def black_digital_delta(forward_value, strike_value, implied_vol_value,
time_value, is_call_bool):
"""
Black-76 delta sensitivity for digital payoff.
:param float forward_value: forward price of underlying at exercise date
:param float strike_value: strike price
:param float implied_vol_value: volatility of underlying price
:param float time_value: year fraction until exercise date
:param boolean is_call_bool: call -> True, put -> False
:return: float
"""
sigma, fms, d0, d1 = _black_param(forward_value, strike_value,
implied_vol_value, time_value)
if sigma == 0.0:
return 0.0
if is_call_bool:
# call
return normal_density(d1) / (sigma * forward_value)
else:
# put
return -normal_density(-d1) / (sigma * forward_value)
def bachelier(forward_value, strike_value, implied_vol_value, time_value,
is_call_bool):
"""
Bachelier formula (Black formula for normal underlying distribution).
:param float forward_value: forward price of underlying at exercise date
:param float strike_value: strike price
:param float implied_vol_value: volatility of underlying price
:param float time_value: year fraction until exercise date
:param boolean is_call_bool: call -> True, put -> False
:return: float
"""
sigma, fms, d = _bachelier_param(forward_value, strike_value,
implied_vol_value, time_value)
if sigma == 0.0:
return option_payoff(forward_value, strike_value, is_call_bool)
# call
# call_value = fms * normal_cdf(d) + exp(-d ** 2 / 2) * sigma / sqrt(2 * pi) # intern version
call_value = fms * normal_cdf(d) + sigma * normal_density(d)
put_value = -fms * normal_cdf(-d) + sigma * normal_density(d)
return call_value if is_call_bool else call_value - fms # put
def black_straddle_vega(forward_value, strike_value, implied_vol_value,
time_value, is_call_bool):
"""
Black-76 vega sensitivity for straddle payoff.
:param float forward_value: forward price of underlying at exercise date
:param float strike_value: strike price
:param float implied_vol_value: volatility of underlying price
:param float time_value: year fraction until exercise date
:param boolean is_call_bool: call -> True, put -> False
:return: float
"""
sigma, fms, d0, d1 = _black_param(forward_value, strike_value,
implied_vol_value, time_value)
if sigma == 0.0:
return 0.0
return 2.0 * forward_value * sqrt(time_value) * normal_density(d1)
def bachelier_gamma(forward_value, strike_value, implied_vol_value, time_value,
is_call_bool):
"""
gamma sensitivity for Bachelier formula.
:param float forward_value: forward price of underlying at exercise date
:param float strike_value: strike price
:param float implied_vol_value: volatility of underlying price
:param float time_value: year fraction until exercise date
:param boolean is_call_bool: call -> True, put -> False
:return: float
"""
sigma, fms, d = _bachelier_param(forward_value, strike_value,
implied_vol_value, time_value)
if sigma == 0.0:
return 0.0
return normal_density(d) / sigma
def black_digital_vega(forward_value, strike_value, implied_vol_value,
time_value, is_call_bool):
"""
Black-76 vega sensitivity for digital payoff.
:param float forward_value: forward price of underlying at exercise date
:param float strike_value: strike price
:param float implied_vol_value: volatility of underlying price
:param float time_value: year fraction until exercise date
:param boolean is_call_bool: call -> True, put -> False
:return: float
"""
sigma, fms, d0, d1 = _black_param(forward_value, strike_value,
implied_vol_value, time_value)
if sigma == 0.0:
return 0.0
# call
call_value = d1 * normal_density(d0) / implied_vol_value
return call_value if is_call_bool else -call_value # put
def bachelier_digital_vega(forward_value, strike_value, implied_vol_value,
time_value, is_call_bool):
"""
vega sensitivity for Bachelier formula for digital option.
:param float forward_value: forward price of underlying at exercise date
:param float strike_value: strike price
:param float implied_vol_value: volatility of underlying price
:param float time_value: year fraction until exercise date
:param boolean is_call_bool: call -> True, put -> False
:return: float
"""
sigma, fms, d = _bachelier_param(forward_value, strike_value,
implied_vol_value, time_value)
if sigma == 0.0:
return 0.0
# call
call_value = -d * normal_density(d) / implied_vol_value
return call_value if is_call_bool else -call_value # put
