models simply by using robust hedging principles like hedging options with options to truncate your exposure. But yes if your alternative is the naïve Black-Scholes-Merton approach of thinking you can hedge out almost all risk all the time by delta hedging yes then you should keep digging into stochastic volatility models and jump diffusion models.

But there is much more to this, and I will tell you much more later (probably).

Balanced Option Hedge published to Wall Street pre-1973

The company Arnold Bernhard & Co. Inc. distributed to Wall Street firms in the late 1960 sheets with Balanced Hedge Ratios on a series of warrants and convertible bonds, what we today would call delta neutral hedge. Personally I like the term "balanced hedge" better than market neutral delta hedge. We do not know the standard deviation of the lifetime of the option (and even less about the future tails), and as the delta often is sensitive to the volatility used in modern delta calculations (watch your DDeltaDVol as described in my Know Your Weapon) we simply do not know the market neutral delta hedge. And even if you knew the standard deviation of the future, delta hedging would still fail dramatically as an argument for risk neutral valuation due to discreteness and in particular due to price-jumps in practice.

Arnold Bernhard & Co in 1970 also published a booklet describing how one could run a very low risk often good upside portfolio by going long warrants or convertible bonds and putting on a balanced hedge using the underlying asset. Such a portfolio was in many ways immune for the errors in delta hedging (delta replication risk is not symmetrical). Any big jump and you made profit. This was not enough to give you an edge, you could simply not buy options and put on balanced hedge (except as pure insurance policy), it was also important to try to distinguish between what likely was over or undervalued options (easier said than done), but if successful at it this was a rather robust strategy, at least you could not blow up.

It is also a myth that knowledgeable traders fully aware of the large replication error in the BSM delta hedging argument simply are buying options and hoping for a Black-Swan-Event. All we say is to truncate your tails. When options according to your analysis are extremely overpriced there exist several ways to take in option premium and still truncate your tail.

Arnold Bernhard & Co. 1970: More Profit and Less Risk --- Convertible Securities and Warrants.

Order and Chaos

order is the birth of chaos

chaos is the birth of order

said a famous philosopher around 430 B.C.

At that time it was not normal to cite others (among the philosophers in that area). Could the reason be because they actually understood the true meaning of these words at that time. Much goes lost in translation, try to read this sentence in its original form as it first was written and you will hopefully understand more.

Nonsystematic risk should not count, but dose it count?

Nonsystematic risk should according to academics not count, but dose it count in practice? This is one of the many topics I touch upon in Models on Models: here some of my thought on this subject:

Let us for a moment assume a gold option market maker that has sold a lot of short-term out-of-the-money puts that he is delta hedging. Then suddenly the market is gapping down and down, as we know his delta hedging will not work that well, his losses are increased further by the implied volatility exploding to the upside. He is losing millions and millions and millions and soon blowing through his risk limits. Soon enough he is called into the head of precious metals and commodity trading:

• The Boss: What on earth is going on? You have been blowing through your risk limits, why did you not cover your tails by buying back some of these puts on the way down!

• Market maker: Don’t worry Sir, what I have lost someone else in our bank must for sure have made. When I got hired you specifically told me that the bank is extremely well diversified in all types of markets and businesses. You should thank me for not wasting money on protecting us for unsystematic risk by paying up for those puts. The other banks have been driving up the prices on these put options to unrealistic levels and are clearly not acting rational. Actually my diversification model told me to sell more puts on the way down, and I did. I expect a raise in salary and a good bonus, on aggregate the bank is probably making loads of money, thanks to their well-diversified portfolio and traders like me!

• The Boss: Guards get this nut out of our building now!

Most individuals working as market makers in options are typically only managing a book of options on a few underlying assets. For example, one individual can be a market maker in options on gold and possibly also other precious metals, another market maker on crude oil options, another a market maker in options on Scandinavian currencies and so on. In few if any big banks will you find an individual that is a market maker in a well-diversified portfolio of all types of underlying assets. Further, the currency desk is typically separated from the equity desk, the fixed income desk from the energy desk, etc. (and market makers often also takes considerabely with "calculated" risk)

A large investmentbank as a whole is typically very well diversified and is well aware of the benefits of diversification. If someone loses 50 million on gold options and another trader makes 50 million the same day on some equity option trading, the CEO of the big bank would possibly not even be informed, or at least not worried, she is mainly interested in aggregated trading results. CEO’s are typically not daily decision makers at the trading floor, except for possibly being involved in setting some major risk limits. Inside their risk limits (that can be considerable) traders and market makers rule the trading floors. Sometimes top traders even get paid more than their CEO.

Proprietary traders and market makers in most big Wall Street banks are mainly rewarded in terms of bonus based on the performance in their own portfolio (trading book), and typically only based slightly on overall performance of the whole investment bank. The market maker is an individual and not a computer trying to optimize risk reward for the whole portfolio of the bank. Even on the same small trading desk one trader making lots of money trading in a few underlying assets can get paid several million dollars in bonus while someone sitting next to him/her trading some other assets, but with moderate losses can get zero bonus or even get fired.

Individual traders in general simply do not get paid based on returns from the banks well diversified portfolio, and often not even much on the desk’s performance, but from their own specific trading portfolios. May be they should get paid much more based on the whole of the bank’s performance (and this varies among banks)? This is a completely different discussion, as a trader you have to trade based on how markets are (and how you get paid), not on how some equilibrium model tells you that the market (and bonus system) should be based on a series of strict theoretical assumptions.

Ericsson's Massive Intradya Gap and the Failure of BSM Delta Hedging

Yesterdays massive intraday gap in Ericsson (25%) was just a good reminder on how the idea of continuous time delta hedging to remove all the risk all the time to argue for risk-neutral valuation is not even a good approximation, it is simply flawed for any practical purpose. And in particular when you really need to remove risk.

If you had been selling otm options and delta hedging them before the massive intraday gap you would got rid of a minimal amount of risk, if this was a large part of your position you would have been blowing up.

Knowledgeable option traders have known this for a long time pre-dating Black-Scholes and Merton, they use much more robust hedging techniques, at least truncating their tail exposure (options against options).

Stochastic volatility models if they relay on delta hedging to remove most risk will unfortunately not help much in cases like this. Yes the SV model will fit the volatility smile and possibly give slightly higher delta for your otm options. When the market gap like this you will even be fried with your SV model (if you not have taken the appropriate steps described in chapter 2 Models on Models ;-). Knowledgeable option traders using options against option hedges will do fine!

Delta hedging removes quite some risk in many cases, and discrete delta hedging was known before Black-Scholes-Merton. But non of these partly ignored practical researchers ever claimed delta hedging could be used as argument for risk neutral valuation.

I am sure some market maker or option trader that actually believed in the delta hedging argument to remove almost all risk got fried in the Ericsson gap. Some long options naturally made a lot of money, staying long options did not make delta hedging work any better, but the risk from its large hedging errors are not symmetric (but did not necessary give you edge). Black-Scholes-Merton did nothing wrong, they pointed out some theoretical academic idea, with nice mathematical result (that not was close hold in practice). That so many believed in their argument was the mistake that potentially have slowed the evolution of quantitative finance, and cost many naiive traders coming out from business school (often brain washed) millions of dollars, or at least for their employer. Well in derivatives ones loss is often anothers gain ;-)

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