Collector's Blog - Trading versus current Theory
http://www.wilmott.com/blogs/collector/index.cfm
en-usSat, 05 Sep 2015 17:13:18 --0100Fri, 28 Mar 2014 00:32:00 --0100BlogCFChttp://blogs.law.harvard.edu/tech/rssblogs@wilmott.comblogs@wilmott.comTaleb's Proof
http://www.wilmott.com/blogs/collector/index.cfm/2014/3/28/Taleb-Proof-of-Risk-Neutral-Valuation-of-Options-Without-Relying-on-Dynamic-Delta-Hedging
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.
<a href=" http://www.fooledbyrandomness.com/OptionPricing.pdf
" > Risk Neutral Option Pricing Without Dynamic Hedging, A Measure-Theoretic Proof, by Nassim Taleb
</a>
??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.
<a href=" http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1012075
" > Option Traders Use (very) Sophisticated Heuristics, Never the Black?Scholes?Merton Formula
</a>
Trading versus current TheoryFri, 28 Mar 2014 00:32:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2014/3/28/Taleb-Proof-of-Risk-Neutral-Valuation-of-Options-Without-Relying-on-Dynamic-Delta-HedgingDemand and Supply
http://www.wilmott.com/blogs/collector/index.cfm/2013/5/23/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 !
Trading versus current TheoryThu, 23 May 2013 15:23:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2013/5/23/Demand-and-SupplyOption traders use (very) sophisticated heuristics, never the Black?Scholes?Merton formula
http://www.wilmott.com/blogs/collector/index.cfm/2011/1/12/Option-traders-use-very-sophisticated-heuristics-never-the-BlackScholesMerton-formula
The paper draws on historical trading methods and 19th and early 20th century references ignored by the finance literature.
<a href=" http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V8F-518TDK9-1&_user=597823&_coverDate=02%2F28%2F2011&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000030758&_version=1&_urlVersion=0&_userid=597823&md5=6dd75f7fc5d1d0d0867332372f079e78&searchtype=a
" >It is time to stop using the wrong designation for option pricing. </a>
Trading versus current TheoryWed, 12 Jan 2011 13:43:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2011/1/12/Option-traders-use-very-sophisticated-heuristics-never-the-BlackScholesMerton-formulaSchrödinger Banks are they dead or alive?
http://www.wilmott.com/blogs/collector/index.cfm/2009/11/6/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.
Trading versus current TheoryFri, 06 Nov 2009 13:26:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2009/11/6/Schrödinger-Banks-are-they--dead-or-aliveThe Duality Bird
http://www.wilmott.com/blogs/collector/index.cfm/2009/8/30/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.
Trading versus current TheorySun, 30 Aug 2009 13:02:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2009/8/30/The-Duality-BirdStochastic Volatility Models
http://www.wilmott.com/blogs/collector/index.cfm/2009/7/8/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).
Trading versus current TheoryWed, 08 Jul 2009 18:12:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2009/7/8/Stochastic-Volatility-ModelsBalanced Option Hedge published to Wall Street pre-1973
http://www.wilmott.com/blogs/collector/index.cfm/2008/5/12/Balanced-Option-Hedge-published-to-Wall-Street--pre1973
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.
Trading versus current TheoryMon, 12 May 2008 11:37:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2008/5/12/Balanced-Option-Hedge-published-to-Wall-Street--pre1973Order and Chaos
http://www.wilmott.com/blogs/collector/index.cfm/2008/3/28/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.
Trading versus current TheoryFri, 28 Mar 2008 02:47:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2008/3/28/Order-and-ChaosNonsystematic risk should not count, but dose it count?
http://www.wilmott.com/blogs/collector/index.cfm/2007/10/24/Nonsystematic-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.
Trading versus current TheoryWed, 24 Oct 2007 00:32:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2007/10/24/Nonsystematic-should-not-count-but-dose-it-countEricsson's Massive Intradya Gap and the Failure of BSM Delta Hedging
http://www.wilmott.com/blogs/collector/index.cfm/2007/10/17/The-Ericsson-Massive-Intradya-Gap-and-the-Failure-of-BSM-Delta-Hedging
Yesterdays massive intraday <td><a href="http://www.espenhaug.com/EricssonGap.gif" >gap in Ericsson</a> (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 ;-)
Trading versus current TheoryWed, 17 Oct 2007 17:00:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2007/10/17/The-Ericsson-Massive-Intradya-Gap-and-the-Failure-of-BSM-Delta-HedgingNo Risk
http://www.wilmott.com/blogs/collector/index.cfm/2007/9/28/No-Risk
Assume you hedge away all risk all the time by hedging some risky assets with a perfect hedge in derivatives. Assume that this hedge works perfectly as assumed. (for example delta hedging to remove all risk all the time works poorly in practice, but that is not my point this time).
Do this mean you have No Risk. Not at all, well only if you like to eat, drink and take your daily bath in dollar bills and coins.
