Narrow Results

We employ neural networks to understand volatility surface movements. We first use daily data on options on the S&P 500 index to derive a relationship between the expected change in implied volatility and three variables: the return on the index, the moneyness of the option, and the remaining life of the option. This model provides an improvement of 10.72% compared with a simpler analytic model. We then enhance the model with an additional feature: the level of the VIX index prior to the change being observed. This produces a further improvement of 62.12% and shows that the expected response of the volatility surface to movements in the index is quite different in high and low volatility environments.

In this chapter, we investigate how different measures of volatility influence bank’s capital structure beside mandatory capital requirements. We study the relationship between four volatility risk measures (volatility skew and spread, variance risk premia, and realized volatility) and bank’s market leverage and we analyze if banks adjust their capital needs in response to significant increase of risk premia discounting from traders. Among the four volatility measures, volatility skew (defined as the difference between OTM put and ATM call implied volatility and representing the perceived tail risk by traders) affects bank’s leverage the most. As volatility skew increases — hence OTM put became more expensive than ATM call — banks deleverage their assets structure. One plausible explanation relates to the higher costs of equity issuance that a bank will face during a period of distress. As the possibility to incur in expensive equity issuance increases the bank prefers to deleverage its balance sheet and create a capital buffer.

Previous studies of the limit order book report that low depths accompany wide spreads and that spreads widen and depths fall in response to higher volume, but some postulate a positive relationship between spreads and depth during normal trading periods. We calculate the option value of the limit order book at 11:00 a.m. for 10 actively traded firms listed on the Australian Stock Exchange. Simultaneously this approach enables us to consider the spread and depth of the limit order book. We find that 33.1% of the option value of the limit order book is provided at the best ask and 34.7% at the best bid. We find that the option value of the limit order book is greatest at the best bid price and the best ask price and is more stable through time than the option value of individual shares or share quantities in the book. Also, consistent with the arguments of Cohen et al. (1981), we find evidence of equilibrium in the supply and demand of liquidity.

This paper investigates the significance of using a variable default boundary when pricing European Black-Scholes options that are subjected to credit risks. We apply numerical method and combine Klein [1] and Johnson and Stulz [2] to link the payout ratio proportionally to the assets of the option writer. We also link the payout ratio to the value of the assets of the writer. Numerical examples compare our results with Klein [1] and Johnson and Stulz [2] based on alternative assumptions, and illustrate when the application of variable default boundary becomes important.

This chapter proposes a theoretical model of initial public offering by taking into account the uncertainty in security pricing and the underwriting process. Three major factors are shown to affect the IPO pricing: (1) the uncertainty of stock price, (2) the legal requirement that restricts underwriters from selling any part of the issue above the fixed offering price, and (3) the underwriters' risk tolerance. It is shown that underpricing is negatively related to the underwriter's risk tolerance, and positively related to the length of premarket period and market price volatility.

The following sections are included:

• INTRODUCTION
• COMPUTING THE MODEL-FREE VOLATILITY
• CHOICE OF INDEX OPTION AND DATA
• INFORMATION IN MODEL-FREE VOLATILITY
• ANALYSES OF THE VOLATILITY TERM STRUCTURE
• PROBLEM SET
• FURTHER RECOMMENDED READINGS