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.
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 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.
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 ;-)
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.
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.
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.
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 ...
How big was the jump for a given security on a day the asset felt down for example 10% intraday, but in multiple steps? One researcher once told me to simply get hold of all intraday data including volumes and you will know. Sorry it is not that easy, the truth is it depends on what trading seat you are sitting in, and if you not are sitting in a trading seat you basically have no clue at all, I love academics but in this respect you need a trading seat, and every seat will have a different view of reality.
I will like to call this the Many-World's Interpretion of Jump-Diffusion:
No this is not about quantum-finance (that I will leave for another time) . To illustrate my point let’s look at a a naïve example: assume a stock in the morning open at same price as it closed the day before, let us say 100, it trades there for a while then starts to drop, 800 shares trade at 97, 3000 shares at 96, 1500 shares at 94, 450 shares at 92, 4000 shares at 91, 70 000 shares at 90 then the market closes.
How big was the largest intraday jump? The largest jump/gap according to the tape was 3%. But again it depends on what seat you are sitting in, if you have a few hundred shares to sell and got filled at 97 and then looked at the tape you would tell everybody that the largest jump was 3% that day, this is your reality.
If you had one million shares to sell and only got filled on a few thousand hitting all the bids on the way down, and possibly did not even want to show your full size to the market because you knew that would take the market even further down, then the jump was at least 10% today and possibly even larger tomorrow if no sizable bids are coming in.
Even having all intraday trades and all volumes traded the reality of the jump size will be different for different traders. Some factors will always be hidden for researchers that not are closely connected to a trading desk, and even the trader himself can only know the reality of the jump from his own position size and only vague for others.
So how do we then calibrate jump-diffusion models to historical data? It is "impossible" for a outside quant to calibrate your model for you if you are a big player (by massaging the data you can probably calibrate it yoursef), for small players probably okay, and the size do not need to be that big, during my many years in trading (I started as a kid) I have seen some of the world most liquid markets dried up faster than you can count to 3. The market traded but only in the absolute minimum size the market maker had to quote, and some market makers even did not pick up the phone, they were probably hiding in the bath room. What looked like something close to diffusion for some was a massive jump. Only the trader and possibly the market makers could smell what was going on.
Diffusion for some can be jumps for others; it all depends on your trading seat! Historical intraday data can "only "detect minimum jump size they can say littel about maximum jump size, except of course in super liquid market situation where anyone can get filled at almost any price...but in such markets periods there is not much of jumps anyway.
The Many-World's Interpretation applies to many factors in quantitative finance, if you ever run a sizable book in an illiquid market (and even some of largest market's can dry up quickly) you will find out.
