# Multiperiod Hedging using Futures: Mean Reversion and the Optimal Hedging Path

## Abstract

**:**

**ACKNOWLEDGEMENTS:**I would like to thank Mike Sher and an anonymous referee for their comments on an earlier draft of this paper. Errors, of course, remain my own responsibility.

## 1. INTRODUCTION

## 2, LITERATURE REVIEW

## 3. MODEL

_{t}− p

_{t−1}= (1 − ϕ)(μ − p

_{t −1}) + u

_{t}

_{t}is the value of the process in period t. If ϕ = 1, then p

_{t}follows a random walk. If ϕ is strictly between 0 and 1, µ is the long run mean of p

_{t}and (1-φ) is the speed of adjustment of p

_{t}to the long run mean. If ϕ = 0, then p

_{t}is independently and identically distributed (i.i.d.) each period. Thus, higher the value of ϕ, lower is the degree of mean reversion. Similar to the model in HD, it is assumed that u

_{t}is i.i.d. with variance ${\sigma}_{u}^{2}$. The process for p

_{t}may be equivalently expressed as

_{t}= (1 − ϕ)μ + ϕp

_{t−1}+ u

_{t}

_{t}denote the price process of the underlying of the futures contract (the “hedging process”).

_{t}− y

_{t−1}= (1 − θ)(κ − y

_{t−1}) + ξ

_{t}

_{t}is the value of the process in period t, κ is the long run mean, and (1-θ) is the rate of mean-reversion. ξ

_{t}is assumed to be i.i.d. with variance ${\sigma}_{\xi}^{2}$. Further, u

_{t}and ξ

_{t}are assumed to be contemporaneously correlated with covariance σ

_{uξ}, but all noncontemporaneous covariances are assumed to be zero.

_{t}is the expectations operator conditional on information available in period t.

#### **3.1 Hedging with Matched-Maturity contracts**

_{T-k}denote the hedge ratio (i.e., futures position per unit of the spot position).

_{uξ}/${\sigma}_{u}^{2}$, derived by Ederington (1979). Second, the hedging strategy derived above is dynamically optimal. Each period's hedge position is chosen after taking into account optimal hedging behavior in the future. Third, the model can be used to not only hedge a long or short cash position, but can also be readily adapted to hedging a future cash flow (such as a future period’s revenues or costs or operating cash flows) by letting the hedged process, p, denote the relevant cash flow, setting the non-stochastic cash position x equal to unity, and interpreting b

_{t}as the futures position rather than the hedge ratio. Fourth, the model can be (obviously) used for hedging (either a fixed cash position or a future cash flow) not just on a one-time basis, but on an on-going basis. The optimal hedging strategy can then be viewed not just as minimizing the variance of a single future cash flow, but as minimizing the volatility of the time series of cash flows (provided the spot price process does not follow a random walk). For example, the model can be applied to a firm trying to minimize the volatility of its monthly foreign currency revenues (input costs) by hedging each month’s foreign currency receipts (consumption) over “N” prior months. Fifth, the model can be readily adapted to hedging using forward contracts. As shown in part B of the Appendix, the optimal, cumulative hedge ratio at time T-k for hedging using forward contracts is given by

**INSERT TABLE 1 AND GRAPH 1 ABOUT HERE**

_{uξ}/${\sigma}_{u}^{2}$ (the ratio of the covariance between the shocks affecting the hedged and hedging processes to the variance of the hedged process) is set equal to unity. For simplicity, the interest rate, r, is set equal to zero. Therefore, the results in the table may be interpreted as relating to hedging using forward contracts rather than futures. Thus, the table contains optimal, cumulative hedge positions in forward contracts for a few selected values of the ratio of ϕ to θ. It is assumed in the calculations that the number of prior periods over which hedging is possible is twelve (i.e., N = 12).

