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Cash-or-nothing put

Binary Options: Black-Scholes Valuation of Binary Options

by R. MAK. on August 15, 2009 · 0 comments

in Currency Trade,Forex Basics,Forex Market,Forex trading,Interesting Facts,Trading,currency

In the Black-Scholes model, we can find the price of the option by the formulas below. In these, stock price is denoted by S, strike price is denoted by K, time to maturity is denoted by T , dividend rate is denoted by q, risk-free interest rate is denoted by r and volatility is denoted by σ . the cumulative distribution function of the normal distribution is denoted by Φ,

$\Phi(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^x e^{-\frac{1}{2} z^2} dz.$

and,

$d_1 = \frac{\ln\frac{S}{K} + (r-q+\sigma^{2}/2)T}{\sigma\sqrt{T}},\,d_2 = d_1-\sigma\sqrt{T}. \,$

Black-Scholes Valuation

Cash-or-nothing call

If the spot is above the strike at maturity then this pays out one unit of cash. Now its value is given by,

$C = e^{-rT}\Phi(d_2). \,$

Cash-or-nothing put

If the spot is below the strike at maturity then this pays out one unit of cash. Now its value is given by,

$P = e^{-rT}\Phi(-d_2). \,$

Asset-or-nothing call

If the spot is above the strike at maturity then this pays out one unit of asset. Now its value is given by,

$C = Se^{-qT}\Phi(d_1). \,$

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Tagged as: Asset-or-nothing call, Asset-or-nothing put, Binary Options, Black-Scholes model, Black-Scholes Valuation of Binary Options, Cash-or-nothing call, Cash-or-nothing put, digital call, foreign currency, foreign exchange, foreign exchange market, foreign interest rate, forex, Forex trading, FX market, Risk-Free Interest Rate, Skew, skew slope, stock price, vanilla European call, vanilla options, Vega of the vanilla call