Abstract
This paper considers fundamental questions of arbitrage pricing that arises when the uncertainty model incorporates ambiguity about risk. This additional ambiguity motivates a new principle of risk- and ambiguity-neutral valuation as an extension of the paper by Ross (1976) (Ross, Stephen A. 1976. The arbitrage theory of capital asset pricing. Journal of Economic Theory 13: 341–60). In the spirit of Harrison and Kreps (1979) (Harrison, J. Michael, and David M. Kreps. 1979. Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20: 381–408), the paper establishes a micro-economic foundation of viability in which ambiguity-neutrality imposes a fair-pricing principle via symmetric multiple prior martingales. The resulting equivalent symmetric martingale measure set exists if the uncertain volatility in asset prices is driven by an ambiguous Brownian motion.
Highlights
One cornerstone of modern neoclassical finance is the fundamental theorem of asset pricing (FTAP)
In its simplest form and if uncertainty is modeled by a probability measure P, the theorem states equivalence between: part (a) the absence of P-arbitrage; and part (b) the existence of a
Based on parts (a)–(d), the FTAP of the present paper is established in a continuous-time setup, in which the risky and ambiguous asset price is driven by a Brownian motion B = ( Bσ ) with uncertain volatility σ
Summary
One cornerstone of modern neoclassical finance is the fundamental theorem of asset pricing (FTAP). Based on parts (a)–(d), the FTAP of the present paper is established in a continuous-time setup, in which the risky and ambiguous asset price is driven by a Brownian motion B = ( Bσ ) with uncertain volatility σ This is a zero-mean and stationary process with novel N(0, [σ, σ ])-normally distributed independent increments. A Samuelson (1965)-type model incorporates this kind of volatility uncertainty in a risky and ambiguous asset price process that follows the stochastic differential equation dSt = μt dt + Vt dBt. As in the classic single-prior setting, the increment dSt in Equation (1) is divided into a locally certain part and a locally uncertain part Vt dBt. The interpretation is d varPr (St ). Discussed the relation between arbitrage and pricing measures in a discrete-time framework
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