Carolyn W. Chang works in the Department of Finance, California State University, Fullerton. Jack S. K. Chang and Min-Ming Wen work with the Department of Finance & Law, California State University, Los Angeles. The authors can be contacted via e-mail: cchang@fullerton.edu. We gratefully acknowledge the financial assistance from the Center for Insurance Studies at California State University, Fullerton, for a CIS Faculty Research Award.

Feature Articles

# Optimum Hurricane Futures Hedge in a Warming Environment: A Risk–Return Jump-Diffusion Approach

Article first published online: 8 NOV 2012

DOI: 10.1111/j.1539-6975.2012.01492.x

© The Journal of Risk and Insurance

Additional Information

#### How to Cite

Chang, C. W., Chang, J. S. K. and Wen, M.-M. (2014), Optimum Hurricane Futures Hedge in a Warming Environment: A Risk–Return Jump-Diffusion Approach. Journal of Risk and Insurance, 81: 199–217. doi: 10.1111/j.1539-6975.2012.01492.x

#### Publication History

- Issue published online: 13 FEB 2014
- Article first published online: 8 NOV 2012

- Abstract
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- References
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### Abstract

- Top of page
- Abstract
- Introduction
- Risk–Return Optimum Futures Hedge in the Presence of Firm's Costs of Hedging
- Numerical Analysis
- Conclusion
- References

We develop an optimum risk–return hurricane hedge model in a doubly stochastic jump-diffusion economy. The model's concave risk–return trade-off dictates that a higher correlation between hurricane power and insurer's loss, a smaller variable hedging cost, and a larger market risk premium result in a less costly but more effective hedge. The resulting hedge ratio comprises of a positive diffusion, a positive jump, and a negative hedging cost component. Numerical results show that hedging hurricane jump risks is most crucial with jump volatility being the dominant factor, and the faster the warming the more pronounced the jump effects.

### Introduction

- Top of page
- Abstract
- Introduction
- Risk–Return Optimum Futures Hedge in the Presence of Firm's Costs of Hedging
- Numerical Analysis
- Conclusion
- References

Economic losses from climate change are substantial and on the rise.1 The risk of property damage and loss from hurricanes has been progressively magnified by the ongoing growth of coastal population and property values.2 Scientific evidence3 further finds that an increasing level of greenhouse gas concentrations would lead to increased severity and frequency of tropical cyclones. These bleak projections, combined with the limited capacity of the insurance industry due to regulations, transaction costs, and geographic concentration of insured risk, subject insurers to facing a new era of climate risks management needing new hedging strategies.

Traditionally, insurance companies seek protection against underwriting risk through the mechanism of reinsurance, but worldwide capacity shortage coupled with the rising number of catastrophes in recent years have made securitization of catastrophic losses on both exchanges and over-the-counter markets a timely and desirable functional alternative. A number of catastrope (CAT)-linked instruments, such as Hurricane Risk Landfall Options (HuRLOs) launched by Weather Risk Solutions in 2008, Hurricane CAT Bonds (e.g., one launched by USAA Inc. in 1997), Hurricane CatEPuts, and the exchange-traded Hurricane Futures and Futures Options have been developed as a result. Among them, the exchange-traded hurricane futures market took off after the 2004–2005 storms saddled the insurance industry. For insurance companies, hurricane futures can provide additional coverage even if they have already bought traditional backup insurance. Similarly, banks, utilities, energy companies, state governments, and pension funds have strong motivations to seek additional coverage if a massive hurricane or a series of storms make landfall in their region.

One of the important issues in futures hedging is the variables considered in the objective function of the optimization problem. The traditional risk-minimization futures hedging approach developed by Johnson (1960) and Edrington (1979) formulates optimum futures hedge by assuming a diffusion underlying process, and then minimizing the one-period risk of the hedge portfolio using the least square regression or its variations.4 Corporate hedging literature has, however, consistently found that actual hedge ratios for commodity firms are significantly lower than those prescribed under the risk-minimization approach, because while making a hedging decision, firms consider not only risk minimization but also the hedging costs incurred. These hedging costs comprise of the fixed cost of setting up and maintaining a hedge program and the variable costs from the forgone expected return and from servicing portfolio rebalancing and daily resettlements.5 These findings suggest an optimal futures hedge model to be set in a risk–return trade-off framework.

Another critical issue in hedging using weather derivatives lies in the specification of physical weather and climate processes. Unlike the norm in perfectly functioning and complete financial markets where the no-arbitrage condition prevails and processes can be random, such as the common assumption of a diffusion underlying process in the traditional risk-minimization hedging models, climate processes are set in missing and incomplete markets such that their characteristic structure need be included in the pricing models. For example, stylized facts of the discrete, sporadic and jump nature of hurricane arrivals are considered by Wu and Chung (2010) who model hurricane arrival process as a jump-diffusion with a mean-reverting stochastic arrival rate.

A third critical issue here is which hurricane futures contract is most suitable for our hedging program. Candidate contracts that firms can adopt into their hedging program include the CME Hurricane Index (CHI) futures contract traded on the Chicago Mercantile Exchange (CME), the hurricane futures contracted listed on Iowa Electronic Markets (IEM), and the more recent Eurex hurricane futures contracts launched on June 29, 2009. The IEM contracts are based on tracking the location where a given hurricane will make its first landfall, whereas the Eurex contracts are settled based on actual insurance industry losses with a lengthy reporting lag as compiled by ISO's Property Claim Services (PCS) unit. Compared to the above two contracts, the more liquid CHI contract turns out to be the most suitable one to be incorporated into our hedging program because it provides a direct measure of the destructive power of a hurricane, thus can be used to better hedge total insured losses. The CHI, measured based on a hurricane's velocity (maximum sustained wind speed in miles per hour) and radius, is superior to the commonly used Saffir-Simpson Hurricane Scale (SSHS) in measuring the destructive power of a hurricane as the latter does not consider the radius of a storm. For example, when Hurricane Katrina made landfall in Louisiana in 2005, it was described as a weak Category 4 storm based on the SSHS, but this classification failed to provide a realistic estimate of the actual physical impact exerted. Hurricane Katrina, by having a very wide coverage area, had a CHI value as high as 19.0 and exerted the most physical damage in the U.S. history.

