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
"""
D2 = d2(S, K, t, r, sigma, q)
if flag == 'c':
return t * K * numpy.exp(-r*t) * cnd(D2) * .01
else:
return -t * K * numpy.exp(-r*t) * cnd(-D2) * .01
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
"""
D2 = d2(S, K, t, r, sigma, q)
if flag == 'c':
return t * K * numpy.exp(-r * t) * cnd(D2) * .01
else:
return -t * K * numpy.exp(-r * t) * cnd(-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
"""
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) * cnd(D1)
third_term = r * K * numpy.exp(-r*t) * cnd(D2)
return - (first_term + second_term + third_term) / 365.0
else:
second_term = -q * S * numpy.exp(-q*t) * cnd(-D1)
third_term = r * K * numpy.exp(-r*t) * cnd(-D2)
return (-first_term + second_term + third_term) / 365.0
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
"""
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) * cnd(D1)
third_term = r * K * numpy.exp(-r * t) * cnd(D2)
return -(first_term + second_term + third_term) / 365.0
else:
second_term = -q * S * numpy.exp(-q * t) * cnd(-D1)
third_term = r * K * numpy.exp(-r * t) * cnd(-D2)
return (-first_term + second_term + third_term) / 365.0