Where Quants Go Wrong - Institutional Money
Where Quants Go WrongA dozen basic lessons in commonsense forquants and risk managers and the traderswho rely on themPaul WilmottPaul Wilmott is a financial consultant, specializing in derivatives, risk management and quantitative finance. Paul is the author of Paul Wilmott Introduces Quantitative Finance (Wiley2007), Paul Wilmott On Quantitative Finance (Wiley 2006), Frequently Asked Questions inQuantitative Finance (Wiley 2006) and other financial textbooks. He has written over 100research articles on finance and mathematics.Paul Wilmott was a founding partner of the volatility arbitrage hedge fund Caissa Capitalwhich managed 170million. His responsibilities included forecasting, derivatives pricing, andrisk management.Dr Wilmott is the proprietor of www.wilmott.com, the popular quantitative finance community website, the quant magazine Wilmott and is the Course Director for the Certificate inQuantitative Finance www.7city.com/cqf. The CQF teaches how to apply mathematics tofinance, focusing on implementation and pragmatism, robustness and transparency.c Paul Wilmott 1
Quants and risk managers keep making the same mistakes overand over again. Each time this happens the losses increase invalue. A small amount of commonsense and some basic mathematics can stop this happening. However, this requires quants,risk managers and their bosses to move their attention away fromcomplexity towards robustness of models. Keywords: Jensen’s inequality; Sensitivity to parameters; Correlation; Diversification; Dynamic hedging; Feedback; Supply anddemand; Closed-form solutions; Calibration; Modelling; Precision; Nonlinearity; CSI Miamic Paul Wilmott Thispaper was borne out of a CQF lecture (www.7city.com/cqf) and aseries of blogs (www.wilmott.com/blogs/paul).2
Topics Lack of diversification Supply and demand Jensen’s inequality arbitrage Sensitivity to parameters Correlation Reliance on continuous hedging (arguments) Feedback Reliance on closed-form solutions Valuation is not linear Calibration Too much precision Too much complexityc Paul Wilmott 3
But first a simple test-yourself quiz.c Paul Wilmott 4
QuizQuestion 1: What are the advantages of diversification amongproducts, or even among mathematical models?Question 2: If you add risk and curvature what do you get?Question 3: If you increase volatility what happens to the valueof an option?Question 4: If you use ten different volatility models to valuean option and they all give you very similar values what can yousay about volatility risk?Question 5: One apple costs 50p, how much will 100 applescost you?c Paul Wilmott 5
Lesson 1: Lack of DiversificationExample: It’s your first day as a trader in a bank. You’re freshout of an Ivy League Masters program. You’re keen and eager,you want to do the best you can in your new job, you want toimpress your employer and make your family proud. So, what doyou trade? What strategies should you adopt? Having been welleducated in theoretical finance you know that it’s important todiversify, that by diversifying you can increase expected returnand decrease your risk. Let’s put that into practice. Traders coin tossing Banks copying each otherc Paul Wilmott 6
Keynes said, “It is better to fail conventionally than to succeedunconventionally.”There is no incentive to diversify while you are playing with OPM(Other People’s Money).Example: Exactly the same as above but replace ‘trades’ with‘models.’ There is also no incentive to use different models fromeveryone else, even if your are better.c Paul Wilmott 7
There’s a timescale issue here as well. Anyone can sell deepOTM puts for far less than any ‘theoretical’ value, not hedgethem, and make a fortune for a bank, which then turns into abig bonus for the individual trader. You just need to be able tojustify this using some convincing model. Eventually things willgo pear shaped and you’ll blow up. However, in the meantime everyone jumps onto the same (temporarily) profitable bandwagon,and everyone is getting a tidy bonus. The moving away from unprofitable trades and models seems to be happening slower thanthe speed at which people are accumulating bonuses from saidtrades and models!c Paul Wilmott 8
Envy:We all know of behavioural finance experiments such as the following two questions.First question, people are asked to choose which world theywould like to be in, all other things being equal, World A orWorld B whereA. You have 2 weeks’ vacation, everyone else has 1 weekB. You have 4 weeks’ vacation, everyone else has 8 weeksThe large majority of people choose to inhabit World B. Theyprefer more holiday to less in an absolute sense, they do notsuffer from vacation envy.c Paul Wilmott 9
