Black-Scholes Option Valuation

#include <stdio.h>
#include <math.h>
#include <gsl/gsl_randist.h>

double gsl_cdf_gaussian_P(double x, double sigma);

double d1(double stock, double strike, double risk_free, double vol, double time)
  {
return (log(stock/strike) + (risk_free + (vol * vol) / 2) * time) / (vol * sqrt(time));
  }

double d2(double d1_output, double vol, double time)
  {
  return d1_output - vol * sqrt(time);
  }

double black_scholes_call(double stock, double pd1, double pd2, double strike, double discount)
  {
    return (stock * pd1 - strike * discount * pd2);
  }


int
main (void)
{

  double stock = 50.0;
  double strike = 40.0;
  double risk_free = 0.01;
  double vol = 0.5;
  double time = 1;

  double d1_output = d1(stock, strike, risk_free, vol, time);
  double pd1 = gsl_cdf_gaussian_P(d1_output, 1);
  double d2_output = d2(d1_output, vol, time);
  double pd2 = gsl_cdf_gaussian_P(d2_output, 1);
  double discount = exp(-1 * risk_free * time);
  double call_value = black_scholes_call(stock, pd1, pd2, strike, discount);

  printf("The value of the call is $%.2f.\n", call_value);

  return 0;
}

The value of the call is $14.96.

The risk free rate should be the annualized continuously-compounded rate on a default free security with the same maturity as the expiration data of the option.

• For example, if the option expired in 3 months, you can use the continuously compounded annual rate for a 3-month Treasury Bill.

Remember that a stock price is adjusted downward by the dividend amount when the dividend is paid. For example, say before a \$1 dividend is paid the stock is \$50. Immediately after the dividend is paid the stock's price will be \$49 (otherwise there would be an arbitrage).

• However, stock option contract terms (such as the strike price) are not adjusted for cash dividends, or stock dividends under 10\%.
• So paying dividend reduces the value of a call option and increases the value of a put.
• Note, because firms often increase or decrease their dividend payments, we can only estimate an expected dividend yield or payment.

In the Black-Scholes world (where the option is European) we can reduce the stock price by the present value of all the dividends during the life of the option.

• The discounting is done from the ex-dividend date to the present.
• We can use the risk-free rate, though this assumes we are certain about the amount of the dividend payment.
• Note, we only include the dividend to be paid during the life of the option. So if the option expires in a month and the next dividend paid by the stock is in two months, we do not include a dividend adjustment.

Volatility (the standard deviation of log-returns) is not directly observable, and it is the toughest input to determine. Two common ways to estimate volatility:

• Use historical data
• Extracting volatility from other options

Important Note: Volatility is assumed to be constant in the Black-Scholes model. This is why you can estimate volatility over a historical period and use that volatility over a later period. But this assumption was made for mathematical ease, and it is not realistic.

So in the model's world, using historical volatility is fine, even though in the real world it is a poor approach.

We can also extract volatility from similar options and use that number as the volatility for our options.

• This is often referred to as 'calibrating to the market'.
• It has the benefit of being a forward-looking measure, which will take into account market expectations of the volatility from future events. Historical volatility, on the other hand, only looks backwards.

This distinction is particularly important if there is an event which will take place during the life of the option, which hasn't happened historically.

• Consider, for example, what the consequences might be if you own options expiring in 3 months on Chevron (CVX) and in one month Congress will vote on legislation to allow unfettered exports of crude oil from the U.S. (exports are now substantially limited).

We can extract a volatility estimate from traded options by plugging the option price into the Black-Scholes formula and solving for volatility. This volatility estimate is called the option's 'implied volatility'.