Transcript Slide 1

Optimal Trading Rules
Ok, there is an arbitrage here. So
what?
Michael Boguslavsky, Pearl Group
[email protected]
Quant Congress Europe ’05, London, October 31 – November 1, 2005
This talk:
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is partly based on joint work with Elena Boguslavskaya
reflects the views of the authors and not of Pearl Group
or any of its affiliates
Slides available at
http://www.boguslavsky.net/fin/quant05.pps
A Christmas story (real)
Ten days before Christmas, a
salesman (S) comes to a trader (T).
S: - Look, my customer is ready to sell
a big chunk of this [moderately
illiquid derivative product] at this
great level!
T: - Yes the level is great, but it is the
end of the year, the thing is risky…
Let’s wait two weeks and I will be
happy to take it on.
What is going on here?
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The trader forfeits a good but potentially noisy piece of
P/L this year, in exchange for a similar P/L next year
Eventual
convergence
Fair value
Current level
offered
Risk of potential
loss: may be forced
to cut the position
Is this an agency problem?

A negative personal discount rate? Is next year’s P/L is more valuable
than this year’s?
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Weird incentive structure? The conventional trader’s “call on P/L” is ITM
now, will be OTM in two weeks, so is the trader waiting for its delta to drop?
Trader’s value function vs. trading account balance
This year
Next year
Terminal utility
Current value
Delta=1
Delta<1
0
P/L to date
0
Potential
new P/L
P/L to date
Potential
new P/L
Not very unusual
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Is this trader just irrational?
This behavior does not seem to be that rare: liquidity
is very poor in many markets for the last few weeks
of the year
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Spreads widen for OTC equity options and CDS
Liquidity premium increases (“flight to quality”)
“January effect”
Actually, there is a plausible model where this
behavior is rational and is a sign of risk aversion. If a
trader is more risk averse than a log-utility one then
he can become less aggressive as his time horizon
gets nearer
Topics
1.
2.
3.
4.
A Christmas story
The basic reversion model
Consequences
Refinements
Two sources of gamma
1. Optimal positions
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Portfolio optimization (Markowitz,…):
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Several assets with known expected returns and
volatilities, need to know how to combine then
together optimally
We need something different: a dynamic
strategy to trade a single asset which has a
certain predictability
Liu&Longstaff, Basak&Croitoru,
Brennan&Schwartz, Karguin, Vigodner,Morton,
Boguslavsky&Boguslavskaia…
1. Modeling reversion trading
Two approaches:
 Known convergence date (usually modeled by a
Brownian bridge) + margin or short selling constraints
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Some hedge fund strategies, private account trading:
margin is crucial
Short futures spreads, index arbitrage, short-term volatility
arbitrage
Unknown or very distant sure convergence date +
“maximum loss” constraint
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Bank prop desk: margin is usually not the binding constraint
Fundamentally-driven convergence plays, statistical
arbitrage, long-term volatility arbitrage
1.The basic model
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A tradable Ornstein-Uhlenbeck process
with known constant parameters
The trader controls position size αt
Wealth Wt>0
Fixed time horizon T: maximizing utility
of the terminal wealth WT
Zero interest rates, no market frictions,
no price impact
Xt is the spread between a tradable
portfolio market value and its fair value
dXt  kXt dt  dBt
dWt   t dXt
1.Example: pair trading
1.Trading rules
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One wants to have a
short position when Xt>0
and a long position when
Xt<0
A popular rule of thumb:
open a position
whenether Xt is outside
the one standard
deviation band around 0
StD( X t  s | X t )  
1  exp(2ks)
2k
1.Log-utility
U (WT )  ln(WT )
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The utility is defined over terminal wealth
as Xt changes, the trader may trade for two reasons
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to exploit the immediate trading opportunity
to hedge against expected changes in future trading
opportunity sets
Log-utility trader is myopic: he does not hedge
intertemporarily (Merton). This feature simplifies the
analysis quite a bit.
1.Power utility
U (WT ) 
1
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(WT  1),
    1
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Special cases:
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γ=0: log-utility
γ=1: risk-neutrality
Generally, log-utility is a rather bold choice: same
strength of emotions for wealth halving as for doubling
Interesting case:
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γ<0: more risk averse than log-utility
1.Optimal strategy: log-utility
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Renormalizing to k=σ=1
Morton; Morton, Mendez-Vivez,
Naik:
Optimal position
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is linear in wealth and price
Given wealth and price, does not
depend on time t
t  Wt X t
1.Optimal strategy: power utility
 t   D( )Wt X t ,
where  T  t is the timeleft for trading and
D( ) is a simplefunctionof  and risk aversionparameter

