Bachelier

The Bachelier (normal) model — the forward is arithmetic Brownian rather than geometric, so the price can be negative and the volatility sigmaN is quoted in price units per sqrt(year) rather than as a fraction. This is the market model for interest-rate options in a low- or negative-rate regime, where a lognormal forward is inadmissible.

d    = (F - K) / (sigmaN sqrt(T))
call = e^{-rT} [ (F - K) N(d) + sigmaN sqrt(T) phi(d) ]
put  = e^{-rT} [ (K - F) N(-d) + sigmaN sqrt(T) phi(d) ]

In AnalyticGreeks: delta() is dV/dF, vega() is dV/dsigmaN (per unit of normal vol).

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Methods

static AnalyticGreeks of(OptionType type, double forward, double strike, double maturity, double normalVol)

static AnalyticGreeks of(OptionType type, double forward, double strike, double maturity, double rate, double normalVol)

forward: forward price / rate F
strike: strike K
maturity: time to expiry in years T
rate: continuously-compounded discount rate r
normalVol: absolute (normal) volatility sigmaN, price units per sqrt(year)

static double price(OptionType type, double forward, double strike, double maturity, double rate, double normalVol)

Bare price.