Strategic commodity allocation

This article extends the study of the financialization of commodities (Rouwenhorst and Tang [Annu. Rev. Financ. Econ., 2012, 4, 447–467]) by considering an investment in the term structure of commodity futures prices. Specifically, we analyse the benefits of adding a distant commodity futures contract and/or a spot commodity (near month futures contract) to a portfolio of bonds and stocks in a setting similar to Brennan and Schwartz [The use of treasury bill futures in strategic asset allocation programs. In Worldwide Asset and Liability Modeling, edited by W.T. Ziemba and J.M. Mulvey, pp. 205–230, 1998 (Cambridge University Press: Cambridge)]. Our analysis employs an empirical study that covers the post financial crisis period. We show that the spot commodity considerably improves the value of the portfolio. However, an investment in the whole term structure of futures contracts is optimally achieved through high opposite positions in the spot commodity and distant futures contracts. We find that these extreme calendar spreads can result in an inappropriate investment.     

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.     

An endogenous volatility approach to pricing and hedging call options with transaction costs

Standard delta hedging fails to exactly replicate a European call option in the presence of transaction costs. We study a pricing and hedging model similar to the delta hedging strategy with an endogenous volatility parameter for the calculation of delta over time. The endogenous volatility depends on both the transaction costs and the option strike prices. The optimal hedging volatility is calculated using the criterion of minimizing the weighted upside and downside replication errors. The endogenous volatility model with equal weights on the up and down replication errors yields an option premium close to the Leland [J. Finance, 1985 Leland, HE. 1985. Option pricing and replication with transaction costs. J. Finance, 40: 1283–1301. [Crossref], [Web of Science ®] , [Google Scholar], 40, 1283–1301] heuristic approach. The model with weights being the probabilities of the option's moneyness provides option prices closest to the actual prices. Option prices from the model are identical to the Black–Scholes option prices when transaction costs are zero. Data on S&P 500 index cash options from January to June 2008 illustrate the model.     

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.     

On accurate and provably efficient GARCH option pricing algorithms

The GARCH model has been very successful in capturing the serial correlation of asset return volatilities. As a result, applying the model to options pricing attracts a lot of attention. However, previous tree-based GARCH option pricing algorithms suffer from exponential running time, a cut-off maturity, inaccuracy, or some combination thereof. Specifically, this paper proves that the popular trinomial-tree option pricing algorithms of Ritchken and Trevor (Ritchken, P. and Trevor, R., Pricing options under generalized GARCH and stochastic volatility processes. J. Finance, , 54(1), 377–402.) and Cakici and Topyan (Cakici, N. and Topyan, K., The GARCH option pricing model: a lattice approach. J. Comput. Finance, , 3(4), 71–85.) explode exponentially when the number of partitions per day, n, exceeds a threshold determined by the GARCH parameters. Furthermore, when explosion happens, the tree cannot grow beyond a certain maturity date, making it unable to price derivatives with a longer maturity. As a result, the algorithms must be limited to using small n, which may have accuracy problems. The paper presents an alternative trinomial-tree GARCH option pricing algorithm. This algorithm provably does not have the short-maturity problem. Furthermore, the tree-size growth is guaranteed to be quadratic if n is less than a threshold easily determined by the model parameters. This level of efficiency makes the proposed algorithm practical. The surprising finding for the first time places a tree-based GARCH option pricing algorithm in the same complexity class as binomial trees under the Black–Scholes model. Extensive numerical evaluation is conducted to confirm the analytical results and the numerical accuracy of the proposed algorithm. Of independent interest is a simple and efficient technique to calculate the transition probabilities of a multinomial tree using generating functions.     

