Fast estimation of true bounds on Bermudan option prices under jump-diffusion processes

Fast pricing of American-style options has been a difficult problem since it was first introduced to the financial markets in 1970s, especially when the underlying stocks’ prices follow some jump-diffusion processes. In this paper, we extend the ‘true martingale algorithm’ proposed by Belomestny et al. [Math. Finance, 2009, 19, 53–71] for the pure-diffusion models to the jump-diffusion models, to fast compute true tight upper bounds on the Bermudan option price in a non-nested simulation manner. By exploiting the martingale representation theorem on the optimal dual martingale driven by jump-diffusion processes, we are able to explore the unique structure of the optimal dual martingale and construct an approximation that preserves the martingale property. The resulting upper bound estimator avoids the nested Monte Carlo simulation suffered by the original primal–dual algorithm, therefore significantly improving the computational efficiency. Theoretical analysis is provided to guarantee the quality of the martingale approximation. Numerical experiments are conducted to verify the efficiency of our algorithm.     

A maximum likelihood approach to volatility estimation for a Brownian motion using high,low and close price data

Abstract

Volatility plays an important role in derivatives pricing, asset allocation, and risk management, to name but a few areas. It is therefore crucial to make the utmost use of the scant information typically available in short time windows when estimating the volatility. We propose a volatility estimator using the high and the low information in addition to the close price, all of which are typically available to investors. The proposed estimator is based on a maximum likelihood approach. We present explicit formulae for the likelihood of the drift and volatility parameters when the underlying asset is assumed to follow a Brownian motion with constant drift and volatility. Our approach is to then maximize this likelihood to obtain the estimator of the volatility. While we present the method in the context of a Brownian motion, the general methodology is applicable whenever one can obtain the likelihood of the volatility parameter given the high, low and close information. We present simulations which indicate that our estimator achieves consistently better performance than existing estimators (that use the same information and assumptions) for simulated data. In addition, our simulations using real price data demonstrate that our method produces more stable estimates. We also consider the effects of quantized prices and discretized time.     

Time-varying long-run mean of commodity prices and the modeling of futures term structures

The exploration of the mean-reversion of commodity prices is important for inventory management, inflation forecasting and contingent claim pricing. Bessembinder et al. [J. Finance, 1995, 50, 361–375] document the mean-reversion of commodity spot prices using futures term structure data; however, mean-reversion to a constant level is rejected in nearly all studies using historical spot price time series. This indicates that the spot prices revert to a stochastic long-run mean. Recognizing this, I propose a reduced-form model with the stochastic long-run mean as a separate factor. This model fits the futures dynamics better than do classical models such as the Gibson–Schwartz [J. Finance, 1990, 45, 959–976] model and the Casassus–Collin-Dufresne [J. Finance, 2005, 60, 2283–2331] model with a constant interest rate. An application for option pricing is also presented in this paper.     

A Simple Way to Estimate Bid‐Ask Spreads from Daily High and Low Prices

We develop a bid‐ask spread estimator from daily high and low prices. Daily high (low) prices are almost always buy (sell) trades. Hence, the high–low ratio reflects both the stock's variance and its bid‐ask spread. Although the variance component of the high–low ratio is proportional to the return interval, the spread component is not. This allows us to derive a spread estimator as a function of high–low ratios over 1‐day and 2‐day intervals. The estimator is easy to calculate, can be applied in a variety of research areas, and generally outperforms other low‐frequency estimators.     

The systematic pricing of market sentiment shock

ABSTRACT

We show that market sentiment shocks create demand shocks for risky assets and a systematic risk for assets. We measure a market sentiment shock as the unexpected portion of the University of Michigan Consumer Sentiment Index’s growth. This shock prices stock returns in arbitrage pricing theory framework at 1% after controlling for market, size, value, momentum, and liquidity risk factors. Its premium lowered the implied risk aversion by 97.9% to 11.46 between 1978 and 2009 in our sentiment consumption-based capital-asset-pricing model. Merton’s [1973. “An Intertemporal Capital Asset Pricing Model.” Econometrica 41: 867–887]. intertemporal capital-asset-pricing model reconfirms our finding that this market sentiment shock is a systematic risk factor that provides investment opportunities.     

