Valuing Binary Calls from a Skewed Implied Volatility Curve
Summary
The document examines how implied volatility skew affects the valuation of a binary call and addresses conflicting signs in two published formulas. Its central method derives the binary price from the negative strike derivative of the vanilla call price, evaluated using the strike-dependent implied volatility. Applying the chain rule separates the no-skew binary value from a correction involving call vega and the slope of implied volatility across strikes.
The discussion also relates binary valuation to a narrow call spread: long the lower-strike call and short the higher-strike call, with the spread scaled to approximate the strike derivative. The included answers agree on the importance of differentiating the call price with respect to strike, though one explicitly disputes a cited formula on sign grounds. The material is a short forum exchange, not a complete treatment of volatility-surface conventions, numerical differentiation, or market frictions; the sign of a skew adjustment must be interpreted consistently with how skew is defined.
Key ideas
- A binary call price can be obtained from the negative strike derivative of a vanilla call price.
- When implied volatility varies by strike, the chain rule adds a term involving call vega and the volatility slope.
- A narrow call spread can approximate the strike derivative used to value the binary.
- Formula signs depend on the definition and direction of the quoted volatility skew.
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Full text
# Binary Option Valuation With Skew
# Binary Option Valuation With Skew
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.)
## Answer by DeepInTheQF (score 1, accepted)
https://quant.stackexchange.com/a/54995
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 :
$$P_{Binary}=-\frac{dP_{call}(S_0,K,T,\sigma^{imp}(K))}{dK}\\=-\frac{\partial P_{call}(S_0,K,T,\sigma^{imp}(K))}{\partial K} |_{\sigma^{imp}(K)}-\frac{\partial \sigma^{imp}(K)}{\partial K}*\frac{\partial P_{call}(S_0,K,T,\sigma^{imp}(K))}{\partial (\sigma^{imp}(K))} \\ =P_{Binary}^{NoSkew}-Skew*CallVega_{Black-Scholes}$$
## Answer by Arshdeep (score 1)
https://quant.stackexchange.com/a/54946
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.