FEYNMAN-KAC FORMULAS FOR BLACK-SCHOLESTYPE OPERATORSSVANTE JANSON∗AND JOHAN TYSK∗Abstract. There are many references showing that a classical solutionto the Black–Scholes equation is a stochastic solution. However, it is theconverse of this theorem which is most relevant in applications and theconverse is also more mathematically interesting. In the present articlewe establish such a converse. We find a Feynman–Kac type theoremshowing that the stochastic representation yields a classical solution tothe corresponding Black–Scholes equation with appropriate boundaryconditions under very general conditions on the coefficients. We alsoobtain additional regularity results in the one-dimensional case.1. IntroductionStochastic formulas for option prices are often easy to formulate and toimplement in for instance Monte Carlo algorithms. It is natural from thepoint of view of applications to let zero be an absorbing barrier for processesdescribing the risky assets. In Section 3 we discuss such a representationin the case of time- and level-dependent volatilities. However, it is oftenadvantageous to instead solve the corresponding Black–Scholes equation andthus be able to use results from the theory of partial differential equations.In the literature one often omits specifying the boundary conditions on thelateral part of the parabolic boundary. This causes no problem when therisky assets are modelled with geometric Brownian motion, since assets inthis model reach zero with probability zero. However, for many models,such as the constant elasticity of variance models, the values of the assetscan vanish with positive probability and boundary conditions need to bespecified. Furthermore, in numerical applications one is often helped byknowing the boundary behavior of the solution even if these conditions aremathematically redundant to specify.In fact, there are many references showing that a classical solution to theBlack–Scholes equation is a stochastic solution, compare Theorem 2.5. How-ever, it is the converse of this theorem which is most relevant in applicationsas described above and the converse is also more mathematically subtle. Inthe present article we establish such a converse. We find a a Feynman–Kactype theorem showing that the stochastic representation yields a classical so-lution to the corresponding Black–Scholes equation with appropriate bound-ary conditions, compare Theorem 5.5. We also obtain additional regularityresults in the one-dimensional case.Date: July 2, 2004; revised October 28, 2004.Key words and phrases. Parabolic equations; stochastic representation.∗Partially supported by the Swedish Research Council.12 SVANTE JANSON AND JOHAN TYSKOne should note that in the standard theory of parabolic equations,boundary regularity is only obtained for operators that are uniformly par-abolic near the boundary. In fact, basic results from that theory fail inthis more general setting of operators degenerating at the boundary. Forinstance, the standard regularity result saying that if the initial (or terminalcondition in a financial setting) condition is continuous and if the opera-tor has smooth coefficients the solution is smooth for any positive time (orany time strictly before expiration), fails. This is seen by considering con-tract functions of the form xαand the stock price modelled by geometricBrownian motion. Here the solution will have the form f(t)xα,forsomefunction f of time, see Example 6.4, and this solution is not smooth if α isnot a non-negative integer. Another example is the Hopf boundary pointlemma, see page 10 of [7]. This lemma says that at a boundary minimumof a solution to a parabolic equation the inner normal derivative must bestrictly positive. This fails for instance in such a well-known example as theBlack–Scholes formula for the call option: the inner normal derivative atthe origin is zero even though this is a minimum point for the option price.In this formula the stock price is modelled by geometric Brownian motion,so the corresponding parabolic operator does degenerate at the boundary.On the other hand, a standard tool such as the maximum principle, is stillavailable in our setting. We will indeed use this fact below.It is somewhat surprising how little attention that has been paid to theissues desribed above given the importance of the type of equation underconsideration and the general mathematical interest of existence and regu-larity questions also for this class of operators. However, there are of coursereferences dealing with this type of problem. A general treatment can befound in Chapter 15 of Friedman [2] and an example closer to the present ar-ticle is given by the work of Heath and Schweizer [3]. In the latter referencethe processes are assumed to reach the boundary with zero probability andin the book by Friedman [2] the coefficients of the equation are assumedto be continuous up to and including the boundary. In contrast to thesereferences, we will allow processes that reach the boundary with positiveprobability as well as coefficients that are not continuous at the boundary,compare Section 3.Our result should be applicable also in biology and chemistry when mod-eling systems on a meso scale, compare [4], and more generally wheneverone uses systems of stochastic processes of non-negative values. Stock pricesshould then be replaced by the number of molecules of various compounds.What is crucial for the results of this paper to hold is the absorbing propertyof the boundary, i.e. once a compound has vanished it does not reappear,and further that on the boundary the process is governed by the number ofremaining molecules and their stochastic properties.Finally we remark that the notation we use for stochastic processes is in-fluenced by the various needs we have in the article but also by the standardnotation in the theory of stochastic processes and financial applications, re-spectively. Thus, when noting that the process X depends on the timevariable t we write Xt. However, when the point in time is of some specialsignificance, such as the expiration date T of an option, we write X(T ), asFEYNMAN-KAC FORMULAS FOR BLACK-SCHOLES TYPE OPERATORS 3is common in finance. Also, when noting that the process X at time t,say,is at some specified point x,wewriteXx,t.Acknowledgement. We thank a referee for careful reading and insightfulcriticism.2. Classical and stochastic solutionsIn this section we collect some