The put call parity
The reflection principle
Put call duality
Mirror mirror on the wall what is the most important symmetry in the world? When the student is ready the teacher will appear!
The put call parity
The reflection principle
Put call duality
Mirror mirror on the wall what is the most important symmetry in the world? When the student is ready the teacher will appear!
I was straight out from business school and naturally knew some about the put-call parity. We also had a brand new system called Knight Ridder in addition to our Reuters and Bloomberg that were the main price-feed system back then (there is nothing new in competing about the speed of information). Knight Ridder was known for its very fast price feed. Back then for some markets Knight Ridder was faster than Reuters and Bloomberg. Knight Ridder also had some type of simple “programing” capabilities, more like a built in primitive spread sheet. We "programmed" in the put-call parity for a long series of Eurodollar option and futures contracts trading in the Chicago pit. If the put-call parity were broken the relevant contracts would be blinking in red on the Knight Ridder screen.
Not often, but every now and then the Knight Ridder screen blinked in red and I hit the the direct line and got fresh trade prices from our broker in the Chicago pit that I could hit. Every time I was on the line the arbitrage was to my disappointment gone. Possibly some locals standing in pit trading had already picked it up, or in many cases it was likely just apparent screen arbitrage simply due to slightly different time stamp on the contracts.
In my many years on various trading floors I have only been able to do put-call arbitrage a couple of times in the OTC market.
The put-call parity has been known in great detail at least for 100 years by traders (see the book of Nelson and Higgins published 1902 and 1904, and more info also in my Models on Models). It is a very robust arbitrage principle, but also more than an arbitrage principle as it can be used to convert calls into puts and puts into calls etc. something that can be important in particular for market makers trying to hedge options with options.
1 use the put-call parity as an arbitrage constrain + to hedge options with options even when no arbitrage opportunity: YES
2 hedge options with options: YES
3 take into account the supply and demand in the options themselves for pricing and hedging options: YES
4 use initial market neutral delta hedging to remove some risk: YES
5 use discrete dynamic delta hedging to remove some risk: YES
6 construct their option portfolios in such a way to protect them from blowing up because they know delta hedging to remove even close to most risk some times fails big time. For example when we have jumps, and we have jumps. YES
7 use close to continuous dynamic delta hedging to remove close to all risk all the time so they can close their eyes and relay on risk-neutrality. NO
8 completely ignore supply and demand for options when pricing and hedging options because thy can replicate any option close to perfectly using dynamic replication with the underlying asset. NO
9. Where point 1 to 6 described and used by knowledgeable option traders before 1973: YES
10. Did any of the authors describing delta hedging before 1973 claim you could remove all the risk all the time using delta hedging : NO
11. Did many of the option formulas published before 1973 relay and evolve around the principles described in 1 to 6 but not on 7 and 8: YES
12. Do knowledgeable option traders also today relay on 1 to 6 and not on 7 and 8: YES
Do you still think knowledgeable option traders use the Black-Scholes-Merton formula (continuous hedging argument) for option pricing and hedging?
You do! Congratulations you just passed the test showing you believe in Santa Claus!
I used to do ;-)
A) It gives robust arbitrage constrains on pricing put versus call. Clearly understood by Nelson 1904...and possibly much earlier, (put-call parity very diffusely indicated by De La Vega 1688).
B) but also as a tool to create calls out of puts, puts out of calls and straddles out of calls or puts for the purpose of hedge options with options. In other words more than simply arbitrage constrain, but a very important tool to transfer risk in optimal and robust way between options. See Nelson (1904) and Reinach (1961).
In Black-Scholes-Merton world only point A is of importance, point B has no significance and is even inconsistent with the BSM model as it means supply and demand for options will play a role. If no pure arbitrage possibilities according to A then there is no need to use put-call parity as described under B.
Example: If you as market maker has loads of customers coming and wants to buy puts in BSM world you can simply manufacture them risk-free based on BSM dynamic delta replication argument. You would not care if there were someone you could get lots of calls from (except in pure arbitrages situation). In real world where dynamic delta replication fails to remove most risk it would be important to get hold of calls if available and convert them to puts from a risk reduction perspective. And if not available you would need to rise put price to get paid for risk you not can hedge (even if you delta hedge to remove some risk).
The original invention and USE of the put-call parity is fully consistent and even "predicts" that supply and demand for options will affect option prices. The BSM way of reducing importance of put-call parity to a pure arbitrage constrain is not consistent with this view.
Any knowledgeable option trader or market maker knows the importance of taking into consideration supply and demand for options. Option traders that don’t are doomed to blow up sooner or later. Delta hedging to remove risk (but not to argue for risk neutral valuation under a series of unrealistic assumptions that fails in practice) as described before BSM are fully consistent with original USE (and knowledgeable traders use today) of put-call parity.