But then the second question is to choose between World A andWorld B in whichA. You earn 50,000 per year, others earn 25,000 on averageB. You earn 100,000 per year, others earn 200,000 on averageGoods have the same values in the two worlds. Now most peoplechoose World A, even though you won’t be able to buy as much‘stuff’ as in World B. But at least you’ll have more ‘stuff’ thanyour neighbours. People suffer a great deal from financial envy.In banking the consequences are that people feel the need to dothe same as everyone else, for fear of being left behind. Again,diversification is just not in human nature.c Paul Wilmott 10
Lesson 2: Supply and DemandSupply and demand is what ultimately drives everything! Butwhere is the supply and demand parameter or variable in Black–Scholes?A trivial observation: The world is net long equities after youadd up all positions and options. So, net, people worry aboutfalling markets. Therefore people will happily pay a premiumfor out-of-the-money puts for downside protection. The resultis that put prices rise and you get a negative skew. That skewcontains information about demand and supply and not aboutthe only ‘free’ parameter in Black–Scholes, the volatility.The complete-market assumption is obviously unrealistic, andimportantly it leads to models in which a small number of parameters are used to capture a large number of effects.c Paul Wilmott 11
The price of milk is a scalar quantity that has to capture in a single number all the behind-the-scenes effects of, yes, production,but also supply and demand, salesmanship, etc. Perhaps thepint of milk is even a ‘loss leader.’ A vector of inputs producesa scalar price.So, no, you cannot back out the cost of production from a singleprice.Similarly you cannot back out a precise volatility from the price ofan option when that price is also governed by supply anddemand, fear and greed, not to mention all the imperfectionsthat mess up your nice model (hedging errors, transactioncosts, feedback effects, etc.).c Paul Wilmott 12
Supply and demand dictate prices, assumptions and models impose constraints on the relative prices among instruments. Thoseconstraints can be strong or weak depending on the strength orweakness of the assumptions and models.c Paul Wilmott 13
Lesson 3: Jensen’s Inequality ArbitrageJensen’s Inequality states that if f (·) is a convex function and xis a random variable thenE [f (x)] f (E[x]) .This justifies why non-linear instruments, options, have inherentvalue.c Paul Wilmott 14
Example: You roll a die, square the number of spots you get,and you win that many dollars. How much is this game worth?(Assuming you expect to break even.) We know that the averagenumber of spots on a fair die is 3 12 but the fair ‘price’ for thisbet is not 2132 .For this exercise f (x) is x2, it is a convex function. SoE[x] 3 12and 21f (E[x]) 3 2 12 14.ButE [f (x)] 1 4 9 16 25 36 15 1 f (E[x]) .66The fair price is 15 16.c Paul Wilmott 15
Jensen’s inequality and convexity can be used to explain therelationship between randomness in stock prices and the valueinherent in options, the latter typically having some convexity.Suppose that a stock price S is random and we want to considerthe value of an option with payoff P (S).If the payoff is convex thenE[P (ST )] P (E[ST ]).c Paul Wilmott 16
We can get an idea of how much greater the left-hand side isthan the right-hand side by using a Taylor series approximationaround the mean of S. WriteS S̄ ,where S̄ E[S], so E[ ] 0. Then 2 E [f (S)] E f (S̄ ) E f (S̄) f (S̄) 12 f (S̄) · · · 1 f (S̄) 2 f (S̄)E 2 f (E[S]) 1f2 (E[S])E 2 .Therefore the left-hand side is greater than the right by approximately1 f (E[S]) E2 2 .c Paul Wilmott 17
This shows the importance of two concepts f (E[S]): This is the convexity of an option. As a rule thisadds value to an option. It also means that any intuition wemay get from linear contracts (forwards and futures) mightnot be helpful with non-linear instruments such as options. 2 : This is the variance of the return on the random un- Ederlying. Modelling randomness is the key to valuing options.The lesson to learn from this is that whenever a contract hasconvexity in a variable or parameter, and that variable or parameter is random, then allowance must be made for this in thepricing.c Paul Wilmott 18
Example: Anything depending on forward rates. If you price afixed-income instrument with the assumption that forward ratesare fixed (the deterministic models of yield, duration, etc.) andthere is some nonlinearity in those rates then you are missingvalue. How much value depends on the convexity with respectto the forward rates and forward rate volatility. Example: Some things are tricky to model and so one tends toassume they are deterministic. Mortgage-backed securities havepayoffs, and therefore values, that depend on prepayment. Oftenone assumes prepayment to be a deterministic function of interest rates, this can be dangerous. Try to quantify the convexitywith respect to prepayment and the variance of prepayment.c Paul Wilmott By‘convexity with respect to forward rates’ I do not mean the curvature inthe forward rate curve, I mean the second derivative of the contract withrespect to the rates.19
