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# Quiz 7 - The problem with "drift" (The 1st who gets it wins a copy of The Black Swan )

When you ask someone the following question: A currency has 5% interest rates (can be generalized to any security). The base currency (costs of funds) is 5%.

The underlying moves up 1% a day for 22 days in a row. How do you compute volatility (Standard Deviation) for the PURPOSE of decision-making (option pricing)?

Almost everyone I've quizzed throughout my career answers: 0% volatility. Their spreadsheet functions using series of log returns also erroneously provide: 0% volatility.

Nonsense.

The real answer is 16% annualized.

Why? STD = Sqrt[(E[X-E[x])^2] MAD =E[|X-E[x]|]

When you are facing an uncertain outcome you do not expect the mean return to be 1% a day. You simply expect 0% drift. Therefore you should not center volatility around the ex post drift but the ex ante one.

In other words, the options would produce the P/L of 0 volatility if and only if the drift is expected to be 1%

The classical anticipating-nonanticipating strategy. AN OPTION BREAKS EVEN AT 16% VOL (+- some adjustment) NOT 0.

Corrollary

A currency has 100% annual interest rates [paid daily]. Base currency is 5%. The exchange rate does not move for a month. What is volatility (monthly, annualized)? Easy...

The winner gets a copy of The Black Swan . I will not offer to sign the copy (I hate to offer to sign my book ... my signature has nothing special ... )

Simply raise vol twice this time by increment DV. Compare the following Portfolio values (hence PV) PV[V] - PV [V+DV] and PV[V+DV]-PV[V+2DV] . If the second value is smaller (larger) than the first and one is long volatility, then he is short (long) the tails. Is the second value is smaller (larger) and he is short volatility, he is long (short) the tails.

Model Risk Effectively this exercise reveals more than fat tails --sensitivity to model errors, sensitivity to problems of distribution. In a way, everything starts and ends with NonGaussianism.

# Quiz 6

How can you figure out if an option book is short the tails? Move ONE single parameter, but twice. Hint: not complicated at all. Please send answer to gamma [at] fooledbyrandomness [dot] com

Simple: Just raise implied volatility and look at the delta. If the delta rises, then the book is look OTM calls and shorter OTM puts, i.e. long skewness --you want Expectation of the cubic returns E[DS^3] to be >0. In other words look at the sign of the DDelta-Dsigma.

The problem is that the analytical derivative is not sufficient since the effect might flip if you have way out of the money options that might "wake up" at higher volatility. So you should make sure that the reaction is monotonic. In other words you might have exposures to higher ODD moments of the distribution.

# Quiz 5 (very simple)- Ferreting Out Asymmetries

How can you figure out by MOVING A SINGLE PARAMETER if a collection of options has an exposure to the third moment of the distribution (i.e. short or long skewness)?

Please send replies to gamma [at] fooledbyrandomness [dot] com

I mean ONE SINGLE parameter.

# Marks-to-Market, Risk, Fraud, Accounting Opacity, & Black Swans

I recall in my past life as a trader working for financial institutions that some desks, mostly bank units, did not want to deal with the volatility of the marks-to-market on a daily or monthly basis and ran "accrual" books. An accrual book is gradually marked throughout its life --so the trader knew pretty much, baring a "black Swan", what his P/L was going to be. Some also tried to escape the marks-to-market when they engaged in some arbitrages that should "converge" at expiration. They claimed to know what the value of the trade would be at expiration time T, so there was no need to mark immediately and deal with the vagaries of the marketplace. In their mind letting the market value these trades did not reflect the economic value of their books. The argument offered was "I will only unwind at expiration, not before". They were certain about it. They were as certain about it as people tying the know are certain that they are united forever.

