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)
cnd_d2 = cnd(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 * cnd(D1)
third_term = r * K * e_to_the_minus_rt * cnd(D2)
return -(first_term + second_term + third_term) / 365.0
else:
second_term = -r * F * e_to_the_minus_rt * cnd(-D1)
third_term = r * K * e_to_the_minus_rt * cnd(-D2)
return (-first_term + second_term + third_term) / 365.0
return (first_term - second_term) / 365.0
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)
cnd_d2 = cnd(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 * cnd(D1)
third_term = r * K * e_to_the_minus_rt * cnd(D2)
return -(first_term + second_term + third_term) / 365.
else:
second_term = -r * F * e_to_the_minus_rt * cnd(-D1)
third_term = r * K * e_to_the_minus_rt * cnd(-D2)
return (-first_term + second_term + third_term) / 365.0
return (first_term - second_term) / 365.0
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 gamma(flag, S, K, t, r, sigma):
"""Return Black-Scholes gamma of an option.
:param S: underlying asset price
:type S: float
:param K: strike price
:type K: float
:param sigma: annualized standard deviation, or volatility
:type sigma: float
:param t: time to expiration in years
:type t: float
:param r: risk-free interest rate
:type r: float
:param flag: 'c' or 'p' for call or put.
:type flag: str
Example 17.4, page 364, Hull:
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> gamma_calc = gamma(flag, S, K, t, r, sigma)
>>> # 0.0655453772525
>>> gamma_text_book = 0.066
>>> abs(gamma_calc - gamma_text_book) < .001
True
"""
d_1 = d1(S, K, t, r, sigma)
v_squared = sigma**2
return pdf(d_1) / (S * sigma * numpy.sqrt(t))
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
"""
D1 = d1(S, K, t, r, sigma, q)
return S * numpy.exp(-q * t) * pdf(D1) * numpy.sqrt(t) * 0.01
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 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
"""
D1 = d1(S, K, t, r, sigma, q)
return S * numpy.exp(-q*t) * pdf(D1) * numpy.sqrt(t) * 0.01
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 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 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
def vega(flag, S, K, t, r, sigma):
"""Return Black-Scholes vega of an option.
:param S: underlying asset price
:type S: float
:param K: strike price
:type K: float
:param sigma: annualized standard deviation, or volatility
:type sigma: float
:param t: time to expiration in years
:type t: float
:param r: risk-free interest rate
:type r: float
:param flag: 'c' or 'p' for call or put.
:type flag: str
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
>>> sigma = 0.2
>>> flag = 'c'
>>> vega_calc = vega(flag, S, K, t, r, sigma)
>>> # 0.121052427542
>>> vega_text_book = 0.121
>>> abs(vega_calc - vega_text_book) < .01
True
"""
d_1 = d1(S, K, t, r, sigma)
return S * pdf(d_1) * numpy.sqrt(t) * 0.01
def gamma(flag, S, K, t, r, sigma):
"""Return Black-Scholes gamma of an option.
:param S: underlying asset price
:type S: float
:param K: strike price
:type K: float
:param sigma: annualized standard deviation, or volatility
:type sigma: float
:param t: time to expiration in years
:type t: float
:param r: risk-free interest rate
:type r: float
:param flag: 'c' or 'p' for call or put.
:type flag: str
Example 17.4, page 364, Hull:
>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> gamma_calc = gamma(flag, S, K, t, r, sigma)
>>> # 0.0655453772525
>>> gamma_text_book = 0.066
>>> abs(gamma_calc - gamma_text_book) < .001
True
"""
d_1 = d1(S, K, t, r, sigma)
v_squared = sigma**2
return pdf(d_1)/(S*sigma*numpy.sqrt(t))
def theta(flag, S, K, t, r, sigma):
"""Return Black-Scholes theta of an option.
:param S: underlying asset price
:type S: float
:param K: strike price
:type K: float
:param sigma: annualized standard deviation, or volatility
:type sigma: float
:param t: time to expiration in years
:type t: float
:param r: risk-free interest rate
:type r: float
:param flag: 'c' or 'p' for call or put.
:type flag: str
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
>>> sigma = 0.2
>>> flag = 'c'
>>> annual_theta_calc = theta(flag, S, K, t, r, sigma) * 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) * 365
>>> # -1.8530056722
>>> annual_theta_reference = -1.8530056722
>>> abs(annual_theta_calc - annual_theta_reference) < .000001
True
"""
two_sqrt_t = 2 * numpy.sqrt(t)
D1 = d1(S, K, t, r, sigma)
D2 = d2(S, K, t, r, sigma)
first_term = (-S * pdf(D1) * sigma) / two_sqrt_t
if flag == 'c':
second_term = r * K * numpy.exp(-r*t) * cnd(D2)
return (first_term - second_term)/365.0
if flag == 'p':
second_term = r * K * numpy.exp(-r*t) * cnd(-D2)
return (first_term + second_term)/365.0