Lesson 4: Sensitivity To ParametersIf volatility goes up what happens to the value of an option?Did you say the value goes up? Oh dear, bottom of the class foryou! I didn’t ask what happens to the value of a vanilla option,I just said “an” option, of unspecified terms. Pricing an up-and-out call option Early-morning panic Vega Getting firedc Paul Wilmott 20
What went wrong was that you assumed volatility to be constant in the option formula/model and then you changed thatconstant. This is only valid if you know that the parameter isconstant but are not sure what that constant is. But that’s nota realistic scenario in finance. In fact, I can only think of onescenario where this makes sense. . . A hot tip from Godc Paul Wilmott 21
By varying a constant parameter you are effectively measuring V. parameterThis is what you are doing when you measure the ‘greek’ vega:vega V. σBut this greek is misleading. Those greeks which measure sensitivity to a ‘variable’ are fine, those which supposedly measuresensitivity to a ‘parameter’ are not. Plugging different constantsfor volatility over the range 17% to 23% is not the same as examining the sensitivity to volatility when it is allowed to roam freelybetween 17 and 23% without the constraint of being constant.I call such greeks “bastard greeks” because they are illegitimate.c Paul Wilmott 22
An example:2.5Barrier value, various vol scenarios2.01.523% volV17% vol1.00.50.0020406080100120140SThe value of some up-and-out call option using volatilities 17%and 23%.c Paul Wilmott 23
The problem arises because this option has a gamma that changessign.The relationship between sensitivity to volatility and gamma isbecause they always go together. In the Black–Scholes equationwe have a term of the form21σ S2 V .2 2 S 2The bigger this combined term is, the more the option is worth.But if gamma is negative large volatility makes this big in absolute value, but negative, so it decreases the option’s value.c Paul Wilmott 24
2.5Barrier value, various vol scenarios2.0Worst price, 17% vol 23%Best price, 17% vol 23%23% vol17% volV1.51.00.50.0020406080100120140SUncertain volatility model, best and worst cases.c Paul Wilmott 25
Meaninglessness of implied volatility:0.8Barrier value versus constant latilityValue versus constant volatility.c Paul Wilmott 26
Lesson 5: Correlation Quant toolboxWhen we think of two assets that are highly correlated then weare tempted to think of them both moving along side by sidealmost. Surely if one is growing rapidly then so must the other?This is not true.c Paul Wilmott 27
40Perfect positive correlation35AB30Asset value2520151050051015202530TimeTwo perfectly correlated assets.c Paul Wilmott 28
And if two assets are highly negatively correlated then they goin opposite directions? No, again not true.40Perfect negative correlation35AC30Asset value2520151050051015202530TimeTwo perfectly negatively correlated assets.c Paul Wilmott 29
If we are modelling using stochastic differential equations thencorrelation is about what happens at the smallest, technicallyinfinitesimal, timescale. It is not about the ‘big picture’ direction.This can be very important and confusing. For example, if weare interested in how assets behave over some finite time horizonthen we still use correlation even though we typically don’t careabout short timescales only our longer investment horizon (atleast in theory).However, if we are hedging an option that depends on two ormore underlying assets then, conversely, we don’t care aboutdirection (because we are hedging), only about dynamics overthe hedging timescale. The use of correlation may then be easierto justify. But then we have to ask how stable is this correlation.So when wondering whether correlation is meaningful in anyproblem you must answer two questions (at least), one concerning timescales (investment horizons or hedging period) andanother about stability.c Paul Wilmott 30
Running shoes500RegimesAB450Regime 3400Regime 4350Regime 2Asset value300250Regime 120015010050002468101214TimeTwo assets, four regimes.c Paul Wilmott 31
As you can see, the dynamics between just two companies canbe fascinating. And can be modelled using all sorts of interestingmathematics. One thing is for sure and that is such dynamicswhile fascinating are certainly not captured by a correlation of0.6!c Paul Wilmott 32
Example: Synthetic CDOs suffer from problems with correlation. People typically model these using a copula approach, andthen argue about which copula to use. Finally because thereare so many parameters in the problem they say “Let’s assumethey are all the same!” Then they vary that single constantcorrelation to look for sensitivity (and to back out implied correlations). Where do I begin criticizing this model? Let’s saythat just about everything in this model is stupid and dangerous.The model does not capture the true nature of the interactionbetween underlyings, correlation never does, and then makingsuch an enormously simplifying assumption about correlation isjust bizarre. (I grant you not as bizarre as the people who lapthis up without asking any questions.)c Paul Wilmott 33
