The fundamental theorem of asset pricing under proportional transaction costs

We extend the fundamental theorem of asset pricing to a model where the risky stock is subject to proportional transaction costs in the form of bid-ask spreads and the bank account has different interest rates for borrowing and lending. We show that such a model is free of arbitrage if and only if one can embed in it a friction-free model that is itself free of arbitrage, in the sense that there exists an artificial friction-free price for the stock between its bid and ask prices and an artificial interest rate between the borrowing and lending interest rates such that, if one discounts this stock price by this interest rate, then the resulting process is a martingale under some non-degenerate probability measure. Restricting ourselves to the simple case of a finite number of time steps and a finite number of possible outcomes for the stock price, the proof follows by combining classical arguments based on finite-dimensional separation theorems with duality results from linear optimisation. The fundamental theorem of asset pricing characterises models of financial markets without arbitrage or free lunch, i.e. the making of risk-free profit without initial investment. It is well known that a classical friction-free model containing a risky stock and a bank account admits no arbitrage if and only if there exists a probability measure on the model under which the stock price, discounted by the interest rate on the bank account, is a martingale. In turn, the collection of such probability measures play a fundamental role in the pricing of contingent claims and derivative securities. The aim of this paper is to extend the fundamental theorem to a model where the risky stock is subject to proportional transaction costs in the form of bid-ask spreads, so that a share can always be sold for the bid price and bought for the ask price, and the bank account has different interest rates for borrowing and lending. We show that such a model is free of arbitrage if and only if one can embed in it a friction-free model that is itself free of arbitrage, in the sense that there exists an artificial friction-free price for the stock between its bid and ask prices and an artificial interest rate between the borrowing and lending interest rates such that, if one discounts this stock price by this interest rate, then the resulting process is a martingale under some non-degenerate probability measure. Section 1 is a formal introduction to a model of a financial market consisting of a risky asset (a stock) and a risk-free asset (a bond, referred to as cash). Department of Mathematics, University of York. Email: ar521york.ac.uk.