BS Fn
BS Fn is a direct subtype of Real To Real Function
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TYPE INCLUSION RELATIONSHIPS
AVAILABLE FUNCTIONS
AVAILABLE CREATE FUNCTION KEYS
TYPICAL OBJECTS OF TYPE BS Fn
This type represents a function that maps the 6 variables S, K, σ, r, q, t to the option price y, according to the Black Scholes formula, as described further below.
The 6 variables must be supplied in the indicated order and are interpreted as follows:
S is the spot price of one share of the option's underlying. For example, S = 130 if the underlying is the Miscrosoft stock, or S = 1.3 if the underlying is the GBP currency from the point of view of a US investor.
K is the strike of the option.
σ is the volatility of the underlying price. For example 0.2 for a vol of 20%.
r is the continuously compounded risk-free discounting interest rate with respect to the expiry of the option. For example 0.01 for a rate of 1%.
δ is the continuously compounded dividend yield assumed to be paid by the underlying until the expiry of the option. For example 0.03 for a yield of 3%.
t is the time to expiry of the option in annual units. For example, 1 for an option expiring in one year.
Case vanilla call:
y = P[FN(d₁) - KN(d₂)]
Case vanilla put:
y = P[KN(-d₂) - FN(-d₁)]
Case cash-or-nothing call:
y = PN(d₂)
Case cash-or-nothing put:
y = PN(-d₂)
Case asset-or-nothing call:
y = PFN(d₁)
Case asset-or-nothing put:
y = PFN(-d₁)
The function N() appearing in the above formulas is the cummulative standard normal probability function, so that N(x) = PROB[G < x], where G is any normally distributed random variable.
The quantities P, F, d₁, d₂ are defined as follows:
P = e⁻ʳᵗ
F = Se⁽ʳ⁻ᵟ⁾ᵗ
d₁ = ln(F/K)/(σt¹ᐟ²) + ½σt¹ᐟ²
d₂ = d₁ - σt¹ᐟ²