It has often been argued that there exists an underlying biological basis of utility functions. Taking this line of argument a step further in this paper, we have aimed to computationally demonstrate the biological basis of the Black-Scholes functional form as applied to classical option pricing and hedging theory. The evolutionary optimality of the classical Black-Scholes function has been com...

We compute a sharp small-time estimate for the price of a basket call under a bi-variate SABR model with both β parameters equal to 1 and three correlation parameters, which extends the work of Bayer,Friz&Laurence[BFL14] for the multivariate Black-Scholes flat vol model. The result follows from the heat kernel on hyperbolic space for n = 3 combined with the Bellaiche[Bel81] heat kernel expansio...

We recover the properties of a wide class of far from extremal charged black branes from properties of near extremal black branes, generalizing the results of Danielsson, Guijosa and Kruczenski.

When we studied discrete-time models we used martingale pricing to derive the Black-Scholes formula for European options. It was clear, however, that we could also have used a replicating strategy argument to derive the formula. In this part of the course, we will use the replicating strategy argument in continuous time to derive the Black-Scholes partial differential equation. We will use this...

This paper proposes a simple modification of the Black–Scholes model by assuming that the volatility of the stock may jump at a random time τ from a value σa to a value σb. It shows that, if the market price of volatility risk is unknown, but constant, all contingent claims can be valued from the actual price C0, of some arbitrarily chosen “basis” option. Closed form solutions for the prices of...

The widespread practice of quoting option prices in terms of their Black-Scholes implied volatilities (IVs) in no way implies that market participants believe underlying returns to be lognormal. On the contrary, the variation of IVs across option strike and term to maturity, which is widely referred to as the volatility surface, can be substantial. In this brief review, we highlight some empiri...

We consider that the surplus of an insurer follows compound Poisson process and the insurer would invest its surplus in risky assets, whose prices satisfy the Black-Scholes model. In the risk process, we decompose the ruin probability into the sum of two ruin probabilities which are caused by the claim and the oscillation, respectively. We derive the integro-differential equations for these rui...

The formation of supermassive black holes (SMBH) is intimately related to galaxy formation, although precisely how remains a mystery. I speculate that formation of, and feedback from, SMBH may alleviate problems that have arisen in our understanding of the cores of dark halos of galaxies.

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...