Andreasen - Derivatices He View From the Trenches

Derivatives -- the View from the Trenches

October 2003

Jesper AndreasenHead of Product DevelopmentNordea Markets, Copenhagen

2

Outline

� Background

� The first fundamental theorem of derivatives trading

� The second fundamental theorem of derivatives trading

� The difference between P and Q

� Arbitrage and efficiency

� Model philosophy � Models that work at work

� Conclusion

3

My Life as a Quant

� CV:

- 1997: PhD, Department of Operations Research, Instituteof Mathematics, Aarhus University.

- 1997-1997: Senior Analyst, Bear Stearns, London.

- 1997-2000: Vice President, Quantitative ResearchDepartment, General Re Financial Products, London.

- 2000-2002: Principal, Head of Quantitative Research atBank of America, London.

- 2002-present: Head of Product Development Group inNordea Markets, Copenhagen.

� Job description

- Develop and implement pricing and hedging models forexotic derivatives.

- Including: interest rates, credit derivatives, equity, foreignexchange and hybrids.

4

� Career highlights

- Solved the passport option problem and implemented theworld's first live model. Did 10 trades.

- Developed Gen Re's Japanese "Power Reverse Dual"business. 350 20-30y trades on 40 Pentium CPUs. Takes6h to run risk reports.

- Sorted out the modeling of Band of America's 1500 tradeportfolio of Bermudan swaptions.

- 2001 Risk Magazine Quant-of-the-Year.

� Career the-opposite

- Realising my first ever serious screw up (USD -450,000).

- Warren Buffet closing down Gen Re Finanical Products.

� Purpose of this talk: to entertain with what I've learned so far.

5

The First Fundamental Theorem of Derivatives Trading

� We assume zero rates and dividends and standard frictionlessmarkets.

� Assume that the underlying stock evolves continuously.

� So there exists two stochastic processes ,� � so that

( ) ( ) ( ) ( )( )

dS t t dt t dW tS t

� �� �

under the real measure P .

� Let V be the value of an option book on S priced on a modelwith constant volatility � .

� Assume that the book is delta hedged, i.e. we dynamicallytrade the stock to keep

0SV �

� Theorem 1: The value of the option book evolves according

2 2 21( ) ( ( ) ) ( ) ( )2 SSdV t t S t V t dt� �� �

� Proof: Follows from ( , ( ))V V t S t� and Ito's lemma that

2

2 2

12

12

t S SS

t SS

dV V dt V dS V dS

V dt V S dt�

� � �

� �

Using 2 2102t SSV S V�� � we get the result

6

Theorem 1 and Trading

� If you're gamma long, 0SSV � , and realised volatility is higherthan pricing volatility you make money.

� The option trader's job is really about balancing realisedagainst implied (or pricing) volatility.

realised vol > implied vol => go long gammarealised vol < implied vol => go short gamma

� In essence this is what all trading is about: buy low -- sellhigh.

� In practice it is of course not that easy to predict how realisedand implied volatility are going to relate to each other over agiven period.

� In the context of the derivation of the Black-Scholes' formula,one can see Theorem 1 as an investigation of the self-financing condition.

7

Theorem 1 and Option Markets

� Black-Scholes implied volatility has to satisfy

2 2 20

1[ ( ( ) ) ( ) ( ) ] 02

TQSSimpliedE u S u V u du� �� ��

� So implied volatility is a weighted average of Q expectedvolatility.

� Consider an option seller that delta hedges his short optionposition, i.e. 0SSV � , with the implied volatility. His totalprofit

2 2 20

1( ) (0) ( ( ) ) ( ) ( )2

TSSimpliedV T V t S t V t dt� �� � ��

is positive if

( )implied t� ��

"most of the time".

� If option sellers are risk averse (and they are) it is unlikelythat they are willing to be short gamma without taking apremium. We should expect

implied volatility > historical volatility

� Implied volatility gaps for big ccys and index

- Rate options: 0.5-1.0%- Equity options: 3.0-5.0%- Foreign exchange: 0.5-1.0%

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� The implied volatility gap in equities is a bit higher than inrates and fx. The reason for this is jumps but we will return tothis.

� The difference in historical and implied volatility does notindicate that there is an arbitrage -- just that there is a riskpremium on volatility.

� Ie, volatility is stochastic and market participants are riskaverse.

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Theorem 1 and Models

� Theorem 1 extends to other parameters such as for examplecorrelation, and more sophisticated models than the constantvolatility Black-Scholes model, generally

2 2

,

1 ( )2 i jx x ij ij

i jdV V dt� �� ��

where ( )ix are your risk factors and i j ijdx dx dt�� � .

