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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≤TSt, and sell it at the

terminal price ST. ◮ Similarly, a fixed-strike lookback call allows its owner to buy

• ne share at a fixed price K, and sell it at the historical

maximum M := max

1≤t≤TSt.

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◮ 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

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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%

• f wealth.
• Dr. Alex Garivaltis

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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.

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◮ 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|.

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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

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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.

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Point of Departure.

◮ Ordentlich and Cover (1998) only priced the rebalancing

• ption at time-0, and only for unlevered hindsight
• ptimization over a single underlying stock. They did not

derive the ∆-hedging strategy. ◮ The last result in their (1998) paper is the formula C0 = 1 + σ

• T

2π , (1) where σ is the volatility and T is the horizon.

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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

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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

• ptimization. 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.

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One Underlying: the Black-Scholes (1973) Market.

◮ For simplicity, we start with a single stock whose price St follows the geometric Brownian motion dSt := St × (µ dt + σ dWt) , (2) where µ is the drift, σ is the log-volatility, and Wt is a standard Brownian motion. ◮ There is a risk-free bond (or bank account) whose price Bt := ert evolves according to dBt = rBt dt. (3)

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Payoff Computation.

◮ We let Vt(b) denote the wealth that accrues at t to a $1 deposit into the rebalancing rule b. ◮ Vt(b) evolves according to dVt(b) Vt(b) = bdSt St + (1 − b)dBt Bt = [r + (µ − r)b] dt + bσ dWt. (4) ◮ Since Vt(b) is a geometric Brownian motion, we can solve to get Vt(b) = exp

• r + (µ − r)b − σ2

2 b2

• t + bσWt
• .

(5)

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◮ For levered hindsight optimization, the payoff of Cover’s rebalancing option is V ∗

t :=

max

−∞<b<∞ Vt(b).

(6) ◮ We need to express the option payoff in terms of the

• bservable St (instead of the Wiener process Wt). This can

be done by using the formula σWt = log St S0

• −
• µ − σ2

2

• t.

(7) ◮ Substituting this expression into (5), we get Vt(b) = exp

• r − σ2

2 b2

• t + b
• log

St S0

• −
• r − σ2

2

• t
• .

(8)

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◮ More compactly, we have Vt(b) = exp

• r − σ2

2 b2

• t + bσ

√ t × zt

• ,

(9) where zt := log(St/S0) − (r − σ2/2)t σ√t . (10) ◮ We write Vt(b) this way because zt 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 Vt(b). The final wealth of b is path-independent: all that matters is St and t.

◮ This convenient property does not obtain for discrete-time rebalancing.

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Computing the Best Rebalancing Rule in Hindsight.

◮ We let b(S, t) denote the best rebalancing rule in hindsight

• ver the known history [0, t].

◮ Maximizing the (quadratic) exponent of Vt(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 V ∗

t = exp

• r + σ2

2 b(S, t)2

• t
• = exp
• rt + z2

2

• .

(12)

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Figure: The payoff of Cover’s derivative for different stock prices and volatilities, T := 5, S0 := 100, r := 0.

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Arbitrage Pricing.

◮ To find the no-arbitrage price C(S, t) of Cover’s rebalancing

• ption, we must solve the Black-Scholes (1973) equation

σ2 2 S2 ∂2C ∂S2 + rS ∂C ∂S + ∂C ∂t − rC = 0. (13) ◮ The associated boundary condition is C(S, T) = V ∗

T(S) = exp

• rT + z2

T/2

• .
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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) × EQ

t

• exp
• rT + z2

T/2

• ,

(14) which is the expected present value of the final payoff with respect to Q and the information available at t. ◮ That is, C(S, t) =

∞

• −∞

ert+z2

T /2f (zT|zt)dzT,

(15) where f (•|zt) is the conditional density of zT. ◮ zT is conditionally normal, with mean zt

• t/T and variance

1 − t/T.

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The Cost of Achieving the Best Rebalancing Rule in Hindsight.

C(S, t) =

• T

t × V ∗

t =

• T

t × exp

• rt + z2

t /2

• .

