To provide one motivation for the development of ARCH models (next handout), we briefly discuss here some difficulties associated with the Black Scholes formula, which is widely used to calculate the price of an option. For example, consider a European call option for a stock. This is the right to buy a specific number of shares of a specific stock on a specific date in the future, at a specifi...

This paper investigates Black–Scholes call and put option thetas, and derives upper and lower bounds for thetas as a function of underlying asset value. It is well known that the maximum time premium of an option occurs when the underlying asset value equals the exercise price. However, we show that the maximum option theta does not occur at that point, but instead occurs when the asset value i...

In the Black-Scholes option price model Brownian motion and the underlying Normal distribution play a fundamental role. Empirical evidence however shows that the normal distribution is a very poor model to fit real-life data. In order to achieve a better fit we replace the Brownian motion by a special Lévy process: the Meixner process. We show that the underlying Meixner distribution allows an ...

Interviewer: First I must mention that I cannot recall ever having heard of a self-interview. Would you mind saying how you came to interview yourself about your new theory? Donald Black: I decided to interview myself partly because I have never been entirely satisfied with the questions interviewers have asked me in the past (see, e.g., Black, 2002a; 2010b). I thought I might be able to ask be...

We investigate when a hedger who over-estimates the volatility will superreplicate a convex claim on several underlying assets. It is shown that the classical Black and Scholes model is the only model, within a large class, for which over-estimation of the volatility yields the desired superreplication property. This is in contrast to the onedimensional case, in which it is known that over-esti...

References [1] F. Black and P. Karasinski, Bond and option pricing when short-term rates are lognormal, Financial Analysts Journal, Vol. 47, No. 4, 52-59, 1991. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, Vol. 81, 637-654, 1973. [3] J. C. Cox, J. E. Ingersoll, and S.A. Ross, An intertemporal general equilibrium model of asset pric...

ii Acknowledgments I am grateful to my supervisor, Wulin Suo, for giving me the opportunity to evaluate the empirical performance of the Black-Scholes Model and the GARCH option pricing model and for his exceptional comments on my manuscript. Also, I dedicate this paper to my family and my love, who always supported and encouraged me in the writing process.

Black and Scholes (1973) proved that under certain assumptions about the market place, the value of a European option, as a function of the current value of the underlying asset and time, verifies a Cauchy problem. We give new conditions for the existence and uniqueness of the value of a European option by using semigroup theory. For this, we choose a suitable space that verifies some condition...

We consider two families of D1-D5-P states and find their gravity duals. In each case the geometries are found to ‘cap off’ smoothly near r = 0; thus there are no horizons or closed timelike curves. These constructions support the general conjecture that the interior of black holes is nontrivial all the way up to the horizon.