Is the information obtained from European options on equally weighted baskets enough to determine the prices of exotic derivatives such as worst-of options?

In recent years there has been a remarkable growth of multi-asset options. These options exhibit sensitivity to the volatility of the underlying assets, as well as to their correlations. The call versus call is a product commonly used to trade correlation within the inter-dealer broker markets. The buyer of correlation buys a European call on the equally weighted basket option and sells a weighted average of European calls on each asset. In this case, the following important question arises: Is the information provided by equally weighted basket options enough to price other European multi-asset exotic derivatives such as worst-of or outperformance options? This article investigates this issue under a stochastic correlation framework. Importantly, this article shows that, when pricing multi-asset exotic derivatives, matching the prices of European equally weighted basket options, quoted in the market, does not guaranty the absence of model risk even in the case where the exotic payoff is observed only at maturity.     

Exchange option pricing under stochastic volatility: a correlation expansion

Efficient valuation of exchange options with random volatilities while challenging at analytical level, has strong practical implications: in this paper we present a new approach to the problem which allows for extensions of previous known results. We undertake a route based on a multi-asset generalization of a methodology developed in Antonelli and Scarlatti (Finan Stoch 13:269–303, 2009) to handle simple European one-asset derivatives with volatility paths described by Ito’s diffusive equations. Our method seems to adapt rather smoothly to the evaluation of Exchange options involving correlations among all the financial quantities that specify the model and it is based on expanding and approximating the theoretical evaluation formula with respect to correlation parameters. It applies to a whole range of models and does not require any particular distributional property. In order to test the quality of our approximation numerical simulations are provided in the last part of the paper.     

The pricing of basket-spread options

Since the pioneering paper of Black and Scholes was published in 1973, enormous research effort has been spent on finding a multi-asset variant of their closed-form option pricing formula. In this paper, we generalize the Kirk [Managing Energy Price Risk, 1995] approximate formula for pricing a two-asset spread option to the case of a multi-asset basket-spread option. All the advantageous properties of being simple, accurate and efficient are preserved. As the final formula retains the same functional form as the Black–Scholes formula, all the basket-spread option Greeks are also derived in closed form. Numerical examples demonstrate that the pricing and hedging errors are in general less than 1% relative to the benchmark results obtained by numerical integration or Monte Carlo simulation with 10 million paths. An implicit correction method is further applied to reduce the pricing errors by factors of up to 100. The correction is governed by an unknown parameter, whose optimal value is found by solving a non-linear equation. Owing to its simplicity, the computing time for simultaneous pricing and hedging of basket-spread option with 10 underlying assets or less is kept below 1 ms. When compared against the existing approximation methods, the proposed basket-spread option formula coupled with the implicit correction turns out to be one of the most robust and accurate methods.     

Unbiased and efficient Greeks of financial options

The price of a derivative security equals the discounted expected payoff of the security under a suitable measure, and Greeks are price sensitivities with respect to parameters of interest. When closed-form formulas do not exist, Monte Carlo simulation has proved very useful for computing the prices and Greeks of derivative securities. Although finite difference with resimulation is the standard method for estimating Greeks, it is in general biased and suffers from erratic behavior when the payoff function is discontinuous. Direct methods, such as the pathwise method and the likelihood ratio method, are proposed to differentiate the price formulas directly and hence produce unbiased Greeks (Broadie and Glasserman, Manag. Sci. 42:269–285, 1996). The pathwise method differentiates the payoff function, whereas the likelihood ratio method differentiates the densities. When both methods apply, the pathwise method generally enjoys lower variances, but it requires the payoff function to be Lipschitz-continuous. Similarly to the pathwise method, our method differentiates the payoff function but lifts the Lipschitz-continuity requirements on the payoff function. We build a new but simple mathematical formulation so that formulas of Greeks for a broad class of derivative securities can be derived systematically. We then present an importance sampling method to estimate the Greeks. These formulas are the first in the literature. Numerical experiments show that our method gives unbiased Greeks for several popular multi-asset options (also called rainbow options) and a path-dependent option.     