(16) ◮ This matches (like gangbusters) the discrete-time O √ T

• “cost of universality” derived by Cover in his “universal

portfolio theory.” ◮ Since the (European-style) price C(S, t) is equal to

• T/t

times intrinsic value, the option is “worth more alive than dead.” The American-style version of the option is never exercised early in equilibrium. ◮ We have C(S, 0) = ∞, in contrast to the finite time-0 price under unlevered hindsight optimization.

• Dr. Alex Garivaltis

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Delta-Hedging the Option.

Differentiating the price, we find ∆ := ∂C ∂S = C × z Sσ√t = C × b(S, t) S . (17)

Theorem

The replicating strategy for Cover’s rebalancing option will always bet the fraction b(S, t) of wealth on the stock in state (S, t). Thus, to replicate Cover’s derivative, one just uses the best rebalancing rule in hindsight as it is known at time t. ◮ The corresponding statement does not hold for the unlevered payoff max

0≤b≤1 VT(b), even when b(S, t) ∈ [0, 1].

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Implied Volatility.

◮ There are generally two implied volatilities σ that rationalize an observed price C of Cover’s derivative. ◮ We start by expressing z2 = 2 (log C − rt) + log t T

• .

(18) ◮ By definition of z, we also have z2 =

• log(S/S0) − (r − σ2/2)t

2 σ2t . (19) ◮ Equating the right-hand-sides of (18) and (19) and simplifying, we get a quadratic equation in the variance σ2.

• Dr. Alex Garivaltis

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Figure: The dual implied volatilities that rationalize any

• bserved price of Cover’s rebalancing option,

t := 0.5, T := 1, r := 0.03, S0 := 100, St := 105.

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Several Underlyings.

◮ The foregoing analysis is extensible to the general asset market with n correlated underlyings, called i ∈ {1, ..., n}. ◮ We have the stock price vector St := (S1t, ..., Snt)′, where dSit := Sit × (µi dt + σi dWit) . (20) The parameters (µi, σi) are the drift and volatility of asset i respectively, and the (Wit)n

i=1 are correlated standard

Brownian motions, with ρij := Corr (dWit, dWjt) . (21)

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Notation.

◮ We let R := [ρij]n×n be the correlation matrix of instantaneous returns, where R is assumed to be invertible. ◮ We let Σ denote the covariance matrix of instantaneous returns per unit time, where Σij = ρijσiσj = Cov dSit Sit , dSjt Sjt dt. (22) ◮ M := diag (σ1, ..., σn) is the (diagonal) matrix of volatilities, and we have Σ = MRM.

• Dr. Alex Garivaltis

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Continuously-Rebalanced Portfolios.

◮ A rebalancing rule is now a vector b := (b1, ..., bn)′ ∈ Rn, where bi is the fraction of wealth to be continuously maintained in stock i. ◮ The rebalancing rule b keeps the fixed fraction 1 −

n

• i=1

bi of wealth in cash. b uses margin loans if this number is negative. ◮ Vt(b) evolves according to dVt(b) Vt(b) =

• r + (µ − r1)′b
• dt +

n

• i=1

biσidWit, (23) where µ := (µ1, ..., µn)′ is the drift vector and 1 := (1, ..., 1)′ is an n × 1 vector of ones.

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◮ Using Itˆ

• ’s Lemma for several diffusion processes (cf. with

Tomas Bj¨

• rk 1998), one can solve the stochastic differential

equation (23) for Vt(b), obtaining Vt(b) = exp

• r + (µ − r1)′b − 1

2b′Σb

• t +

n

• i=1

biσiWit

• .

(24) ◮ We need to express Vt(b) in terms of the observables (Sit)n

i=1

(rather than the Wiener processes (Wit)n

i=1). Thus, we

substitute the expression σiWit = log Sit Si0

• −
• µi − σ2

i

2

• t

(25) into (24).

