In searching for methods of valuation of Binary options with skew, I have found two formulas which are at odds. I cannot find any other references to this valuation formula. Should Vega be positive or negative?:
https://en.wikipedia.org/wiki/Binary_option#Skew
$C = C_{noskew} - Vega_{v} * Skew$
https://www.cboe.com/institutional/pdf/listedbinaryoptions.pdf
$c = Binary_{No-Skew} + Vega_{Black-Scholes} * Skew $
(Comedically, I don't know which to trust more; Lehman or Wikipedia.)
The Price of a Binary Call Option is given by : $$P_{Binary}=-\frac{dP_{call}(S_0,K,T,\sigma^{imp}(K))}{dK}$$ Where $\sigma^{imp}(K)$ is the implied Black-scholes volatility. In fact, since the real market corresponds to a smiled volatility, the correct Black-scholes volatility to be used depends on the option strike K.
Hence we obtain that :
$$P_{Binary}=-\frac{dP_{call}(S_0,K,T,\sigma^{imp}(K))}{dK}\\=-\frac{\partial P_{call}(S_0,K,T,\sigma^{imp}(K))}{\partial K} |_{\sigma^{imp}(K)}-\frac{\partial \sigma^{imp}(K)}{\partial K}*\frac{\partial P_{call}(S_0,K,T,\sigma^{imp}(K))}{\partial (\sigma^{imp}(K))} \\ =P_{Binary}^{NoSkew}-Skew*CallVega_{Black-Scholes}$$
In the second link, the 'no skew' call price is negative - call prices actually decrease as strike increases. So it is clearly absurd. I'd go with wikipedia.
If I need to be a bit mathematical, the first derivative of the call option payoff w.r.t strike is exactly the NEGATIVE OF the random variable that represents the payoff of the binary - this should be obvious once you write the at expiry payoff (not today's price) of the call and differentiate w.r.t strike. Go to the T forward measure, take expectations and you find that you can price (to the extent that your first derivative is accurate) the binary as a call spread, with short the higher strike and long the lower strike.