This research contributes to existing literature by developing a new optimum risk–return hurricane futures hedging model, considering the low intensity/high severity nature of physical hurricane phenomenon, and using the CHI contracts for insurers and other related parties to optimally mitigate the impacts of hurricane risk on their total insured losses.6 Under the context of an incomplete market where hurricane arrivals are not traded thus cannot be arbitraged away, we apply a doubly stochastic jump-diffusion process with mean-reverting stochastic arrival intensity rate to describe the discrete, sporadic jump nature of hurricane arrivals as suggested by Wu and Chung (2010). After deriving the hedged portfolio processes, we construct the hedging model taking both the cost of hedging and the insurers' risk minimization motives into consideration to arrive at a risk–return optimum futures hedge ratio. Sensitivity analysis concludes the research by exploring the impact of basis risk, variable hedging cost, jump volatility, and jump intensity on hedging effectiveness. It also explores the effect of global warming, and highlights the importance of considering the jump component in a hedging program.

Our derivation of a new three-component optimum risk–return hedge ratio is based upon the optimality condition at which the firm's marginal cost of hedging equals its marginal benefit of hedging at the optimum. This cost–benefit trade-off relationship is shown to be concave under which more risk reduction is accompanied with an increasing marginal cost of hedging. As a result, a firm reaches an optimum hedge ratio that is less, and often substantially less, than the corresponding risk-minimization counterpart. Our model shows that this hedge ratio is positively related to hedging effectiveness, whereas negatively related to firm's marginal cost of hedging. Moreover, it can be represented by a weighted summation of three component hedge ratios, namely, a positive diffusion, a positive jump, and a negative cost component, with the respective weight being the ratio of the component risk to the total futures risk.

The rest of the article is organized as follows. In the second section, we set up a correlated CHI and CAT loss process with stochastic intensity, derive the corresponding futures price process, construct the corresponding risk–return optimum futures hedge model, and finally derive and analyze a three-component optimum hedge ratio. In the third section, we implement numerical analyses to validate the model's predictions and explore the global warming effect on optimum hurricane futures hedge. In the fourth section, we conclude the article.

### Risk–Return Optimum Futures Hedge in the Presence of Firm's Costs of Hedging

- Top of page
- Abstract
- Introduction
- Risk–Return Optimum Futures Hedge in the Presence of Firm's Costs of Hedging
- Numerical Analysis
- Conclusion
- References

#### Deriving the Hurricane Futures Price Process

A critical issue in hedging using weather derivatives lies in the specification of physical weather and climate processes. Unlike the norm in perfectly functioning and complete financial markets where the underlying asset/index is traded and the no-arbitrage condition prevails in deriving the price process, weather derivatives are set in missing and incomplete markets with nontraded underlying weather index such that physical weather and climate characteristics need be included in process specifications. Wu and Chung (2010) model and empirically test catastrophe arrivals using a doubly Poisson process with a mean-reverting stochastic arrival rate as a flexible model to describe different types of catastrophe loss processes. In this vein, we assume the following correlated jump-diffusion processes with mean-reverting stochastic intensity regarding the changes of the CHI value (*dc _{t}*) and the changes of insurer's CAT loss (

*dS*) are exogenously given as in Equations (1) and (2):

_{t}- (1)

and

- (2)

where Μ_{c} and Σ_{c}, and Μ_{s} and Σ_{s} denote the respective instantaneous expected value and standard deviation conditional on no jumps for the changes of CHI value and the changes of insurer's CAT loss, respectively; they are functions of state variables the information set at time *t*; *Z*_{c}, and *Z*_{s} denote the standard Wiener processes with instantaneous diffusion covariance () between asset return and CHI change. is defined as ; Π is the Poisson process with stochastic intensity parameter *j* representing the jump component of the hurricane arrival process; *A* and *X* denote the gross jump sizes of CHI and loss triggered by this arrival, respectively, with lognormal distributions represented by ln*A* ∼ *N*(Μ_{a}(*Ω,t*),Σ_{a}(*Ω,t*)) and ln*X* ∼ *N*(Μ_{x}(*Ω,t*), Σ_{x}(*Ω,t*)); and finally, Π, *A*, and *X* are assumed to be independent of *Z*_{c} and *Z*_{s}. For ease of exposition, we will denote *c*_{t} as *c* and *S*_{t} as *S* in the remainder of the article.

Furthermore, the change of the intensity parameter *j* follows a mean-reverting Ornstein-Uhlenbeck process defined as follows:

- (3)

where *j* denotes the jump arrival intensity, *Κ _{j}* is the speed of adjustment,

*m*is the long-run mean rate,

_{j}*Σ*is the instantaneous variance, and

_{j}2*Z*represents the standard Wiener process.

_{j}The solution of this process is known to be

- (4)

Integrating the expected value of (4) over a time period Τ yields the expected intensity over Τ,

- (5)

where

This stochastic intensity specification indicates that CAT shocks may arrive in clusters around the time of important news announcements such that the arrival intensity will be high to reflect the lumpiness of the jump arrival. After the release of the news, however, it will revert to a lower long-run mean level *m _{j}*, and vice versa. Although

*j*denotes the intensity of arrival, the speed of adjustment parameter

*Κ*governs the level of persistence of the intensity process—higher values of

_{j}*Κ*implying that the intensity process leaves the high state more quickly, and vice versa.

