def rho(flag, S, K, t, r, sigma, q):
"""Returns the Black-Scholes-Merton rho 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 rho is defined as the change in price
for each 1 percent change in r, hence we multiply by 0.01.
Example 17.7, page 368, Hull:
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> q = 0
>>> sigma = 0.2
>>> flag = 'c'
>>> rho_calc = rho(flag, S, K, t, r, sigma, q)
>>> # 0.089065740988
>>> rho_text_book = 0.0891
>>> abs(rho_calc - rho_text_book) < .0001
True
"""
D2 = d2(S, K, t, r, sigma, q)
if flag == 'c':
return t * K * numpy.exp(-r * t) * N(D2) * .01
else:
return -t * K * numpy.exp(-r * t) * N(-D2) * .01
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