Bivariate normal mixture spread option valuation

We discuss the pricing and hedging of European spread options on correlated assets when the marginal distribution of each asset return is assumed to be a mixture of normal distributions. Being a straightforward two-dimensional generalization of a normal mixture diffusion model, the prices and hedge ratios have a firm behavioural and theoretical foundation. In this ‘bivariate normal mixture’ (BNM) model no-arbitrage option values are just weighted sums of different ‘2GBM’ option values that are based on the assumption of two correlated lognormal diffusions, and likewise for their sensitivities. The main advantage of this approach is that BNM option values are consistent with both volatility smiles and with the implied correlation ‘frown’. No other ‘frown consistent’ spread option valuation model has such straightforward implementation. We apply analytic approximations to compare BNM valuations of European spread options with those based on the 2GBM assumption and explain the differences between the two as a weighted sum of six second-order 2GBM sensitivities. We also examine BNM option sensitivities, finding that these, like the option values, can sometimes differ substantially from those obtained under the 2GBM model. Finally, we show how the correlation frown that is implied by the BNM model is affected as we change (a) the correlation structure and (b) the tail probabilities in the joint density of the asset returns.     

Monte Carlo valuation of natural gas investments

In this evaluation of energy assets related to natural gas, our particular focus is on a base load natural gas combined cycle power plant and a liquefied natural gas facility in a realistic setting. We also value several American-type investment options following the least squares Monte Carlo approach. We calibrate mean-reverting stochastic processes for gas and electricity prices by using data from NYMEX NG futures contracts and the Spanish wholesale electricity market, respectively. Additional sources of uncertainty concern the initial investment outlay, or the option's time to maturity, or the cost of CO₂ emission permits.     

Continuous rainbow options on commodity outputs: what is the real value of switching facilities?

We develop real rainbow option models to value an operating asset with the flexibility to choose between two commodity outputs. We provide a quasi-analytical solution and a numerical lattice solution to a model with continuous switching opportunities between two commodity outputs, taking into account operating and switching costs. The models are applied to an illustrative case, demonstrating that the quasi-analytical solution and the lattice approach provide near identical results for the asset valuation and optimal switching boundaries. We find that the switching boundaries generally narrow as prices decline. In the presence of operating costs and temporary suspension, however, the thresholds diverge for low enough prices. A fertilizer plant with flexibility between selling ammonia and urea is valued in an empirical section using our real option models. Despite the high correlation between the two alternative commodities, ammonia and urea, there is significant value in the flexibility to choose between the two. Both strategic and policy implications for stakeholders in flexible assets are discussed, with some generalisations outside the fertilizer industry.     

Valuing an Early‐Stage Biotechnology Investment as a Rainbow Option

In an article published in this journal in 2003, Richard Shockley and three of his students presented a detailed valuation of an early‐stage biotechnology investment using a binomial lattice option pricing model. The article demonstrates how investments with multiple stages can be treated as “compound sequential options”—that is, as series of options in which investments in one option provide the opportunity to invest in the next in the series. In this article, the author uses the same business case analyzed by Shockley et al. to demonstrate how to value this early‐stage biotechnology investment by separately modeling the two types of risks: technology and product market. An option that has two distinct kinds of risk that develop differently over time is known as a “rainbow option.” The key adjustment to the option pricing model required to value such an option is that, instead of the standard binomial option pricing model with two outcomes at each point in time, the author uses a “quadranomial” option pricing model with four outcomes at each point in time. By distinguishing technology risks from product market risks and allowing them to develop differently over time, the author's analysis leads to a very different valuation and, indeed, a different decision about the initial investment than the one produced by Shockley's model.     

The quality option implicit in futures contracts

The quality option implicit in futures contracts allows the short position to satisfy the contract by delivering one of a variety of specified assets. If, at the time the contract is purchased, knowledge of which of the allowed assets will be cheapest at maturity is uncertain, then the quality option will have value. The greater the value of this option, the lower will be the futures price. This paper presents, and tests, a futures pricing model that incorporates the quality option aspect of commodity futures contracts. Our research shows that the quality option has a significant impact on futures prices.     

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.     

The Quality Option and Timing Option in Futures Contracts

Often futures contracts contain quality options whereby the short position has the choice of delivering one of an acceptable set of assets. We explore the implications of the quality option on the futures price. We develop a method for pricing the quality option for the general case of n deliverable assets and provide numerical illustrations of its significance. Even when the asset prices are very highly correlated, this option can have nontrivial value, especially when there is a large number of deliverable assets. We analyze the impact of the timing option and its interaction with the quality option. A procedure is developed for valuing the timing option in the presence of the quality option, and some numerical estimates are obtained.     

