"Frontmatter". In: Analysis of Financial Time Series

BLACK–SCHOLES FORMULA 235risk preferences cannot affect the solution of the equation. A nice consequence ofthis property is that one can assume that investors are risk-neutral. In a risk-neutralworld, we have the following results:• The expected return on all securities is the risk-free interest rate r, and• The present value of any cash flow can be obtained by discounting its expectedvalue at the risk-free rate.6.6.2 FormulasThe expected value of a European call option at maturity in a risk-neutral world isE ∗ [max(P T − K, 0)],where E ∗ denotes expected value in a risk-neutral world. The price of the call optionat time t isc t = exp[−r(T − t)]E ∗ [max(P T − K, 0)]. (6.18)Yet in a risk-neutral world, we have µ = r, and by Eq. (6.10), ln(P T ) is normallydistributed as[ ( )]ln(P T ) ∼ N ln(P t ) + r − σ 2(T − t), σ 2 (T − t) .2Let g(P T ) be the probability density function of P T . Then the price of the call optionin Eq. (6.18) isc t = exp[−r(T − t)]∫ ∞K(P T − K )g(P T )dP T .By changing the variable in the integration and some algebraic calculations (detailsare given in Appendix A), we havec t = P t (h + ) − K exp[−r(T − t)](h − ), (6.19)where (x) is the cumulative distribution function (CDF) of the standard normalrandom variable evaluated at x,h + = ln(P t/K ) + (r + σ 2 /2)(T − t)σ √ T − th − = ln(P t/K ) + (r − σ 2 /2)(T − t)σ √ T − t= h + − σ √ T − t.In practice, (x) can easily be obtained from most statistical packages. Alternatively,one can use an approximation given in Appendix B.