_{j}Intuitively, weather risks would have zero correlations with risks on the general financial market and thus weather derivatives are essentially zero-beta assets bearing no systematic risk. This hypothesis has been empirically verified by Hoyt and McCullough (1999) and others who show that the arrival of a hurricane and the change of its physical power upon arrival exhibit no correlation to the changes in financial prices. As explained in Hull (2012, p. 760), this feature helps explain the trading success of catastrophe bonds and some other CAT products in capital markets—for offering fund managers/buyers diversification benefits with possible above normal returns and improved risk–return trade-off. It also helps simplify valuation of weather derivatives. With a zero risk premium, risk neutral pricing can be directly applied to pricing weather derivatives using historical/real-world data estimates. To sum up, because today's CHI futures price embeds no risk premiums, it is essentially a forward-looking unbiased market predictor of the expected CHI value on the expected landfall date. As such the futures price process is a martingale where the martingale probability measures of news arrival and intensity change equal to their physical counterparts. This is tantamount to saying that the risk-neutralized processes of the futures' two underlying factors, *c* and *j*, are identical to their physical processes represented by Equations (1) and (3), respectively.

As a follow-up of the above discussions, note that because CME CHI futures contracts are event driven and settled based on the closing CHI value when a hurricane makes landfall, its expiration date is uncertain, rendering the contract to have a random maturity. Nevertheless as this maturity risk bears no correlation with the market risk and thus bears no risk premium, we can use the expected maturity, that is, the expected landfall date provided by weather forecast, in actual risk-neutral pricing applications.7

Now assuming there is no basis risk between the change in the cash and the futures prices to circumvent the issue of stochastic interest rates and that the futures price function is twice continuously differentiable in the underlying factor values, *c* and *j*, we next derive the futures price process below by applying the generalized Ito's lemma to the function with respect to Equations (1) and (3),

- (6)

where *F* denotes the futures price, denotes the instantaneous expected price change due to the arrival of a jump event, denotes the instantaneous standard deviation of futures returns with *dZ _{f}* the increment to a standard Wiener process and ; the expected total price change is zero due to continuous marking-to-market; the Poisson process

*dΠ*and random gross jump size

*A*are the same as in Equation (1) due to the fact that the Poisson event for the futures price occurs if and only if it occurs for the CHI value. Moreover, when the Poisson event occurs, the jump in the futures price should be equal to the jump in the expected CHI value because the futures price is a forward-looking unbiased predictor of the CHI.

#### Deriving the Hedged Portfolio Process

Next, with a given hedge ratio, *h*, we consider a short spot/long futures hedging portfolio *P* defined as the dollar amount of long futures per dollar of a fixed short spot position. Given the CAT loss and the CHI futures price processes, we form the price dynamics of the hedge portfolio described as,

- (7)

*Ω(t)*, the instantaneous total hedge portfolio variance at time *t*, is given in Equation (8) below:

- (8)

where *Ρ _{sf}* denotes the diffusion correlation coefficient, measuring the correlation coefficient between spot diffusion return and futures diffusion return, and

*Ρ*denotes the jump correlation coefficient, measuring the correlation coefficient between spot jump return and futures jump return. These two coefficients measure the basis risk in our cross-hedging effectiveness evaluation as the changes of the firm's total loss and the CHI value would not be perfectly correlated when a catastrophic event occurs—the former depends on both physical and nonphysical factors whereas the latter depends on physical factors of the hurricane only.8 The higher the value of these coefficients, the more correlated is the firm's total loss with the CHI value, the smaller the basis risk and the more effective is the futures hedge. In the event when the correlations are perfect, the basis risk would be zero.

_{XA}#### The Marginal Cost of Hedging

Typical hedging costs to the insurers include the fixed costs of setting up and managing a hedging program and variable costs that include the foregone reduction in loss when a hurricane arrives in a weaker form than expected, the transaction costs, and the costs of servicing variation margin due to daily resettlement. As justified by Haushalter's (2000) finding that though the economy of scale in hedging is particularly relevant for the fixed cost component, it is not significant for the marginal costs of increasing the extent of hedging, we thus employ the following linear cost structure9 to specify the firm's total hedging cost *TC*:10

- (9)

where *h* is the hedge ratio, *a* represents the fixed cost, and *b* stands for the variable hedging cost.

The marginal cost of hedging, *MC*, is defined as *MC = dTC/dΩ(t)*. In other words, *MC* is the cost of obtaining the next unit of risk reduction for the hedge portfolio and this marginal risk reduction is given by Equation (10):

- (10)

Because both *TC* and *Ω(t)* are functions of the hedge ratio *h*, we can express *MC* as the ratio of the derivative of *TC* with respect to *h* () to the derivative of *Ω(t)* with respect to *h* (). Thus, MC is described as,

- (11)

Because *dΩ(t)*/dh < 0 (the larger the hedge ratio, the lower the portfolio risk), the concavity of the market risk–return trade-off indicates that *MC* is a negative increasing function of the hedge ratio—the larger the hedge ratio the larger the variable hedging cost. As a result, the absolute value of *MC* is minimized when *h = 0* and maximized when *h = h _{rm}*, where

*h*denotes the corresponding risk-minimization optimum hedge ratio.

_{rm}#### The Marginal Benefit of Hedging

In the risk–return optimum hedging setup, the benefit of hedging (*ϕ)* in risk reduction can be measured by the market risk premium earned per unit of risk reduction by dividing the total spot risk premium by the total spot risk as follows:

- (12)

where *rp* denotes the total spot risk premium that can be determined in a jump-diffusion framework as demonstrated in Chang (1995) and Framstad, Øksendal, and Sulem (2001), and is the total instantaneous spot variance of return.