• Dr. Alex Garivaltis

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◮ Define the auxiliary vector zt := (z1t, ..., znt)′, where zit := log(Sit/Si0) − (r − σ2

i /2)t

σi √t . (26) ◮ In the risk-neutral (Q) world, zt is a vector of unit normals whose correlation matrix is equal to R. ◮ Thus, we have the compact expression Vt(b) = exp

• r − 1

2b′Σb

• t +

√ t · b′Mzt

• .

(27)

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Payoff Computation.

◮ Maximizing the (quadratic) exponent of Vt(b) with respect to b, we find that the best rebalancing rule in hindsight over [0, t] is b(S, t) = 1 √t · M−1R−1zt . (28) ◮ The payoff of Cover’s rebalancing option is V ∗

t = exp

• rt + 1

2z′

tR−1zt

• = exp
• r + 1

2b′Σb

• t
• ,

(29) where b on the right-hand-side denotes b(S, t).

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The Black-Scholes Equation for Several Underlyings.

◮ The arbitrage-free price C(S1, ..., Sn, t) is now governed by the Black-Scholes equation for several underlyings (cf. with Paul Wilmott 2001): 1 2

n

• i=1

n

• j=1

ρijσiσjSiSj ∂2C ∂Si∂Sj + r

n

• i=1

Si ∂C ∂Si + ∂C ∂t − rC = 0. (30) ◮ The associated boundary condition is C(S, T) = exp

• rT + 1

2z′ TR−1zT

• .
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Risk-Neutral Pricing.

◮ As usual, we solve the Black-Scholes equation by putting C(St, t) = e−r(T−t) × EQ

t

• exp
• rT + 1

2z′

TR−1zT

• ,

(31) which is the Q-expected present value of the final payoff conditional on St. ◮ We can write C =

∞

• −∞

· · ·

∞

• −∞

exp

• rt + 1

2z′

TR−1zT

• f (zT|zt)dz1T · · · dznT,

(32) where f (•|zt) is the conditional density of zT. ◮ zT is conditionally multivariate normal, with mean vector

• t/T · zt and covariance matrix (1 − t/T)R.
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Pricing Formula.

C(S, t) = T t n

2

× V ∗

t =

T t n

2

× exp

• rt + 1

2z′

tR−1zt

• .

(33) ◮ Again, we see that the American-style version of the option is never exercised early in equilibrium. ◮ Our proportionality relation C ∝ T n/2 agrees with the O

• T n/2

combinatorial bound from Cover’s discrete-time universal portfolio theory.

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Deltas.

◮ Differentiating, we find that the replicating strategy will own ∆i := ∂C ∂Si = C · (R−1zt)i Siσi √t = C · bi(S, t) Si (34) shares of stock i in state (S, t), where (R−1zt)i denotes the ith coordinate of the vector R−1zt

(n×1)

.

Theorem

The replicating strategy for Cover’s rebalancing option will always bet the fraction bi(S, t) of wealth on stock i in state (S, t). Thus, in order to delta-hedge Cover’s derivative, one just uses the best rebalancing rule in hindsight as it is known at time t.

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Asymptotic Performance of the Delta-Hedging Strategy.

◮ The practitioner invests $1 into the replicating strategy at some time t > 0. This buys him 1/C(St, t) units of the

• ption. (He cannot buy the option at t = 0, since

C(S0, 0) = ∞). ◮ Since C(St, t) > 1, the trader’s dollar buys him less than 1 unit of the option. ◮ At T, the option pays off V ∗

T, and since he owns 1/C(St, t)

units of it, his wealth at T is V ∗

T/C(St, t).

◮ Note that we guarantee to achieve the deterministic (!!) fraction 1/C(St, t) of the final wealth of the best rebalancing rule in hindsight over [0, T].

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◮ The excess compound-growth rate of the best rebalancing rule in hindsight (over and above the replicating strategy) is log V ∗

T

T − log {V ∗

T/C(St, t)}

T = log C(St, t) T = 1 T

• rt + 1

2z′

tR−1zt + n

2 log T t

• ,

(35) which tends to 0 as T → ∞. ◮ The excess wealth of the best rebalancing rule in hindsight is {1 − 1/C(St, t)} V ∗

T, which tends to ∞ as T → ∞.

