optimal trading rules ok, there is an arbitrage here. so what? michael boguslavsky, pearl group...
TRANSCRIPT
Optimal Trading RulesOk, there is an arbitrage here. So
what?
Michael Boguslavsky, Pearl Group
Quant Congress Europe ’05, London, October 31 – November 1, 2005
This talk:
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?
The trader forfeits a good but potentially noisy piece of P/L this year, in exchange for a similar P/L next year
Current level offered
Fair value
Eventual convergence
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?
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?
P/L to date
This year
0Potential new P/L
Delta=1
P/L to date
Next year
0Potential new P/L
Delta<1
Terminal utility
Current value
Trader’s value function vs. trading account balance
Not very unusual 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• 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
A Christmas story
1. The basic reversion model
2. Consequences
3. Refinements
4. Two sources of gamma
1. Optimal positions
Portfolio optimization (Markowitz,…):• 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• 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• Bank prop desk: margin is usually not the binding constraint
• Fundamentally-driven convergence plays, statistical arbitrage, long-term volatility arbitrage
1.The basic model 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
ttt dBdtkXdX
ttt dXdW
1.Example: pair trading
1.Trading rules 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
k
ksXXStD tst 2
)2exp(1)|(
1.Log-utility
The utility is defined over terminal wealth as Xt changes, the trader may trade for two reasons
• 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.
)ln()( TT WWU
1.Power utility
Special cases: • 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:
• γ<0: more risk averse than log-utility
1
),1(1
)(
TT WWU
1.Optimal strategy: log-utility
Renormalizing to k=σ=1 Morton; Morton, Mendez-Vivez,
Naik: Optimal position
• is linear in wealth and price
• Given wealth and price, does not depend on time t
ttt XW
1.Optimal strategy: power utility
parameter aversion risk and offunction simple a is )(
and gfor tradinleft time theis where
,)(
D
tT
XWD ttt
)(
)(')(
,coshsinh)('
,sinhcosh)(
,1
1
2
C
CD
C
C
Boguslavsky&Boguslavskaya, ‘Arbitrage under Power’, Risk, June 2004
1.Optimal strategy: power utility
ttt XWD )(
Optimal position• is linear in wealth
and price
• depends on time left T-t
1.How to prove it
Value function J(Wt,Xt,t): expected terminal utility conditional on information available at time t
Hamilton-Jacobi-Bellman equation
First-order optimality condition on α
PDE on J
)1(1
Esup),,( Tttt WtXWJ
0)2
1
2
1(sup 2 xwwwxxwxt JJJxJxJJ
ww
xw
ww
w
J
J
J
Jxtxw ),,(*
0)(2
1
2
1 2 ww
xw
ww
wwwxxxt J
J
J
JxJJxJJ
1.An interesting bit
ww
xw
ww
w
J
J
J
Jxtxw ),,(*
Myopic demand
Hedging the changes in the
future investment opportunity set
1.A sample trajectory
2.A possible answer to the Christmas puzzle
May be that trader was just a bit risk-averse:• 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?
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. When is it optimal to start cutting a losing position?
2. When the spread widens, does the trader
• get sad because he is losing money on his existing positions or
• get happy because of new better trading opportunities?
2.Q1. Cutting losses
)(/1 whenethernegative is
))(1()(),( covariance theSo
))(( is of termdiffusion The
2
2
DX
DXWDdXdCov
XWDdα
t
t
tttt
t
•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
0 whenethernegative si
))(1(),( covariance theSo
))(1( is of termdiffusion The
t
tt
ttt
X
DXJdXdJCov
DXJdJ
A power utility trader with the optimal position is never happy with spread widening
3.Refinements
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
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
Gamma from option positions: positive gamma when hedging concave payoffs, negative when hedging convex payoffs
Gamma from dynamic strategies: • positive gamma when playing antimartingale
strategies, negative when playing martingale strategies
• positive when trend-chasing, negative when providing liquidity (e.g. marketmaking or trading mean-reversion)
4.Example: short gamma in St. Petersburg paradox
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
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
• 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
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