Dynamic Hedge Ratios

on another correlated asset to minimize price risk. The asset prices follow geometric brownian motions (GBMs), such that the prices are positively correlated through the two wiener processes. The hedger’s problem is to consider a futures strategy θ to maximize expected utility. θ generates profits or losses of G(θ)_t = Rt

0θ_sdF_s at any time t, and total final wealth is a function of which strategy θ is chosen:

W(θ) = kS_t+G(θ)_T, where k is the number of units in the future commitment.

The problem is formulated as

max

θ E(u[W(θ)]), (3.3.30)

where u(w) = w−cw² for some constant c. The strategy θ^∗ that maximizes the expression in (3.3.30) can also be found by minimizingE([W(θ)−L]²) wrt. θ, where L is some target level of final wealth. The traditional minimum variance objective is solved by minimizing the variance of W(θ) wrt. θ. Some numerical examples are provided, and the continuous-time hedges are compared to those that are fixed. The continuous-time hedges also include discrete-time hedges.

Kroner and Sultan (1993) account for the long-term cointegration relationship and the dynamic nature of assets by assuming a bivariate ECM with GARCH errors.

The ECM deals with the long-term relationship, whilst the GARCH errors permits the distribution’s second moments to change over time. Dynamic hedging is then pursued for foreign currency futures. If an investor has a fixed long position of one unit in the spot market, and is short −b units in the futures market, the return on the portfolio is x = s−bf, where s and f are changes in the spot and futures prices. But since the distribution of spot and futures prices is time-varying, this relationship should be xt = st−bt⁰ft, where t⁰ < t, st and ft are changes in prices between timet⁰ and t and b_t⁰ is the futures position at time t⁰. The optimal futures position is chosen at each time t by maximizing the expected utility function with the following conditional moment

E_t[U(x_t+1)] =E_t(x_t+1)−γσ_t²(t_t+1). (3.3.31) The optimal time-varying hedge ratio that maximizes this utility function is given by

b^∗_t = Et(ft+1) + 2γσt(st+1, ft+1)

2γσ²_t(f_t+1) . (3.3.32)

If futures prices were to be martingales, this would lead to the standard MV hedge b^∗_t = ^σt(s_σ2^t+1,ft+1)

t(ft+1) , but with time-varying conditional moments instead of fixed uncon-ditional moments. b^∗ is estimated by using a bivariate GARCH(1,1) ECM model.

The model is used for their dynamic hedging strategy, where rebalancing of the hedge ratio only happens if the utility gains from doing so offsets the losses due to transaction costs. Their results indicate that dynamically updating the hedge ratio given transaction costs of doing so improves hedging performance compared to the classic static hedge.

Ankirchner, Dimitroff, Heyne, and Pigorsch (2012) investigate the optimal time-varying hedge ratio when the basis is stationary. Bertus, Godbey, and Hilliard (2009) look at a quite similar problem. Both of these articles will be used heavily in later chapters, and will not be covered to a great extent in this chapter.

Investors and hedgers would normally want to hedge their exposures through a multi-period horizon. Howard and D’Antonio (1991) are two amongst many that consider a multi-period hedging problem. They assume that the spot prices of the asset being hedged shows autocorrelation, and further states that if the autocor-relation is zero, then a single-period hedge is appropriate, and if it is non-zero, a multi-period hedge should be considered. The spot return process follows an au-toregressive model of order 1 (AR(1)), and the percentage change in futures prices is i.i.d. and N(0, σ_t) ∀ t. There is thus no autocorrelation in the futures return series. Furthermore, it is assumed that the quantity being hedged grows at the rate g. The multi-period hedge ratio is larger than the single-period hedge ratio, and it increases with the number of periods being hedged.

Lien and Luo (1993) formulate the problem in the same way. They divide the hedging horizon into T periods, and minimize the variance of the wealth at the end of the hedging horizon, time T. They assume a cointegration process between spot and futures prices, and use an ECM to estimate the hedge ratios. Again, we follow the structure of Chen, Lee, and Shrestha (2003). N_s,t is the spot position at the beginning of t, andNf,t=−htNs,t is the futures position taken at timet. The final wealth at timeT is

W_T =W₀+

T−1

X

t=0

N_s,t[S_t+1−S_t+h_t(F_t+1−F_t)]

=W₀+

T−1

X

t=0

N_s,t[∆S_t+1+h_t∆F_t+1].

(3.3.33)

When minimizing the variance ofW_T, the optimal hedge ratios are given by h_t=−Cov(∆St+1,∆Ft+1)

V ar(∆F_t+1) −

T−1

X

i=t+1

Ns,i

N_s,t

Cov(∆Ft+1,∆Si+1+hi∆Fi+1)

V ar(∆F_t+1) . (3.3.34)

In this case, the optimal hedge ratio h_t will vary over time, and the difference from the static hedge ratio is the second term on the right-hand side. Interesting to notice, is the fact that if changes in the current futures prices are uncorrelated to changes in the future spot prices or future futures prices, thenh_twill be no different than the static MV hedge ratio. This shows that correlation between the prices of the asset or commodity being hedged and the hedging instrument will affect the optimal hedge ratio.

