Commodity Spread Option with Cointegration

We derive the valuation formula of a European call option on the spread of two cointegrated commodity futures prices, based on the Gibson–Schwartz with cointegration (GSC) model. We also analyze the American commodity spread option including the early exercise premium representation and an analytical approximation valuation formulae with cointegration. In the numerical analysis, we compare the spread option values calculated by the GSC model and the Gibson–Schwartz (GS) model that ignores cointegration. Consistent with the intuition that the cointegration prevents the prices from diverging, the GSC model prices the commodity spread option lower than the GS model which have longer maturity of more than 6 years. In other words, the GS model may overprice the commodity spread options for those with longer maturity without taking account of cointegration. Thus, incorporating cointegration is important for valuation and hedging of long-term commodity spread options such as large scale oil refining plant developments.     

No-arbitrage conditions for storable commodities and the modeling of futures term structures

One distinguishable feature of storable commodities is that they relate to two markets: cash market and storage market. This paper proves that, if no arbitrage exists in the storage-cash dual markets, the commodity convenience yield has to be non-negative. However, classical reduced-form models for futures term structures could allow serious arbitrages due to the high volatility of the convenience yield. To avoid negative convenience yield, this paper proposes a semi-affine arbitrage-free model, which prices futures analytically and fits futures term structures reasonably well. Importantly, our model prices commodity-related contingent claims (such as calendar spread options) quite differently with classical models.     

Convenience yields

This paper revisits the notion of a convenience yield in the context of modern option pricing theory. We show that, with a proper specification of the cash flows to holding a commodity, a convenience yield as a separate concept does not exist. Rather, a convenience yield is best viewed as a label given to certain cash flows generated from storing a commodity. In particular, it represents the payoffs from two embedded options which we call the scarcity and usage options. This characterization of a convenience yield is new to the literature, although consistent with its existing interpretations and uses.     

A Simple Approach to Valuing Risky Fixed and Floating Rate Debt

We develop a simple approach to valuing risky corporate debt that incorporates both default and interest rate risk. We use this approach to derive simple closed-form valuation expressions for fixed and floating rate debt. The model provides a number of interesting new insights about pricing and hedging corporate debt securities. For example, we find that the correlation between default risk and the interest rate has a significant effect on the properties of the credit spread. Using Moody's corporate bond yield data, we find that credit spreads are negatively related to interest rates and that durations of risky bonds depend on the correlation with interest rates. This empirical evidence is consistent with the implications of the valuation model.     

The stochastic behavior of commodity prices with heteroskedasticity in the convenience yield

We document a new stylized fact regarding the dynamics of the commodity convenience yield: the volatility of the convenience yield is heteroskedastic for industrial commodities; specifically, the volatility (variance) of the convenience yield depends on the convenience yield level. To explore the economic and statistical significance of the improved specification of the convenience yield process, we propose an affine model with three state variables (log spot price, interest rate, and the convenience yield). Our model captures three important features of commodity futures—the heteroskedasticity of the convenience yield, the positive relationship between spot-price volatility and the convenience yield and the dependence of futures risk premium on the convenience yield. Moreover our model predicts an upward sloping implied volatility smile, commonly observed in commodity option market.     

The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging

In this article we compare three models of the stochastic behavior of commodity prices that take into account mean reversion, in terms of their ability to price existing futures contracts, and their implication with respect to the valuation of other financial and real assets. The first model is a simple one-factor model in which the logarithm of the spot price of the commodity is assumed to follow a mean reverting process. The second model takes into account a second stochastic factor, the convenience yield of the commodity, which is assumed to follow a mean reverting process. Finally, the third model also includes stochastic interest rates. The Kalman filter methodology is used to estimate the parameters of the three models for two commercial commodities, copper and oil, and one precious metal, gold. The analysis reveals strong mean reversion in the commercial commodity prices. Using the estimated parameters, we analyze the implications of the models for the term structure of futures prices and volatilities beyond the observed contracts, and for hedging contracts for future delivery. Finally, we analyze the implications of the models for capital budgeting decisions.     

Long term spread option valuation and hedging

This paper investigates the valuation and hedging of spread options on two commodity prices which in the long run are in dynamic equilibrium (i.e., cointegrated). The spread exhibits properties different from its two underlying commodity prices and should therefore be modelled directly. This approach offers significant advantages relative to the traditional two price methods since the correlation between two asset returns is notoriously hard to model. In this paper, we propose a two factor model for the spot spread and develop pricing and hedging formulae for options on spot and futures spreads. Two examples of spreads in energy markets – the crack spread between heating oil and WTI crude oil and the location spread between Brent blend and WTI crude oil – are analyzed to illustrate the results.     