The value of embedded real options: Evidence from consumer automobile lease contracts—A note

Giaccotto et al. [2007. Journal of Finance 62, 411–445] provide a simple model for pricing the cancellation and the purchase options typically embedded in automobile lease contracts, assuming constant interest rates. They show that the cancellation option is worthless because of a penalty applied if the lease is terminated before maturity. We extend their results by developing a model with stochastic interest rates, and show that the cancellation option has a significant value also in presence of the penalty. We provide sufficient conditions to make the cancellation option worthless in our more general framework.     

American options: the EPV pricing model

Summary We explicitly solve the pricing problem for perpetual American puts and calls, and provide an efficient semi-explicit pricing procedure for options with finite time horizon. Contrary to the standard approach, which uses the price process as a primitive, we model the price process as the expected present value of a stream, which is a monotone function of a Lévy process. Certain processes exhibiting mean-reverting, stochastic volatility and/or switching features can be modeled this way. This specification allows us to consider assets that pay no dividends at all when the level of the underlying stochastic factor is too low, assets that pay dividends at a fixed rate when the underlying stochastic process remains in some range, or capped dividends.The authors are grateful to the anonymous referees for valuable comments and suggestions.     

A Stochastic Correlation Model with Mean Reversion for Pricing Multi-Asset Options

We set up a new kind of model to price the multi-asset options. A square root process fluctuating around its mean value is introduced to describe the random evolution of correlation between two assets. In this stochastic correlation model with mean reversion term, the correlation is a random walk within the region from −1 to 1, and it is centered around its equilibrium value. The trading strategy to hedge the correlation risk is discussed. Since a solution of high-dimensional partial differential equation may be impossible, the Quasi-Monte Carlo and Monte Carlo methods are introduced to compute the multi-asset option price as well. Taking a better-of two asset rainbow as an example, we compare our results with the price obtained by the Black–Scholes model with constant correlation.     

Real term structure forecasts of consumption growth

This paper employs an empirically tractable affine term structure model of real interest rates to examine the predictive ability of the real short-term interest rate and its term spread with a longer-term interest rate to predict future real consumption growth. The estimates of the model provide support of the consumption smoothing hypothesis. The paper shows that the real term structure is spanned by two mean-reverting state variables. The mean-reverting property of these variables can consistently explain the forecasting ability of the short-term real rate and term spread to forecast future consumption growth rate, over different horizons ahead. Although the risks associated with changes in these variables are both priced in the market, they are not volatile enough to obscure the information of the real term structure about future real consumption growth.     

Equilibrium in a Dynamic Limit Order Market

We model a dynamic limit order market as a stochastic sequential game with rational traders. Since the model is analytically intractable, we provide an algorithm based on Pakes and McGuire (2001) to find a stationary Markov‐perfect equilibrium. We then generate artificial time series and perform comparative dynamics. Conditional on a transaction, the midpoint of the quoted prices is not a good proxy for the true value. Further, transaction costs paid by market order submitters are negative on average, and negatively correlated with the effective spread. Reducing the tick size is not Pareto improving but increases total investor surplus.     

可转换债券在公司连续融资中作用机理的研究——以万科股份有限公司为例

本文通过模型分析了可转换债券在公司连续融资中的作用机理;并结合万科公司利用可转换债券融资的案例,分析了万科转债在万科公司连续融资中发挥的作用。在万科公司以未来投资期权价值不确定为基础的连续融资过程中,当投资期权项目被证实有价值时,通过可转换债券的赎回(强制转换)把初始项目积累的资金转向有价值的投资期权项目,有利于万科公司有价值投资期权项目的后续融资,验证了Mayers(1998)的连续融资假说。     

Pairs trading

‘Pairs Trading’ is an investment strategy used by many Hedge Funds. Consider two similar stocks which trade at some spread. If the spread widens short the high stock and buy the low stock. As the spread narrows again to some equilibrium value, a profit results. This paper provides an analytical framework for such an investment strategy. We propose a mean-reverting Gaussian Markov chain model for the spread which is observed in Gaussian noise. Predictions from the calibrated model are then compared with subsequent observations of the spread to determine appropriate investment decisions. The methodology has potential applications to generating wealth from any quantities in financial markets which are observed to be out of equilibrium.     