Synthetic CDO90807060Price50Equity tranche 0-3%Junior3-6%Mezzanine %45%CorrelationVarious tranches versus correlation.c Paul Wilmott 34
Lesson 6: Reliance on Continuous Hedging (Arguments)One of the most important concepts in quantitative finance isthat of delta or dynamic hedging. This is the idea that you canhedge risk in an option by buying and selling the underlying asset.This is called delta hedging since ‘delta’ is the Greek letter usedto represent the amount of the asset you should sell. Classicaltheories require you to rebalance this hedge continuously. Insome of these theories, and certainly in all the most popular,this hedging will perfectly eliminate all risk. Once you’ve got ridof risk from your portfolio it is easy to value since it should thenget the same rate of return as putting money in the bank.c Paul Wilmott 35
This is a beautiful, elegant, compact theory, with lots of important consequences. Two of the most important consequences(as well as the most important which is. . . no risk!) are that,first, only volatility matters in option valuation, the direction ofthe asset doesn’t, and, second, if two people agree on the levelof volatility they will agree on the value of an option, personalpreferences are not relevant.The assumption of continuous hedging seems to be crucial tothis theory. But is this assumption valid?c Paul Wilmott 36
The figure shows a comparison between the values of an at-themoney call, strike 100, one year to expiration, 20% volatility,5% interest rate, when hedged at fixed intervals (the red line)and hedged optimally (the green line). The lines are the meanvalue plus and minus one standard deviation. All curves converge to the Black-Scholes complete-market, risk-neutral, priceof 10.45, but hedging optimally gets you there much faster. Ifyou hedge optimally you will get as much risk reduction fromjust 10 rehedges as if you use 25 equally spaced rehedges.c Paul Wilmott 37
1412Mean /- SD1086Mean /- 1 SD, optimal dynamic hedgingMean /- 1 SD, hedging at fixed intervals4200510152025Number of hedgesRisk reduction when hedging discretely.c Paul Wilmott 38
From this we can conclude that as long as people know the bestway to dynamically hedge then we may be able to get away withusing risk neutrality even though hedging is not continuous.But do they know this?Everyone is brought up on the results of continuous hedging, andthey rely on them all the time, but they almost certainly do nothave the necessary ability to make those results valid!c Paul Wilmott 39
Lesson 7: FeedbackAre derivatives a good thing or a bad thing? Their origins arein hedging risk, allowing producers to hedge away financial riskso they can get on with making pork bellies or whatever. Nowderivatives are used for speculation, and the purchase/sale ofderivatives for speculation outweighs their use for hedging.Does this matter? We know that speculation with linear forwardsand futures can affect the spot prices of commodities, especiallyin those that cannot easily be stored. But what about all thenew-fangled derivatives that are out there?c Paul Wilmott 40
A simplistic analysis would suggest that derivatives are harmless,since for every long position there is an equal and opposite shortposition, and they cancel each other out. But this misses theimportant point that usually one side or the other is involved insome form of dynamic hedging used to reduce their risk. Oftenone side buys a contract so as to speculate on direction of theunderlying. The seller is probably not going to have exactly theopposite view on the market and so they must hedge away riskby dynamically hedging with the underlying. And that dynamichedging using the underlying can move the market. This is thetail wagging the dog!c Paul Wilmott 41
There are two famous examples of this feedback effect: Convertible bonds—volatility decrease 1987 crash and (dynamic) portfolio insurance—volatility increasec Paul Wilmott 42
1400.3AssetRealized vol1200.251000.20.15VolatilityAsset price80600.1400.052000012345678910TimeSimulation when hedging long gamma.c Paul Wilmott 43
0.8140AssetRealized vol0.71200.61000.50.4VolatilityAsset price80600.3400.2200.100012345678910TimeSimulation when hedging short gamma.c Paul Wilmott 44
Lesson 8: Reliance on Closed-form SolutionsExample: You need to value a fixed-income contract and soyou have to choose a model. Do you (a) analyze historical fixedincome data in order to develop an accurate model, this is thensolved numerically, and finally back tested using a decade’s worthof past trades to test for robustness, or (b) use Professor X’smodel because the formulæ are simple and, quite frankly, youdon’t know any numerical analysis, or (c) do whatever everyoneelse is doing? Typically people will go for (c), partly for reasonsalready discussed, which amounts to (b).c Paul Wilmott 45