Pricing American options written on two underlying assets

This paper extends the integral transform approach of McKean [Ind. Manage. Rev., 1965, 6, 32–39] and Chiarella and Ziogas [J. Econ. Dyn. Control, 2005, 29, 229–263] to the pricing of American options written on more than one underlying asset under the Black and Scholes [J. Polit. Econ., 1973, 81, 637–659] framework. A bivariate transition density function of the two underlying stochastic processes is derived by solving the associated backward Kolmogorov partial differential equation. Fourier transform techniques are used to transform the partial differential equation to a corresponding ordinary differential equation whose solution can be readily found by using the integrating factor method. An integral expression of the American option written on any two assets is then obtained by applying Duhamel’s principle. A numerical algorithm for calculating American spread call option prices is given as an example, with the corresponding early exercise boundaries approximated by linear functions. Numerical results are presented and comparisons made with other alternative approaches.     

Drift Estimation of Generalized Security Price Processes from High Frequency Derivative Prices

This paper presents a framework for using high frequency derivative prices to estimate the drift of generalized security price processes. This work may be seen more generally as a quasi-likelihood approach to estimating continuous-time parameters of derivative pricing models using discrete option data. We develop a generalized derivative-based estimator for the drift where the underlying security price process follows any arbitrary state-time separable diffusion process (including arithmetic and geometric Brownian motion as special cases). The framework provides a method to measure premia in derivative prices, test for risk-neutral pricing and leads to a new empirical approach to pricing derivative contingent claims. A sufficient condition for the asymptotic consistency of the generalized estimator is also obtained. A study based on generating the S&P500 index and calls shows that the estimator can correctly estimate the drift parameter. This revised version was published online in November 2006 with corrections to the Cover Date.     

Search and herding effects in peer-to-peer lending: evidence from prosper.com

I examine loan data from Prosper.com—a website which allows borrowers to post loans and for lenders to bid on those loans. The Prosper market somewhat resembles the theoretical model of search, herding, and crowding in a large market described in Berkovich and Tayon (Phd. dissertation—Essays on search and herding. University of Pennsylvania, Pennsylvania, 2009). That model predicts that assets with high and low prices have high variance in the difference between price and true value. These extreme price regions of the asset-space are where private information provides excess returns. In the Prosper market, I find some evidence for the model since loans with low ex post returns show higher variance of the difference between price and ex post return. I also find that high-priced loans provide excess returns even after accounting for risk-aversion.     

Random walks,liquidity molasses and critical response in financial markets

Stock prices are observed to be random walks in time despite a strong, long-term memory in the signs of trades (buys or sells). Lillo and Farmer have recently suggested that these correlations are compensated by opposite long-ranged fluctuations in liquidity, with an otherwise permanent market impact, challenging the scenario proposed in Quantitative Finance, 2004, 4, 176, where the impact is instead transient, with a power-law decay in time. The exponent of this decay is precisely tuned to a critical value, ensuring simultaneously that prices are diffusive on long time scales and that the impact function is nearly lag independent. We provide new analysis of empirical data that confirm and make more precise our previous claims. We show that the power-law decay of the bare impact function comes both from an excess flow of limit order opposite to the market order flow, and to a systematic anti-correlation of the bid–ask motion between trades, two effects that create a ‘liquidity molasses’ which dampens market volatility.     

The power of patience: a behavioural regularity in limit-order placement

Abstract

In this paper we demonstrate a striking regularity in the way people place limit orders in financial markets, using a data set consisting of roughly two million orders from the London Stock Exchange. We define the relative limit price as the difference between the limit price and the best price available. Merging the data from 50 stocks, we demonstrate that for both buy and sell orders, the unconditional cumulative distribution of relative limit prices decays roughly as a power law with exponent approximately –1.5. This behaviour spans more than two decades, ranging from a few ticks to about 2000 ticks. Time series of relative limit prices show interesting temporal structure, characterized by an autocorrelation function that asymptotically decays as C(τ)～τ^?0.4. Furthermore, relative limit price levels are positively correlated with and are led by price volatility. This feedback may potentially contribute to clustered volatility.     

Endogenous market segmentation for lemons

Information asymmetry between sellers and buyers often prevents socially desirable trade. This article presents a new mechanism that mitigates the inefficiencies caused by information asymmetry. I consider decentralized markets under adverse selection and show that such markets can be endogenously segmented in a way that improves social welfare. Endogenous segmentation is driven by low‐quality sellers’ incentive to attract more buyers by separating from high‐quality sellers. The mechanism helps us understand the roles of several real‐world institutions, such as multiple marketplaces, costless advertisements, and nonbinding list prices.     