#### The Risk–Return Optimum Hedge Ratio

The concavity of the market risk–return trade-off dictates that a firm is only willing to pay up to the market risk premium when determining the optimum hedge ratio. Passing the optimum point will make the foregone marginal cost of hedging too large to bear vis-À-vis the market risk premium. In other words, the optimum hedge ratio is obtained under the optimality condition at which the firm's marginal cost of hedging (*MC*) equals its marginal benefit of hedging (*ϕ)*—the market risk premium.

Having derived the expressions for both the marginal cost and marginal benefit of hedging, we now apply the above optimality condition to derive the optimum hedge ratio. Because *MC* < 0, adding a negative sign to Equation (11), equating it to Equation (12) and rearranging the terms, we derive Equations (13a) and (13b) as follows:

- (13a)

which can be alternatively expressed in risk unit as

- (13b)

Equation (13a) is the optimality condition stated in return that at the optimum point, the marginal cost per unit of risk reduction equals the market risk premium. Equation (13b), on the other hand, utilizes our risk–return trade-off relationship to state the optimality condition in risk unit. In the context of the market risk–return trade-off, the firm will hedge to the point where the marginal risk reduction through hedging equals the marginal risk increase due to the increase in hedging cost. In contrast, should the hedging be motivated for risk minimization only, the investor would determine the optimum hedge ratio as if either the variable cost of hedging was zero or the market risk premium was infinite.

By solving Equation (13a) for *h*, we derive the optimum hedge ratio as described in Equation (14a), which can be further decomposed into three hedge ratio components—a diffusion, a jump, and a hedging cost component, as described in Equation (14b):

- (14a)

- (14b)

Equation (14a) reveals that a higher correlation, a smaller value of variable hedging cost, and a larger market risk premium can result in a less costly but more effective hedge, leading to a more valuable hedging strategy. These properties are consistent with the empirical findings in Haushalter (2000), which suggest that the extent to which a firm hedge is negatively related to hedging costs but positively related to hedging effectiveness.

Equation (14b) reveals that the optimum hedge ratio can be decomposed as a weighted sum of three component hedge ratios, namely, a positive diffusion, a positive jump, and a negative hedging cost ratio. The respective weight is the ratio of the corresponding risk component to total futures risk. The hedge ratio for the respective diffusion and jump component is equivalent in form to the commonly defined market Β risk for risk minimization, whereas the hedge ratio for cost component is –1, attributed to the fact that the hedging cost structure is characterized as a linear function in the hedge ratio.

### Numerical Analysis

- Top of page
- Abstract
- Introduction
- Risk–Return Optimum Futures Hedge in the Presence of Firm's Costs of Hedging
- Numerical Analysis
- Conclusion
- References

Based on a series of sensitivity analyses, we first analyze Equations (14a) and (14b) and verify the model's predictions to highlight the effects of basis risk, variable hedging cost, jump arrival intensity, and jump volatility on the determination of the optimum hedging ratio and its components. We then turn to the global warming effect—how global warming with increasing hurricane intensity of stronger hurricanes affects optimum futures hedge.

#### Numerical Validation

We consider a typical September East Coast hurricane contract with annual volatility of 60 percent. We break down the 60 percent annual volatility as *Σ _{s}* =

*Σ*= 0.1, Σ

_{f}_{A}= Σ

_{x}= 0.0913, and

*j*= 30. We choose

*mj*= 29.6219 and

*Κ*= 0.4671, the respective long-term mean rate and speed of adjustment for the jump arrival intensity, according to the findings of Wu and Chung (2010). We first explore the basis risk effect on the hedging performance that because the changes of the firm's loss and the CHI value are not perfectly correlated when an event occurs, the hedging effectiveness may be less than desired. We apply the value between 0 and 1 for both jump correlation (

_{j}*Ρ*) and diffusion correlation (

_{XA}*Ρ*) but with the constraint that

_{sf}*Ρ*>

_{XA}*Ρ*to incorporate that when a Poisson event occurs the correlation accentuates. Finally, we choose

_{sf}*b*= 0.05 and ϕ = 0.5 to model the hedging cost component. The results are presented in Table 1. Under this scenario, the average hedge ratio is about 0.521, and the respective ratios for diffusion hedging and jump hedging are 0.313 and 0.735. The optimum hedge ratios present an increasing trend with the increases in both jump correlation (

*Ρ*) and diffusion correlation (

_{XA}*Ρ*), indicating a lower degrees of basis risk leads to higher degrees of hedging effectiveness.

_{SF}Ρ_{sf} | Ρ_{XA} | h* | Ρ_{sf} | Ρ_{XA} | h* |
---|---|---|---|---|---|

Notes ^{}The optimal hedge ratio, as derived in Equation (14b), is a weighted average of the diffusion, jump, and cost hedge ratios, with the respective weight being the ratio of the corresponding risk component to the total futures risk. As *Ρ*and_{sf}*Ρ*increase, basis risk decreases, leading to increasing diffusion and jump hedge ratios, and thus increasing overall optimum hedge ratio. Basis risk bears no effect on the hedging weights, which are, respectively, 0.039, 0.961, and 0.193, for the diffusion, jump, and cost components. Given the assumption that_{XA}*Σ*=_{s}*Σ*and Σ_{f}_{A}= Σ_{x}, the diffusion and jump hedge ratios are always equal to the chosen correlation levels, whereas the cost hedge ratio is always –1 because of the assumed linear hedging cost structure. The asterisk (*) indicates average values.
| |||||

