def vega(flag, S, K, t, r, sigma, q):
"""Returns the Black-Scholes-Merton vega of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
:param S: underlying asset price
:type S: float
:param K: strike price
:type K: float
:param t: time to expiration in years
:type t: float
:param r: annual risk-free interest rate
:type r: float
:param sigma: volatility
:type sigma: float
:param q: annualized continuous dividend yield
:type q: float
:returns: float
The text book analytical formula does not multiply by .01,
but in practice vega is defined as the change in price
for each 1 percent change in IV, hence we multiply by 0.01.
Example 17.6, page 367, Hull:
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> q = 0
>>> sigma = 0.2
>>> flag = 'c'
>>> vega_calc = vega(flag, S, K, t, r, sigma, q)
>>> # 0.121052427542
>>> vega_text_book = 0.121
>>> abs(vega_calc - vega_text_book) < .01
True
"""
D1 = d1(S, K, t, r, sigma, q)
return S * numpy.exp(-q * t) * pdf(D1) * numpy.sqrt(t) * 0.01
def delta(flag, S, K, t, r, sigma, q):
"""Returns the Black-Scholes-Merton delta of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
:param S: underlying asset price
:type S: float
:param K: strike price
:type K: float
:param t: time to expiration in years
:type t: float
:param r: annual risk-free interest rate
:type r: float
:param sigma: volatility
:type sigma: float
:param q: annualized continuous dividend yield
:type q: float
:returns: float
Example 17.1, page 355, Hull:
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> q = 0
>>> sigma = 0.2
>>> flag = 'c'
>>> delta_calc = delta(flag, S, K, t, r, sigma, q)
>>> # 0.521601633972
>>> delta_text_book = 0.522
>>> abs(delta_calc - delta_text_book) < .01
True
"""
D1 = d1(S, K, t, r, sigma, q)
if flag == 'p':
return -numpy.exp(-q * t) * N(-D1)
else:
return numpy.exp(-q * t) * N(D1)
def gamma(flag, S, K, t, r, sigma, q):
"""Returns the Black-Scholes-Merton gamma of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
:param S: underlying asset price
:type S: float
:param K: strike price
:type K: float
:param t: time to expiration in years
:type t: float
:param r: annual risk-free interest rate
:type r: float
:param sigma: volatility
:type sigma: float
:param q: annualized continuous dividend yield
:type q: float
:returns: float
Example 17.4, page 364, Hull:
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> q = 0
>>> sigma = 0.2
>>> flag = 'c'
>>> gamma_calc = gamma(flag, S, K, t, r, sigma, q)
>>> # 0.0655453772525
>>> gamma_text_book = 0.066
>>> abs(gamma_calc - gamma_text_book) < .001
True
"""
D1 = d1(S, K, t, r, sigma, q)
numerator = numpy.exp(-q * t) * pdf(D1)
denominator = S * sigma * numpy.sqrt(t)
return numerator / denominator
def theta(flag, S, K, t, r, sigma, q):
"""Returns the Black-Scholes-Merton theta of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
:param S: underlying asset price
:type S: float
:param K: strike price
:type K: float
:param t: time to expiration in years
:type t: float
:param r: annual risk-free interest rate
:type r: float
:param sigma: volatility
:type sigma: float
:param q: annualized continuous dividend yield
:type q: float
:returns: float
The text book analytical formula does not divide by 365,
but in practice theta is defined as the change in price
for each day change in t, hence we divide by 365.
Example 17.2, page 359, Hull:
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> q = 0
>>> sigma = 0.2
>>> flag = 'c'
>>> annual_theta_calc = theta(flag, S, K, t, r, sigma, q) * 365
>>> # -4.30538996455
>>> annual_theta_text_book = -4.31
>>> abs(annual_theta_calc - annual_theta_text_book) < .01
True
Using the same inputs with a put.
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'p'
>>> annual_theta_calc = theta(flag, S, K, t, r, sigma, q) * 365
>>> # -1.8530056722
>>> annual_theta_reference = -1.8530056722
>>> abs(annual_theta_calc - annual_theta_reference) < .000001
True
"""
D1 = d1(S, K, t, r, sigma, q)
D2 = d2(S, K, t, r, sigma, q)
first_term = (S * numpy.exp(-q * t) * pdf(D1) * sigma) / (2 *
numpy.sqrt(t))
if flag == 'c':
second_term = -q * S * numpy.exp(-q * t) * N(D1)
third_term = r * K * numpy.exp(-r * t) * N(D2)
return -(first_term + second_term + third_term) / 365.0
else:
second_term = -q * S * numpy.exp(-q * t) * N(-D1)
third_term = r * K * numpy.exp(-r * t) * N(-D2)
return (-first_term + second_term + third_term) / 365.0
