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Optimal Trading Rules Ok, there is an arbitrage here. So what? Michael Boguslavsky, Pearl Group Quant Congress Europe ’05, London, October 31 – November 1, 2005

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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

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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.

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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

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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 0 Potential new P/L Delta=1 P/L to date Next year 0 Potential new P/L Delta<1 Terminal utility Current value Trader’s value function vs. trading account balance

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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

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Topics A Christmas story 1. The basic reversion model 2. Consequences 3. Refinements 4. Two sources of gamma

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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…

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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

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1.The basic model A tradable Ornstein-Uhlenbeck process with known constant parameters The trader controls position size α t Wealth W t >0 Fixed time horizon T: maximizing utility of the terminal wealth W T Zero interest rates, no market frictions, no price impact X t is the spread between a tradable portfolio market value and its fair value

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1.Example: pair trading

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1.Trading rules One wants to have a short position when X t >0 and a long position when X t <0 A popular rule of thumb: open a position whenether X t is outside the one standard deviation band around 0

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1.Log-utility The utility is defined over terminal wealth as X t 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.

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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

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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

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1.Optimal strategy: power utility Boguslavsky&Boguslavskaya, ‘Arbitrage under Power’, Risk, June 2004

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1.Optimal strategy: power utility Optimal position is linear in wealth and price depends on time left T-t

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1.How to prove it Value function J(W t,X t,t): expected terminal utility conditional on information available at time t Hamilton-Jacobi-Bellman equation First-order optimality condition on α PDE on J

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1.An interesting bit Myopic demand Hedging the changes in the future investment opportunity set

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1.A sample trajectory

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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!

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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

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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?

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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 W t start cutting position simultaneously If γ are different, same W t does not mean same W 0

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2.Q2. Sad or Happy A power utility trader with the optimal position is never happy with spread widening

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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

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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

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4.Long/short gamma

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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)

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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

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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

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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

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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

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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

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4.Contingent claim payoff at T

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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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