Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models

In this paper we propose two efficient techniques which allow one to compute the price of American basket options. In particular, we consider a basket of assets that follow a multi-dimensional Black–Scholes dynamics. The proposed techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are both based on Machine Learning, exploited together with binomial trees or with a closed form formula for integration. Moreover, these two methods solve the backward dynamic programing problem considering a Bermudan approximation of the American option. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. The two methods mainly differ in the approach used to compute the continuation value: a single step of the binomial tree or integration according to the probability density of the process. Numerical results show that these two methods are accurate and reliable in handling American options on very large baskets of assets. Moreover we also consider the rough Bergomi model, which provides stochastic volatility with memory. Despite that this model is only bidimensional, the whole history of the process impacts on the price, and how to handle all this information is not obvious at all. To this aim, we present how to adapt the GPR-Tree and GPR-EI methods and we focus on pricing American options in this non-Markovian framework.     

THE DESIGN OF MULTI-YEAR STOCK OPTION PLANS

Despite the explosion in the corporate use of stock options, the incentives created by stock options are not well understood by either the boards who grant them or the executives who are meant to be motivated by them. A major source of confusion stems from the corporate practice of using multi-year stock option plans. Such multi-year grants create subtle, potentially important links between current performance and future grants that can significantly dilute incentives for better performance.
For example, so-called "fixed value" plans provide very weak, even perverse, incentives ex ante since the value of future option grants is completely insulated from current performance. Under such plans, an executive's reward for superior performance is to receive fewer options, and to receive more options for substandard performance. In contrast, the fixed number plan creates an intrinsic link between changes in this year's stock price and changes in the value of future option grants.
The author also reports the findings of new empirical research that shows that stock option plans, taken as a whole, have a pay-to-performance correlation that is eight times stronger than that of salary and bonus. But, consistent with the analysis above, fixed value option plans have pay-to-performance that is only six times that of salary and bonus, as compared to ten times for fixed number plans.     

Expensing stock options: a fair-value approach

Now that companies such as General Electric and Citigroup have accepted the premise that employee stock options are an expense, the debate is shifting from whether to report options on income statements to how to report them. The authors present a new accounting mechanism that maintains the rationale underlying stock option expensing while addressing critics' concerns about measurement error and the lack of reconciliation to actual experience. A procedure they call fair-value expensing adjusts and eventually reconciles cost estimates made at grant date with subsequent changes in the value of the options, and it does so in a way that eliminates forecasting and measurement errors over time. The method captures the chief characteristic of stock option compensation--that employees receive part of their compensation in the form of a contingent claim on the value they are helping to produce. The mechanism involves creating entries on both the asset and equity sides of the balance sheet. On the asset side, companies create a prepaid-compensation account equal to the estimated cost of the options granted; on the owners'-equity side, they create a paid-in capital stock-option account for the same amount. The prepaid-compensation account is then expensed through the income statement, and the stock option account is adjusted on the balance sheet to reflect changes in the estimated fair value of the granted options. The amortization of prepaid compensation is added to the change in the option grant's value to provide the total reported expense of the options grant for the year. At the end of the vesting period, the company uses the fair value of the vested option to make a final adjustment on the income statement to reconcile any difference between that fair value and the total of the amounts already reported.     

THE CROSS-SECTIONAL EFFECTS OF OPTION LISTING ON FIRM STOCK RETURN VARIANCES

We develop a set of hypotheses to explain cross-sectional differences in variance changes associated with option listing. Transactions variance is decomposed into three components: the bid-ask spread, return autocorrelations, and intrinsic variance. Each is investigated separately. We find support for hypotheses that link: (1) changes in dealer transactions costs to changes in the bid-ask spread following option listing; (2) changes in the quantity and quality of information and the value of new information to movements of the return autocorrelation structure toward zero; and (3) changes in trading volume and the clientele that trades the underlying security to changes in intrinsic variance following option listing.     

Growth through rigidity: An explanation for the rise in CEO pay

The dramatic rise in CEO compensation during the 1990s and early 2000s is a longstanding puzzle. In this paper, we show that much of the rise can be explained by a tendency of firms to grant the same number of options each year. Number-rigidity implies that the grant-date value of option awards will grow with firm equity returns, which were very high on average during the tech boom. Further, other forms of CEO compensation did not adjust to offset the dramatic growth in the value of option pay. Number-rigidity in options can also explain the increased dispersion in pay, the difference in growth between the US and other countries, and the increased correlation between pay and firm-specific equity returns. We present evidence that number-rigidity arose from a lack of sophistication about option valuation that is akin to money illusion. We show that regulatory changes requiring transparent expensing of the grant-date value of options led to a decline in number-rigidity and helps explain why executive pay increased less with equity returns during the housing boom in the mid-2000s.     