A novel Monte Carlo approach to hybrid local volatility models

We present in a Monte Carlo simulation framework, a novel approach for the evaluation of hybrid local volatility [Risk, 1994, 7, 18–20], [Int. J. Theor. Appl. Finance, 1998, 1, 61–110] models. In particular, we consider the stochastic local volatility model—see e.g. Lipton et al. [Quant. Finance, 2014, 14, 1899–1922], Piterbarg [Risk, 2007, April, 84–89], Tataru and Fisher [Quantitative Development Group, Bloomberg Version 1, 2010], Lipton [Risk, 2002, 15, 61–66]—and the local volatility model incorporating stochastic interest rates—see e.g. Atlan [ArXiV preprint math/0604316, 2006], Piterbarg [Risk, 2006, 19, 66–71], Deelstra and Rayée [Appl. Math. Finance, 2012, 1–23], Ren et al. [Risk, 2007, 20, 138–143]. For both model classes a particular (conditional) expectation needs to be evaluated which cannot be extracted from the market and is expensive to compute. We establish accurate and ‘cheap to evaluate’ approximations for the expectations by means of the stochastic collocation method [SIAM J. Numer. Anal., 2007, 45, 1005–1034], [SIAM J. Sci. Comput., 2005, 27, 1118–1139], [Math. Models Methods Appl. Sci., 2012, 22, 1–33], [SIAM J. Numer. Anal., 2008, 46, 2309–2345], [J. Biomech. Eng., 2011, 133, 031001], which was recently applied in the financial context [Available at SSRN 2529691, 2014], [J. Comput. Finance, 2016, 20, 1–19], combined with standard regression techniques. Monte Carlo pricing experiments confirm that our method is highly accurate and fast.     

Pricing variance and volatility swaps in a stochastic volatility model with regime switching: discrete observations case

This study presents a set of closed-form exact solutions for pricing discretely sampled variance swaps and volatility swaps, based on the Heston stochastic volatility model with regime switching. In comparison with all the previous studies in the literature, this research, which obtains closed-form exact solutions for variance and volatility swaps with discrete sampling times, serves several purposes. (1) It verifies the degree of validity of Elliott et al.'s [Appl. Math. Finance, 2007, 14(1), 41–62] continuous-sampling-time approximation for variance and volatility swaps of relatively short sampling periods. (2) It examines the effect of ignoring regime switching on pricing variance and volatility swaps. (3) It contributes to bridging the gap between Zhu and Lian's [Math. Finance, 2011, 21(2), 233–256] approach and Elliott et al.'s framework. (4) Finally, it presents a semi-Monte-Carlo simulation for the pricing of other important realized variance based derivatives.     

Valuation of forward start options under affine jump-diffusion models

Under the general affine jump-diffusion framework of Duffie et al. [Econometrica, 2000, 68, 1343–1376], this paper proposes an alternative pricing methodology for European-style forward start options that does not require any parallel optimization routine to ensure square integrability. Therefore, the proposed methodology is shown to possess a better accuracy–efficiency trade-off than the usual and more general approach initiated by Hong [Forward Smile and Derivative Pricing. Working paper, UBS, 2004] that is based on the knowledge of the forward characteristic function. Explicit pricing solutions are also offered under the nested jump-diffusion setting proposed by Bakshi et al. [J. Finance, 1997, 52, 2003–2049], which accommodates stochastic volatility and stochastic interest rates, and different integration schemes are numerically tested.     

An improved convolution algorithm for discretely sampled Asian options

We suggest an improved FFT pricing algorithm for discretely sampled Asian options with general independently distributed returns in the underlying. Our work complements the studies of Carverhill and Clewlow [Risk, 1990, 3(4), 25–29], Benhamou [J. Comput. Finance, 2002, 6(1), 49–68], and Fusai and Meucci [J. Bank. Finance, 2008, 32(10), 2076–2088], and, if we restrict our attention only to log-normally distributed returns, also Ve?e? [Risk, 2002, 15(6), 113–116]. While the existing convolution algorithms compute the density of the underlying state variable by moving forward on a suitably defined state space grid, our new algorithm uses backward price convolution, which resembles classical lattice pricing algorithms. For the first time in the literature we provide an analytical upper bound for the pricing error caused by the truncation of the state space grid and by the curtailment of the integration range. We highlight the benefits of the new scheme and benchmark its performance against existing finite difference, Monte Carlo, and forward density convolution algorithms.     

