Christmas QuestSticky strike Black-Scholes only with Transcendental and imaginary numbers?You use the Black-Scholes model. In practice we naturally have a volatility smile. Assume sticky strike vol smile (fixed vol for each strike), further assume different maturities are priced with different vols. In other words sticky strike vol surface !!!! Black-Scholes with different vol for each strike is naturally not consistent, but still "close enough" to what people do in practice(sticky delta or whatever)....good enough for a Christmas quest.For the special case when the strike is = forward price (at-the-money forward)A) Find a Super Super simple expression for the Black-Scholes formula ONLY using Transcendental Numbers (hint pi and e) and imaginary numbers (hint i). The asset price, the strike price and the volatility (sigma) as well as any Real numbers or any functions are NOT allowed to enter the option formula. Also show the volatility function. For this you can in addition to Transcendental and imaginary numbers use the strike X, and time to maturity T + the square root function. (forget about arbitrage free for a while, only requierment same output as Black-Scholes)B) Even if the option formula you get is "perfectly okay?": gives same result as the Black-Scholes as long as same input parameters (in according to vol function that leads to simplified solution), the volatility function you get will lead to a special form of arbitrage opportunities, how can you spot this from the option prices (very simple) and how from the vol output (not very simple, but quite simple).The first to solve A will get the English edition of my book, the first one to solve B will get the Chinese edition of my book.!! The questions are probably to simple for a math professor. Omar have you got the result? Possibly several solutions, but only one as neat/compact as the one I have in mind.(PS: Hope I got it right myself, and that my question actually has the solution I have in mind)Marry Christmas Guys and Girls (Friends? If you want a friend buy a dog)

"Merry Christmas Guys and Girls (Friends? If you want a friend buy a dog)"Hedgers buy two dogs.

Hedgers buy two dogs. >>Of uncorrelated breeds.

Let me make sure I understand the question. The Black-Scholes price for an at-the-forward-money call option is:S * (2*pi)^-.5 * Integral from - sigma*T^.5/2 to + sigma*T^.5/2 of exp(-x^2/2)dxYou want choose sigma as a function of S, T and any transcendental or imaginary (but not rational) numbers; with square root as an allowed function; such that the call price above can be written as a simple expression of transcendental and imaginary numbers; with no functions or parameters.For example, if we defined sigma as:2 * t^-.5 * Inverse Cumulative Normal Function [0.5 + (pi/2S)^.5]then the call option price would be 1. This is not an allowed solution because I have used rational numbers (2, 0.5) and a function other than square root (inverse cumulative normal) in the definition of sigma; and because the answer (1) is a rational number. Is this all correct?

You are getting very close Aaron, but no rational numbers...

Collector, I guess it's time to explain...P