def gamma(flag, F, K, t, r, sigma):
"""Returns the Black gamma of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
:param F: underlying futures price
:type F: 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
:returns: float
>>> F = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> v1 = gamma(flag, F, K, t, r, sigma)
>>> # 0.0640646705882
>>> v2 = 0.0640646705882
>>> abs(v1-v2) < .000001
True
"""
D1 = d1(F, K, t, r, sigma)
return pdf(D1) * numpy.exp(-r * t) / (F * sigma * numpy.sqrt(t))
def theta(flag, F, K, t, r, sigma):
"""Returns the Black theta of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
:param F: underlying futures price
:type F: 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
:returns: float
>>> F = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> v1 = theta(flag, F, K, t, r, sigma)
>>> v2 = -0.00816236877462
>>> abs(v1-v2) < .000001
True
>>> flag = 'p'
>>> v1 = theta(flag, F, K, t, r, sigma)
>>> v2 = -0.00802799155312
>>> abs(v1-v2) < .000001
True
"""
e_to_the_minus_rt = numpy.exp(-r * t)
two_sqrt_t = 2 * numpy.sqrt(t)
D1 = d1(F, K, t, r, sigma)
D2 = d2(F, K, t, r, sigma)
pdf_d1 = pdf(D1)
N_d2 = N(D2)
first_term = F * e_to_the_minus_rt * pdf(D1) * sigma / two_sqrt_t
if flag == 'c':
second_term = -r * F * e_to_the_minus_rt * N(D1)
third_term = r * K * e_to_the_minus_rt * N(D2)
return -(first_term + second_term + third_term) / 365.
else:
second_term = -r * F * e_to_the_minus_rt * N(-D1)
third_term = r * K * e_to_the_minus_rt * N(-D2)
return (-first_term + second_term + third_term) / 365.0
return (first_term - second_term) / 365.0
def vega(flag, F, K, t, r, sigma):
"""Returns the Black vega of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
:param F: underlying futures price
:type F: 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
:returns: float
::
==========================================================
Note: 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.
==========================================================
>>> F = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> v1 = vega(flag, F, K, t, r, sigma)
>>> # 0.118317785624
>>> v2 = 0.118317785624
>>> abs(v1-v2) < .000001
True
"""
D1 = d1(F, K, t, r, sigma)
return F * numpy.exp(-r * t) * pdf(D1) * numpy.sqrt(t) * 0.01
def delta(flag, F, K, t, r, sigma):
"""Returns the Black delta of an option.
:param flag: 'c' or 'p' for call or put.
:type flag: str
:param F: underlying futures price
:type F: 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
:returns: float
>>> F = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> v1 = delta(flag, F, K, t, r, sigma)
>>> v2 = 0.45107017482201828
>>> abs(v1-v2) < .000001
True
"""
D1 = d1(F, K, t, r, sigma)
if flag == 'p':
return -numpy.exp(-r * t) * N(-D1)
else:
return numpy.exp(-r * t) * N(D1)