� The general points is that if your model , ( )ij� , is misspecifiedyou are going to make daily

( )O dt

losses, not ( )O dW .

� So contrary to common belief: bad models cause bleeding --not blow-ups.

� So it may take a while before you realise that your model iswrong.

� That is, you can pile up a lot of trades, make a lot of "modelvalue", and many fine bonuses can be paid, before you realisethat your book is bleeding.

� …and this has (involuntarily) been discovered by manybanks:

- UBS remark of IPS portfolio in the early 90s.

- Commerzbank (and other's) remark of Bermudan swaptionbook in the late 90s.

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- Interest rate option problems at NatWest (and other places)in mid 90s.

- Several banks' current problem with exotic equity optionbooks.

- …

� All sad examples of taking it a bit too easy on the modelfront.

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Theorem 1 and Technical Results

� A lot of technical results can be derived from the Theorem 1:

� For example, set 0� � and let

log0

log0

1( ) ln ( ) ( )( )

contractpriced at delta hedge

of contractat

V t S t S tS t

�

�

�

�

� � ������

�����

then

20 0

1 1ln ( ) ln (0) ( ) ( )2 ( )

T TS T S u du dS u

S u�� � � �� �

� Hence a contract paying the realised variance can bereplicated with a simple delta strategy combined with acontract paying ln ( )S T , which in turn can be staticallyreplicated with positions in European options.

� Another example is 0� � and

( )

00

( ) ( ( ) ) 1 ( )S t Kcall option delta hedgepriced at of call option

at

V t S t K S t

�

�

�

�

�

�

� � � ��������

�������

� Theorem 1 and a few manipulations yield

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( , )[ ( ) | ( ) ] 2( , )

Q T

KK

C T KE t S t KK C T K

� � �

where

( , ) [( ( ) ) ]QC T K E S T K �� �

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� Most of this sort of hocus-pocus was derived by BrunoDupire in early 90s, but has received far too little recognitionand attention in textbooks and academic circles.

� The French banks produce a lot of very good quants and theyare all breed on a solid dose of Theorem 1 and all thecorollaries.

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Theorem 2: The Gospel of the Jump

� You can get a long way of understanding the world withTheorem 1, but there is one thing missing and that is jumps.

� Suppose the stock evolves according to

( ) ( ) ( ) ( ) ( ) ( )( )

dS t t dt t dW t I t dN tS t

� �� � �

�

where ,� � are stochastic processes and N is a Poissonprocess with stochastic intensity ( )t� , and the jump size isstochastic with distribution given by

( )~ ( ; )I t t� �

� Suppose we price according to the model

( ) ( ) ( ) ( )( )

dS t dW t I t dN t mdtS t

� �� � �

�

under Q , where

[ ( ) 1] , ( )~ ( ), [ ( )]QQ dN t dt I t m E I t� �� � � �

� Theorem 2: A delta neutral trading book will accumulategains at

2 2 21 ( ) [ ]2

QSSdV S V dt VdN E V dt� � �� � �� � �

Proof: Using the pricing equation

2 210 [ ]2

Qt S SSV mSV S V E V� � �� � � � �

and Ito's lemma yields the result.

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Theorem 2 and Trading

� This tells us that if realised volatility is the same as pricingvolatility and if we're gamma short ( 0SSV � ) and delta neutral( 0SV � ) then 0V� � and thereby

[ ] 0QdV VdN E V dt�� � � � �

� So on a short option book you can sit and collect "jumppremium" and look like an absolute hero -- until a jumpoccurs.

� This is essentially what is called "picking up pennies in frontof a steam engine" or "the trader's option".

� No matter how people look at this themselves, this isessentially the strategy that a lot of hedge funds follow.

� A couple of spectacular examples:

- The blow up of LOR and portfolio insurance industry in1987.

- The collapse of CRT in 1987.

- The blow up of LTCM in 1998.

- …

� Many of these funds had prominent academics involved, sofollowing the actual disasters we heard a lot of good storieslike

- "19 standard deviation event..."- "Liquidity squeeze…" - …

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� But in essence the strategies involved were more or lessasking for it:

- Delta hedging a put option notional of USD 70bn.

- Hedging S&P vol with Microsoft puts bought fromMicrosoft.

- Buying half of the Danish mortgage bonds.

- …

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Theorem 2 and the Equity Derivatives Markets

� The implied volatility has to satisfy

2 2 2, 00

, 00

0 [ ( ) (0)]1 [ ( ( ) ) ( ) ( ) | ]2

[ ( ) | ( )]

Q

TQSS

TQ

E V T V

E u S u V u du

E V u dN u du

� �

� �

� ��

�

� �

� �

� �

�

�

where Q in this case is the "market" risk neutral measure.

