We examine how price impact in the underlying asset market affects the replication of a European contingent claim. We obtain a generalized Black-Scholes pricing PDE and establish the existence and uniqueness of a classical solution to this PDE. We show that unlike the case with transaction costs, replication in the presence of price impact is always cheaper than superreplication. This model imp...

In this short paper, we study the asymptotics for the price of call options for very large strikes and put options for very small strikes. The stock price is assumed to follow the Black–Scholes models. We analyze European, Asian, American, Parisian and perpetual options and conclude that the tail asymptotics for these option types fall into four scenarios.

This paper aims to investigate the assumptions under which the binomial option pricing model converges to the Black-Scholes formula. The results are not original; the paper mostly follows the outline of Cox, Ross, and Rubenstein[1]. However, the convergence is treated in greater detail than I have found elsewhere in the literature. This exercise clarifies the assumptions behind the binomial mod...

Using the solitonic solution-generating technique we rederived the one-rotational five-dimensional black ring solution found by Emparan and Reall. The seed solution is not the Minkowski metric, which is the seed of S-rotating black ring. The obtained solution has more parameters than the Emparan and Reall’s S-rotating black ring. We found the conditions of parameters to reduce the solution to t...

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...

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...