Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1027value [Hull (2000)]O(0) = e −rT E[O(T )], (6.40)where r is the risk–free interest and E[·] indicates the mean value, whichcan be computed only if a model for the asset underlying the option isunderstood. For example, the value O of an European call option at thematurity T will be max{S T −X, 0}, where X is the strike price, while for anEuropean put option the value O at the maturity will be max{X − S T , 0}.It is worth emphasizing, for what follows, that the case of an Europeanoption is particularly simple, since in such a situation the price of theoption can be evaluated by means of analytical formulae, which are get bysolving the BSM partial differential equation with the appropriate boundaryconditions [Hull (2000); Paul and Baschnagel (1999)]. On the other hand,many further kinds of options are present in the financial markets, suchas American options (options which can be exercised at any time up tothe expiration date) and exotic options [Hull (2000)], i.e., derivatives withcomplicated payoffs or whose value depend on the whole time evolutionof the underlying asset and not just on its value at the end. For suchoptions with path-dependent and early exercise features no exact solutionsare available and pricing them correctly is a great challenge.In the case of options with possibility of anticipated exercise before theexpiration date, the above discussion needs to be generalized, by introducinga slicing of the time interval T . Let us consider, for definiteness, thecase of an option which can be exercised within the maturity but only atthe times t 1 = ∆t, t 2 = 2∆t, . . . , t n = n∆t = T. At each time slice t i−1the value O i−1 of the option will be the maximum between its expectationvalue at the time t i scaled with e −r∆t and its value in the case of anticipatedexercise Oi−1 Y . If S i−1 denotes the price of the underlying asset atthe time t i−1 , we can thus write for each i = 1, . . . , nO i−1 (S i−1 ) = max { O Y i−1(S i−1 ), e −r∆t E[O i |S i−1 ] } , (6.41)where E[O i |S i−1 ] is the conditional expectation value of O i , i.e., its expectationvalue under the hypothesis of having the price S i−1 at the time t i−1 .In this way, to get the actual price O 0 , it is necessary to proceed backwardin time and calculate O n−1 , . . . , O 1 , where the value O n of the optionat maturity is nothing but On Y (S n ). It is therefore clear that evaluatingthe price of an option with early exercise features means to simulate theevolution of the underlying asset price (to get the Oi Y ) and to calculate a