Accounting for Option–based Compensation: The Economic Cost Approach

The intrinsic value approach amortizes over the life of the option, the difference between the stock price on the date of the grant and the exercise price of the option. The fair market value approach amortizes over the life of the option, the market value of stock options on the date of the grant. These approaches do not reflect the changes in the option–based compensation cost after the grant date. This paper proposes an economic cost approach that not only adjusts for the changes in the value of the options during its life but also records the issuance of the stock at fair market value on the exercise date.     

Valuation of Contingent Pension Liabilities and Guarantees Under Sponsor Default Risk

This article analyzes the relationship between a pension fund with contingently indexed defined benefit liabilities and its sponsor, using contingent claims analysis. As pension funds generally choose to run a mismatch risk, future surpluses and deficits will occur. Surpluses are divided between beneficiaries and sponsor through contingent indexation of the benefits and refunding. Covering a deficit at the pension fund level is a function of the sponsor's financial ability to do so. This article suggests that this system creates an asymmetric allocation of the residual risk between sponsor and beneficiaries. The optimal investment policy for the pension fund in this context can be found by reverse engineering option valuation formulas. The main conclusion is that sponsor default risk negatively impacts the optimum risk profile and thereby the market value of contingent pension liabilities.     

Valuing the Option to Purchase an Asset at a Proportional Discount

I analyze the value of a nonstandard call option that allows the holder to purchase an underlying asset at a discount proportional to the asset's market price. Several applications for this type of option exist, including its use in employee compensation contracts. I derive the value of this option for a dividend-paying asset and for an option whose exercise price reflects a time-varying discount factor. The derived value incorporates the optimal time at which the option should be exercised. One application of this option relates to a residential real estate program in China.     

Optimal Bond Trading with Tax Clienteles: A Discrete-Time Dynamic Trading Model

This study develops a discrete-time dynamic trading model for bond pricing under differential taxation. The model incorporates both the tax-timing option effect and the tax-clientele effect. Investors from all tax brackets have a chance to bid for a bond, and the marginal investor is the one who is willing to pay the highest price. Simulation results show that inter-bracket trading occurs frequently as the interest rate changes, which enhances the value of the tax option. These results are shown to be robust to changes in interest rate process and tax regimes.     

A Simple Formula for the Expected Rate of Return of an Option over a Finite Holding Period

Under conditions consistent with the Black-Scholes formula, a simple formula is developed for the expected rate of return of an option over a finite holding period possibly less than the time to expiration of the option. Under these conditions, surprisingly, the expected future value of a European option, even prior to expiration, is shown equal to the current Black-Scholes value of the option, except that the expected future value of the stock at the end of the holding period replaces the current stock price in the Black-Scholes formula and the future value of a riskless invesment of the striking price replaces the striking price. An extension of this result is used to approximate moments of the distribution of returns from an option portfolio.     

A Proposal for Expensing Employee Compensatory Stock Options for Financial Reporting Purposes

This article presents an accounting approach for employee stock options based on the insight that the currentperiod compensation expense should reflect only that part of the option value that is earned independent of the obligation of continued employment. Given that the maturity of vested options is typically shortened to 90 days when an employee resigns or is terminated, this method views the employee as owning a 90-day option (even if the stated maturity of the option is ten years) and earning a 90-day extension to that option each quarter as a result of the employee's continued employment. In the case of vested options, the compensation expense in each quarterly accounting period is thus the value of the 90-day extension of the option's maturity. There is no option expense in the quarter when the option is either exercised or expires.
In the case of unvested options, the expected option value at vesting should be estimated quarterly starting at the time of grant and the corresponding estimated expense should be revised and allocated as a pro rata accrual each quarter over the vesting period. The cumulative expense over the entire vesting period will equal the fair market value of the option at its vesting date.
Besides reflecting the economics of the exchange of value for labor involved in stock option grants, this approach has a number of practical advantages:

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  The 90-day maturity permits the use of publicly traded options to determine fair market value and makes Black-Scholes and other (lattice) pricing models more reliable.

       
  
  

HOW TO USE EVA IN THE OIL AND GAS INDUSTRY

The use of EVA in the oil industry has lagged behind that in most other industries because the accounting information reported by oil and gas concerns does such a poor job of representing management's effectiveness in adding value for shareholders. The essence of the problem is that the exploration activities of oil companies create assets whose changes in value are recognized by the stock market long before they are reflected on income statements or balance sheets. As a result, all accountingbased performance measures, including generic measures of EVA (which are derived from accounting information), fail to provide meaningful goals, decision tools, or compensation benchmarks.
This article provides a new, EVAbased framework for performance measurement and incentive compensation for oil and gas firms—and for companies in extractive industries in general. The authors show that, when adjusted by a publicly available measure of hydrocarbon reserve value known as "SEC-10," EVA's ability to explain annual stock returns rises from under 10% to almost 50%. Moreover, because SEC-10 has several important limitations as a measure of reserve value, there is considerable additional room for improving EVA's explanatory power. And the actual implementation of an EVA financial management system for an individual oil company can and should be based on more precise estimates of reserve value than those provided by SEC-10.
To this end, the authors provide an approach to hydrocarbon reserve valuation that captures the "real option" value of undeveloped reserves. By incorporating real option values, this new EVA financial management system for oil companies aligns management's incentives with the goal of creating shareholder wealth by rewarding managers for creating real option value as well as current cash flow—and by forcing managers to consider the optimal "exercise" of such strategic options.     