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Figure: Simulation 1. One underlying, for the parameter values σ := 0.7, r := 0.02, T := 200, ν := 0.04, µ := ν + σ2/2.

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Figure: Simulation 2. One underlying, for the parameter values σ := 0.17, r := 0.02, T := 200, ν := 0.08, µ := ν + σ2/2.

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Figure: Simulation 3. Two underlyings, σ := (0.55, 0.7)′, r := 0.02, T := 200, ν := (0.03, 0.08)′, µi := νi + σ2

i /2, ρ := 0.2.

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Epilogue.

(Cover’s Rebalancing Option With Discrete Hindsight Optimization.)

◮ We can attempt to reduce the option price by hindsight-optimizing over a finite set B := {b(1), b(2), ..., b(L)}

• f continuously-rebalanced portfolios. In so doing, we can

guarantee to achieve a higher final percentage of the (less powerful) benchmark.

◮ This leads to an entirely different option price and ∆-hedging strategy (parameterized by the set B), which learns to grow its capital at the same asymptotic rate as the best element of B in hindsight. ◮ The discrete approach reveals a deeper connection between rebalancing options and the Margrabe-Fischer theory of exchange options, which confer the right to exchange one geometric Brownian motion for another. ◮ In fact, the no-exercise theorem holds for any subset B of Rn, and for any exercise price, K.

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Epilogue (2).

(The Margin Loan Trilogy.)

◮ In practice, “beat the market asymptotically” requires a ready supply of inexpensive margin loans.

◮ Two Resolutions of the Margin Loan Pricing Puzzle seeks to rationalize the (tremendously high) margin loan interest rates that are charged by some stock brokers, which seem to violate the no-arbitrage axiom (since these brokers could presumably eliminate risk by shorting their clients’ portfolios). ◮ Nash Bargaining Over Margin Loans to Kelly Gamblers derives exact formulas for a Nash Bargaining solution whereby a stock broker and a wealthy investor bargain over a contract that jointly specifies the client’s portfolio, the quantity of margin loans, and the margin loan interest rate. ◮ The Laws of Motion of the Broker Call Rate in the United States contains an applied time series analysis of the U.S. broker call rate, which is the rate at which stock brokers can borrow money (in an organized market) for the expressed purpose of funding their margin loans to retail clients.

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References I

[1] Bj¨

• rk, T., 1998. Arbitrage Theory in Continuous Time.

New York: Oxford University Press. [2] Black, F. and Scholes, M., 1973. The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), pp.637-654. [3] Cover, T.M., 1991. Universal Portfolios. Mathematical Finance, 1(1), pp.1-29. [4] Cover, T.M. and Gluss, D.H., 1986. Empirical Bayes Stock Market Portfolios. Advances in Applied Mathematics, 7(2), pp.170-181. [5] Cover, T.M. and Ordentlich, E., 1996. Universal Portfolios with Side Information. IEEE Transactions on Information Theory, 42(2), pp.348-363.

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References II

[6] Cox, J.C., Ross, S.A. and Rubinstein, M., 1979. Option Pricing: A Simplified Approach. Journal of Financial Economics, 7(3), pp.229-263. [7] Jamshidian, F., 1992. Asymptotically Optimal Portfolios. Mathematical Finance, 2(2), pp.131-150. [8] Kelly, J.L., 1956. A New Interpretation of Information

• Rate. Bell System Technical Journal, 35(4), pp.917-926.

[9] Luenberger, D.G., 1998. Investment Science. New York: Oxford University Press. [10] Ordentlich, E. and Cover, T.M., 1998. The Cost of Achieving the Best Portfolio in Hindsight. Mathematics of Operations Research, 23(4), pp.960-982. [11] Reiner, E. and Rubinstein, M., 1992. Exotic Options. Working Paper.

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References III

[12] Wilmott, P., 1998. Derivatives: The Theory and Practice

• f Financial Engineering. Chichester: John Wiley & Sons.

[13] Wilmott, P., 2001. Paul Wilmott Introduces Quantitative

• Finance. Chichester: John Wiley & Sons.
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