def black_scholes(spot_value,
strike_value,
vol_value,
time_value,
is_call_bool,
rate=0.):
"""
Black Scholes option pricing formula on log-normal spot value
:param forward_value: forward price of underlying at exercise date
:type forward_value: real
:param strike_value: strike of the option
:type strike_value: real
:param vol_value: volatility of underlying price
:type vol_value: non-negative real
:param time_value: year fraction until exercise date
:type time_value: non-negative real
:param is_call_bool: call -> True, put -> False
:type is_call_bool: boolean
:param rate: risk free rate
:type rate: real
:return: option price
:rtype: real
:return:
"""
if not vol_value >= 0:
raise AssertionError("Negative vol in %s" % __name__)
sigma = vol_value * math.sqrt(time_value)
if sigma == 0:
# non-random option value, e.g. on exercise date
return option_payoff(spot_value, strike_value, is_call_bool)
d1 = (math.log(spot_value / strike_value) + rate * time_value +
0.5 * sigma**2) / sigma
d2 = d1 - sigma
if is_call_bool:
return spot_value * normal_cdf(d1) - strike_value * math.exp(
-rate * time_value) * normal_cdf(d2)
else:
return strike_value * math.exp(-rate * time_value) * normal_cdf(
-d2) - spot_value * normal_cdf(-d1)
def hw_discount_bond_option(forward_value, strike_value, implied_vol_value, time_value, is_call_bool,
time_to_bond_value, mean_reversion_value, maturity_discount_value):
"""
discount bond option pricing formula in the Hull White framework
:param float forward_value: forward price of underlying (discount bond) at bond maturity (D(t,Tau,r))
:param float strike_value: strike price
:param float implied_vol_value: volatility of the spot rate
:param float time_value: year fraction until exercise date (option maturity date)
:param float bool is_call_bool: call -> True, put -> False
:param float time_to_bond_value: year fraction between option's maturity and bond's maturity
:param float maturity_discount_value: forward price of underlying (discount bond) at option maturity date (D(t,T,r))
:param float mean_reversion_value: mean reversion / alpha
:return: float
discount bond option pricing formula in the Hull White framework
as described in A. Pelsser, *Efficient Methods for Valuing Interest Rate Derivatives*, 2000, pp. 50
"""
if time_value == 0:
return 0.0
else:
strike = maturity_discount_value * strike_value
A = forward_value / strike
B = (1 - math.exp(-mean_reversion_value * (time_to_bond_value))) / mean_reversion_value
var = ((implied_vol_value ** 2) / (2 * mean_reversion_value)) * \
(1 - math.exp(-2 * mean_reversion_value * time_value))
sigma = B * math.sqrt(var)
h = (math.log(A) / sigma) + 0.5 * sigma
if is_call_bool:
# call
option_bond_price = forward_value * normal_cdf(h) - strike * normal_cdf(h - sigma)
else:
# put
option_bond_price = strike * normal_cdf(-h + sigma) - forward_value * normal_cdf(-h)
return option_bond_price