I am writing this note because one day, in 2005, at a panel discussion, a board member of FNMA and an advocate of "modern finance" got emotional about the bad press related to some accounting irregularities that was supposed to have taken place at that firm. The panel discussion had nothing directly to do with FNMA or with accounting policies. It was about risk management. I thought of the argument proposed and realized that by not marking to market every single item in one's book one fell prey to model risk. It was the same type of epistemic arrogance that was behind the central planner: you set an equality between A and B by fiat. [It is not just some unpleasant member of the board of FNMA that falls prey to epistemic arrogance. Merton Miller took similar arguments when he defended the Metalgesselschaft traders who went bust trading short term futures against long term forwards. His argument was that "long term" things should be OK and that we should not have paid attention to the "short term" differences in market values].

Why Model Risk? The simplest of securities embeds model risks: the way the contract is described in your system may be missing a minor component. Minor, except... Say that I have the simplest of trades deemed fungible on my books: I am long a forward with Bank A and short the exact same one with Bank B. I may be hedged, but I have at least a credit risk there. Sometimes the smallest variations in the contracts can be significant. No two contracts will ever be exactly fungible unless they are legally offsetting.

Now the market knows that these contracts are not as identical as they are thought to be. Markets discover things faster than some slow-thinking regulator or overmathematized risk manager. When Russian options traded at 5 implied when supplied by a Russian bank and at 11 with a nonRussian institution, you had a marks-to-market risk not accounted for by models. The market knew it, not the banks.

Another sucker's problem is the classical forward-future "mispricing". The forward IS NOT a future, be it only because a future has cash-flow elements throughout its life, something the models miss severely. Many blew up on this.

A Safer System Many corporations do the following arbitrage. They buy plenty of companies, say n units. Say half the companies do well, the other half do poorly. All of them will be marked at cost on your books. You have a bad quarter: no problem. Just sell those that fetch a price higher than acquisition (i.e. books) and you will show a profit. GE does that routinely (Jack Welsh admits it in his memoirs).

The same with traders. When you let them "accrue" you end up having the books doing worse than market. If the trader has a profit, he takes it. A loss becomes accrued. It is like traders becoming "long term investors" when their positions are under water.

A system without opacity will be like Japanese institutions: seemingly less volatile, but exposed to large losses. (I compare this in The Black Swan to a dictatorship that shows political stability but incurs the risk of revolution compared to a country like Italy with a smaller risk of reolution but more fluctuations. Fluctuat nec mergitur.)

One good thing about hedge funds: unlike FNMA they mark to market. They have such bad press that they are forced to do so. Unlike banks and corporations they cannot play nasty games and fool their shareholders. Recall what Enron did with its "contracts". They may have other problems, but, at least, they are transparent.

Reduction and Platonicity

This problem resembles the more general one of the creation of categories and mental representations that simplify and reduce --it is necessary to simplify. Except that we forget that they aqre just simplifications.

# Path Dependence and Volatility

Introduction: A Garlic-Oriented Meeting

The first time I met Emanuel Derman, it was in the summer of 1996, at Uncle Nick's on 48th street and 9th Avenue. Stan Jonas paid, I remember (it is sometimes easier to remember who paid than the exact conversation). Derman and Dupire had come up with the local volatility model and I was burning to talk to Emanuel about it. I was writing Dynamic Hedging and in the middle of an intense intellectual period (I only experienced the same intellectual intensity in 2005-2006 as I was writing The Black Swan). I was tortured with one aspect to the notion of volatility surface. I could not explain it then. I will try now.

First, note the following. Local volatility does not mean what you expect volatility to be along a stochastic sample path that delivers a future price-time pair. It is not necessarily the mean square variation along a sample path. Nor is it the expected mean-square variation along a sample path that allows you to break-even on a dynamic hedge. It is the process that would provide a break even P/L for a strategy.

The resulting subtelty will take more than one post to explain (or I may expand in Dynamic Hedging 2). But I will try to explain as much as I can right here.

The first problem is that options are not priced off a mean-square variation in the underlying, but off a mean variation in the underlying. I cover the point elsewhere --the use of L2 norm is not adequate. Skip this for now.

