2007/12/09 by Alan Brace, A. Brace, Giorgio Fabbri +6
Economics, Econometrics and Finance · Mathematics · Social Sciences · #35R60 #37L55 #60H15 #Computational Finance (q-fin.CP) #FOS: Economics and business #FOS: Mathematics #Financial Risk and Volatility Modeling #Insurance, Mortality, Demography, Risk Management #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:35R60 #msc:37L55 #msc:60H15 #q-fin.CP
paper · pdf · doi:10.48550/arxiv.0712.1343
21 pages
openalex publication_date 2007/12/09 · arxiv created 2007/12/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present an Hilbert space formulation for a set of implied volatility models introduced in \citeBraceGoldys01 in which the authors studied conditions for a family of European call options, varying the maturing time and the strike price T an K, to be arbitrage free. The arbitrage free conditions give a system of stochastic PDEs for the evolution of the implied volatility surface σt(T,K). We will focus on the family obtained fixing a strike K and varying T. In order to give conditions to prove an existence-and-uniqueness result for the solution of the system it is here expressed in terms of the square root of the forward implied volatility and rewritten in an Hilbert space setting. The existence and the uniqueness for the (arbitrage free) evolution of the forward implied volatility, and then of the the implied volatility, among a class of models, are proved. Specific examples are also given.