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## Optimal Trading Rules

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### Optimal Trading Rules

Ok, there is an arbitrage here. So what?

Michael Boguslavsky, Pearl Group

michael@boguslavsky.net

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

Eventual

convergence

Fair value

Risk of potential

loss: may be forced

to cut the position

Current level offered

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?

Trader’s value function vs. trading account balance

This year

Next year

Terminal utility

Current value

Delta=1

Delta<1

0

0

Potential new P/L

Potential new P/L

P/L to date

P/L to date

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

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

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 Xtis outside the one standard deviation band around 0

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.

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.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

1.Optimal strategy: power utility

Boguslavsky&Boguslavskaya, ‘Arbitrage under Power’, Risk, June 2004

1.Optimal strategy: power utility

- 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

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

- When is it optimal to start cutting a losing position?
- 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

- 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 γare different, same Wt does not mean same W0

2.Q2. Sad or Happy

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.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

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

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