The second problem is that options have a vega variation ALONG THE PATH so the PL for a strategy is decomposed as PL from variation in S (asset price), and P/L from changes in "volatility" (or expected mean deviation) --what the option is reflecting about future additional variations in the price of S between that point of evaluation and some terminal expiration. The P/L from changes in both implied and delivered volatility is path dependent. Severely path dependent.

Look at the graph. Take S an equity index. Assume that you Start at S0, at time t0. At time t2 you are at S2. Fine. But you can get there by two ways. The first way, path 1, is through S1a. The second one, path 2, is through S1b. Note that path 2 is more volatile than sample path 1 (mean square or mean absolute deviations). Consider an option valued at time to and at time t2 (it expires sometimes in the future, say t3).

Now take the standard "implied volatility" (what we call implied volatility by inverting the commonly used Bachelier-Thorp model, what I used to formerly & mistakenly call "Black Scholes" and is still called so by those who have not spoken to Espen Haug). Paradoxically, path 2 will lead to a lower terminal implied volatility at time t2, although that sample path was more volatile. The "retracement" brought a lower volatility. It is not just implied volatility that is lower at S2b (2nd path), but the future variations in the underlying S are expected to be dampened. You can check it empirically by taking volatility at new lows (say with S at the min three months window) and comparing it to situations in which you recover from new lows.

In my discussion of barrier options I talk about prices in areas with stops and a high density of barriers that have not been knocked out. This is far worse.

So we see a path dependence, a strong memory for the route taken by a price, etc. Mathematically, it means that you cannot easily work with a process --transition probabilities are not unconditional.

But the equation is fine and useable; it is the naive interpretation that is often wrong. I initially thought that if we had sigma0 at (S0,t0) and sigma 2 at (S2, t2), the local volatility should be ON AVERAGE approximately (sigma0 +sigma 2)/2. That is very rough approximation. It will be lower, much lower for large deviations, higher for smaller ones.

Forward volatility is what it takes to break even in a strategy along the average of all routes followed by the underlying security; there is a stochastic element in it and nonlinearities in option reactions to these variations. Because of such stochastic element, it will be HIGHER than the average sample paths for out of the money options, and lower for at the money ones. Why lower for the at the money ones? Because the collection of paths that will end unchanged are far less troublesome than the ones that stray.

# "successful" models & simple backtests

Seven Years Later I was reading (for the new edition Dynamic Hedging) an article by Leland and Rubinstein on portfolio insurance when I stumbled upon the following footnote:

M. Rubinstein and H. Leland, "Replicating Options with Positions in Stocks and Cash," Financial Analysts Journal 37, No. 4 (July-August 1981), pp. 63-72. [Added Note: This article was reprinted in the 50th Anniversary Issue of the Financial Analysts Journal (January/February 1995), having been selected as one of the 22 best articles out of the 3,200 published in the Journal during its 50 year history.]

I went to the paper and it was, of course, proposing their dangerously misleading method. But it was selected in 1995, after the crash of 1987, as one of the best articles.

Further a journalist tried to argue with the great Benoit M. about the "success" of Markowitz "successful" formula ( it is its 50th anniversary). Using success in being used --not in empirical tests --as a criterion is a fraud. Astrology has been so successful (much more than 50 years, perhaps 3300 years!).Should we use such popularity as a criterion for election. "Successful".

Now How Do We BackTest?

Simply you take the model by professor Rubinstein and the other idiot and run them through history. Further assume the nontrivial fact that people will not produce bids to be good citizens, but will back-off when you supply them with SP500 futures. ("Sunshine trading" has never worked, it is some normative economist's contraptionl; if you don't know what it means, no problem, it is something neclassical economists came up with to save their theories and you do not need to lknow anything about it).

Even without feedback effects, will the revision policy reduce risk? Of course not: you don't have a clue about the value of that option that you think that you have and did not buy because of fat tails. By fat tails I mean real fat tails, scalable fat tails. See the graph in an earlier post about the payoff of option contracts and how "smooth" it gets.