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.     

Are tightened trading rules always bad? Evidence from the Chinese index futures market

This paper investigates the impact of tightened trading rules on the market efficiency and price discovery function of the Chinese stock index futures in 2015. The market efficiency and the price discovery of Chinese stock index futures do not deteriorate after these rule changes. Using variance ratio and spectral shape tests, we find that the Chinese index futures market becomes even more efficient after the tightened rules came into effect. Furthermore, by employing Schwarz and Szakmary [J. Futures Markets, 1994, 14(2), 147–167] and Hasbrouck [J. Finance, 1995, 50(4), 1175–1199] price discovery measures, we find that the price discovery function, to some extent, becomes better. This finding is consistent with Stein [J. Finance, 2009, 64(4), 1517–1548], who documents that regulations on leverage can be helpful in a bad market state, and Zhu [Rev. Financ. Stud., 2014, 27(3), 747–789.], who finds that price discovery can be improved with reduced liquidity. It also suggests that the new rules may effectively regulate the manipulation behaviour of the Chinese stock index futures market during a bad market state, and then positively affect its market efficiency and price discovery function.     

The Stochastic Seasonal Behaviour of Natural Gas Prices

Previous studies have explored the seasonal behaviour of commodity prices as a deterministic factor. This paper goes further by proposing a general (n+2m)‐factor model for the stochastic behaviour of commodity prices, which nests the deterministic seasonal model by Sorensen (2002) . We consider seasonality as a stochastic factor, with n non‐seasonal and m seasonal factors. The non‐seasonal factors are as defined in Schwartz (1997) , Schwartz and Smith (2000) and Cortazar and Schwartz (2003) . The seasonal factors are trigonometric components generated by stochastic processes. The model has been applied to the Henry Hub natural gas futures contracts listed by NYMEX. We find that models allowing for stochastic seasonality outperform standard models with deterministic seasonality. We obtain similar results with other energy commodities. Moreover, we find that stochastic seasonality implies that the volatility of futures returns follows a seasonal pattern. This result has important implications in terms of option pricing.     

On VIX futures in the rough Bergomi model

The rough Bergomi model introduced by Bayer et al. [Quant. Finance, 2015, 1–18] has been outperforming conventional Markovian stochastic volatility models by reproducing implied volatility smiles in a very realistic manner, in particular for short maturities. We investigate here the dynamics of the VIX and the forward variance curve generated by this model, and develop efficient pricing algorithms for VIX futures and options. We further analyse the validity of the rough Bergomi model to jointly describe the VIX and the SPX, and present a joint calibration algorithm based on the hybrid scheme by Bennedsen et al. [Finance Stoch., forthcoming].     

Turbocharging Monte Carlo pricing for the rough Bergomi model

The rough Bergomi model, introduced by Bayer et al. [Quant. Finance, 2016, 16(6), 887–904], is one of the recent rough volatility models that are consistent with the stylised fact of implied volatility surfaces being essentially time-invariant, and are able to capture the term structure of skew observed in equity markets. In the absence of analytical European option pricing methods for the model, we focus on reducing the runtime-adjusted variance of Monte Carlo implied volatilities, thereby contributing to the model’s calibration by simulation. We employ a novel composition of variance reduction methods, immediately applicable to any conditionally log-normal stochastic volatility model. Assuming one targets implied volatility estimates with a given degree of confidence, thus calibration RMSE, the results we demonstrate equate to significant runtime reductions—roughly 20 times on average, across different correlation regimes.     