Boguslavsky&Boguslavskaya, ‘Arbitrage
under Power’, Risk, June 2004
1
,
1 
C ( )  cosh   sinh ,
C ' ( )   sinh   2 cosh ,
D ( ) 
C ' ( )
C ( )
1.Optimal strategy: power utility
t  D( )Wt X t
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Optimal position
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is linear in wealth
and price
depends on time left
T-t
1.How to prove it
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Value function J(Wt,Xt,t):
1
expected terminal utility
J (Wt , X t , t )  sup  E t (WT  1)
conditional on

information available at
time t
Hamilton-Jacobi-Bellman sup ( J  xJ  xJ  1 J  1  J  J )  0
2
2
equation
J
J
First-order optimality
 * ( w, x, t )  x w  xw
J ww J ww
condition on α
PDE on J
J
J 2
1
1
2
t
J t  xJ x 
x
w
xx
ww
xw
J xx  J ww ( x w  xw )  0
2
2
J ww J ww
1.An interesting bit
J w J xw
*
 ( w, x, t )  x

J ww J ww
Myopic
demand
Hedging the
changes in the
future investment
opportunity set
1.A sample trajectory
2.A possible answer to the
Christmas puzzle
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May be that trader was just
a bit risk-averse:
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Assuming that
reversion period k = 8 times a
year, volatility σ = 1, two weeks
before Christmas, inverse
quadratic utility γ=-2:
Position multiplier D(τ)
jumps 50% on January, 1!
2. Or is it?
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This effect is not likely to be the only cause of
the liquidity drop
About 30% of the Christmas liquidity drop can
be explained by holidays (regression of
normalized volatility spreads for other holidya
periods) and by year end
Liquidity drop is self-maintaining: you do not
want to be the only liquidity provider on the
street
2.Interesting questions
1.
2.
When is it optimal to start cutting a
losing position?
When the spread widens, does the
trader
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get sad because he is losing money on his
existing positions or
get happy because of new better trading
opportunities?
2.Q1. Cutting losses
T hediffusion termof dαt is  D( )(Wt   t X t )
So thecovarianceCov(d , dX )  D( )Wt (1  X t2 D( ))
is negativewhenetherX t2  1 / D( )
•Another interpretation of this equation is that it is optimal to start
cutting a losing position as soon as position spread exceeds total
wealth
•This result is independent of the utility parameter γ: traders with
different gamma but same wealth Wt start cutting position
simultaneously
•If γ are different, same Wt does not mean same W0
2.Q2. Sad or Happy
T hediffusion termof dJt is J t X t (1  D( ))
So thecovarianceCov(dJ , dX )  J t X t (1  D( ))
i s negativewhenetherX t  0
A power utility trader with the optimal position is
never happy with spread widening
3.Refinements
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Transaction costs: discrete
approximations
The model can be combined with optimal
stopping rules to detect regime changes:
e.g. independent arrivals of jumps in k
Heavy tailed or dependent driving
process
4.Two sources of gamma
The right definition of long/short gamma:
• Gamma is long iff the dynamic position
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returns are skewed to the left: frequent small
losses are balanced with infrequent large
gains
Gamma is short iff the dynamic position
returns are skewed to the right: frequent small
gains are balanced with infrequent large
losses
4.Long/short gamma
4.Sources of gamma
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Gamma from option positions: positive gamma
when hedging concave payoffs, negative when
hedging convex payoffs
Gamma from dynamic strategies:
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positive gamma when playing antimartingale
strategies, negative when playing martingale
strategies
positive when trend-chasing, negative when providing
liquidity (e.g. marketmaking or trading meanreversion)
4.Example: short gamma in St.
Petersburg paradox
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The classical doubling up on
losses strategy when
playing head-or-tail
Each hour we gamble until
either a win or a string of 10
losses
Our P/L distribution over a
year will show strong signs
of negative gamma: many
small wins and a few large
losses
A gamma position achieved
without any derivatives
4. Gamma positions
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Almost every technical analysis or
statistical arbitrage strategy carries a
gamma bias
Usually coming not form doubling-up but
form holding time rules:
• With a Brownian motion, instead of doubling
the position we can just quadruple holding
time
4.Two long gamma strategies
Trend following vs. buying strangles:
 Option market gives one price for the
protection
 Trend-following programs give another
 Some people are arbitraging between the two
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Leverage trend-following program performance
Additional jump risk
Usually ad-hoc modeled with some regression
and range arguments
4. Hedging trend strategy with
options: an example
From: Amenc, Malaise, Martellini, Sfier:
‘Portable Alpha nad portable beta
strategies in the Eurozone,’’ Eurex
publications, 2003
4.Two short gamma strategies
Trading reversion vs. static option
portfolios
 Can be done in the framework described
above
 Gives protection against regime changes
 In equilibrium, yields a static option
position replicating reversion trading
strategy
4.Contingent claim payoff at T
Summary
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The optimal strategy for trading an
Ornstein-Uhlenbeck process for a
general power utility agent
Possible explanation of several market
“anomalies”
Applications to combining option and
technical analysis/statistical arbitrage
strategies