Example: You are an aeronautical engineer designing a newairplane. Boy, those Navier–Stokes equations are hard! How doyou solve non-linear equations? Let’s simplify things, after allyou made a paper plane as a child, so let’s just scale things up.The plane is built, a big engine is put on the front, it’s filledwith hundreds of passengers, and it starts its journey along therunway. You turn your back, without a thought for what happensnext, and start on your next project.One of those examples is fortunately not real. Unfortunately, theother is.c Paul Wilmott 46
Quants love closed-form solutions. The reasons are1. Pricing is faster2. Calibration is easier3. You don’t have to solve numericallyc Paul Wilmott 47
Popular examples of closed-form solutions/models are, in equityderivatives, the Heston stochastic volatility model, and in fixedincome, Vasicek, Hull & White, etc.Models with closed-form solutions have several roles in quantitative finance. Closed-form solutions are useful for preliminary insight good for testing your numerical scheme before going on tosolve the real problem for examining second-year undergraduate mathematiciansc Paul Wilmott Tobe fair to Vasieck I’m not sure he ever claimed he had a great model,his paper set out the general theory behind the spot-rate models, with whatis now known as the Vasicek model just being an example.48
Lesson 9: Valuation is Not LinearYou want to buy an apple, so you pop into Waitrose. An applewill cost you 50p. Then you remember you’ve got friends comingaround that night and these friends really adore apples. Maybeyou should buy some more? How much will 100 apples cost?c Paul Wilmott 49
Here’s a quote from a well-known book: “The change of numeraire technique probably seems mysterious. Even though onemay agree that it works after following the steps in the chapter,there is probably a lingering question about why it works. Theauthor’s opinion is that it may be best simply to regard it as a‘computational trick’. Fundamentally it works because valuationis linear. . . . The linearity is manifested in the statement thatthe value of a cash flow is the sum across states of the world ofthe state prices multiplied by the size of the cash flow in eachstate. . . . After enough practice with it, it will seem as naturalas other computational tricks one might have learned.”Note it doesn’t say that linearity is an assumption, it is casuallytaken as a fact. Valuation is apparently linear. Now there’ssomeone who has never bought more than a single apple!c Paul Wilmott 50
Example: The same author may be on a sliding royalty scale sothat the more books he sells the bigger his percentage. How cannonlinearity be a feature of something as simple as buying applesor book royalties yet not be seen in supposedly more complexfinancial structured products?c Paul Wilmott 51
Example:A bank makes a million dollars profit on CDOs.Fantastic! Let’s trade 10 times as much! They make 10million profit. The bank next door hears about this and decidesit wants a piece of the action. They trade the same size. Between the two banks they make 18million profit. Where’d the 2million go? Competition between them brings the price andprofit margin down. To make up the shortfall, and because ofsimple greed, they increase the size of the trades. Word spreadsand more banks join in. Profit margins are squeezed. Totalprofit stops rising even though the positions are getting biggerand bigger. And then the inevitable happens, the errors in themodels exceed the profit margin (margin for error), and betweenthem the banks lose billions. “Fundamentally it works becausevaluation is linear.” Oh dear!c Paul Wilmott 52
To appreciate the importance of nonlinearity you have to understand that there is a difference between the value of a portfolioof contracts and the sum of the values of the individual contracts. In pseudomath, if the problem has been set up properly,you will getValue(A B) Value(A) Value(B).c Paul Wilmott WhatI call, to help people remember it, the ‘Beatles effect.’ The Fab Fourbeing immeasurably more valuable as a group than as individuals. . . Wings,Thomas the Tank Engine,. . .53
Static hedging of a barrier optionc Paul Wilmott 54
Here is a partial list of the advantages to be found in somenon-linear models. Perfect calibration Speed Easy to add complexity to the model Optimal static hedging Can be used by buy and sell sidesc Paul Wilmott 55
Reading list: Hoggard, Whalley & Wilmott (1994) on costs Avellaneda, Levy & Parás (1995) on the Uncertain VolatilityModel (UVM) In Hua & Wilmott on modelling crashes and CrashMetrics In Ahn & Wilmott on mean-variance pricing and hedgingc Paul Wilmott 56
Lesson 10: Calibration“A cynic is a man who knows the price of everything and thevalue of nothing,” said Oscar Wilde.Example: Wheels cost 10 each. A soapbox is 20. How muchis a go-cart? The value is 60.Price?Worth?c Paul Wilmott 57