Two Closed-Form Formulas for the Futures Price in the Presence of a Quality Option

The paper derives closed-form formulas for the futures price in the presence of a multi-asset quality option. This is done for two cases: In the first one the underlying assets are zero coupon bonds with different maturities in the single-factor Vasicek model. In the second one these are commodities in a multi-factor setting, again with Vasicek interest rate uncertainty.     

Mean Reversion and Momentum: Another Look at the Price-Volume Correlation in the Real Estate Market

Based on behavioral finance and economics literature, we construct a theoretical framework in which consumers of newly constructed housing units perceive prices to follow a stochastic mean reversion pattern. Given this belief and the high carrying cost maintained by real estate developers, potential buyers opt to either exercise immediately or defer the purchase. We simulate the model within a real option framework by which we show that the optimal time to wait before exercising a purchase is positively related to the price level; hence, a negative (positive) correlation between transaction volume and price level (yield) emerges. Observing data on housing prices and new construction sales in Israel for the years 1998–2007, we apply an adaptive expectation regression model to test consumers’ belief in both mean reversion and momentum price patterns. The empirical evidence shows that while consumers’ demand pattern is simultaneously consistent with the belief in both momentum and mean reversion processes, the effect of the latter generally dominates. Moreover, while the data does not allow for testing the volume and price-level correlation, it does provide support to the positive volume-price yield correlation.     

Does more information in stock price lead to greater or smaller idiosyncratic return volatility?

We investigate the relation between price informativeness and idiosyncratic return volatility in a multi-asset, multi-period noisy rational expectations equilibrium. We show that the relation between price informativeness and idiosyncratic return volatility is either U-shaped or negative. Using several price informativeness measures, we empirically document a U-shaped relation between price informativeness and idiosyncratic return volatility. Our study therefore reconciles the opposing views in the following two strands of literature: (1) the growing body of research showing that firms with more informative stock prices have greater idiosyncratic return volatility, and (2) the studies arguing that more information in price reduces idiosyncratic return volatility.     

Alternative Methods for Projecting Equity Returns: Implications for Evaluating Social Security Reform Proposals

The effect upon future Social Security benefits resulting from the introduction of individual accounts depends on both the potential risks and returns of private equities, yet the historical evidence about the determinants of stock market risks and returns is mixed. In particular, correlations between equity returns and market fundamentals (such as the dividend–price ratio) are weak at annual frequencies, which has led some to conclude that a random returns (fixed mean and variance) model is the preferred specification for simulating the future path of equity returns. Although choosing between the random returns model and models based on market fundamentals does equally well for explaining variation of equity returns in the short run, the distinction is important when projecting equity returns over longer periods, as shown here in the context of a Monte Carlo simulation of Social Security reform. If equity returns are even weakly correlated with market fundamentals then (1) the expected future average return may be a function of the starting values for market fundamentals, and (2) the overall range of cumulative outcomes is narrower than the random returns model suggests.     

基于蒙特卡罗模拟的多标的资产期权定价研究

在单标的资产价格随机模型的基础上,推导了具相关性的多标的资产价格的随机过程公式,以此构造蒙特卡罗模拟高维欧式期权定价的随机模型,给出模拟算法,并分析了影响蒙特卡罗模拟效果的几个关键因素。模拟算例的结果显示模拟效果较好。     

Asset price bubbles from heterogeneous beliefs about mean reversion rates

Harrison and Kreps showed in 1978 how the heterogeneity of investor beliefs can drive speculation, leading the price of an asset to exceed its intrinsic value. By focusing on an extremely simple market model—a finite-state Markov chain—the analysis of Harrison and Kreps achieved great clarity but limited realism. Here we achieve similar clarity with greater realism, by considering an asset whose dividend rate is a mean-reverting stochastic process. Our investors agree on the volatility, but have different beliefs about the mean reversion rate. We determine the minimum equilibrium price explicitly; in addition, we characterize it as the unique classical solution of a certain linear differential equation. Our example shows, in a simple and transparent manner, how heterogeneous beliefs about the mean reversion rate can lead to everlasting speculation and a permanent “price bubble.”     