0 | 0.3 | 0.095 | 0.3 | 0.6 | 0.395 |

0 | 0.4 | 0.191 | 0.3 | 0.7 | 0.491 |

0 | 0.5 | 0.287 | 0.3 | 0.8 | 0.587 |

0 | 0.6 | 0.383 | 0.3 | 0.9 | 0.683 |

0 | 0.7 | 0.480 | 0.3 | 1 | 0.780 |

0 | 0.8 | 0.576 | 0.4 | 0.5 | 0.303 |

0 | 0.9 | 0.672 | 0.4 | 0.6 | 0.399 |

0 | 1 | 0.768 | 0.4 | 0.7 | 0.495 |

0.1 | 0.2 | 0.003 | 0.4 | 0.8 | 0.591 |

0.1 | 0.3 | 0.099 | 0.4 | 0.9 | 0.687 |

0.1 | 0.4 | 0.195 | 0.4 | 1 | 0.783 |

0.1 | 0.5 | 0.291 | 0.5 | 0.6 | 0.403 |

0.1 | 0.6 | 0.387 | 0.5 | 0.7 | 0.499 |

0.1 | 0.7 | 0.483 | 0.5 | 0.8 | 0.595 |

0.1 | 0.8 | 0.580 | 0.5 | 0.9 | 0.691 |

0.1 | 0.9 | 0.676 | 0.5 | 1 | 0.787 |

0.1 | 1 | 0.772 | 0.6 | 0.7 | 0.503 |

0.2 | 0.3 | 0.103 | 0.6 | 0.8 | 0.599 |

0.2 | 0.4 | 0.199 | 0.6 | 0.9 | 0.695 |

0.2 | 0.5 | 0.295 | 0.6 | 1 | 0.791 |

0.2 | 0.6 | 0.391 | 0.7 | 0.8 | 0.603 |

0.2 | 0.7 | 0.487 | 0.7 | 0.9 | 0.699 |

0.2 | 0.8 | 0.583 | 0.7 | 1 | 0.795 |

0.2 | 0.9 | 0.680 | 0.8 | 0.9 | 0.703 |

0.2 | 1 | 0.776 | 0.8 | 1 | 0.799 |

0.3 | 0.4 | 0.203 | 0.9 | 1 | 0.803 |

0.3 | 0.5 | 0.299 | 0.313* | 0.735* | 0.521* |

Next, we explore the effects of variable hedging cost on the determination of the optimum hedge ratios. We apply standardized values between 0 and 0.1 to the variable hedging costs (*b*) and choose *Ρ _{sf}* = 0.6 and

*Ρ*= 0.8,

_{XA}*ceteris paribus*. We document the results in Table 2, which shows that, consistent with the findings in the corporate hedging literature, the optimum hedge ratios present a decreasing trend with higher hedging cost, and the average hedge ratio of 0.583 is lower than the corresponding risk-minimization hedge ratio of 0.792, which occurs when the variable hedging cost

*b*is zero. As the variable hedge cost increases from zero, the weight of the hedging cost component in the overall optimum hedge ratio steadily increases from 0 to 0.387. Figure 1 also illustrates that a higher level of hedging cost leads to a lower degree of hedge ratio.

b | h* | Cost Hedging Weight |
---|---|---|

Notes ^{}As the variable hedging cost *b*increases, the cost hedging weight increases, making hedging more expensive and leading to decreasing overall optimum hedge ratio. It bears no effect on other variables. The respective diffusion and jump hedging weights are 0.039 and 0.961, whereas the respective diffusion, jump, and cost hedge ratios are 0.6, 0.8, and –1. The asterisk (*) indicates average values.
| ||

0 | 0.792 | 0.000 |

0.01 | 0.754 | 0.039 |

0.02 | 0.715 | 0.077 |

0.03 | 0.676 | 0.116 |

0.04 | 0.638 | 0.155 |

0.05 | 0.599 | 0.193 |

0.06 | 0.560 | 0.232 |

0.07 | 0.522 | 0.271 |

0.08 | 0.483 | 0.309 |

0.09 | 0.444 | 0.348 |

0.1 | 0.406 | 0.387 |

0.050* | 0.583* | 0.193* |

To investigate how sensitive are the optimum hedge ratios to the jump arrival intensity, we choose *j* = [10, 50], *ceteris paribus*. The results of this sensitivity analysis are reported in Table 3 and Figure 2 and present the trend that an increase of jump arrival intensity causes a larger hedge ratio because of the increase in jump hedging weight relative to other components. In addition, the hedge ratio shows a strong correlation with jump arrival intensity with a high correlation coefficient value of 0.982 (computed using the corresponding paired values of *j* and hedge ratios in the table), suggesting that the jump arrival intensity is likely to have dominating effects on the determination of the optimum hedge ratio.

j | h* | Diffusion Hedging Weight | Jump Hedging Weight | Cost Hedging Weight | j | h* | Diffusion Hedging Weight | Jump Hedging Weight | Cost Hedging Weight |
---|---|---|---|---|---|---|---|---|---|

Notes ^{}As the jump arrival intensity *j*increases, the jump hedging weight increases whereas other weights decrease. The overall optimum hedge ratio, however, increases, indicating the jump arrival intensity is likely to have a dominating effect on the determination of the overall optimum hedge ratio. There is no effect on other variables. The respective diffusion, jump, and cost hedge ratios are 0.6, 0.8, and –1. The asterisk (*) indicates average values.
| |||||||||