Pricing catastrophe options with stochastic claim arrival intensity in claim time

We model claim arrival and loss uncertainties jointly in a doubly-binomial framework to price an Asian-style catastrophe (CAT) option with a non-traded underlying loss index using the no-arbitrage martingale pricing methodology. We span these uncertainties by benchmarking to the shadow price of a one-claim bond and the premium of a reinsurance contract. We implement a stochastic time change from calendar time to claim time to more efficiently price the CAT option as a random sum – a binomial sum of claim time binomial Asian option prices. This choice of the operational time dimension allows us to incorporate different patterns of catastrophe arrivals by adjusting the claim arrival probability. We demonstrate this versatility by incorporating a mean-reverting Ornstein-Uhlenbeck intensity arrival process. Simulation results verify our model predictions and demonstrate how the claim arrival probability varies with the expected claim arrival intensity.     

Growth to value: Option exercise and the cross section of equity returns

We propose a general equilibrium model to study the link between the cross section of expected returns and book-to-market characteristics. We model two primitive assets: value assets and growth assets that are options on assets in place. The cost of option exercise, which is endogenously determined in equilibrium, is highly procyclical and acts as a hedge against risks in assets in place. Consequently, growth options are less risky than value assets, and the model features a value premium. Our model incorporates long-run risks in aggregate consumption and replicates the empirical failure of the conditional capital asset pricing model (CAPM) prediction. The model also quantitatively accounts for the pattern in mean returns on book-to-market sorted portfolios, the magnitude of the CAPM-alphas, and other stylized features of the cross-sectional data.     

Time diversification: Definitions and some closed-form solutions

We establish general conditions under which younger investors should invest a larger proportion of their wealth in risky assets than older ones. In the finite horizon dynamic setting, we show that such phenomenon, known as ‘‘time diversification,” can occur in the presence of human wealth, guaranteed consumption, or mean-reverting stock returns. We formalize two alternative notions of time diversification commonly confounded in the literature. Analytic solutions are provided for both time-series and cross-sectional forms of time diversification. To our best knowledge, this paper is the first to solve in closed-form the hedging demand for a CARA investor with inter-temporal consumption and a finite horizon, facing mean-reverting expected returns. Our results indicate that horizon can have a significant effect on the portfolio demand of a CARA investor due to inter-temporal hedging.     

Analysis of embedded options in individual pension schemes in Germany

Newly introduced government-subsidized pension products in Germany are required to contain a promise by the seller to provide a “money-back guarantee” at the end of the term. The client is also given the right to stop paying premiums at any time (paid-up option). In this case, the amount of all premiums paid must also be guaranteed by the seller at maturity, no matter when the client stopped paying the premiums. Previous analyses of guarantees in such government-subsidized pension products have ignored this additional option. Within a generalized Black/Scholes framework, we analyze the value of the paid-up option for different products, market scenarios, and client behavior. Our results indicate that the paid-up option significantly increases the value of the money-back guarantee. Furthermore, we find that reducing volatility by shifting the client’s assets from stocks to bonds as maturity approaches is a suitable means of reducing the risk arising from the “pure” money-back guarantee but much less effective in reducing the risk arising from the paid-up option. JEL Classification G13 · G23 · G28     

CREDIT SPREADS AND THE ZERO‐COUPON TREASURY SPOT CURVE

We examine the relation between credit spreads on industrial bonds and the underlying Treasury term structure. We use zero‐coupon spot rates to eliminate the coupon bias and to allow for a consistent study both within and across the different credit ratings. Our results indicate that the level and slope of the Treasury term structure are negatively correlated with changes in the credit spread on investment‐grade corporate bonds. We also find that the relation between credit spreads and the Treasury term structure is relatively stable through time. This is good news for value‐at‐risk calculations, as this suggests that the correlations among assets of different credit classes are stable; therefore use of historic correlations to model spread relations can be valid.     

商业银行贷款定价行为与房地产价格泡沫

房地产信贷是基于潜在资产的看跌期权,贷款人是卖方,借款人是买方。贷款人对看跌期权价值的低估会导致资产价格膨胀、偏离基础价值,而银行家和股东均存在低估看跌期权的正向激励。开放经济条件下,贷款利率、存款利率与汇率三个因素增加资产市场价格与基础价值的差距。基于对PW模型的扩展分析,中国2005年7月至2007年12月的数据也证实了房地产价格与利差和汇率负相关,与存款利率正相关。