Derivatives -- the View from the Trenches

Derivatives -- the View from the Trenches
October 2003
Jesper Andreasen
Head of Product Development
Nordea Markets, Copenhagen

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
2

My Life as a Quant
CV:
- 1997: PhD, Department of Operations Research, Institute
of Mathematics, Aarhus University.
- 1997-1997: Senior Analyst, Bear Stearns, London.
- 1997-2000: Vice President, Quantitative Research
Department, General Re Financial Products, London.
- 2000-2002: Principal, Head of Quantitative Research at
Bank of America, London.
- 2002-present: Head of Product Development Group in
Nordea Markets, Copenhagen.
Job description
- Develop and implement pricing and hedging models for
exotic derivatives.
- Including: interest rates, credit derivatives, equity, foreign
exchange and hybrids.
3

Career highlights
- Solved the passport option problem and implemented the
world's first live model. Did 10 trades.
- Developed Gen Re's Japanese "Power Reverse Dual"
business. 350 20-30y trades on 40 Pentium CPUs. Takes
6h to run risk reports.
- Sorted out the modeling of Band of America's 1500 trade
portfolio 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.
4

The First Fundamental Theorem of Derivatives Trading
We assume zero rates and dividends and standard frictionless
markets.
Assume that the underlying stock evolves continuously.
So there exists two stochastic processes , so that
dS()
t
() tdt() tdWt ()
St ()
under the real measure P .
Let V be the value of an option book on S priced on a model
with constant volatility .
Assume that the book is delta hedged, i.e. we dynamically
trade the stock to keep
V 0
S
Theorem 1: The value of the option book evolves according
1
dV t t
S t V t dt
2 2 2
() ( () ) ()
SS
()
2
Proof: Follows from V V(, t S())
t and Ito's lemma that
1
dV Vtdt VSdS VSSdS
2
1 2 2
Vdt t VSS
S dt
2
1 2 2
Using 0 Vt
S VSS
we get the result
2
2
5

Theorem 1 and Trading
If you're gamma long, VSS
0, and realised volatility is higher
than pricing volatility you make money.
The option trader's job is really about balancing realised
against implied (or pricing) volatility.
realised vol > implied vol => go long gamma
realised vol < implied vol => go short gamma
In essence this is what all trading is about: buy low -- sell
high.
In practice it is of course not that easy to predict how realised
and implied volatility are going to relate to each other over a
given period.
In the context of the derivation of the Black-Scholes' formula,
one can see Theorem 1 as an investigation of the selffinancing
condition.
6

Theorem 1 and Option Markets
Black-Scholes implied volatility has to satisfy
1
E u S u V u du
2
Q
T
2 2 2
[ ( ( )
) ( ) ( ) ] 0
0
implied
SS
So implied volatility is a weighted average of Q expected
volatility.
Consider an option seller that delta hedges his short option
position, i.e. VSS
0, with the implied volatility. His total
profit
1 T
2 2 2
VT ( ) V(0) ( ( ) ) ( ) ( )
2
t
0
implied
St VSS
tdt
is positive if
implied
() t
"most of the time".
If option sellers are risk averse (and they are) it is unlikely
that they are willing to be short gamma without taking a
premium. 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%
7

The implied volatility gap in equities is a bit higher than in
rates and fx. The reason for this is jumps but we will return to
this.
The difference in historical and implied volatility does not
indicate that there is an arbitrage -- just that there is a risk
premium on volatility.
Ie, volatility is stochastic and market participants are risk
averse.
8

Theorem 1 and Models
Theorem 1 extends to other parameters such as for example
correlation, and more sophisticated models than the constant
volatility Black-Scholes model, generally
1
dV V
dt
2 2
xx ( )
i j ij ij
2 ij ,
where ( x i ) are your risk factors and dxidx
j ijdt
.
The general points is that if your model ,( ij)
, is misspecified
you are going to make daily
Odt ( )
losses, not OdW ( ).
So contrary to common belief: bad models cause bleeding --
not blow-ups.
So it may take a while before you realise that your model is
wrong.
That is, you can pile up a lot of trades, make a lot of "model
value", and many fine bonuses can be paid, before you realise
that your book is bleeding.
…and this has (involuntarily) been discovered by many
banks:
- UBS remark of IPS portfolio in the early 90s.
- Commerzbank (and other's) remark of Bermudan swaption
book in the late 90s.
9

- Interest rate option problems at NatWest (and other places)
in mid 90s.
- Several banks' current problem with exotic equity option
books.
- …
All sad examples of taking it a bit too easy on the model
front.
10

