On the Relation Between Option and Stock Prices: A Convex Optimization Approach

@article{Bertsimas2002OnTR,
  title={On the Relation Between Option and Stock Prices: A Convex Optimization Approach},
  author={Dimitris Bertsimas and Ioana Popescu},
  journal={Oper. Res.},
  year={2002},
  volume={50},
  pages={358-374}
}
The idea of investigating the relation of option and stock prices based just on the no-arbitrage assumption, but without assuming any model for the underlying price dynamics, has a long history in the financial economics literature. We introduce convex and, in particular semidefinite optimization methods, duality, and complexity theory to shed new light on this relation. For the single stock problem, given moments of the prices of the underlying assets, we show that we can find best-possible… 

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References

SHOWING 1-10 OF 33 REFERENCES

Bounds on contingent claims based on several assets

Option Prices and the Underlying Asset's Return Distribution

This work examines the relation between option prices and the true, as opposed to risk-neutral, distribution of the underlying asset. If the underlying asset follows a diffusion with an instantaneous

Semi-parametric upper bounds for option prices and expected payoffs

Martingales and Arbitrage in Securities Markets with Transaction Costs

Abstract We derive the implications from the absence of arbitrage in dynamic securities markets with bid-ask spreads. The absence of arbitrage is equivalent to the existence of at least an equivalent

Pricing with a Smile

prices as a function of volatility. If an option price is given by the market we can invert this relationship to get the implied volatility. If the model were perfect, this implied value would be the

Implied Binomial Trees

This article develops a new method for inferring risk-neutral probabilities (or state-contingent prices) from the simultaneously observed prices of European options. These probabilities are then used

The valuation of options for alternative stochastic processes

Generalized Chebychev Inequalities: Theory and Applications in Decision Analysis

TLDR
A very general framework for analyzing these kinds of problems where, given certain "moments" of a distribution, the authors can compute bounds on the expected value of an arbitrary "objective" function.

Options and Efficiency

This paper argues that in an uncertain world options written on existing assets can improve efficiency by permitting an expansion of the contingencies that are covered by the market. The two major

Some NP-complete problems in quadratic and nonlinear programming

TLDR
A special class of indefinite quadratic programs is constructed, with simple constraints and integer data, and it is shown that checking (a) or (b) on this class is NP-complete.