Exact Replication of the Best Rebalancing Rule in Hindsight [Garivaltis, A., 2019. The Journal of Derivatives, 26 (4), pp.35-53.] Alex Garivaltis Assistant Professor of Economics, Northern Illinois University November 14, 2019 Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 1/43
Introduction: Lookback Options. ◮ The exotic option literature has several examples of derivatives with “lookback” or “no-regret” features (cf. with Paul Wilmott 1998). ◮ For instance, a floating-strike lookback call allows its owner to look back at the price history of a given stock, buy a share at the realized minimum m := min 1 ≤ t ≤ T S t , and sell it at the terminal price S T . ◮ Similarly, a fixed-strike lookback call allows its owner to buy one share at a fixed price K , and sell it at the historical maximum M := max 1 ≤ t ≤ T S t . Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 2/43
◮ This paper prices and replicates a markedly different type of lookback option, whose payoff is the final wealth that would have accrued to a $1 deposit into the best continuous rebalancing rule (or fixed-fraction betting scheme) determined in hindsight. ◮ This contingent claim has been studied by Thomas Cover and his collaborators (1986, 1991, 1996, 1998) who used it as a performance benchmark for discrete-time portfolio algorithms. Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 3/43
Continuous Rebalancing Rules. Definition In the context of one underlying stock, a rebalancing rule b ∈ R continuously maintains the fraction b of wealth in the stock at all times, and keeps the fraction 1 − b of wealth in cash. Example The rule b = 0 . 7 trades continuously so as to maintain 70% of wealth in the stock. When the stock outperforms cash over [ t , t + dt ], the trader must sell a precise number of shares so as to restore the target allocation. Similarly, when the stock underperforms bank interest over [ t , t + dt ], the scheme dictates that he buy additional shares to bring his exposure back up to 70% of wealth. Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 4/43
Taxonomy of Rebalancing Rules. (for a single underlying stock) ◮ 0 < b < 1: volatility harvesting scheme (cf. with David Luenberger 1998) that mechanically forces you to buy low and sell high. ◮ b = 1: buy-and-hold. ◮ b > 1: uses margin loans. The scheme continuously maintains a fixed debt-to-assets ratio of ( b − 1) ÷ b . When the stock rises, you instantly adjust by borrowing additional cash against your new wealth. Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 5/43
◮ b = 0: all cash. ◮ b < 0: continuously maintains a short position in the stock with dollar value equal to | b | times the current bankroll. When the stock falls, the scheme must short a little more so as to restore the target proportion | b | . Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 6/43
Figure: b := 2 for Vanguard S&P 500 index ETF under monthly rebalancing, Jan 2012-Aug 2018. ◮ In hindsight, b := 2 (rebalanced monthly) would have compounded at 31 . 8% a year versus 15 . 6% a year for the S&P. Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 7/43
Figure: b := 0 . 5 for AMD shares under monthly rebalancing, Apr 1986-Aug 2018. ◮ In hindsight, b := 0 . 5 (rebalanced monthly) would have compounded at 7 . 79% a year versus 1 . 77% a year for AMD. Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 8/43
Point of Departure. ◮ Ordentlich and Cover (1998) only priced the rebalancing option at time-0, and only for unlevered hindsight optimization over a single underlying stock. They did not derive the ∆-hedging strategy. ◮ The last result in their (1998) paper is the formula � T C 0 = 1 + σ 2 π , (1) where σ is the volatility and T is the horizon. Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 9/43
Thomas Cover (1938-2012) was an information theorist and statistician (at Stanford University) who also contributed to portfolio theory. Among other things, he did seminal work (Cover and Hart 1967) on the k -Nearest Neighbors algorithm used in Machine Learning. Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 10/43