� Time for a few (very) rough calculations: Set lnV S� , that iswe hold a log-contract. We get

2 2

2 2

2 2 2

2 2 2

1 10 ( ) ( [ ln ] [ ])21 ( ) ( [ln(1 )] [ ])21 1( )2 2

( )

Q Q

Q Q

E S E SS

E I E I

m

m

� � �

� � �

� � �

� � �

� � � � � � �

� � � � � �

� � � �

� � �

� So if for S&P 500

- Implied volatility is 0.20� �

- Historical volatility is 0.17� �

then we have

� m0.1 +/-33%1 +/-11%

10 +/-3%

� Quiz: which one?

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Theorem 2 and the Equity Volatility Smile

� To get the answer to our quiz we need to look at the impliedBlack volatility smile (or skew rather) in S&P500 options.

� The model uses 0.15, 0.18, 0.30, 0.15m� � �� � � � � .

� …and it provides a very good fit to the market -- full data:

expiry\strike 75% 80% 85% 90% 95% 100% 105% 110% 115% 120%obs-1m 41.72% 36.82% 32.05% 27.38% 22.91% 19.15% 17.87% 19.06% 21.16% 23.51%obs-3m 32.34% 29.51% 26.78% 24.15% 21.67% 19.42% 17.61% 16.54% 16.30% 16.65%obs-6m 28.76% 26.73% 24.79% 22.94% 21.19% 19.59% 18.19% 17.07% 16.30% 15.92%obs-1y 26.08% 24.67% 23.33% 22.06% 20.88% 19.78% 18.79% 17.92% 17.20% 16.64%obs-2y 23.77% 22.91% 22.10% 21.35% 20.64% 19.98% 19.37% 18.81% 18.29% 17.84%obs-3y 22.87% 22.24% 21.64% 21.08% 20.56% 20.07% 19.61% 19.19% 18.79% 18.42%mdl-1m 44.96% 38.20% 30.84% 23.91% 20.24% 19.15% 18.80% 18.65% 18.57% 18.53%mdl-3m 33.97% 30.03% 26.00% 22.68% 20.57% 19.42% 18.79% 18.42% 18.19% 18.03%mdl-6m 30.18% 27.70% 25.12% 22.77% 20.91% 19.59% 18.69% 18.07% 17.63% 17.32%mdl-1y 26.40% 24.98% 23.50% 22.09% 20.83% 19.78% 18.93% 18.26% 17.73% 17.31%mdl-2y 24.05% 23.32% 22.53% 21.69% 20.83% 19.98% 19.17% 18.43% 17.77% 17.18%mdl-3y 22.75% 22.24% 21.73% 21.19% 20.64% 20.07% 19.49% 18.92% 18.36% 17.83%

15.00%

17.00%

19.00%

21.00%

23.00%

25.00%

27.00%

75.00%

80.00%

85.00%

90.00%

95.00%

100.00%

105.00%

110.00%

115.00%

120.00%

obs-1ymdl-1yobs-2ymdl-2yobs-3ymdl-3y

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Theorem 2 and Risk Aversion

� Clearly the parameters

0.18, 0.30m� � � �

seem extreme. Does the market really expect market crashesof -30% every fifth year?

� It is important to note, however, that these parameters are notthe historical parameters -- the are market Q measureparameters and therefore include a healthy dose of riskpremium.

� Suppose

- Historical jump intensity of 0.02� � .- Historical mean jump of 0.25m � � .

� Then in equilibrium the market jump parameters are afunction of the relative risk aversion (RRA) of therepresentative agent:

RRA � m �

1 3% -27% 15%2 4% -28% 15%5 8% -33% 15%7 15% -36% 15%

10 36% -40% 15%

� So we are in RRA = 5-7 territory which by no means isextreme.

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� So in front of you, you have an option quant who uses utilityfunctions and relative risk aversion to consider variousproblems -- probably not what you expected.

� The main problem in using utility theory for riskconsiderations is not that you can't get people to specify theirrelative risk aversion. The real problem is that all utilitytheory depends crucially on the P distribution and that isalmost impossible to get with any decent accuracy.

� If jumps happen twice every century, estimating the meanjump and standard deviation is going to be quite difficult.

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The Difference between P and Q

� Suppose we know that under P

dS dt dWS � �� �

� and assume that 0.17� � . Then we wish to estimate the driftas

21 ( ) 1ˆ (ln )(0) 2

S t tt S

� �� �

� We have

ˆ[ ]stdt�

� �

� This gives us the following table

horizon std[ �̂ ]1 17.0%

10 5.4%20 3.8%50 2.4%

100 1.7%200 1.2%400 0.9%

� To get within 1% error you need about 400 years of data.

