def bsm_pricing_put_return(S0, K, sigma, t, div):
T = t / 365
r = fred_1r_ir_today()
d1 = (np.log(S0 / K) + (r + sigma**2 / 2) * T) / (sigma * np.sqrt(T))
d2 = (np.log(S0 / K) + (r - sigma**2 / 2) * T) / (sigma * np.sqrt(T))
price = K * np.exp(-r * T) * ss.norm.cdf(-d2) - S0 * np.exp(
-div * T) * ss.norm.cdf(-d1)
# Calculate put delta
put_delta = np.exp(-div * T) * (ss.norm.cdf(d1) - 1)
# Calculate put gamma
put_gamma = np.exp(-div * T) / (S0 * sigma * np.sqrt(T)) * (
1 / np.sqrt(2 * np.pi)) * np.exp(-(d1**2) / 2)
# Calculate put theta:
put_theta = (1 / 365) * (-S0 * sigma * np.exp(-div * T) /
(2 * np.sqrt(T)) *
(1 / np.sqrt(2 * np.pi)) * np.exp(-(d1**2) / 2) +
r * K * np.exp(-r * T) * ss.norm.cdf(-d2) -
div * S0 * np.exp(-div * T) * ss.norm.cdf(d1))
# Calculate vega:
vega = (1 / 100) * (S0 * np.exp(-div * T) * np.sqrt(T)) * (
1 / np.sqrt(2 * np.pi)) * np.exp(-(d1**2) / 2)
# Calculate put rho:
rho = -(1 / 100) * K * T * np.exp(-r * T) * ss.norm.cdf(-d2)
return price, put_delta, put_gamma, put_theta, vega, rho
def bsm_pricing_call_return(S0, K, sigma, t, div):
T = t / 365
r = fred_1r_ir_today()
d1 = (np.log(S0 / K) + (r + sigma**2 / 2) * T) / (sigma * np.sqrt(T))
d2 = (np.log(S0 / K) + (r - sigma**2 / 2) * T) / (sigma * np.sqrt(T))
price = S0 * np.exp(-div * T) * ss.norm.cdf(d1) - K * np.exp(
-r * T) * ss.norm.cdf(d2)
# Calculate call delta
call_delta = np.exp(-div * T) * ss.norm.cdf(d1)
# Calculate call gamma
call_gamma = np.exp(-div * T) / (S0 * sigma * np.sqrt(T)) * (
1 / np.sqrt(2 * np.pi)) * np.exp(-(d1**2) / 2)
# Calculate put gamma
put_gamma = call_gamma
# Calculate call theta:
call_theta = (1 / 365) * (-S0 * sigma * np.exp(-div * T) /
(2 * np.sqrt(T)) *
(1 / np.sqrt(2 * np.pi)) * np.exp(-(d1**2) / 2) -
r * K * np.exp(-r * T) * ss.norm.cdf(d2) +
div * S0 * np.exp(-div * T) * ss.norm.cdf(d1))
# Calculating vega
vega = (1 / 100) * (S0 * np.exp(-div * T) * np.sqrt(T)) * (
1 / np.sqrt(2 * np.pi)) * np.exp(-(d1**2) / 2)
# Calculating call Rho
rho = (1 / 100) * K * T * np.exp(-r * T) * ss.norm.cdf(d2)
return price, call_delta, call_gamma, call_theta, vega, rho
def bsm_put(S0, K, sigma, t, div):
T = t / 365
r = fred_1r_ir_today()
d1 = (np.log(S0 / K) + (r + sigma**2 / 2) * T) / (sigma * np.sqrt(T))
d2 = (np.log(S0 / K) + (r - sigma**2 / 2) * T) / (sigma * np.sqrt(T))
price = K * np.exp(-r * T) * ss.norm.cdf(-d2) - S0 * np.exp(
-div * T) * ss.norm.cdf(-d1)
return price
def bsm_pricing_call(S0, K, sigma, t, div):
T = t / 365
r = fred_1r_ir_today()
d1 = (np.log(S0 / K) + (r + sigma**2 / 2) * T) / (sigma * np.sqrt(T))
d2 = (np.log(S0 / K) + (r - sigma**2 / 2) * T) / (sigma * np.sqrt(T))
price = S0 * np.exp(-div * T) * ss.norm.cdf(d1) - K * np.exp(
-r * T) * ss.norm.cdf(d2)
# Calculate call delta
call_delta = np.exp(-div * T) * ss.norm.cdf(d1)
# Calculate call gamma
call_gamma = np.exp(-div * T) / (S0 * sigma * np.sqrt(T)) * (
1 / np.sqrt(2 * np.pi)) * np.exp(-(d1**2) / 2)
# Calculate put gamma
put_gamma = call_gamma
# Calculate call theta:
call_theta = (1 / 365) * (-S0 * sigma * np.exp(-div * T) /
(2 * np.sqrt(T)) *
(1 / np.sqrt(2 * np.pi)) * np.exp(-(d1**2) / 2) -
r * K * np.exp(-r * T) * ss.norm.cdf(d2) +
div * S0 * np.exp(-div * T) * ss.norm.cdf(d1))
# Calculating vega
vega = (1 / 100) * (S0 * np.exp(-div * T) * np.sqrt(T)) * (
1 / np.sqrt(2 * np.pi)) * np.exp(-(d1**2) / 2)
# Calculating call Rho
rho = (1 / 100) * K * T * np.exp(-r * T) * ss.norm.cdf(d2)
print("S0\tstock price at time 0:", S0)
print("K\tstrike price:", K)
print("r\tcontinuously compounded risk-free rate:", r)
print("sigma\timplied volatility of the stock price per year:", sigma)
print("T\ttime to maturity in trading years: {:0.04F}".format(T))
print("Call_price: {:0.04F}".format(price))
print("Delta: {:0.04F}".format(call_delta))
print("Gamma: {:0.04F}".format(call_gamma))
print("Theta: {:0.04F}".format(call_theta))
print("Vega: {:0.04F}".format(vega))
print("Rho: {:0.04F}".format(rho))