In other words, a soft option (a dynamic strategy) will NEVER replace a hard option (a real contract) in the real world because the production costs are severely stochastic.

Figure 1 : Term Structure of the Underlying Security. S is the spot, F the forward.

(This idea came to me when I saw in London earlier this month a hotshot from UCLA who spent his life talking about "pricing" American options, but never tried to look outside the Gaussian "Black-Scholes" based models. As Yogi Berra says: "Don't tell them if they don't know").

War Story 1 : The Currency Interest rate Flip

I recall in the 1980s the German currency carried lower interest rates than the US. When rate 1 is lower than rate 2, then, on regular pricing systems, for vanilla currency options, the American Put is higher than the European Put, but American Call =European Call. At some point the rates started converging; they eventually flipped as the German rates rose a bit after the reunification of Deutschland. I recall the trade in which someone who understood model error (not a finance professor) trying to buy American Calls Selling European Calls and paying some trader who got an immediate marks-to-market P/L (from the mark-to-model). The systems gave an identical value to these -it looked like free money, until the trader blew up. Nobody could initially figure out why they were losing money after the flip --the systems were missing on the difference. There was no big liquidity but several billions went through. Eventually the payoff turned out to be big.

We repreated the game a few times around devaluations as interest rates would shoot up and there was always some sucker with a math degree willing to do the trade.

War Story 2: The Stock Squeeze Spitz called me once in during the 2000 Bachelier conference to tell me that we were in trouble. We were long listed American calls on some Argentinian stock and short the delta in stock. The stock was some strange ADR that got delisted and we had to cover our short ASAP. Somehow we could not find the stock, and begging Bear Stearns failed to help. The solution turned out to be trivial: exercise the calls, enough of them to get the stock. We were lucky that our calls were American, not European, otherwise we would have been squeezed to tears. Moral: an American call has hidden optionality on model error.

These hiden optionalities on model errors are more numerous than the ones in the two examples I just gave. I kept discovering new ones.

Misplaced Precision

So many "rigorous" research papers have been involved in the “exact” pricing of American options, though within model when in fact their most interesting attribute is that they benefit from the breakdown of models. Indeed an interesting test to see if someone understand quantitative finance is to quiz him on American options. If he answers by providing a “pasting boundary” story but using a Black-Scholes type world, then you can safely make the conclusion that he represents an intellectual and financial danger. Furthermore, with faster computers, a faster pricing algorithm does not carry large advantages. The problem is in the hidden optionality...

Major points to know

An American option is always worth equally or more than the European option of the same nominal maturity. An American option has always a shorter or equal expected life than a European option. An American option’s value increases with the following factors: Higher volatility of interest rates. Higher volatility of volatility. Higher instability of the slope of the volatility curve. DANGER: A conventional pricing system will trick you into using the wrong parameter for the American option, as we will see.

Rule: The major difference between an American and European option is that the holder of the American option has the right to decide on whether the option is worth more dead or alive. In other words is it worth more held to expiration or immediately exercised ?

War Story 3: American Option and The Squeeze

I recall in the late 1990s seeing a strange situation: Long dated over-the-counter call options on a European Equity index were priced exceedingly below whatever measure of historical volatility one can think of. What happened was that traders were long the calls, short the future, and the market had been rallying slowly. They were losing on their future sales and had to pay for it –without collecting on their corresponding profits on the option side. The calls kept getting discounted; they were too long-dated and nobody wanted to toutch them. What does this mean?

Consider that a long term European option can trade below intrinsic value! I mean intrinsic value by the forward! You may not have the funds to arb it... The market can become suddenly inefficient and bankrupt you on the marks as your options can be severely discounted. I recall seing the cash-future discount reach 10% during the crash of 1987.

But with an American option you have a lower bound on how much you can be squeezed.

Let us look for cases of differential valuation.