Contributions of Garivaltis (2018). ◮ I price Cover’s derivative at any time t (not just at t = 0). One cannot properly hedge (or replicate) the option without the general price C ( S , t ). ◮ By contrast, I also solve the problem for levered hindsight optimization. When leverage is allowed in hindsight, the replicating strategy becomes especially simple. ◮ Rather than just a single underlying stock, I generalize to the multivariate “correlation” or “rainbow” version of the option, that considers continuously-rebalanced portfolios over any number of correlated stocks. ◮ I analyze implied volatility and the option sensitivities (e.g. the option Greeks), and I derive the (multivariate) performance guarantees for the corresponding ∆-hedging strategy. The reader also has the pleasure of seeing these guarantees borne out in simulation. Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 11/43
One Underlying: the Black-Scholes (1973) Market. ◮ For simplicity, we start with a single stock whose price S t follows the geometric Brownian motion dS t := S t × ( µ dt + σ dW t ) , (2) where µ is the drift, σ is the log-volatility, and W t is a standard Brownian motion. ◮ There is a risk-free bond (or bank account) whose price B t := e rt evolves according to dB t = rB t dt . (3) Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 12/43
Payoff Computation. ◮ We let V t ( b ) denote the wealth that accrues at t to a $1 deposit into the rebalancing rule b . ◮ V t ( b ) evolves according to dV t ( b ) V t ( b ) = bdS t + (1 − b ) dB t S t B t = [ r + ( µ − r ) b ] dt + b σ dW t . (4) ◮ Since V t ( b ) is a geometric Brownian motion, we can solve to get r + ( µ − r ) b − σ 2 �� � � 2 b 2 V t ( b ) = exp t + b σ W t . (5) Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 13/43
◮ For levered hindsight optimization, the payoff of Cover’s rebalancing option is V ∗ t := −∞ < b < ∞ V t ( b ) . max (6) ◮ We need to express the option payoff in terms of the observable S t (instead of the Wiener process W t ). This can be done by using the formula µ − σ 2 � S t � � � σ W t = log − t . (7) S 0 2 ◮ Substituting this expression into (5), we get r − σ 2 r − σ 2 �� � � � S t � � � �� 2 b 2 V t ( b ) = exp t + b log − t . S 0 2 (8) Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 14/43
◮ More compactly, we have √ r − σ 2 �� � � 2 b 2 V t ( b ) = exp t + b σ t × z t , (9) where z t := log( S t / S 0 ) − ( r − σ 2 / 2) t σ √ t . (10) ◮ We write V t ( b ) this way because z t is distributed unit normal with respect to the risk-neutral measure Q (but not the physical measure, P ). ◮ Note that the drift µ has disappeared from the expression for V t ( b ). The final wealth of b is path-independent: all that matters is S t and t . ◮ This convenient property does not obtain for discrete-time rebalancing. Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 15/43
Computing the Best Rebalancing Rule in Hindsight. ◮ We let b ( S , t ) denote the best rebalancing rule in hindsight over the known history [0 , t ]. ◮ Maximizing the (quadratic) exponent of V t ( b ) with respect to b , we find the vertex b ( S , t ) = z ( S , t ) σ √ t . (11) ◮ After simplification, the hindsight-optimized wealth is given by r + σ 2 rt + z 2 �� � � � � 2 b ( S , t ) 2 V ∗ t = exp t = exp . (12) 2 Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 16/43
Figure: The payoff of Cover’s derivative for different stock prices and volatilities, T := 5 , S 0 := 100 , r := 0 . Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 17/43
Arbitrage Pricing. ◮ To find the no-arbitrage price C ( S , t ) of Cover’s rebalancing option, we must solve the Black-Scholes (1973) equation σ 2 2 S 2 ∂ 2 C ∂ S 2 + rS ∂ C ∂ S + ∂ C ∂ t − rC = 0 . (13) ◮ The associated boundary condition is C ( S , T ) = V ∗ � rT + z 2 � T ( S ) = exp T / 2 . Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 18/43
Risk-Neutral Pricing. ◮ From the martingale approach to option pricing, we know that (cf. with Paul Wilmott 1998) the Black-Scholes equation is uniquely solved by putting C ( S , t ) = e − r ( T − t ) × E Q rT + z 2 � � �� exp T / 2 , (14) t which is the expected present value of the final payoff with respect to Q and the information available at t . ◮ That is, ∞ � e rt + z 2 T / 2 f ( z T | z t ) dz T , C ( S , t ) = (15) −∞ where f ( •| z t ) is the conditional density of z T . ◮ z T is conditionally normal, with mean z t � t / T and variance 1 − t / T . Dr. Alex Garivaltis Exact Replication of the Best Rebalancing Rule in Hindsight 19/43
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