The true meaning of No Risk is the topic of my latest column article:
"The Illusion of Risk-Free and the Deeper Meaning of Risk-Neutral Valuation? Wilmott Magazine (September):
There is actually several indications that our grandparents and parents know/knew more about how to minimize risk than finance professor specializing in Risk-neutral valuation, it is all in my article.
Trading versus current TheoryFri, 28 Sep 2007 18:18:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2007/9/28/No-RiskArbitrage and Scalability
http://www.wilmott.com/blogs/collector/index.cfm/2007/4/13/Arbitrage-and-Scalability
There are surprisingly often close to risk-free arbitrage opportunities in the world, the problem is that the relatively few traders that notice such arbitrage opportunities often are recently well off and they only look for reasonably scalable arbitrage opportunities.
Even risk-free arbitrage takes time to execute. So all arbitrages are relative to your alternative cost (your time). The higher your alternative cost the larger the arbitrage must be for you to spend time on it (if not it is not an arbitrage). For example in the next hour I know for sure I can pick off $50 risk-free in the market, is it worth it, probably not. No for me this is not an arbitrage, because I cannot scale the risk-free arb up to cover or beat my alternative time. For someone else making $50 a week it is a great arb. Of course the world is not fair, such people typically do not have access or knowledge about such arbs, if not such arbs would not persist for so long time.
Would I tell you about this arbitrage, no I am trying to figure how to scale one of the worlds most persistent arb opportunities. If you suddenly can do a $50 arbitrage 1000 or 1000000 times over at the same time it takes to execute a $50 arb you have a great business going!
Arbitrage trading is often not about finding risk-free arbs, it is about figuring out how to make them scalable! If I dont figure out how to scale this persisten arb I will write a paper about it ;-) One of the worlds most persitent and least understood arb opportunites.
Since I am bored today I will do this $50 arb once today, then at least I have a real example to use for my paper.
Conclusion:
There is no such thing as absolute arbitrage; all arbitrages are relative to your alternative time/cost.
Trading versus current TheoryFri, 13 Apr 2007 18:05:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2007/4/13/Arbitrage-and-ScalabilityArbitrage
http://www.wilmott.com/blogs/collector/index.cfm/2007/1/4/Arbitrage
Some people do not believe in Arbitrage opportunities, in particular many academics do not believe in such things.
The last few days I found incredible arbitrage opportunities, one that made me a small fortune; sorry I can not tell you about it, as also I think arbitrage opportunities will close down if too many know about them. The problem is most people close their eyes, they are not on alert. That brings me to the second arbitrage opportunity I found, that not was great but still an arbitrage. I just checked in at the airport, at the security gate there was 3 long lines, then I looked around and there was a forth empty line, I walked over too it, and even then nobody followed me. I looked closely at the people in the long lines, I was wondering if they all where finance professors? This is not risk free arbitrage, I am not selling and buying at different price, but this is still arbitrage if you take into account time is money.
The other arbitrage that made me a small forutne was basically the same, it was just in front of the eyes of a series of people, but most people simply ignored it (at least for some time). It had nothing to do with complex calculations, it had to do with staying alert always looking for arbitrage opportunites.
Trading versus current TheoryThu, 04 Jan 2007 09:06:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2007/1/4/ArbitrageEarly Discoveries and Mathematical Proofs
http://www.wilmott.com/blogs/collector/index.cfm/2006/6/29/Early-Discoveries-and-Mathematical-Proofs
Let us look at 3 ways of describing the same discovery:
a) Two plus two equals four.
b) 2 + 2 = 4
c) Let us assume A is a constant variable represent an integer containing the number 2, and that B represent a constant variable containing an integer with number 2. C represents constant variable containing an unknown integer with sum equals to A + B we can therefore prove that
A + B = C
Based on the assumptions that the variables not are stochastic but constants, this lead us to the conclusion that C must equals 4. For full proof and axioms see appendix 1.5.6 and 1.5.7.
Why do many researcher only seems to understand solution c) and they even ignores referring to work before them published in the form of b) and particular if it is published in form a) .
Some very interesting proofs of this exist in the literature and some of the old masters will soon be back ...
Trading versus current TheoryThu, 29 Jun 2006 01:06:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2006/6/29/Early-Discoveries-and-Mathematical-ProofsOvertrading and Overtraining
http://www.wilmott.com/blogs/collector/index.cfm/2006/6/8/Overtrading-and-Overtraining
Time is money!
Time is muscles!
But don?t forget:
Overtrading will cost you money!
Overtraining will cost you muscles!
Most traders are overtrading and undertraining, but by stop overtrading you can get more of both, this is simple relative value arbitrage!
Here transforming time into mucles in my sigma shirt!
"Everything that can be counted does not necessarily count; everything that counts cannot necessarily be counted. " Albert E.
Trading versus current TheoryThu, 08 Jun 2006 21:49:00 --0100http://www.wilmott.com/blogs/collector/index.cfm/2006/6/8/Overtrading-and-Overtraining