Example 8 - Black-Scholes The value of a European “call option,” c (s,t ) , with exercise price e and expiration date T , satisfies the “asset-or-nothing payoff ” c (s,T ) = s, s ≥ e;= 0, s e . Prior to expiration c (s,t ) is estimated by the Black-Scholes differential equation ( ) ( ) 2 2 2 2 2 0 2 2 t ss s t s s s c s c rsc rc c s c r sc rc σ σ + + − ≡ + + −σ − = . The parameters in the model are the risk-free interest rate, r , and the stock volatility,σ . The boundary conditions are c (0,t ) = 0 and ( , ) 1, s c s t ≈ s →∞ . This development is described in Wilmott, et al. (1995), pages 41-57. There are explicit solutions for this equation based on the Normal Curve of Probability. The normal curve, and the solution itself, can be efficiently computed with the IMSL function ANORDF, IMSL (1994), page 186. With numerical integrationthe equation itself or the payoff can be readily changed to include other formulas, c (s,T ) , and corresponding boundary conditions. We use e = 100,r = 0.08,T − t = 0.25,σ2 = 0.04,sL = 0, and sR = 150. P1069——Fortran—Math
Points to consider when reading a published paper (1) Does the paper involve the development of a theoretical model or is it merely a technique looking for an application so that the motivation for the whole exercise is poor? (2) Are the data of ‘good quality’? Are they from a reliable source? Is the size of the sample sufficiently large for the model estimation task at hand? (3) Have the techniques been validly applied? Have tests been conducted for possible violations of any assumptions made in the estimation of the model? (4) Have the results been interpreted sensibly? Is the strength of the results exaggerated? Do the results actually obtained relate to the questions posed by the author(s)? Can the results be replicated by other researchers? (5) Are the conclusions drawn appropriate given the results, or has the importance of the results of the paper been overstated?
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