Option pricing under regime switching

Abstract

This paper develops a family of option pricing models when the underlying stock price dynamic is modelled by a regime switching process in which prices remain in one volatility regime for a random amount of time before switching over into a new regime. Our family includes the regime switching models of Hamilton (Hamilton J 1989 Econometrica 57 357–84), in which volatility influences returns. In addition, our models allow for feedback effects from returns to volatilities. Our family also includes GARCH option models as a special limiting case. Our models are more general than GARCH models in that our variance updating schemes do not only depend on levels of volatility and asset innovations, but also allow for a second factor that is orthogonal to asset innovations. The underlying processes in our family capture the asymmetric response of volatility to good and bad news and thus permit negative (or positive) correlation between returns and volatility. We provide the theory for pricing options under such processes, present an analytical solution for the special case where returns provide no feedback to volatility levels, and develop an efficient algorithm for the computation of American option prices for the general case.     

Implied Foreign Exchange Risk Premia

This paper uses implied volatilities from foreign exchange option prices and the results of no‐arbitrage theory to estimate foreign exchange risk premia. In particular, under the assumption of no‐arbitrage, the foreign exchange risk premium is driven by the difference between investors’ market prices of risk in the two currencies. In an international economy with three currencies, sterling, US dollar and Deutschemark, we can use the information on implied volatilities of the three cross rates to derive estimates of implied or ex ante market prices of risk and of foreign exchange risk premia. The foreign exchange risk premia estimates are then compared to survey‐based risk premia.     

The US 2000–2002 market descent: clarification

Didier Sornette and Wei-Xing Zhou respond to the issues raised by Anders Johansen in his comment ‘An alternative view’ published in Quantitative Finance 3 C6.     

Nils Ekholm

Abstract

The problem of χ² tests of a linear hypothesis H₀ for ‘matched samples’ in attribute data has been discussed earlier by the author (Bennett, 1967, 1968). This note presents corresponding results for the hypothesis that the multinomial probabilities p satisfy (c ?1) functional restrictions: F ₁(p) = 0, ... , F _C?1(p) = 0. An explicit relationship between the usual ‘goodness-of-fit’ χ² and the modified minimum χ² (=χ*²) of Jeffreys (1938) and Neyman (1949) is demonstrated for this situation. An example of the test for the 2 × 2 × 2 contingency table is given and compared with the solution of Bartlett (1935).     

One-state variable binomial models for European-/American-style geometric Asian options

Abstract

This paper is concerned with geometric Asian options whose pay-offs depend on the geometric average of the underlying asset prices. Following the Cox et al (1979 J. Financial Economics 7 229-63) arbitrage arguments, we develop one-state variable binomial models for the options on the basis of the idea of Cheuk and Vorst (1997 J. Int. Money Finance 16 173-87). The models are more efficient and faster than those lattice methods (for the options) proposed by Hull and White (1993 J. Derivatives 1 21-31), Ritchken et al (1993 Manage. Sci. 39 1202-13), Barraquand and Pudet (1996 Math. Finance 6 17-51) and Cho and Lee (1997 J. Financial Eng. 6 179-91). We also establish the equivalence of the models and certain difference schemes.     

Consistent pricing and hedging for a modified constant elasticity of variance model

Abstract

This paper considers a modification of the well known constant elasticity of variance model where it is used to model the growth optimal portfolio (GOP). It is shown that, for this application, there is no equivalent risk neutral pricing measure and therefore the classical risk neutral pricing methodology fails. However, a consistent pricing and hedging framework can be established by application of the benchmark approach.

Perfect hedging strategies can be constructed for European style contingent claims, where the underlying risky asset is the GOP. In this framework, fair prices for contingent claims are the minimal prices that permit perfect replication of the claims. Numerical examples show that these prices may differ significantly from the corresponding ‘risk neutral’ prices.     

Nonlinearities in the Brazilian Yield Curve

ABSTRACT

The article indicates the yield curve can be modeled using a continuous estimator as smooth transition regression, instead of traditional switch models, because bonds are traded continuously in the financial market. The results indicate that nonlinearity in the yield curve explains the pitfalls of monetary policy. The positive correlation between inflation and spread is consistent with a rise on uncertainty due to inflation risk or seems to indicate Brazilian Central Bank’s monetary policy credibility in the sample period. Therefore, if dependence on international capital exists, the Brazilian economic policy makers must monitor the movements in yield and analyze its feedback frequently in order to guide their plans and decisions.