Robust Control Approach to Option Pricing: A Representation Theorem and Fast Algorithm

Abstract

The so-called "interval model" for security prices, together with a robust control approach, allows one to construct a consistent theory of option pricing, including discrete time trading and arbitrary transaction costs. In this context, a new representation theorem for the viscosity solution of the relevant Isaacs (differential) quasivariational inequality leads to simple formulas and fast numerical algorithms to compute a hedging portfolio. We argue that in spite of a less satisfactory market model, the overall theory is not much less realistic than the classical Black and Scholes theory, but rather, it only shifts from the portfolio model to the market model the place where the model is violated when sudden large price changes occur on the market.

A deterministic‐control‐based approach to fully nonlinear parabolic and elliptic equations

“…Comparing with (2.5), we recognize that u " .x 0 ; t 0 / is the price of the option. This robust-control interpretation of the Black-Scholes-Merton theory was previously noted in [10].…”

“…So an investor such as Helen-whose goal is robust control (optimizing the worstcase outcome)-will make the same choices as one who adopts the Black-ScholesMerton viewpoint. This robust-control perspective on the Black-Scholes-Merton theory of option pricing was previously noted in [10].…”

Preliminary remarks on option pricing and dynamic hedging

“…Remark 2.2: Note that S TS,fluct (t) is analogous to "noises" in engineering according to the analysis of [17]. 4 See [25] and [36] for the estimation of S trend (t) and of its derivatives. See [22], and the references therein, for convincing numerical experiments including forecasting results which are deduced from the trends.…”

“…PRICING WITHOUT TRENDS We limit ourselves for simplicity's sake to European call options, which are options for the right to buy a stock or an index at a certain price at a certain maturity date. 4 The notion of "noise" has sometimes a quite different meaning in quantitative finance ( [5]).…”

“…• learning techniques (see, e.g., [27], [31]), • control theory (see, e.g., [2], [4], [10], [14], [37], [41], [43], [49]). In 1997, Scholes and Merton won the Nobel Prize in economics -Black died in 1995 -not for the discovery of the pricing formulas which were already known ( [8], [44], [45], [48]), but for the methods they introduced for deriving them.…”