Líkön og mælingar – Fjármálaafleiður 3.1 Financial derivatives The Black-Scholes stock model

Líkön og mælingar – Fjármálaafleiður 3.2 The Black-Scholes model Assumptions Stock process is described by Model 3 with  and  constant. No dividends (arðgreiðslur), transaction costs or taxes. The risk-free rate of interest, r, is constant. The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate.

Líkön og mælingar – Fjármálaafleiður 3.3 The Black-Scholes equation Given these assumptions, the value, f, of a derivative is: Example – long forward contract f = S - Ke -r(T-t) so the r.h.s. is

Líkön og mælingar – Fjármálaafleiður 3.4 Exact solution (1) The Black-Scholes equation can be solved in the case of European call options (ECO): –Exercised only on the expiration date, t=T. Boundary conditions: –If S(t)=0 for all t, then f(0,t)=0 for all t. –When S , then f(S,t)  –When t=T, then f(S,T) = max(S-K, 0)

Líkön og mælingar – Fjármálaafleiður 3.5 Exact solution (2) The solution is: where

Líkön og mælingar – Fjármálaafleiður 3.6 Assignment 2 Write programs that calculate and display the value of an European call option according to the Black- Scholes model. Use the exact solution. Use real data and the calculated . Choose two values for K (above and below the mean stock value for the period) and choose two values for r (r=0 and r=0.1 for example) (plot 4 graphs). Describe the effect of K, r and the stock price S, on the ECO value.

Líkön og mælingar – Fjármálaafleiður 3.7 Tips (1) An approximation for N(x) is: where

Líkön og mælingar – Fjármálaafleiður 3.8 Tips (2) PERL subroutines # Defining sub N { $in=$_[0];#Store the value passed to the function $nx=2*$in; $res=$nx/4;#The last expression evaluated is returned } # Calling $val = &N($d1);#If $d1=1, then $val=0.5