A theory of risk disclosure

In this paper, we consider the price effects of risk disclosure. We develop a model in which investors are uncertain about the variance of a firm’s cash flows and the firm releases an imperfect signal regarding this variance. In our model, uncertainty over the riskiness of a firm’s cash flows leads to a variance uncertainty premium in its price. We demonstrate that risk disclosure decreases the firm’s cost of capital by reducing this premium and that the market response to risk disclosure is small when the expected level of risk is high. Moreover, we find that firms acquire and disclose more risk information when their cash flow risk is greater than expected. Finally, we demonstrate that in a multi-asset setting, only risk disclosure concerning systematic risks will impact the cost of capital.     

Clearing price distributions in call auctions

We propose a model for price formation in financial markets based on the clearing of a standard call auction with random orders, and verify its validity for prediction of the daily closing price distribution statistically. The model considers random buy and sell orders, placed employing demand- and supply-side valuation distributions; an equilibrium equation then leads to a distribution for clearing price and transacted volume. Bid and ask volumes are left as free parameters, permitting possibly heavy-tailed or very skewed order flow conditions. In highly liquid auctions, the clearing price distribution converges to an asymptotically normal central limit, with mean and variance in terms of supply/demand-valuation distributions and order flow imbalance. By means of simulations, we illustrate the influence of variations in order flow and valuation distributions on price/volume, noting a distinction between high- and low-volume auction price variance. To verify the validity of the model statistically, we predict a year's worth of daily closing price distributions for five constituents of the Eurostoxx 50 index; Kolmogorov–Smirnov statistics and QQ-plots demonstrate with ample statistical significance that the model predicts closing price distributions accurately, and compares favourably with alternative methods of prediction.     

Performance persistence of closed-end funds

Studies of performance persistence of closed-end funds (CEFs) use two measures of persistence; autocorrelation and rank correlation of performance. The autocorrelation measure offers limited information because it cannot separate persistence relative to the market and to the industry. The rank correlation measure is generally applied to two periods, disregarding multi-period persistence. We investigate performance persistence of CEFs in terms of both market price return and net asset value return using contingency tables and multiple regression models. Jensen’s alpha and the Sharpe ratio are used as measures of risk-adjusted performance. We test three hypotheses: (i) CEFs performing better than the industry median will do so persistently, (ii) CEFs outperform the market persistently; and (iii) performance persistence can be partly explained by dividend yield. The findings are fivefold. First, the number of persistent years varies with the models used to calculate risk-adjusted performance. Second, with 4-index unconditional beta fixed variance model, CEFs persistently beat their industry for six out of 10 years in terms of both market price return and net asset value return. Third, with a 4-index unconditional beta fixed variance model, we find performance persistence relative to market for 6 and 7 years, out of the 10 years considered, in terms of market price return and net asset value return, respectively. Fourth, the disaggregate sample tests show that performance of municipal bond funds is more persistent than equity funds and taxable bond funds. Fifth, dividend patterns can partially explain persistence with liquidity as control.     

Valuation of Residential Mortgage-Backed Securities with Default Risk Using an Intensity-Based Approach

This paper develops a valuation model for fixed-rate mortgages, mortgage pools, and residential mortgage-backed securities (RMBS's) using an intensity-based approach. This model incorporates full prepayment, partial prepayment, and default in valuing a mortgage. Full prepayment is further classified into “refinancing” and “sale of a house” depending on the reason. The time of occurrence of each of these three types of prepayment and default is modeled as the first jump time of a Cox process. Under these conditions, the valuation formula for a mortgage as well as a partial differential equation (PDE) that the mortgage value satisfies is provided. As for implementation of the model, the short-term riskless interest rate and the house price are adopted as state variables. Each intensity process is specified in a manner that allows a jump in intensity depending on the state variables and the borrower's incentive for prepayment or default. Through such specifications, it is shown that our model has characteristics similar to some structural models in previous literature. As for the numerical method for valuation, we propose a simple backward induction technique on a tree instead of the commonly used Monte Carlo method. Additionally, the method for estimating the model is discussed, and the results of numerical simulations are reported.This paper represents the view of the author and does note necessarily the views of the Mitsubishi UFJ Securities Co., Ltd. or members of its staff.     

