Binary Options: Black-Scholes Valuation of Binary Options

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,

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

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,

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

Asset-or-nothing put

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

$P = Se^{-qT}Phi(-d_1). ,$ Services

Foreign Exchange

If we denote by the FOR/DOM exchange rate by S (i.e. 1 unit of foreign currency is worth S units of domestic currency) we can observe that if the spot at maturity is above or below the strike then paying out 1 unit of the domestic currency is exactly like a cash-or nothing call and put respectively.

Similarly, if the spot at maturity is above or below the strike then paying out 1 unit of the foreign currency is exactly like an asset-or nothing call and put respectively. Hence now if we take r_FOR , the foreign interest rate, r_DOM , the domestic interest rate, and the rest as above, then we will be having the following results.

In case of a digital call (this is a call FOR/put DOM) by which one unit of the domestic currency is paid out then we get as present value,

$C = e^{-r_{DOM} T}Phi(d_2) ,$

In case of a digital put (this is a put FOR/call DOM) by which one unit of the domestic currency is paid out then we get as present value,

$P = e^{-r_{DOM}T}Phi(-d_2) ,$

While in case of a digital call (this is a call FOR/put DOM) by which one unit of the foreign currency is paid out then we get as present value,

$C = Se^{-r_{FOR} T}Phi(d_1) ,$

and in case of a digital put (this is a put FOR/call DOM) by which one unit of the foreign currency is paid out then we get as present value,

$P = Se^{-r_{FOR}T}Phi(-d_1) ,$

Skew

In the standard Black-Scholes model, the premium of the binary option can be interpreted by a person in the risk-neutral world as the expected value = probability of being in-the-money * unit, discounted to the present value.

A more sophisticated analysis based on call spreads can be used in order to take volatility skew into account. binaryoption

At long expirations, a binary call option is identical to a tight call spread using two vanilla options. The value of a binary cash-or-nothing option, C, can be model at strike K, as an extremely small tight spread, where C_v is a vanilla European call:

$C = lim_{epsilon to 0} frac{C_v(K-epsilon) - C_v(K)}{epsilon}$

Thus, we can say that the value of a binary call is the negative of the derivative of the price of a vanilla call:

$C = -frac{dC_v}{dK}$

When volatility skew is taken into account by someone, σ is a function of K:

$C = -frac{dC_v(K,sigma(K))}{dK} = -frac{partial C_v}{partial K} - frac{partial C_v}{partial sigma} frac{partial sigma}{partial K}$

The the premium of the binary option is equal to first term ignoring skew:

$-frac{partial C_v}{partial K} = -frac{partial (SPhi(d_1) - Ke^{-rT}Phi(d_2))}{partial K} = e^{-rT}Phi(d_2) = C_{noskew}$

$frac{partial C_v}{partial sigma}$ is known as the Vega of the vanilla call; $frac{partial sigma}{partial K}$ is sometimes referred to as the “skew slope” or just “skew”. As the skew is typically negative, so the value of a binary call is higher when taking skew into account.

C = C_noskew − Vega_v * Skew

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