10 | 0.492 | 0.059 | 0.941 | 0.296 | 31 | 0.602 | 0.038 | 0.962 | 0.190 |

11 | 0.500 | 0.058 | 0.942 | 0.288 | 32 | 0.606 | 0.037 | 0.963 | 0.187 |

12 | 0.508 | 0.056 | 0.944 | 0.281 | 33 | 0.609 | 0.037 | 0.963 | 0.184 |

13 | 0.515 | 0.055 | 0.945 | 0.274 | 34 | 0.612 | 0.036 | 0.964 | 0.181 |

14 | 0.522 | 0.054 | 0.946 | 0.268 | 35 | 0.615 | 0.036 | 0.964 | 0.178 |

15 | 0.528 | 0.052 | 0.948 | 0.261 | 36 | 0.618 | 0.035 | 0.965 | 0.175 |

16 | 0.534 | 0.051 | 0.949 | 0.255 | 37 | 0.621 | 0.034 | 0.966 | 0.172 |

17 | 0.540 | 0.050 | 0.950 | 0.250 | 38 | 0.623 | 0.034 | 0.966 | 0.170 |

18 | 0.546 | 0.049 | 0.951 | 0.244 | 39 | 0.626 | 0.033 | 0.967 | 0.167 |

19 | 0.552 | 0.048 | 0.952 | 0.239 | 40 | 0.629 | 0.033 | 0.967 | 0.165 |

20 | 0.557 | 0.047 | 0.953 | 0.234 | 41 | 0.631 | 0.032 | 0.968 | 0.162 |

21 | 0.562 | 0.046 | 0.954 | 0.229 | 42 | 0.634 | 0.032 | 0.968 | 0.160 |

22 | 0.567 | 0.045 | 0.955 | 0.224 | 43 | 0.636 | 0.032 | 0.968 | 0.158 |

23 | 0.571 | 0.044 | 0.956 | 0.220 | 44 | 0.638 | 0.031 | 0.969 | 0.156 |

24 | 0.576 | 0.043 | 0.957 | 0.216 | 45 | 0.640 | 0.031 | 0.969 | 0.153 |

25 | 0.580 | 0.042 | 0.958 | 0.212 | 46 | 0.643 | 0.030 | 0.970 | 0.151 |

26 | 0.584 | 0.042 | 0.958 | 0.208 | 47 | 0.645 | 0.030 | 0.970 | 0.149 |

27 | 0.588 | 0.041 | 0.959 | 0.204 | 48 | 0.647 | 0.029 | 0.971 | 0.147 |

28 | 0.592 | 0.040 | 0.960 | 0.200 | 49 | 0.649 | 0.029 | 0.971 | 0.145 |

29 | 0.595 | 0.039 | 0.961 | 0.197 | 50 | 0.651 | 0.029 | 0.971 | 0.144 |

30 | 0.599 | 0.039 | 0.961 | 0.193 | 30* | 0.590* | 0.040* | 0.960* | 0.202* |

To further explore jump effects, we next conduct the analysis based on different jump volatility scenarios. We let jump volatility Σ_{A} and Σ_{x} be in the interval of [0.054, 0.154], *ceteris paribus*. Table 4 and Figure 3 illustrate that the optimum hedge ratio is an increasing function of the jump volatility, because of the increase in jump hedging weight relative to other components, suggesting that jump volatility and hedge ratios move in the same direction. Moreover, the hedge ratio shows a strong correlation with jump volatilities with a high correlation coefficient value of 0.92 (computed using the corresponding paired values of Σ_{A} & Σ_{x} and hedge ratios in the table), suggesting that the jump effect has a dominating role in governing the optimum hedge ratio.

Σ & _{A}Σ_{x} | h* | Diffusion Hedging Weight | Jump Hedging Weight | Cost Hedging Weight |
---|---|---|---|---|

Notes ^{}As the jump volatilities *Σ*and_{A}*Σ*increase, the jump hedging weight increases whereas other weights decrease. The overall optimum hedge ratio, however, increases, indicating the jump volatilities are likely to have dominating effects on the determination of the overall optimum hedge ratio. There is no effect on other variables. The respective diffusion, jump, and cost hedge ratios are 0.6, 0.8, and –1. The asterisk (*) indicates average values._{x}
| ||||

0.054 | 0.258 | 0.104 | 0.896 | 0.521 |

0.064 | 0.403 | 0.076 | 0.924 | 0.382 |

0.074 | 0.498 | 0.058 | 0.942 | 0.291 |

0.084 | 0.563 | 0.046 | 0.954 | 0.228 |

0.094 | 0.609 | 0.037 | 0.963 | 0.184 |

0.104 | 0.643 | 0.030 | 0.970 | 0.151 |

0.114 | 0.669 | 0.025 | 0.975 | 0.126 |

0.124 | 0.689 | 0.021 | 0.979 | 0.107 |

0.134 | 0.704 | 0.018 | 0.982 | 0.092 |

0.144 | 0.717 | 0.016 | 0.984 | 0.080 |

0.154 | 0.727 | 0.014 | 0.986 | 0.070 |

0.104* | 0.589* | 0.041* | 0.959* | 0.203* |

#### The Global Warming Effect

Scientific evidence has shown that global warming relates to increased intensity of hurricanes over the past few decades. Elsner, Kossin, and Jagger (2008) detect increasing intensity of the strongest tropical cyclones by using quantile regression with a 30-year trend that has been related to an increase in Sea Surface Temperatures (SST) over the Atlantic Ocean and elsewhere. An increase in SST of 1 °C results in an increase in the global frequency of the strong hurricanes (those with wind speeds exceeding 49 ms−1) from 13 to 17 cyclones per year. Webster et al. (2005) use global tropical cyclone statistics for the satellite era (1970–2004) to identify changes in the number, duration, and intensity of tropical cyclones in a warming environment. They find a large increase in the number and proportion of hurricanes reaching categories 4 and 5 (those with wind speeds exceeding 56 ms−1 according to the Saffir-Simpson scale) although the total number of hurricanes in the same time period has been steady. The global number of hurricanes has almost doubled from 50 per pentad in the 1970s to near 90 per pentad during the past decade, whereas the proportion has increased from around 20 percent to around 31 percent. Emanuel (2005) finds that hurricanes have become 70–80 percent more powerful in their degree of destructiveness in the same time period. Finally, a recent forecast for the 2010 Atlantic hurricane season by the National Oceanic and Atmospheric Administration of 8–14 hurricanes with 3–7 developing into Category 3 or higher has made year 2010 one of the most active ever.