The representation of American options prices under stochastic volatility and jump-diffusion dynamics

This paper considers the problem of pricing American options when the dynamics of the underlying are driven by both stochastic volatility following a square-root process as used by Heston [Rev. Financial Stud., 1993, 6, 327–343], and by a Poisson jump process as introduced by Merton [J. Financial Econ., 1976, 3, 125–144]. Probability arguments are invoked to find a representation of the solution in terms of expectations over the joint distribution of the underlying process. A combination of Fourier transform in the log stock price and Laplace transform in the volatility is then applied to find the transition probability density function of the underlying process. It turns out that the price is given by an integral dependent upon the early exercise surface, for which a corresponding integral equation is obtained. The solution generalizes in an intuitive way the structure of the solution to the corresponding European option pricing problem obtained by Scott [Math. Finance, 1997, 7(4), 413–426], but here in the case of a call option and constant interest rates.     

Hedging jump risk,expected returns and risk premia in jump-diffusion economies

This article presents a pure exchange economy that extends Rubinstein [Bell J. Econ. Manage. Sci., 1976, 7, 407–425] to show how the jump-diffusion option pricing model of Black and Scholes [J. Political Econ., 1973, 81, 637–654] and Merton [J. Financ. Econ., 1976, 4, 125–144] evolves in gamma jumping economies. From empirical analysis and theoretical study, both the aggregate consumption and the stock price are unknown in determining jumping times. By using the pricing kernel, we determine both the aggregate consumption jump time and the stock price jump time from the equilibrium interest rate and CCAPM (Consumption Capital Asset Pricing Model). Our general jump-diffusion option pricing model gives an explicit formula for how the jump process and the jump times alter the pricing. This innovation with predictable jump times enhances our analysis of the expected stock return in equilibrium and of hedging jump risks for jump-diffusion economies.     

Modelling the shape of the limit order book

This article develops a parsimonious way to use the shape of the limit order book to produce an estimate of the asset price. The posited model captures and describes the evolution of the distribution of limit orders on the bid and ask sides of the LOB during the trading session and provides estimates of the execution asset price over time. The performance of the model is evaluated against some existing standards from the market microstructure literature during the trading session. Empirical evidence on listed companies confirm a strong contribution of our methodology to the innovation in asset prices, according to the information share coefficients. We also document a significant improvement relative to the Hasbrouck [J. Finance, 1991, 46, 179–207] model when our model estimates are included as regressors.     

A new closed-form formula for pricing European options under a skew Brownian motion

In this paper, we present a new pricing formula based on a modified Black–Scholes (B-S) model with the standard Brownian motion being replaced by a particular process constructed with a special type of skew Brownian motions. Although Corns and Satchell [2007. “Skew Brownian Motion and Pricing European Options.” The European Journal of Finance 13 (6): 523–544] have worked on this model, the results they obtained are incorrect. In this paper, not only do we identify precisely where the errors in Although Corns and Satchell [2007. “Skew Brownian Motion and Pricing European Options”. The European Journal of Finance 13 (6): 523–544] are, we also present a new closed-form pricing formula based on a newly proposed equivalent martingale measure, called ‘endogenous risk neutral measure’, by which only endogenous risks should and can be fully hedged. The newly derived option pricing formula takes the B-S formula as a special case and it does not induce any significant additional burden in terms of numerically computing option values, compared with the effort involved in computing the B-S formula.     

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.     

Fast Ninomiya–Victoir calibration of the double-mean-reverting model

Abstract

We consider the three-factor double mean reverting (DMR) option pricing model of Gatheral [Consistent Modelling of SPX and VIX Options, 2008], a model which can be successfully calibrated to both VIX options and SPX options simultaneously. One drawback of this model is that calibration may be slow because no closed form solution for European options exists. In this paper, we apply modified versions of the second-order Monte Carlo scheme of Ninomiya and Victoir [Appl. Math. Finance, 2008, 15, 107–121], and compare these to the Euler–Maruyama scheme with full truncation of Lord et al. [Quant. Finance, 2010, 10(2), 177–194], demonstrating on the one hand that fast calibration of the DMR model is practical, and on the other that suitably modified Ninomiya–Victoir schemes are applicable to the simulation of much more complicated time-homogeneous models than may have been thought previously.