Evaluating Natural Resource Investments under Different Model Dynamics: Managerial Insights

We focus on factors that drive the dynamics of commodity prices. We highlight the capital budgeting implications of three highly‐cited, nested, multi‐factor models for commodity prices that have been successful in empirical investigations. Competing assumptions regarding commodity prices and their convenience yields can account for differences close to 40% on average, and in excess of 60% in cases, in the valuation of typical natural resource investments. These value differences are found to increase with the maturity and the intrinsic value of the investment, and also with the level and the volatility of the resource's convenience yield. Resources such as oil or copper, that are used for production purposes, usually exhibit high and volatile convenience yields; thus our findings should be more relevant for decision‐makers in such sectors.     

The analysis and valuation of interest rate options

This paper provides a simple, alternative model for the valuation of European-style interest rate options. The assumption that drives the hedging argument in the model is that the forward prices of bonds follow an arbitrary two-state process. Later, this assumption is made more specific by postulating that the discount on a zero-coupon bond follows a multiplicative binomial process. In contrast to the Black-Scholes assumption applied to zero-coupon bonds, the limiting distribution of this process has the attractive features that the zero-bond price has a natural barrier at unity (thus precluding negative interest rates), and that the bond price is negatively skewed. The model is used to price interest rate options in general, and interest rate caps and floors in particular. The model is then generalized and applied to European-style options on bonds. A relationship is established between options on swaps and options on coupon bonds. The generalized model then provides a computationally simple formula, closely related to the Black-Scholes formula, for the valuation of European-style options on swaps.     

Emission Allowance as a Derivative on Commodity-Spread

We provide a valuation formula for emission allowance. Assuming that the value of emission allowance on the last day of a trading phase is equal to a spread of commodity prices (e.g. electricity and natural gas) when the spread is positive and less than the penalty, we show that the emission allowance price is equal to the value of a portfolio of European call options on the spread of the commodities. Using the formula, we obtain a hedging strategy for emission allowance trading. We also empirically analyze option value embedded in emission allowance, and find by numerical analysis that the option value is relatively large.     

Commodity price modelling that matches current observables: a new approach

We develop a stochastic model of the spot commodity price and the spot convenience yield such that the model matches the current term structure of forward and futures prices, the current term structure of forward and futures volatilities, and the inter-temporal pattern of the volatility of the forward and futures prices. We let the underlying commodity price be a geometric Brownian motion and we let the spot convenience yield have a mean-reverting structure. The flexibility of the model, which makes it possible to simultaneously achieve all these goals, comes from allowing the volatility of the spot commodity price, the speed of mean-reversion parameter, the mean-reversion parameter, and the diffusion parameter of the spot convenience yield all to be time-varying deterministic functions.     

Pricing a class of exotic commodity options in a multi-factor jump-diffusion model

In a recent paper, Crosby introduced a multi-factor jump-diffusion model which would allow futures (or forward) commodity prices to be modelled in a way which captured empirically observed features of the commodity and commodity options markets. However, the model focused on modelling a single individual underlying commodity. In this paper, we investigate an extension of this model which would allow the prices of multiple commodities to be modelled simultaneously in a simple but realistic fashion. We then price a class of simple exotic options whose payoff depends on the difference (or ratio) between the prices of two different commodities (for example, spread options), or between the prices of two different (i.e. with different tenors) futures contracts on the same underlying commodity, or between the prices of a single futures contract as observed at two different calendar times (for example, forward start or cliquet options). We show that it is possible, using a Fourier transform-based algorithm, to derive a single unifying form for the prices of all these aforementioned exotic options and some of their generalizations. Although we focus on pricing options within the model of Crosby, most of our results would be applicable to other models where the relevant ‘extended’ characteristic function is available in analytical form.     

Oil convenience yields estimated under demand/supply shock

This paper extends the call option model of Milonas and Thomadakis (1997) to estimate oil convenience yields with futures prices. We define the business cycle of a seasonal commodity with demand/supply shocks and find that the convenience yield for crude oil exhibits seasonal behavior. The convenience yield for West Texas Intermediate (WTI) crude oil is the highest in the summer, while that for Brent crude oil is the highest in the winter. This implies that WTI crude oil is more sensitive to high summer demand and that Brent crude oil is more sensitive to shortages in winter supply. Convenience yields are negatively related to the inventory level of the underlying crude oil and positively related to interest rates due to the business cycle. We also show that convenience yields may explain price spread between WTI crude oil and Brent crude oil. Our computed convenience yields are consistent with Fama and French (1988) in that oil prices are more volatile than futures prices at low inventory level, verifying the Samuelson (1965) hypothesis that future prices are less variables than spot prices at lower inventory levels.