Theorem 1 and Technical Results
A lot of technical results can be derived from the Theorem 1:
For example, set 0 and let
then
1
Vt () ln St () St
()
St ()
logcontract
priced at
0
delta hedge
of logcontract
at
0
1 T
T
2 1
ln ST ( ) ln S(0) ( ) ( )
2 u du
dSu
0 0
Su ( )
Hence a contract paying the realised variance can be
replicated with a simple delta strategy combined with a
contract paying ln ST ( ) , which in turn can be statically
replicated with positions in European options.
Another example is 0 and
Vt () ( St () K) 1 St
()
call option
priced at 0
St () K
delta hedge
of call option
at 0
Theorem 1 and a few manipulations yield
E t S t K
Q 2
[ ( ) | ( ) ] 2
C ( T, K)
K C ( T, K)
T
2
KK
where
Q
CT ( , K) E [( ST ( ) K)
]
11

Most of this sort of hocus-pocus was derived by Bruno
Dupire in early 90s, but has received far too little recognition
and attention in textbooks and academic circles.
The French banks produce a lot of very good quants and they
are all breed on a solid dose of Theorem 1 and all the
corollaries.
12

Theorem 2: The Gospel of the Jump
You can get a long way of understanding the world with
Theorem 1, but there is one thing missing and that is jumps.
Suppose the stock evolves according to
dS()
t
() tdt() tdWt () ItdNt
() ()
St ( )
where ,
are stochastic processes and N is a Poisson
process with stochastic intensity () t , and the jump size is
stochastic with distribution given by
It ()~ (;) t
Suppose we price according to the model
dS()
t
dW () t I() t dN()
t mdt
St ( )
under Q , where
Q
QdNt [ ( ) 1] dt, It ( )~ (), mE [ It ( )]
Theorem 2: A delta neutral trading book will accumulate
gains at
1 (
2 2 )
2
Q
SS
[ ]
dV S V dt VdN E V dt
2
Proof: Using the pricing equation
1
V mSV S V E V
2
2 2
Q
0 t S
SS
[ ]
and Ito's lemma yields the result.
13

Theorem 2 and Trading
This tells us that if realised volatility is the same as pricing
volatility and if we're gamma short ( VSS
0) and delta neutral
( VS
0) then V
0 and thereby
dV VdN E Q [ V ] dt
0
So on a short option book you can sit and collect "jump
premium" and look like an absolute hero -- until a jump
occurs.
This is essentially what is called "picking up pennies in front
of a steam engine" or "the trader's option".
No matter how people look at this themselves, this is
essentially the strategy that a lot of hedge funds follow.
A couple of spectacular examples:
- The blow up of LOR and portfolio insurance industry in
1987.
- The collapse of CRT in 1987.
- The blow up of LTCM in 1998.
- …
Many of these funds had prominent academics involved, so
following the actual disasters we heard a lot of good stories
like
- "19 standard deviation event..."
- "Liquidity squeeze…"
- …
14

But in essence the strategies involved were more or less
asking for it:
- Delta hedging a put option notional of USD 70bn.
- Hedging S&P vol with Microsoft puts bought from
Microsoft.
- Buying half of the Danish mortgage bonds.
- …
15

Theorem 2 and the Equity Derivatives Markets
The implied volatility has to satisfy
Q
0 E [ V( T) V(0)]
1
E u S u V u du
2
Q
T
2 2 2
[ ( ( ) ) ( ) ( )|
0
SS , 0
]
T
Q
E [ V( u)| dN( u)]
du
0
, 0
where Q in this case is the "market" risk neutral measure.
Time for a few (very) rough calculations: Set
we hold a log-contract. We get
V
ln S, that is
1 1
E S E S
2
S
2 2 Q
Q
0 ( ) ( [ ln ] [ ])
1 (
2 2 ) (
Q [ln(1 )]
Q
E I E [ I ])
2
1 2 2 1 2
( ) m
2 2
2 2 2
m
(
)
So if for S&P 500
- Implied volatility is 0.20
- Historical volatility is 0.17
then we have
Quiz: which one
m
0.1 +/-33%
1 +/-11%
10 +/-3%
16

Theorem 2 and the Equity Volatility Smile
To get the answer to our quiz we need to look at the implied
Black volatility smile (or skew rather) in S&P500 options.
27.00%
25.00%
23.00%
21.00%
19.00%
obs-1y
mdl-1y
obs-2y
mdl-2y
obs-3y
mdl-3y
17.00%
15.00%
75.00
%
80.00
%
85.00
%
90.00
%
95.00
%
100.00
%
105.00
%
110.00
%
115.00
%
120.00
%
The model uses 0.15, 0.18, m 0.30, 0.15.
…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%
17