Chen and Tu (2010) continue this trend of multi-period hedging, but considers a mean-variance objective rather than minimum variance. Both the spot and futures prices are exogenous stochastic processes, and the hedger is assumed to short a certain number of futures each period. The T-period optimal hedging strategy consists of a series of hedging ratios{b₀, b₁, . . . b_T−1}. The proceeds from each period is reinvested at the risk-free rate i and held until the end of period T. Transaction costs are ignored, so the wealth at time T is

W_T =W₀+

T−1

X

t=0

[∆S_t+1−b_t∆F_t+1](1 +i)^T−t−1. (3.3.35) By maximizing the expected utility EU₀(W_T) = E₀(W_T)−γV ar₀(W_T), where γ is the risk aversion parameter, the series of optimal hedge ratios can be solved by backward induction. This starts inT −1, and continues to time 0. They obtain a closed-form solution to the problem, whereb_T−k is the hedge ratio ∀k= 1,2, . . . T :

b_T−k =−E_T−k(∆F_T−k+1) + 2γ(1 +i)^k−1Cov_T−k(∆S_T−k+1,∆F_T−k+1) 2γ(1 +i)^k−1V ar_T−k(∆F_T−k+1)

+2γPT−1

t=T−k+1(1 +i)^T−t−1Cov_T−k(∆S_t+1−b_t∆F_T+1,∆F_T−k+1) 2γ(1 +i)^k−1V arT−k(∆Ft−k+1) .

(3.3.36)

As can be seen, when γ = 1, i = 0 and E_T−k(∆F_T−k+1) = 0, b_T−k becomes the MV dynamic hedge ratio. The hedge ratio given above is hard to estimate since it depends on future hedge ratios, but it can be reduced to a single-period ratio when there is zero serial correlation between spot and futures price changes, and when second moments of the spot and futures price changes contain GARCH effects (Chen and Tu, 2010). Their next proposition is thus that the hedge ratio is reduced to

b_T−k = −ET−k(∆FT−k+1) + 2γ(1 +i)^k−1CovT−k(∆ST−k+1,∆FT−k+1)

2γ(1 +i)^k−1V ar_T−k(∆F_T−k+1) . (3.3.37)

In order to estimate the latter hedge ratio and the hedging effectiveness, they employ a bivariate GARCH model to model spot and futures price changes in the two well known indexes S&P-500 and FTSE-100.

Rao (2011) also examines multi-period hedging, but considers mean-reverting price processes and unbiased futures markets. He further looks at the optimal hedg-ing strategy when hedghedg-ing is done ushedg-ing nearby futures contracts that are behedg-ing rolled over, and when matched-maturity contracts are used. He finds that if both price processes mean-revert at around the same rate and one uses matched-maturity contracts, the optimal hedging strategy is front-loaded, meaning that large hedge positions are entered into long before the hedging horizon is over. If the hedged price process shows higher mean reversion than the futures price process, a single-period hedge seems more appropriate (or back-loaded). When hedging with nearby futures contracts, it depends on the mean reversion of the hedged price process.

Neuberger (1999) specifically looks at an agent who has a long-term commitment to supply a certain commodity in the future, and wishes to hedge using futures with a nearby maturity that are being rolled over. He estimates the model parameters through a cross-sectional regression, and the main assumption is that the expected value of the opening price of a futures contract is a linear function of other futures contract prices.

Low, Muthuswamy, Sakar, and Terry (2002) consider a multi-period MV hedge based on a cost-of-carry model for the futures prices. Rolling hedges are not consid-ered, so futures contracts maturing the closest (but after) to the end of the hedging horizon are used. The static cost-of-carry outperforms the dynamic one in their article, and both of them outperform the cointegrated ECM hedge (Lien and Luo, 1993), the GARCH hedge (Kroner and Sultan, 1993), and the classic MV hedge (Ed-erington, 1979). Static hedging is less expensive since transaction costs are lower, and static hedging models are less prone to estimation errors than are dynamic mod-els. Static hedge ratios might therefore outperform the dynamic ones, even though they should seem more appropriate according to theory. In the period prior to the terminal periodT, the following hedge ratio is calculated

h^∗_T−1 = Cov_T−1(S_T, F_T)

V ar_T−1(F_T) . (3.3.38)

In order to find the optimal hedge ratios in earlier periods, backward recursion is

used with the following formula for the optimal hedge ratio h^∗_T−k = Cov_T−k[F_T−k+1, S_T −PT−1

t=T−k+1h^∗_t(F_t+1−F_t)]

V ar_T−k(F_T−k+1

. (3.3.39)