Case 1 (Simplest, the bang comes from the changes in the carry of the premium) Why do changes in interest rate carry always comparatively benefit the American option ? Take a 1 year European and American options on a forward trading at 100, i.e. with a spot at 100. The American option will be priced on the risk management system at exactly the same value as the European one. S=100, F=100, where S is the spot and F is the forward.

Assume that the market rallies and the spot goes to 140. Both options will go to parity, and be worth \$40.

Case 1 A. Assume that interest rates are longer 0, that both rates go to 10%. F stays equal to S. Suddenly the European option will go from \$40 to the present value of \$40 in one year using 10%, i.e. \$36.36. The American option will stay at \$40, like a rock.

Case 1 B. Assume the domestic rate goes up to 10%, spot unchanged. F will be worth approximately of S. It will go from 140 to 126, but the P/L should be neutral if the option still has no gamma around 126 (i.e. the options trade at intrinsic value). The European option will still drop to the PV of 26, i.e. 23.636, while the American will be at 26.

We can thus see that the changes in carry always work to the advantage of the American option (assuming the trader is properly delta neutral in the forward).

We saw in these two cases the outperformance of the American option. We know the rule that ::

If in all scenarios option A is worth at least the same as option B and, in some scenarios can be worth more than option B, then it is not the greatest idea to sell option A and buy option B at the exact same price.

This tells us something but not too much: we know we need to pay more, but how much more?

Case 2 : Sensitivity (more serious) to changes in the Dividend/Foreign rate

Another early exercise test needs to be in place, now. Say that we start with S = 140 and F = 140 and that we have both rates equal to 0. Let us compare a European and an American option on cash. As before, they will initially bear the same price on the risk management system.

Assume that that the foreign rate goes to 20%. F goes to approximately S, roughly 1.16. The European call option will be worth roughly \$16 (assuming no time value), while the American option will be worth \$40. Why ? because the American option being a very smart option, chooses whatever fits it better, between the cash and the future, and positions itself there.

Case 3: More Complex: Sensitivity to the Slope of the Yield Curve.

Now let us assume that the yield curve has kinks it it, that it is not quite as linear as one would think. We often such niceties around year end events, when interest rates flip, etc.

As Figure 1 shows the final forward might not be the most relevant item. Any bubbling on the intermediate date would affect the value of the American option. Remember that only using the final F is a recipe for being picked-on by a shrewd operator. A risk management and pricing system that uses no full term structure would be considered greatly defective, as it would price both options at the exact same price when clearly the American put is worth more because one can lock-in the forward to the exact point in the middle -- where the synthetic underlying is worth the most. Thus using the final interest rate differential would be totally wrong.

To conclude from these examples, the American option is extremely sensitive to the interest rates and their volatility. The higher that volatility the higher the difference between the American and the European. Pricing Problems

It is not possible to price American options using a conventional Monte Carlo simulator. We can, however, try to price them using a more advanced version -or a combination between Monte Carlo and an analytical method. But the knowledge thus gained would be simply comparative.

Further results will follow. It would be great knowledge to quantify their difference, but we have nothing in the present time other than an ordinal relationship.

The Stopping Time Problem

Another non-trivial problem with American options lies in the fact that the forward hedge is unknown. It resembles the problem with a barrier option except that the conditions of termination are unknown and depend on many parameters (such as volatility, base interest rate, interest rate differential). The intuition of the stopping time problem is as follows: the smart option will position itself on the point on the curve that fits it the best.

Note that the forward maturity ladder in a pricing and risk management system that puts the forward delta in the terminal bucket is WRONG.

Conclusion

All we know is that an American option is worth more than a European. How much more is difficult to quantify.

Traders get get more bang for the complexity by trading barrier options. We suggest avoiding selling these instruments whenever possible and covering them back-to-back in the market. Their return on complexity is smaller than that of barriers, simply because they have more complexity and offer less edge.

# Squared Variations

..are very unstable compared to Absolute Deviations.

Finally they are catching up with our truism

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=929494

(courtesy Gur Huberman).