#### **3.2 Hedging using nearby contracts (stack hedging)**

#### **3.3 Hedging using matched-maturity contracts v/s nearby contracts**

**INSERT TABLE 2 ABOUT HERE**

## 4. CONCLUSION

## FOOTNOTES:

^{1.}The issue of whether forward/futures prices are unbiased remains controversial. See Deaves and Krinsky (1995) for a good summary of early studies of this issue. Chinn and Coibion (2010) is a more recent study that also contains a good review of this topic and their findings appear to support this hypothesis for a range of commodities, especially energy related ones such as crude oil, heating oil, natural gas, and gasoline. They also find that even for non-energy commodities, “with almost no exceptions, we cannot reject the null of unbiasedness for the last five years of our sample, despite numerous departures from the null in the early and middle periods of our sample.” In any case, making any other assumption in its place would only serve to obscure the arguments of this paper.^{2}The term “nearby” contract is being used to mean a contract that will mature in the very next period. This is, of course, an assumption made primarily for convenience. However, it may be a close-enough approximation for practical purposes.

## APPENDIX

## A Hedging with matched-maturity futures contracts

_{T-k}denotes the hedge ratio (i.e., futures position per unit of the spot position taken at time T-k in order to hedge time T cash flows).

_{T}(focusing only on the error terms) is given by

_{T-1}, and setting the derivative equal to zero yields

_{T}is given by

_{T-1}is a known constant and using the first order condition gives the result

## B Hedging with matched-maturity forward contracts

_{t}is the expectation operator conditional on information available in period t.

_{uξ}is the covariance of these error terms.

_{T-k}denote the cumulative hedge ratio (cumulative futures position per unit of spot position) as of time T-k.

_{T}is given by (taking n to be 1 in equation B4)

_{T-1}, and setting the derivative equal to zero gives the result that

_{T}is given by

_{T-2}is

_{T-1}is a known constant given by

## C The Myers and Hanson (1996) framework

_{T}. Each period, the hedger chooses a hedge ratio, b

_{t}, and enters into a futures position, b

_{t}x, at the prevailing futures price, f

_{t}. Futures positions are marked to market at the end of each period. The hedger’s wealth at the end of each period is given by:

_{t}(x) represents non-stochastic costs. The hedger’s problem is to design a strategy to choose the futures position, b

_{t}, each period so as to maximize the expected utility of terminal wealth subject to the wealth constraints above:

_{t}is the expectations operator conditional on information available at time t, and U is an increasing and strictly concave utility function. N is the number of periods prior to the target period that the hedging activity can be initiated, and may be dictated either by internal corporate policy or external constraints such as availability of hedging contracts.

_{t}, follow a martingale with a zero-mean random shock, e

_{t}:

_{t}= f

_{t−1}+ e

_{t}

_{t}(p

_{T}), follows a martingale with a zero-mean random shock, v

_{t}:

_{t}(p

_{T}) = E

_{t−1}(p

_{T}) + v

_{t}

_{t}and v

_{t}are linearly related in the following manner:

_{t}= δ

_{t}e

_{t}+ ε

_{t}

_{t}is an unpredictable, zero-mean error term, and is independent of e

_{t}at all lags. δ

_{t}is the (possibly time-varying) slope coefficient of the linear regression of v

_{t}on e

_{t}, and, as such, is the ratio of the covariance between e

_{t}and v

_{t}to the variance of e

_{t}

_{t}, and the spot price of the underlying of the futures contract follow mean-reverting processes given by equations (1) and (3) in the main body of the current paper. Given the assumption of unbiased futures prices, the futures price as of time t of the contract maturing at T will evolve as follows:

_{t}and ξ

_{t}are contemporaneously correlated, their relationship can be represented as follows:

_{t}is the slope coefficient of the regression of u

_{t}on ξ

_{t}, and is therefore the ratio of the covariance of the two terms to the variance of ξ

_{t}. η

_{t}is a zero-mean random shock and unrelated to ξ

_{t}at all lags.