Quality options and hedging in Japanese Government Bond Futures markets

Quality options for Japanese Government Bond Futures contracts are analysed using a discrete trinomial tree approach based upon a two-factor Heath, Jarrow, and Morton (1990b) model. The impacts of the quality option on hedging effectiveness are investigated. In general, the pure quality option is found to be relatively small and, while the quality option does not have a dramatic impact upon hedging, accounting for the quality option can improve the performance of optimal hedging strategies.     

Levered and inverse VIX ETP option contract adjustments: No harm,no foul?

The terms of exchange-traded stock option contracts are usually adjusted when corporate actions take place. These adjustments are made to safeguard the value of the outstanding option contracts. Recently, a new type of corporate event has appeared − levered and inverse exchange-traded product issuers are reducing leverage ratios with increased frequency. While such changes directly affect option values, no contract adjustments are made, resulting in windfall transfers of wealth from outstanding long to outstanding short option holders. In one instance alone, the transfer was more than $US100 million. To remedy the problem, we offer a simple contract adjustment procedure.     

Stock market momentum, business conditions, and GARCH option pricing models

This paper examines the forecasting performance of GARCH option pricing models from a market momentum perspective, and the possible impacts of financial crises and business conditions are also examined. The empirical results demonstrate that market momentum impacts the forecasting performance of GARCH option pricing models. The EGARCH model performs better under downward market momentum, while the standard GARCH performs better under upward market momentum. In addition, parsimonious models generally outperform richly parameterized ones. The above findings are robust to financial crises, and the results further demonstrate that business conditions influence the forecasting performance of GARCH option pricing models.     

Option Trading and the Intervalling Effect Bias in Beta

Prior studies show that the beta coefficient of a security changes systematically as the length of measurement interval is varied. This phenomenon, which is called the intervalling effect bias in beta, has been attributed to the friction in the trading system that causes the delays in the price-adjustment process. This study shows that option listing is associated with a decline in the beta intervalling effect bias. The decline is most pronounced for small firms. We also find that our sample firms grow significantly after option listing. Since prior research indicates that market value is a major determinant of the magnitude of the intervalling effect, we re-examine our results using a subsample that controls for market value. The results indicate that the decline in the beta bias from the pre-listing to post-listing period is still prevalent after we control for the change in firm size. Overall, the evidence is consistent with the notion that option trading reduces the delays in the price-adjustment process, which in turn reduces the intervalling effect bias in beta.     

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.     

Real Estate Risk,Corporate Investment and Financing Choice

This paper empirically examines how real estate risk impacts corporate investment and financing decisions. Using a panel of United States firms from 1985 to 2013, we document that real estate risk is negatively associated with firms’ long-term investments and long-term external financing in equity and debt. The results are robust to different risk measurements and in particular salient during the financial crisis period when the endogeneity between risk and investment is less of a concern. The effect on firm leverage, however, depends on risk measures. Overall, in contrast to previously documented positive effects of the real estate value, real estate risk exposure exhibits mostly the opposite effects on investment, financing and capital structure. This difference is consistent with option value determinants. Findings in this paper shed new lights on the impact of real estate holding on corporate decisions, offer a new explanation for the underperformance of hedge funds’ real estate strategies, and confirm the theoretical predictions in Deng et al. (2015).     

The impacts of share pledging on firm investment timing and valuation

This paper applies a model in the real options framework to analyze the impacts of controlling shareholder’s share pledging on corporate investment timing and valuation. We find that the optimal investment timing shows an inverted U-shape with the pledge ratio, indicating that share pledging exacerbates firms’ over-investment and worsens firms’ under-investment. Furthermore, share pledging hurts firms’ option value unless active measures are taken to control the pledging risks. The maintenance requirement can keep controlling shareholder from irrational early investments and protect investors from severe wealth losses. In addition, our work can provide testable empirical implications.     

Fair insurance guaranty premia in the presence of risk-based capital regulations,stochastic interest rate and catastrophe risk

A multiperiod model is developed to measure the costs posed to the guaranty fund in a setting that incorporates risk-based capital regulations, interest rate risk and the possibility of catastrophic losses. The guaranty contract is modeled as a put option on the asset of the insurance company with a stochastic strike price and an uncertain maturity. The impacts of the key factors of this model are examined numerically and shown to make material differences in the costs to the guaranty fund.