You cannot derive BSM formula (consistent with BSM model) based on the original intentions of put-call parity without loosing some of the original ideas behind the USE and intentions behind the put-call parity, but possibly you can derive the more robust Bachelier-Thorp formula in this way? (will naturally not be consistent with BSM model, which is the good news). Most option formulas before BSM are anyway consistent with all the intentions behind the put-call parity. As soon as you NOT claim you can remove most risk with dynamic delta replication then supply and demand for options will play important role, and put-call parity will play a role also as a tool for "hedging/risk transfer" in addition to its use as a pure arbitrage constrain.
BSM is consistent with the arbitrage argument behind the put-call parity, but not with important aspects of its original intentions as used back then and today in the marketplace.
In addition to this BSM enforces flat vol smile, relays on the Gaussian, uses a hedging argument that only works at the university campus but fails big time in the market. Traders though they where using it and fudging it, but reading the ancient partly forgotten and ignored history it is clear knowledgeable option traders are relaying on other more robust principles and formulas.
"When Cyrus Field finally succeeded in joining Europe and America by cable in 1866, international arbitrage of securities was made possible. Although American securities had been purchased in considerable volume abroad after 1800, the lack of quick communication placed a definite limit on the amount of active trading in securities between London and the New York markets. But even though the facilities of the cable were availiable, the business of securities arbitrage was somewhat slow in developing. At the outbreak of the World War it had reached fair proportions when arbitrage activities were suspended."
The book is interesting read 241 pages.
Henry Deutsch 1910 is another interesting book titled "Arbitrage" 232 pages, describing put-call parity and arbitrage in a series of securities.
Ernest Brown Skinner 1913 one of my favorite books "The Mathematical Theory of Investments" basically a formula collection 245 pages, describing all types of basic interest rate compounding, discounting, annuities, basic probabilities etc.
Nelson 1904, "The A B C of Option Arbitrage" describes the put-call parity, market neutral delta hedging for atm options, and almost all options in London at that time was issued atm.
The academics re-discovered put-call parity delta hedging and many other principles of arbitrage and trading around 1970s....they ignored the past. They did not miss out of a single source, but of a whole series of sources, the irony is some of the same academics claim other do not cite. You have to cite every paper published in Journal A, but what arbitrage traders and researchers published in books and Journal B long before them is not important....
The Black-Scholes-Merton formula is only consistent with a academic version of the put-call parity. Using the continuous time dynamic delta hedging argument of BSM and getting the option formula consistent with the CAPM put strong constrains on what type of put-call parity that is consistent with the BSM formula. In the BSM world the put-call parity must have the same volatility for each strike. This is not consistent with the original use and development of the put-call parity. Black-Scholes-Merton are putting unnecessary and unrealistic constrains on the put-call parity.
Myth: Put-call parity was first described by Stoll 1969 in Journal of Finance.
Fact: From the known and available literature the put call parity was diffusely touched upon as early as in the 1688 by De-La Vega, and possibly earlier. It was fully described in much more detail by Nelson (1904) that refer to Higgins (1902). Neither Stoll nor Black, Scholes or Merton had any references to Higgins 1902, Nelson 1904, Bronzin 1908 etc. ...
The put-call parity in its original form as described by for example Nelson 1904 was an extremely robust arbitrage constrain that was practical usable and described in the way it still is used by the option market today. The volatility could be different for each strike and in this way be consisted with the fact that also supply and demand for options themselves played an important role and play an important role in option pricing and hedging.
The fact is the Black-Scholes-Merton model is not consistent with the original form of the put-call parity. Only the option formula as used by knowledgeable option traders that developed from a series of inventions by option traders and researches pre-Black Scholes, the Bachelier-Thorp formula is consistent with the original put-call parity.
Please take the time to read some of the ancient masters before you make up your conclusion. Read Bachelier, Nelson 1904.... Bronzin 1908 and several other German sources are under translation to English. Many more references given in our paper: Haug and Taleb 2008 "Why We Never Have Used the Black-Scholes-Merton Option Pricing Formula.
Please stop ignoring the Arrow of time and what can be done in the real market place versus fantasy assumptions that in some cases put strong unrealistic restrictions on well-established methodologies, like the put-call parity. The put-call parity came long long before Black-Scholes-Merton....and they should have tried to make their model consistent with how traders used the put-call parity not to put theoretical restrictions on the put-call parity.
When something not is consistent with academic models then many academics (luckily not all) are happy to put in some non-scientific fudge to save their current view of the model-world and claim it robust instead of seeking the truth. The volatility smile is NOT consistent with BSM but fully consistent with the ORIGINAL put-call parity that came long before both BSM and their strongly restricted version of the put-call parity. Stop ignoring the Arrow of time.