� So with absence very very long time series of data it isextremely difficult to estimate the real distribution.

� This of course has the consequence that there essentially areas many P measures as there are agents in the economy!

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The Difference between P and Q and Market Efficiency

� The difficulty of obtaining the "true" P measure aside, it isclear that the market generally prices in significant riskpremia.

� There are tons of examples:

- Implied volatility > Historical volatility.

- Equity volatility skew.

- Corporate bond spreads are much wider than historicaldefault probabilities.

- The long end of the yield curve is far to volatile to beconsistent with the historical mean reversion of interestrates.

- Long maturity volatilities are far to volatile to beconsistent with historical mean reversion of volatility.

� This does not mean that the market is inefficient.

� On the contrary it means that the market is generally efficientbut that there are significant risk premia.

� So the market is risk averse and financial theory works!

� What hedge funds do is to a 99% extend to collect riskpremia.

� So hedge funds generally run in front of the steam engines!

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Eat like a Chicken and Shit like an Elephant…

� …Naseem Taleb coined that phrase.

� Given that the markets are efficient and risk averse it is goingto be difficult to find cheap options.

� Here are my own attempts:

- Mispricing of skew in exotic equity options: buyguaranteed fund structures and sell put option on theindex.

- Mispricing of equity volatility skew versus credit defaultswaps: buy put options and sell credit default swaps.

- Never seen events: negative interest rates. Buy zero strikefloors.

� There are plenty opportunity to sell expensive options. Myfavourites -- at your won risk:

- High strike caps.

- Buy and hold large diversified portfolios of corporatebonds.

� The first category of strategies are not easy to actually do andrequire careful analysis and execution.

� So essentially there are two ways to get rich:

- Hard and inspired work combined with unconventionalthinking.

- Luck.

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Model Philosophy

� Keep it simple -- but not too simple.

� Focus on the key features of the market and the product thatyou are considering.

� The same recipe will not work across different underlyings:Equities, interest rates, foreign exchange, and commoditiesare fundamentally different.

� So fundamentally different models have to be used for eachmarket.

� Use models that can calibrate with close to closed form.

� Do not attempt to fit the model to all market data. A lot of thedata that some traders will claim is the market is simplyrubbish.

� A 95% (whatever that means) fit is good enough.

� Do not over-complicate: 7 factor yield curve models withstochastic volatility, jumps and ARCH-GARCH are of no useif you can not calibrate quickly or you can not get out anaccurate risk reports in finite time.

� Cross check different models against each other.

� Write public papers about your models.

� Study the literature -- not only Hull's book.

� Use the best possible numerical methods.

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� Models that Work at Work

� Interest rates

- Use normal -- not log normal models.

- Or better yet, models that can slide between normality andlog normality.

- If feasible add stochastic volatility to capture smile.

- Use a moderate number of factors 1-2.

- Have several models and benchmark them against eachother

� Equities

- Use log-normal models with jumps.

- Use common jumps to capture big market moves.

- Only add stochastic volatility if you have nothing else todo.

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� Foreign exchange

- Black-Scholes with some add-hoc adjustments for smileseem to work ok.

- Otherwise stochastic volatility.

� Credit derivatives

- The standard simple default probability stripping stuff forplain CDS.

- Gaussian and other copula models is the way to go oncredit correlation products.

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An Interest Rate Model at Work in January

� Best fit of model to independent swaption volatility smiledata in January 2003.

� On the x-axis we have the strike quoted in terms of Black-Scholes swaption delta.

� The y-axis reports the discrepancy between our official marks(reval) and the model (sv) to the official quotes in terms ofBlack-Scholes volatilities.

-4.00%

-3.00%

-2.00%

-1.00%

0.00%

1.00%

2.00%

3.00%

4.00%

0.00% 20.00% 40.00% 60.00% 80.00% 100.00%reval diffsv diff

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An Interest Rate Model at Work in June

� Using the same parameters in June 2003 against newindependent data.

� So the model is better at following the market smile than thetrader is.

-4.00%

-3.00%

-2.00%

-1.00%

0.00%

1.00%

2.00%

3.00%

4.00%

0.00% 20.00% 40.00% 60.00% 80.00% 100.00%

reval diffsv diff

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Conclusion

� Markets are efficient but risk averse.

� Hedge funds collect risk premia.

� There are not very many cheap options and even far fewerdirect arbitrage opportunities.

� Financial theory works!

� Models work!