Theorem 2 and Risk Aversion
Clearly the parameters
0.18, m 0.30
seem extreme. Does the market really expect market crashes
of -30% every fifth year
It is important to note, however, that these parameters are not
the historical parameters -- the are market Q measure
parameters and therefore include a healthy dose of risk
premium.
Suppose
- Historical jump intensity of 0.02.
- Historical mean jump of m 0.25.
Then in equilibrium the market jump parameters are a
function of the relative risk aversion (RRA) of the
representative 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 is
extreme.
18

So in front of you, you have an option quant who uses utility
functions and relative risk aversion to consider various
problems -- probably not what you expected.
The main problem in using utility theory for risk
considerations is not that you can't get people to specify their
relative risk aversion. The real problem is that all utility
theory depends crucially on the P distribution and that is
almost impossible to get with any decent accuracy.
If jumps happen twice every century, estimating the mean
jump and standard deviation is going to be quite difficult.
19

The Difference between P and Q
Suppose we know that under P
dS
S
dt
dW
and assume that 0.17 . Then we wish to estimate the drift
as
1 St ( ) 1
t S(0) 2
2
ˆ (ln )
t
We have
std[ ˆ
]
t
This gives us the following table
horizon std[ ˆ ]
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 is
extremely difficult to estimate the real distribution.
This of course has the consequence that there essentially are
as many P measures as there are agents in the economy!
20

The Difference between P and Q and Market Efficiency
The difficulty of obtaining the "true" P measure aside, it is
clear that the market generally prices in significant risk
premia.
There are tons of examples:
- Implied volatility > Historical volatility.
- Equity volatility skew.
- Corporate bond spreads are much wider than historical
default probabilities.
- The long end of the yield curve is far to volatile to be
consistent with the historical mean reversion of interest
rates.
- Long maturity volatilities are far to volatile to be
consistent 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 efficient
but 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 risk
premia.
So hedge funds generally run in front of the steam engines!
21

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 going
to be difficult to find cheap options.
Here are my own attempts:
- Mispricing of skew in exotic equity options: buy
guaranteed fund structures and sell put option on the
index.
- Mispricing of equity volatility skew versus credit default
swaps: buy put options and sell credit default swaps.
- Never seen events: negative interest rates. Buy zero strike
floors.
There are plenty opportunity to sell expensive options. My
favourites -- at your won risk:
- High strike caps.
- Buy and hold large diversified portfolios of corporate
bonds.
The first category of strategies are not easy to actually do and
require careful analysis and execution.
So essentially there are two ways to get rich:
- Hard and inspired work combined with unconventional
thinking.
- Luck.
22

Model Philosophy
Keep it simple -- but not too simple.
Focus on the key features of the market and the product that
you are considering.
The same recipe will not work across different underlyings:
Equities, interest rates, foreign exchange, and commodities
are fundamentally different.
So fundamentally different models have to be used for each
market.
Use models that can calibrate with close to closed form.
Do not attempt to fit the model to all market data. A lot of the
data that some traders will claim is the market is simply
rubbish.
A 95% (whatever that means) fit is good enough.
Do not over-complicate: 7 factor yield curve models with
stochastic volatility, jumps and ARCH-GARCH are of no use
if you can not calibrate quickly or you can not get out an
accurate 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.
23

Models that Work at Work
Interest rates
- Use normal -- not log normal models.
- Or better yet, models that can slide between normality and
log normality.
- If feasible add stochastic volatility to capture smile.
- Use a moderate number of factors 1-2.
- Have several models and benchmark them against each
other
Equities
- Use log-normal models with jumps.
- Use common jumps to capture big market moves.
- Only add stochastic volatility if you have nothing else to
do.
24

Foreign exchange
- Black-Scholes with some add-hoc adjustments for smile
seem to work ok.
- Otherwise stochastic volatility.
Credit derivatives
- The standard simple default probability stripping stuff for
plain CDS.
- Gaussian and other copula models is the way to go on
credit correlation products.
25

An Interest Rate Model at Work in January
Best fit of model to independent swaption volatility smile
data in January 2003.
4.00%
3.00%
2.00%
1.00%
0.00%
0.00% 20.00% 40.00% 60.00% 80.00% 100.00%
-1.00%
reval diff
sv diff
-2.00%
-3.00%
-4.00%
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 of
Black-Scholes volatilities.
26

An Interest Rate Model at Work in June
Using the same parameters in June 2003 against new
independent data.
4.00%
3.00%
2.00%
1.00%
0.00%
0.00% 20.00% 40.00% 60.00% 80.00% 100.00%
reval diff
sv diff
-1.00%
-2.00%
-3.00%
-4.00%
So the model is better at following the market smile than the
trader is.
27

Conclusion
Markets are efficient but risk averse.
Hedge funds collect risk premia.
There are not very many cheap options and even far fewer
direct arbitrage opportunities.
Financial theory works!
Models work!
28