Optimal Trading Rules Ok, there is an arbitrage here. So what? Michael Boguslavsky, Pearl Group [email protected] Quant Congress Europe ’05, London,

Optimal Trading RulesOk, 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:

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