In the insurance and urban economics literature, there is a growing interest in investigating the global warming effects. For example, Hallegatte (2007) finds that a 10 percent increase in potential hurricane intensity in the east coast of the United States can cause a 54 percent increase in the mean normalized economic losses. Esteban, Webersik, and Shibayama (2009) investigate the effect of a global warming-induced increase in typhoon intensity on urban productivity in Taiwan and find the loss in GDP could reach 0.7 percent by current trend. In this article, we study the global warming effect on optimum futures hedge. We analyze Equations (14a) and (14b) using sensitivity analysis to highlight the effects of increasing jump arrival intensity (*j*) and jump volatility (*Σ _{A}*,

*Σ*) on the determination of the optimum hedge ratio and its components. We assume that the jump arrival intensity falls in the interval of [10, 50], the jump volatility in the interval of [0.0537, 0.1537], and the correlation parameters measuring the basis risks,

_{x}*Ρ*and

_{sf}*Ρ*, are within simulated values between 0 and 1 with the condition that

_{XA}*Ρ*>

_{XA}*Ρ*,

_{sf}*ceteris paribus*.

The results are presented in Table 5 and show that the average optimum hedge ratio across all jump arrival intensity and jump volatility levels is 0.584. As the arrival of stronger hurricanes become more frequent, the weight on jump hedging component increases relative to other weights, leading to increasingly larger optimum hedge ratios. Table 5 and Figure 4 further compare how the optimum hedge ratio varies with the change of each of the two jump factors—jump arrival intensity and jump volatility. The results show that the change in the optimum hedge ratio is more sensitive to the change in jump volatility than to the change in jump arrival intensity. For example, as the jump arrival intensity increases its value from 10 to 50, the optimum hedge ratio moves up steadily from 0.528 to 0.642, whereas the hedge ratio increases rapidly from 0.262 to 0.723 over the range of jump volatility from 0.0537 to 0.1537. A correlation analysis confirms this finding by showing that the correlation coefficient (computed using the corresponding paired values in the table) between the change in the optimum hedge ratio and the jump volatility is as large as 0.86, whereas it is about 0.28 between the movements of the hedge ratio and the jump arrival intensity. These findings of jump effects indicate that in a warming environment, firms should systematically adjust their jump hedging weights according to their forecasts of jump arrival intensity and volatility to mitigate the impacts of hurricane risk on their total losses.

Scenarios | j | Σ & _{A}Σ_{x} | h* | Diffusion Hedging Weight | Jump Hedging Weight | Cost Hedging Weight |
---|---|---|---|---|---|---|

Notes ^{}Global warming effects are represented by the scenarios of jump arrival intensity ( *j*) and jump volatility (*Σ*,_{A}*Σ*). In the jump volatility scenario, we vary_{x}*j*from 10 to 50 for each volatility level, whereas in the jump arrival intensity scenario, we vary (*Σ*,_{A}*Σ*) from 0.0537 to 0.1537 for each intensity level. The optimal hedge ratio is derived in Equation (14b) as a weighted average of the diffusion, jump and cost hedge ratios, with the respective weight being the ratio of the corresponding risk component to the total futures risk. As the jump volatilities_{x}*Σ*and_{A}*Σ*or the jump arrival intensity_{x}*j*increase, the jump hedging weight increases whereas other weights decrease. The overall optimum hedge ratio however increases, indicating they are likely to have dominating effects on the determination of the overall optimum hedge ratio. The results also show that the change in the optimum hedge ratio is more sensitive to the change in jump volatility than to the change in jump arrival intensity. There is no effect on other variables. The respective diffusion, jump, and cost hedge ratios are 0.6, 0.8, and –1.^{}Symbol “*” indicates average value.
| ||||||

Jump volatility | 0.0537 | 0.262 | 0.103 | 0.897 | 0.517 | |

0.0637 | 0.383 | 0.080 | 0.920 | 0.401 | ||

0.0737 | 0.482 | 0.061 | 0.939 | 0.306 | ||

0.0837 | 0.550 | 0.048 | 0.952 | 0.241 | ||

0.0937 | 0.598 | 0.039 | 0.961 | 0.194 | ||

0.1037 | 0.634 | 0.032 | 0.968 | 0.160 | ||

0.1137 | 0.661 | 0.027 | 0.973 | 0.134 | ||

0.1237 | 0.682 | 0.023 | 0.977 | 0.113 | ||

0.1337 | 0.699 | 0.019 | 0.981 | 0.097 | ||

0.1437 | 0.712 | 0.017 | 0.983 | 0.085 | ||

0.1537 | 0.723 | 0.015 | 0.985 | 0.074 | ||

Jump arrival intensity | 10 | 0.528 | 0.052 | 0.948 | 0.261 | |

15 | 0.517 | 0.054 | 0.946 | 0.272 | ||

20 | 0.546 | 0.049 | 0.951 | 0.244 | ||

25 | 0.569 | 0.044 | 0.956 | 0.222 | ||

30 | 0.589 | 0.041 | 0.959 | 0.203 | ||

35 | 0.605 | 0.037 | 0.963 | 0.187 | ||

40 | 0.620 | 0.035 | 0.965 | 0.173 | ||

45 | 0.632 | 0.032 | 0.968 | 0.162 | ||

50 | 0.642 | 0.030 | 0.970 | 0.151 | ||

0.584* |

### Conclusion

- Top of page
- Abstract
- Introduction
- Risk–Return Optimum Futures Hedge in the Presence of Firm's Costs of Hedging
- Numerical Analysis
- Conclusion
- References