Yes we cannot derive BSM consistent from the original put-call parity because BSM is contradicting the essence of the original put-call parity. We can however derive the market formula (Bachelier-Thorp, the way knowledgeable traders use the pricing formula ) based on the original put-call parity as it originally was developed and still used today together with other robust PRACTICAL arbitrage constraints.
The put-call parity as used by BSM is NOT
C - P = S - X*exp(-rt).
BSM put-call parity is obviously on a restricted form (even if not stated directly in their paper), it is of this type:
C (sigma_X) - P(sigma_X) = S - X*exp(-rt),
That is sigma must be same for all strikes, not stated directly, but everything else will be inconsistent with BSM model, and even if it was consistent with put-call parity with volatility smile that is not enough, but a minimum. For example if you had a stochastic vol model and assumed you still could remove most risk with dynamic delta hedging, the model and argument would blow up in your face for jumps in market, and we have jumps in the market, I am interested in what can be used in the market.
You cannot price up options based on put-call parity and other robust arbitrage principles as well as supply demand of options and now enforce strict BSM methodology on it without changing whole valuation/pricing process to enforce flat vol smile. BSM is consistent with put-call parity, but enforces flat vol smile on everything used in relation to BSM including the put-call parity, this was not original use of put-call parity that was consistent with supply demand based option pricing and not with flat-vol smile.
Recently Myron Scholes attacked Nassim Taleb for not citing academics
” Scholes said academics do not take Taleb seriously because he does not cite previous academic literature in his theories, relegating him to a man who "popularizes ideas and is making money selling books."
Full story here:
So Taleb is not taken seriously by academics (according to Scholes) because he cites plenty of academics, but not every academic Scholes would prefer in the light Scholes would prefer? The Black Swan has 1350 citations to loads of academics and non-academics. I am sure Nassim always could cite more, we all could. However I find it surprising that Scholes are throwing this stone when he is sitting in a glass house. What about all the academics not citing traders that had published great ideas (that actually was used in practice) long before the academics?
For example Professor Stoll published on the put-call parity in Journal of Finance in 1969 without citing several traders that had published the put-cal parity in greater detail than him long before him. Most academics still citing Stoll as the one that came up with the put call parity. What about Myron Scholes himself, why did he not cite for example Filer (1959), Reinach, (1961), Thorp (1969), Bernhard, (1970), Nelson (1904)..... and a whole series of papers that had been looking at discrete market neutral delta hedging as well as the put-cal parity long before their paper in 1973. There is a whole series of academics that did not cite traders (as well as academics) that published ideas or part of their ideas long before them. Traders normally not care if academic cite them or not, they do not bother to scream about it like some academics do, traders have their income from trading and not from academic jobs where it is publish or perish.
There could of course be many reasons academics do not cite traders, they simply did not know about the sources. Still part of an academic job should be at least to do a good attempt to dig out literature that published central ideas before them, even if this means traveling around visiting a few libraries and dig out some books and papers. My point is not that academics did not cite every trader or ancient academics that came up with central ideas long before them. As researchers dig into the literature it will over time be clear who came up with different ideas first, until time again forget/diffuse the originators of an idea.
My point is simply that academics should be careful throwing stones at traders (that often also are academics) for not citing every possible academics they prefer in the light they prefer, at least not before they start to cite properly themselves, in particular when it comes to great ideas like the put-call parity that was invented long before their time.
Many academics (not all) have a tendency to only look into their own defined prestigious journals for who is who, and for who published what and when, they are not looking for the truth (or if their theories has any link to what actually can be done in the markets), but to publish or perish.
He was a professor in Staatswissenschaft at university of Heidelberg. His booklet was published in 1875. I find it interesting that he is referring to option literature all the way back to the 1600. Another ignored and forgotten German source I got hold of today from a library with high security and a great selection of dusty old books mentions option trading all the way back to 1500.
How can it be that options have traded for at least 400 to 500 years, and that both traders and professors have published actively about them for many hundred years and still some of today’s finance professors think people hardly could price or hedge options before 1973?
More on this and other ignored and forgotten sources later on. Financial Archeology is quite interesting indeed.
Leser, E. (1875): “Zur Geschichte Der Pramiengeshafte”. University of Heidelberg
My favorite quote is from Mills (1927) (it tells it all)
”A distribution may depart widely from the Gaussian type because the influence of one or two extreme price change.”
Mandelbrot went deeper into fat-tails in the early 1960s. Then in the 60s and 70s there was a series of so called great discoveries in academic finance, they all based their models on Gaussian. Empirical facts was pushed under the carpet, this to get every theoretical model consistent with each other (CAPM, Black-Scholes-Merton, Sharpe Ratio etc.)
