A wrapped convolution can manufacture option arbitrage

An implied-volatility surface is a map of option prices across strike and maturity. A visually smooth surface can still quote an impossible market: a butterfly can cost less than zero, or a longer-lived option can cost less than its shorter-lived twin. Luo and colleagues avoid policing those failures with a loss penalty. Their decoder emits a probability distribution whose prices are valid by construction.

An option strike is the price at which its holder may buy the asset. Forward normalization divides away carry so the future asset ratio M has expected value one. The normalized call price is then E[(M−z)⁺], where z is strike divided by the forward. Because this payoff is convex, its expectation cannot fall as a martingale gains another independent increment.

AddMart's learned Transformer reads nine sparse quotes and predicts non-negative mixtures for 30 log-price increments. The decoder convolves those increments, exponentiates and normalizes them to mean one, prices calls, then inverts Black's formula to implied volatility. The learning network is flexible; the decoder supplies the financial structure.

Take one log increment on the grid −1, 0, 1 with probabilities 0.25, 0.50, 0.25. After two independent increments, exact convolution places mass on five sums from −2 through 2. The TypeScript core below runs the page; Python and C++ are line-for-line translations of the same index rule.

def convolve(a, b, circular=False):
    size = len(a) if circular else len(a) + len(b) - 1
    out = [0.0] * size
    for i, ai in enumerate(a):
        for j, bj in enumerate(b):
            index = (i + j) % size if circular else i + j
            out[index] += ai * bj
    total = sum(out)
    return [mass / total for mass in out]

The wrap is not harmless compression. It changes which states carry tail probability, then each maturity is renormalized to have mean one. That preserves a valid marginal distribution at each date, but it no longer preserves the coupling that made the dates a martingale.

If L₁,…,Lⱼ are independent log increments, AddMart sets Mⱼ equal to exp(ΣL)/E[exp(ΣL)]. The next multiplicative factor is non-negative, independent of the past, and has mean one. Therefore E[Mⱼ₊₁ | past] = Mⱼ. Jensen's inequality says every convex payoff—including a call— has non-decreasing expected value across that step.

The authors compare exact full-support convolution with two shortcuts: cropping the expanding support and using circular FFT convolution. Reliable-region IV-MAE barely distinguishes them. The arbitrage audit does.

The formal result is global over the modeled strike domain and from time zero through the last maturity bucket. The market data are listed equity and ETF options, but the construction treats liquid out-of-the-money quotes as European-equivalent Black–76 IVs using parity-implied forwards. That limits early-exercise and dividend effects; it does not prove the original American contracts themselves obey a European pricing model.

The main experiment also excludes same-day-expiry options, removes maturities below one day, and defines a reliable accuracy region that excludes contracts with fewer than 14 days remaining. The nine reference quotes use a structured three-maturity by three-delta layout. Extra references help but not monotonically, and zero-shot transfer deteriorates when assets have different volatility levels or smile shapes.

First, audit violations by strike and maturity rather than averaging them: the toy shows why aliases can hide in a wing. Second, report the runtime and memory trade-off of exact support at the same batch size. Third, stress the quote layout, same-day expiries, and dividend-heavy names. Finally, compare tail functionals against later realized outcomes—not only a smoothed option-implied reference.