Calibrating a market model with stochastic volatility to commodity and interest rate risk

Based on the multi-currency LIBOR Market Model, this paper constructs a hybrid commodity interest rate market model with a stochastic local volatility function allowing the model to simultaneously fit the implied volatility surfaces of commodity and interest rate options. Since liquid market prices are only available for options on commodity futures, rather than forwards, a convexity correction formula for the model is derived to account for the difference between forward and futures prices. A procedure for efficiently calibrating the model to interest rate and commodity volatility smiles is constructed. Finally, the model is fitted to an exogenously given correlation structure between forward interest rates and commodity prices (cross-correlation). When calibrating to options on forwards (rather than futures), the fitting of cross-correlation preserves the (separate) calibration in the two markets (interest rate and commodity options), while in the case of futures a (rapidly converging) iterative fitting procedure is presented. The fitting of cross-correlation is reduced to finding an optimal rotation of volatility vectors, which is shown to be an appropriately modified version of the ‘orthonormal Procrustes’ problem in linear algebra. The calibration approach is demonstrated in an application to market data for oil futures.     

The Pricing Formula for Commodity-Linked Bonds with Stochastic Convenience Yields and Default Risk

At the maturity, the owner of a commodity-linked bond has the right to receive the face value of the bond and the excess amount of spot market value of the reference commodity bundle over the prespecified exercise price. This payoff structure is an important characteristic of the commodity-linked bonds. In this paper, we derive closed pricing formulae for the commodity-linked bonds. We assume that the reference commodity price and the value of the firm (bonds' issuer) follow geometric Brownian motions and that the net marginal convenience yield and interest rate follow Ornstein–Uhlenbech processes. In the appendix, we derive pricing formulae for bonds which are the same as the above commodity-linked bonds, except that the reference commodity price in the definition of the payoff at the maturity is replaced by the value of a special asset which depends on the convenience yield. This revised version was published online in August 2006 with corrections to the Cover Date.     

Revisiting the Tourinho real options model: outstanding issues 30 years later

This article presents and extends the first known model in real options, proposed in Tourinho (1979), and provides thoughts on addressing issues that are still outstanding 30 years later. It discusses the need to ensure the existence of market equilibrium when applying real options valuation to price assets, once all agents behave as suggested by the solution to the pricing equation. It argues that this can be achieved by using a stochastic process for the price that is sufficiently general to respond to supply and demand imbalances in the market for the resource. Once the individual decision rules are derived, the parameters of the process must be determined to ensure market equilibrium exists. For reserves of natural resources, this can be done by using a mean-reverting process for the price of the commodity and ensuring that the long-term price to which it reverts equals the trigger price for development of the marginal reserve.     

多资产大宗商品期权定价研究

基于大宗商品收益率与便利收益服从均值回复过程的假设,建立带协整效应的多资产大宗商品期权定价模型,求解多资产大宗商品期权价格的解析解,将大宗商品期权定价推广到更一般情况.结果表明:标的资产收益率增加,期权价格上升,替代品期权价格下降;标的资产的便利收益增加,期权价格下降,相应替代品的期权价格上升.     

中债估值识别了债券信用风险吗？——基于跳跃视角的实证分析

本文以2011—2018年中国A股上市公司发行的一般公司债为样本,探究了中债估值跳跃对债券信用利差的影响及作用机制,以此说明中债估值对债券信用风险的识别作用。研究发现:中债估值跳跃能够显著提高债券信用利差,其中,中债估值上跳降低了信用利差,下跳提高了信用利差,且相对于上跳,下跳对信用利差的作用更大。异质性分析发现:中债估值跳跃对信用利差的作用在机构投资者中较大,同时在信息不对称性较严重、流动性较差及违约风险较高的债券中也较大。进一步研究发现:中债估值跳跃不仅包含了公共信息,还含有私有信息,并能改善股票分析师预测表现。本研究说明中债估值能够识别债券信用风险,具有信息含量,对于债券市场信息环境建设和系统性金融风险防范具有重要意义。     

THE SURVIVAL ZONE FOR A BOND WITH BOTH CALL AND PUT OPTIONS EMBEDDED

We present a numerical method to price bonds that have multiple embedded options with an emphasis on the case with both long call and short put options. The valuation framework is a one-factor model for the term structure of interest rates, where the instantaneous interest rate is allowed to follow a fairly general stochastic process. The equilibrium interest rates that define the free boundaries for the embedded call and put options are given. We demonstrate the survival zone within which a bond with both long call and short put options remains afloat. We show that even moderate levels of transaction costs can have a significant effect on exercise of options.     

A two-factor cointegrated commodity price model with an application to spread option pricing

In this paper, we propose an easy-to-use yet comprehensive model for a system of cointegrated commodity prices. While retaining the exponential affine structure of previous approaches, our model allows for an arbitrary number of cointegration relationships. We show that the cointegration component allows capturing well-known features of commodity prices, i.e., upward sloping (contango) and downward sloping (backwardation) term-structures, smaller volatilities for longer maturities and an upward sloping correlation term structure. The model is calibrated to futures price data of ten commodities. The results provide compelling evidence of cointegration in the data. Implications for the prices of futures and options written on common commodity spreads (e.g., spark spread and crack spread) are thoroughly investigated.