Samuelson 1965 Option Pricing Formula [Loxx]

4 oct 2022

Samuelson 1965 Option Pricing Formula [Loxx] is an options pricing formula that pre-dates Black-Scholes-Merton. This version includes Analytical Greeks.

Samuelson (1965; see also Smith, 1976) assumed the asset price follows a geometric Brownian motion with positive drift, p. In this way he allowed for positive interest rates and a risk premium.

c = SN(d1) * e^((rho - omega) * T) - Xe^(-omega * T)N(d2)

d1 = (log(S / X) + (rho + v^2 / 2) * T) / (v * T^0.5)

d2 = d1 - (v * T^0.5)

where rho is the average rate of growth of the share price and omega is the average rate of growth in the value of the call. This is different from the Boness model in that the Samuelson model can take into account that the expected return from the option is larger than that of the underlying asset omega > rho.

Analytical Greeks
Delta Greeks: Delta, DDeltaDvol, Elasticity
Gamma Greeks: Gamma, GammaP, DGammaDvol, Speed
Vega Greeks: Vega , DVegaDvol/Vomma, VegaP
Theta Greeks: Theta
Rate/Carry Greeks: Option growth rate sensitivity, Share growth rate sensitivity
Probability Greeks: StrikeDelta, Risk Neutral Density

Inputs
S = Stock price.
X = Strike price of option.
T = Time to expiration in years.
omega = Average growth rate option
rho = Average growth rate share
v = Volatility of the underlying asset price
cnd (x) = The cumulative normal distribution function
nd(x) = The standard normal density function
convertingToCCRate(r, cmp ) = Rate compounder

Things to know
Only works on the daily timeframe and for the current source price.
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