In a doubly stochastic jump-diffusion economy where hurricane arrivals exhibit sporadic random jumps with mean-reverting stochastic intensity due to physical weather and climate phenomena, we develop a risk–return jump-diffusing futures hedging model in incomplete markets for a firm to hedge hurricane risks in the presence of hedging costs. Optimal hedging is achieved under a concave trade-off where the marginal cost of hedging equals the marginal benefit of hedging, leading to an optimal hedge ratio significantly less than the traditional risk-minimization counterpart. It reveals that a higher correlation between hurricane power and insurer's loss, a smaller variable hedging cost, and a larger market risk premium can result in a less costly but more effective hedge, leading to a more valuable hedging strategy. The optimum hedge ratio can also be decomposed into a positive diffusion, a positive jump, and a negative hedging cost component, enabling more refined implementation and analysis of the hedge program. Numerical analysis indicates that hedging hurricane arrival jump risks in both arrival and volatility uncertainties is the most crucial, thereby carrying the largest weight in the determination of the overall hedge ratio, whereas hedging diffusion risk contributes the least. Furthermore, global warming with more pronounced jump effects has caused the optimum hedge ratio to rise up—the faster the warming, the larger the increase. The results also indicate that hedging jump volatility is more crucial than hedging jump arrival intensity. These findings of jump effects indicate that in a warming environment, firms should systematically adjust their jump hedging weights according to their forecasts of jump arrival intensity and volatility to mitigate the impacts of hurricane risk on their total losses.

- 1
Catastrophic weather-related insurance losses in the United States have been rising 10 times faster since 1971 than premiums, population, or economic growth.

- 2
- 3
Among the evidence provided is a comprehensive idealized hurricane intensity modeling study by Knutson and Tuleya (2004), which uses future climate projections from nine different global climate models and four different versions of the Geophysical Fluid Dynamics Lab (GFDL) hurricane model. According to this study, an 80-year buildup of atmospheric CO

_{2}at 1 percent/year with compounding leads to roughly a one-half category increase in potential hurricane intensity on the Saffir-Simpson scale and an 18 percent increase in precipitation near the hurricane core. A 1 percent/year CO_{2}increase is an idealized scenario of future climate forcing. An implication of the study is that if the frequency of tropical cyclones remains the same over the coming century, a greenhouse-gas induced warming may lead to an increasing risk in the occurrence of highly destructive Category 5 storms. Elsner, Kossin, and Jagger (2008), Webster et al. (2005), and Emanuel (2005) provide empirical evidence that global warming has been increasing the severity and destructive power of hurricanes over the past few decades. - 4
More recent literature has paid attention to the incorporation of the impacts of random jumps, stochastic volatility, and stochastic basis. Brennan and Schwartz (1990) incorporate stochastic mean-reverting basis when studying the optimum strategy of an arbitrageur. Chang, Chang, and Fang (1996) derive a two-factor optimum futures hedge model in a jump-diffusion framework with stochastic basis. Schwartz (1997) applies a Kalman filter empirical methodology to estimate mean reversion in the basis change to investigate commodity futures pricing and hedging. Recognizing that hedgers are only concerned with downside risk, Lien and Tse (1998) employ a bivariate APARCH-M model for the spot and futures return to derive time-varying lower partial moment hedge ratios. Hilliard and Reis (1998) develop a three-factor jump-diffusion model incorporating stochastic interest rates and stochastic convenience yield to value futures and futures options.

- 5
Hartzmark (1987) and Peck and Nahmias (1989) find that for oat traders, the actual mean hedge ratio in terms of the fraction of spot contracts hedged was 0.45 in 1980 vis-À-vis the optimum risk-minimization hedge ratio of 0.82. For wheat traders, the corresponding hedge ratios were 0.88 and 0.49, respectively. Tufano (1998) finds that for a sample of 48 gold mining firms, the mean hedge ratio over 1990–1993 in terms of next year's expected production was only 0.256. Haushalter (2000) finds that among 100 oil and gas producers, the mean actual hedge ratio over 1992–1994 in terms of next year's expected production was only about 0.15, whereas Jin and Jorion (2006) find that for 119 U.S. oil and gas producers, the corresponding mean hedge ratios over 1998–2001 were 0.33 and 0.41, respectively.

- 6
Pricing of CAT-linked instruments have been widely studied in the insurance and actuarial science literature. For a recent reference, see Chang, Lin, and Yu (2011). To our knowledge however, optimum hurricane futures hedge has not been studied. As discussed in Chang, Lin, and Yu, the recent development of CAT risk has also focused on the convergence of the financial services industry and (re)insurance sector, which has further led to the development of hybrid insurance/financial instruments. Considering the asymmetric information between inside and outside reinsurers about an insurer's risk, Finken and Laux (2009) show how a CAT-linked instrument (e.g., CAT bonds) can be factored in the pricing of reinsurance contracts. Considering the hedging effect between a short reinsurance position and a short CAT bond position, Lee and Yu (2007) explore reinsurance pricing when a firm also sells CAT bonds to lay off the exposure.

- 7
The empirical evidence in the science literature provided by Emanuel et al. (2004) and others, which shows that hurricane forecast models have been increasingly successful in forecasting hurricane tracks, also provides supports for the risk-neutral pricing approach.

- 8
Cummins, Lalonde, and Phillips (2004) point out that in the absence of a traded underlying asset, insurance-linked securities that are structured to pay off on

*parametric*indices based on the physical characteristics of catastrophic events greatly reduce or eliminate moral hazard but expose hedgers to basis risk. - 9
Howard and D'Antonio (1994) adopt this linear cost structure when studying optimum futures hedge in a one-period framework.

- 10
Haushalter (2000) documents that the extent to which a firm hedges is positively related to financial leverage. Because hedging can reduce financing costs (e.g., Stulz, 1996), this finding supports the notion that hedging and financing decisions should be jointly made. Focusing on the question of optimum futures hedge rather than on optimum capital structure, this study assumes a given level of financial leverage.

### References

- Top of page
- Abstract
- Introduction
- Risk–Return Optimum Futures Hedge in the Presence of Firm's Costs of Hedging
- Numerical Analysis
- Conclusion
- References

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