For example the idea behind the Sharpe ratio was great: to get a simple quantitative measure of risk versus returns. The problem was the way the Sharpe ratio measure risk (sigma alone) was basically useless. To use Sharpe ratio is simply dangerous, still most funds use it as marketing device (for often naiive investors). But it must be better than nothing? Well before the Sharpe ratio researcher evidently at least looked at the whole historical distribution (for as much data they had), far from good but at least better (at least it forced you to think).
Do we need a global credit contraction and melt down of finacial markets to make people once agian rember and understand what Mills and others pointed out in early 1900 ? Everybody on Wall Street claim they understand fat-tails, but from recent market blow ups (from only moderate moves) this do not seem to be the reality....
Mandelbrot, B. (1962): “The Variation of Certain Speculative Prices,” Thomas J. Watson Research Center Report NC-87: The International Re- search Center of the International Business Machine Corporation.
Mitchell, Wesley, C. (1915): “The Making and Using of Index Numbers,” Introduction to Index Numbers and Wholesale Prices in the United States and Foreign Countries (published in 1915 as Bul letin No. 173 of the U.S. Bureau of Labor Statistics, reprinted in 1921 as Bul letin No. 284, and in 1938 as Bul letin No. 656).
Mills F. (1927) The Behaviour of Prices, National Bureau of Economic Research.
Oliver, M. (1926): Les Nombres Indices de la Variation des Prix. Paris doc- toral dissertation.
more details also in my chapter 1: Derivatives Models on Models
1688 De La Vega describes active option market and diffusely the put-call parity. See also Knoll (2004) that points out that put-call parity potentially can be traced back 2000 years.
1900 Bachelier (in his Dr thesis) come sup with first option formula (assuming normal distributed asset), and comes with a lot of mathematical insight in option pricing.
1902/1904 Higgins and Nelson: Market neutral delta hedging for at-the-money options well known. Put-call parity fully known.
1908 Vinzenz Bronzin publish book deriving several option pricing formulas. Bronzin based his risk-neutral option valuation on robust arbitrage principles such as the put-call parity and the link between the forward price and call and put options.
1910 Henry Deutsch describes put-call parity but in less detail than Higgins and Nelson.
1936 W. D. Gann describes market neutral delta hedging of at-the-money options, but in less details than Nelson (1904). Gann also indicates more dynamic hedging.
1956 Kruizenga re-discovered the put-call parity, but in less details than was already described by Higgins and Nelson.
1960 Freid describe empirical realtionships between warrants and the common stock price.
1961 Sprenkle publish option formula assuming log-normal distributed asset price.
1961 Reinach describes how market makers (converters) actively use put-call parity as well as how they hedge embedded options in convertible bonds with options.
1964 Boness describes an option formula basically identical to the Black-Scholes-Merton formula. However he do not argue for risk-neutral valuation based on continuous time delta hedging or the Capital Asset Pricing model.
1965 McKean gives closed form solution for American perpetual options, but not under theoretical risk-neutrality.
1967 Ed Thorp and Kausoff extend market neutral delta hedging to options with any strike (moneyness) or time to maturity.
1969 Thorp points to the direction of dynamic delta hedging, and how it will remove more risk than static delta hedging.
1969 Stoll publish a paper in Journal of Finance describing the Put-Call parity. However there was little nothing new, more was known already in the early 1900.
1970Arnold Bernhard & Co describe market neutral delta hedging of convertible bonds and warrants. Describe numerically how to approximate delta. Publish list of hedge ratio (delta) for large numbers of convertible bonds and warrants.
1973 Black-Scholes (1973) and Merton (1973) referring to Thorp and Kassouf extends delta hedging to continuous time delta hedging. It was a brilliant mathematical idea, but trading is not mathematics. Black, Scholes and Merton naturally knew that continuous time delta hedging not was possible, it was naturally only meant as an approximation. However for an approximation to work reasonably well it must be robust. The idea of continuous time delta hedging is far from robust. Even frequent delta-hedging do not remove enough risk to argue for risk neutral valuation, (see Derivatives Models on Models) for a summary on this topic.
Conclusion: Most robust hedging and pricing principles were basically known before Black, Scholes and Merton. The last step of taking market neutral delta hedging to continuous time delta hedging (or making the model consitent with CAPM) is the only step that not is of any practical use. Even frequent dynamic delta hedging cannot be used as an argument for risk-neutral valuation on its own. More on this in my book Derivatives Models on Models Chapter 2
Delta hedging removes a lot of risk, but not enough to argue for risk-neutral valuation (except in a fantasy world). Non of the traders/researchers describing delta hedging before 1973 ever claimed it could be used to remove all the risk all the time.