Appendix B. Call Option Class

Example B-1 contains a class definition for a European call option in the Black-Scholes-

Merton (1973) model (cf. Chapter 3, and in particular Example 3-1).

Example B-1. Implementation of a Black-Scholes-Merton call option class

Valuation of European call options in Black-Scholes-Merton Model

incl. Vega function and implied volatility estimation

— class-based implementation

bsm_option_class.py

from math import log, sqrt, exp
from scipy import stats

class call_option(object):
”’ Class for European call options in BSM model.

            Attributes

            ==========

            S0  :   float

                            initial stock/index level

            K   :   float

                            strike  price

            T   :   float

                            maturity    (in year    fractions)

            r   :   float

                            constant    risk-free   short   rate

            sigma   :   float

                            volatility  factor  in  diffusion   term



            Methods

            =======

            value   :   float

                            return  present value   of  call    option

            vega    :   float

                            return  Vega    of  call    option

            imp_vol:    float

                            return  implied volatility  given   option  quote

            ”’

def init(self, S0, K, T, r, sigma):
self.S0 = float(S0)
self.K = K
self.T = T
self.r = r
self.sigma = sigma

def value(self):
”’ Returns option value. ”’
d1 = ((log(self.S0 / self.K)

• (self.r + 0.5 * self.sigma * 2 ) self.T)
  / (self.sigma * sqrt(self.T)))
  d2 = ((log(self.S0 / self.K)


• (self.r - 0.5 * self.sigma * 2 ) self.T)
  / (self.sigma sqrt(self.T)))
  value = (self.S0 stats.norm.cdf(d1, 0.0, 1.0)

                                            -   self.K  *   exp(-self.r *   self.T) *   stats.norm.cdf(d2,  0.0,    1.0))

return value

def vega(self):
”’ Returns Vega of option. ”’
d1 = ((log(self.S0 / self.K)

• (self.r + 0.5 * self.sigma * 2 ) self.T)
  / (self.sigma sqrt(self.T)))
  vega = self.S0 stats.norm.cdf(d1, 0.0, 1.0) * sqrt(self.T)
  return vega

def imp_vol(self, C0, sigma_est=0.2, it= 100 ):
”’ Returns implied volatility given option price. ”’