The WWVNc Paul Wilmott 58
inverse problems frozen fish CSI Miami crystal ballsc Paul Wilmott 59
Problems with calibration: Over fitting. You lose important predictive information ifyour model fits perfectly. The more instruments you calibrateto the less use that model is. Fudging hides model errors: Perfect calibration makes youthink you have no model risk, when in fact you probablyhave more than if you hadn’t calibrated at all. Always unstable. The parameters or surfaces always changewhen you recalibrate. Confusion between actual parameter values and those seenvia derivatives. For example there are two types of creditrisk, the actual risk of default and the market’s perceivedrisk of default.c Paul Wilmott 60
Why is calibration unstable? Market Price of Riskc Paul Wilmott 61
AR-15-20Market price of interest rate risk versus time.c Paul Wilmott 62
When you calibrate you are saying that whatever the marketsentiment is today, as seen in option prices, is going to pertainforever. So if the market is panicking today it will always panic.But the figure shows that such extremes of emotion are shortlived. And so if you come back a week later you will now becalibrating to a market that has ceased panicking and is perhapsnow greedy!Calibration assumes a structure for the future that is inconsistentwith experience, inconsistent with common sense, and that failsall tests.c Paul Wilmott 63
Lesson 11: Too much precisionGiven all the errors in the models, their unrealistic assumptions,and the frankly bizarre ways in which they are used, it is surprisingthat banks and funds make money at all! Zero sum Who owns the worldc Paul Wilmott 64
1. There is demand for some contract, real or perceived2. The contract must be understood in terms of risk, valuation,potential market, profit, etc.3. A deal gets done with an inbuilt profit margin4. The contract is then thrown into some big pot with lots ofother contracts and they are risk managed together5. A profit is accrued, perhaps marking to model, or perhaps atexpirationc Paul Wilmott 65
Stages 2 and 4 are inconsistent.c Paul Wilmott 66
The point of this lesson is to suggest that more effort is spenton the benefits of portfolios than on fiddly niceties of modellingto an obsessive degree of accuracy. Accept right from the startthat the modelling is going to be less than perfect. It is not truethat one makes money from every instrument because of theaccuracy of the model. Rather one makes money on averageacross all instruments despite the model. These observationssuggest to me that less time should be spent on dodgy models, meaninglessly calibrated, but more time on models that areaccurate enough and that build in the benefits of portfolios.c Paul Wilmott 67
Some models are better than others. Sometimes even workingwith not-so-good models is not too bad. To a large extent whatdetermines the success of models is the type of market. Let megive some examples.Equity, FX and commodity markets: Here the models areonly so-so. There has been a great deal of research on improving these models, although not necessarily productive work.Combine less-than-brilliant models with potentially very volatilemarkets and exotic, non-transparent, products and the result canbe dangerous. On the positive side as long as you diversify acrossinstruments and don’t put all your money into one basket thenyou should be ok.c Paul Wilmott 68
Fixed-income markets: These models are pretty dire. So youmight expect to lose (or make) lots of money. Well, it’s notas simple as that. There are two features of these marketswhich make the dire modelling less important, these are a) theunderlying rates are not very volatile and b) there are plenty ofhighly liquid vanilla instruments with which to try to hedge modelrisk. (I say “try to” because most model-risk hedging is really afudge, inconsistent with the framework in which it is being used.)c Paul Wilmott 69
Correlation markets: Oh, Lord! Instruments whose pricingrequires input of correlation (FI excepted, see above) are accidents waiting to happen. The dynamic relationship between justtwo equities can be beautifully complex, and certainly never tobe captured by a single number, correlation. Fortunately theseinstruments tend not to be bought or sold in non-diversified,bank-destroying quantities. (Except for CDOs, of course!)c Paul Wilmott 70
Credit markets: Single-name instruments are not too bad.Again problems arise with any instrument that has multiple ‘underlyings,’ so the credit derivatives based on baskets. . . you knowwho you are. Bu
Paul Wilmott Paul Wilmott is a financial consultant, specializing in derivatives, risk managementand quan-titative finance. Paul is the author of Paul Wilmott Introduces Quantitative Finance (Wiley 2007), Paul Wilmott On Quantitative Finance (Wiley 2006), Frequently Asked Questions in Quantitative Finance (Wiley 2006) and other financial .
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