def BSM_theta(St, K, t, T, r, sigma):
# change in option price to change in time to maturity
''' Black-Scholes-Merton THETA of European call option.
Parameters
==========
St : float
stock/index level at time t
K : float
strike price
t : float
valuation date
T : float
date of maturity/time-to-maturity if t = 0; T > t
r : float
constant, risk-less short rate
sigma : float
volatility
Returns
=======
theta : float
European call option THETA
'''
d1 = d1f(St, K, t, T, r, sigma)
d2 = d1 - sigma * math.sqrt(T - t)
theta = -(St * dN(d1) * sigma /
(2 * math.sqrt(T - t)) + r * K * math.exp(-r * (T - t)) * N(d2))
return theta
def BSM_gamma(St, K, t, T, r, sigma):
# change in delta to change in underlying
''' Black-Scholes-Merton GAMMA of European call option.
Parameters
==========
St : float
stock/index level at time t
K : float
strike price
t : float
valuation date
T : float
date of maturity/time-to-maturity if t = 0; T > t
r : float
constant, risk-less short rate
sigma : float
volatility
Returns
=======
gamma : float
European call option GAMM
'''
d1 = d1f(St, K, t, T, r, sigma)
gamma = dN(d1) / (St * sigma * math.sqrt(T - t))
return gamma
def BSM_vega(St, K, t, T, r, sigma):
# change in option price to change in volatility
''' Black-Scholes-Merton VEGA of European call option.
Parameters
==========
St : float
stock/index level at time t
K : float
strike price
t : float
valuation date
T : float
date of maturity/time-to-maturity if t = 0; T > t
r : float
constant, risk-less short rate
sigma : float
volatility
Returns
=======
vega : float
European call option VEGA
'''
d1 = d1f(St, K, t, T, r, sigma)
vega = St * dN(d1) * math.sqrt(T - t)
return vega
def BSM_delta(St, K, t, T, r, sigma):
# change in option price to change in underlying
''' Black-Scholes-Merton DELTA of European call option.
Parameters
==========
St : float
stock/index level at time t
K : float
strike price
t : float
valuation date
T : float
date of maturity/time-to-maturity if t = 0; T > t
r : float
constant, risk-less short rate
sigma : float
volatility
Returns
=======
delta : float
European call option DELTA
'''
d1 = d1f(St, K, t, T, r, sigma)
delta = N(d1)
return delta
def BSM_rho(St, K, t, T, r, sigma):
# change in option price to change in interest rate
''' Black-Scholes-Merton RHO of European call option.
Parameters
==========
St : float
stock/index level at time t
K : float
strike price
t : float
valuation date
T : float
date of maturity/time-to-maturity if t = 0; T > t
r : float
constant, risk-less short rate
sigma : float
volatility
Returns
=======
rho : float
European call option RHO
'''
d1 = d1f(St, K, t, T, r, sigma)
d2 = d1 - sigma * math.sqrt(T - t)
rho = K * (T - t) * math.exp(-r * (T - t)) * N(d2)
return rho