Repeat Sales Index for Thin Markets

The repeat sales model is commonly used to construct reliable house price indices in absence of individual characteristics of the real estate. Several adaptations of the original model by Bailey et al. (J Am Stat Assoc 58:933–942, 1963) are proposed in literature. They all have in common using a dummy variable approach for measuring price indices. In order to reduce the impact of transaction price noise on the estimates of price indices, Goetzmann (J Real Estate Finance Econ 5:5–53, 1992) used a random walk with drift process for the log price levels instead of the dummy variable approach. The model that is proposed in this article can be interpreted as a generalization of the Goetzmann methodology. We replace the random walk with drift model by a structural time series model, in particular by a local linear trend model in which both the level and the drift parameter can vary over time. An additional variable—the reciprocal of the time between sales—is included in the repeat sales model to deal with the effect of the time between sales on the estimated returns. This approach is robust can be applied in thin markets where relatively few selling prices are available. Contrary to the dummy variable approach, the structural time series model enables prediction of the price level based on preceding and subsequent information, implying that even for particular time periods where no observations are available an estimate of the price level can be provided. Conditional on the variance parameters, an estimate of the price level can be obtained by applying regression in the general linear model with a prior for the price level, generated by the local linear trend model. The variance parameters can be estimated by maximum likelihood. The model is applied to several subsets of selling prices in the Netherlands. Results are compared to standard repeat sales models, including the Goetzmann model.     

The evaluation of European compound option prices under stochastic volatility using Fourier transform techniques

Compound options are not only sensitive to future movements of the underlying asset price, but also to future changes in volatility levels. Because the Black–Scholes analytical valuation formula for compound options is not able to incorporate the sensitivity to volatility, the aim of this paper is to develop a numerical pricing procedure for this type of option in stochastic volatility models, specifically focusing on the model of Heston. For this, the compound option value is represented as the difference of its exercise probabilities, which depend on three random variables through a complex functional form. Then the joint distribution of these random variables is uniquely determined by their characteristic function and therefore the probabilities can each be expressed as a multiple inverse Fourier transform. Solving the inverse Fourier transform with respect to volatility, we can reduce the pricing problem from three to two dimensions. This reduced dimensionality simplifies the application of the fast Fourier transform (FFT) method developed by Dempster and Hong when transferred to our stochastic volatility framework. After combining their approach with a new extension of the fractional FFT technique for option pricing to the two-dimensional case, it is possible to obtain good approximations to the exercise probabilities. The resulting upper and lower bounds are then compared with other numerical methods such as Monte Carlo simulations and show promising results.     

How much is the gap?—Efficient jump risk-adjusted valuation of leveraged certificates

This paper develops a novel and highly efficient numerical algorithm for the gap risk-adjusted valuation of leveraged certificates. The existing literature relies on Monte Carlo simulations, which are not fast enough to be used in a market-making environment. This is because issuers need to compute thousands of price updates per second. By valuing leveraged certificates as multi-window barrier options, we explicitly model random jumps that occur at known times, such as between the exchange closing and re-opening. Our algorithm combines the one-day transition probability with Simpson’s numerical integration rule. This yields a backward induction scheme which requires a significantly coarser spatial and time grid than finite-difference methods. We confirm its robustness and accuracy through Monte Carlo simulations.     

Housing Price Dynamics in Time and Space: Predictability,Liquidity and Investor Returns

It is widely accepted that aggregate housing prices are predictable, but that excess returns to investors are precluded by the transactions costs of buying and selling property. We examine this issue using a unique data set—all private condominium transactions in Singapore during an eleven-year period. We model directly the price discovery process for individual dwellings. Our empirical results clearly reject a random walk in prices, supporting mean reversion in housing prices and diffusion of innovations over space. We find that, when house prices and aggregate returns are computed from models that erroneously assume a random walk and spatial independence, they are strongly autocorrelated. However, when they are calculated from the appropriate model, predictability in prices and in investment returns is completely absent. We show that this is due to the illiquid nature of housing transactions. We also conduct extensive simulations, over different time horizons and with different investment rules, testing whether better information on housing price dynamics leads to superior investment performance.