## D Hedging with nearby forwards or futures

_{T-k}, using dynamic programming. With one period to go (that is, as of time T-1),

_{T}is given by

_{T-1}is the forward position as at time T-1, ${\sigma}_{u}^{2}$ is the variance of the error term, u, ${\sigma}_{\xi}^{2}$ is the variance of the error term, ξ, and ${\sigma}_{u\xi}$ is the covariance between these error terms. Differentiating with respect to b

_{T-1}, and setting the derivative equal to zero gives the result that

_{T}is given by

_{T-1}is a known constant and using the first order condition gives the result

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**Table 1.**

**Hedging with matched-maturity forward contracts:**

**How the ratio of the autoregressive coefficients (φ/θ) affects the optimal hedging path**Notes:

- Hedge ratios have been calculated for one to twelve periods prior to the terminal date (i.e., N = 12) and ${\sigma}_{u}^{2}$ = σ
_{uξ}= 1. - h
_{T-k}is the (cumulative) forward position as of time T-k for the purpose of hedging cash flows in period T. Thus, k = number of periods remaining.

h_{T-k} | ||||

k | φ/θ = 0.1 | φ/θ = 0.8 | φ/θ = 1 | φ/θ = 1.2 |

1 | 1 | 1 | 1 | 1.0 |

2 | 0.1 | 0.8 | 1 | 1.2 |

3 | 0.01 | 0.64 | 1 | 1.4 |

4 | 0.001 | 0.512 | 1 | 1.7 |

5 | 0.0001 | 0.4096 | 1 | 2.1 |

6 | 0.00001 | 0.3277 | 1 | 2.5 |

7 | 0.00000 | 0.2621 | 1 | 3.0 |

8 | 0.00000 | 0.2097 | 1 | 3.6 |

9 | 0.00000 | 0.1678 | 1 | 4.3 |

10 | 0.00000 | 0.1342 | 1 | 5.2 |

11 | 0.00000 | 0.1074 | 1 | 6.2 |

12 | 0.00000 | 0.0859 | 1 | 7.4 |

Optimal Hedging Strategy | Hedging with nearby futures | Hedging with matched- maturity futures |
---|---|---|

Optimal Hedge Ratio k periods prior to the terminal date | ||

Back-Loaded Hedging Strategy* (Take substantial hedging positions only in periods close to the terminal date) | If ϕ ≈ 0 (Hedged process is close to being i.i.d.) | If ϕ << θ (Hedged process mean-reverts considerably quicker than the hedging process) |

Underhedge and then augment | If 0 < ϕ < 1 | If 0 < ϕ/θ < 1 |

Front-Loaded Hedging Strategy** (Take substantial hedging positions as far ahead aspossible) | If ϕ ≈ 1 (Hedged process is close to a random walk) | If ϕ ≈ θ (Both the hedged and hedging processes mean-revert at roughly the same rate) |

Overhedge and then reduce positions | If ϕ > 1 | If ϕ/θ > 1 |

**Graph 1.**

**Hedging with matched-maturity forward contracts:**

**Cumulative forward position, h**

_{T-k}against number of periods remaining, k for alternative values of the ratio of the autoregressive coefficients, ϕ/θ (assuming ${\mathit{\sigma}}_{\mathit{u}}^{\mathbf{2}}$ = σ_{uξ})## Share and Cite

**MDPI and ACS Style**

Rao, V.K. Multiperiod Hedging using Futures: Mean Reversion and the Optimal Hedging Path. *J. Risk Financial Manag.* **2011**, *4*, 133-161.
https://doi.org/10.3390/jrfm4010133

**AMA Style**

Rao VK. Multiperiod Hedging using Futures: Mean Reversion and the Optimal Hedging Path. *Journal of Risk and Financial Management*. 2011; 4(1):133-161.
https://doi.org/10.3390/jrfm4010133

**Chicago/Turabian Style**

Rao, Vadhindran K. 2011. "Multiperiod Hedging using Futures: Mean Reversion and the Optimal Hedging Path" *Journal of Risk and Financial Management* 4, no. 1: 133-161.
https://doi.org/10.3390/jrfm4010133