Path-dependent exotics: Asians, lookbacks and autocallables

EXO · Chapter 213 min readAsked at Optiver, SIG, Citadel Securities, Akuna

Assumes Digitals and barriers: why the hedging is the hard part.

After this lesson you should be able to

  • Rank Asian, vanilla and lookback options by value and say why.
  • Explain how discrete monitoring changes a barrier price.
  • Describe what a desk selling autocallables ends up holding.

Path dependence means the payoff depends on how the underlying got somewhere, not just where it finished. That single change rules out lattices, makes the hedge dynamic in a new way, and produces a family of products whose risks are concentrated at specific prices and dates.

Proposition 2.1

Asian, vanilla, lookback — in that order

An Asian option settles on the average, a vanilla on the final price, a lookback on the best price achieved. Averaging reduces variance, so the Asian is cheapest; the lookback selects the most favourable point of the path, so it is dearest. Being able to state that ordering and its reason is most of what an interviewer wants here.

Holds when

  • Asian volatility is roughly σ/3\sigma/\sqrt{3} for continuous averaging over the life — about 58%58\% of the spot volatility.
  • Averaging also makes settlement harder to manipulate, which is why commodity and FX contracts use it.
  • Geometric-average Asians have a closed form; arithmetic ones do not, and are priced by simulation or approximation.

Example 2.2

A one-year at-the-money vanilla call on a $100\$100 stock at 20%20\% volatility is worth about $8\$8. Roughly what is the equivalent Asian worth?

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Worked solution

  1. Formula
    CATM≈0.4 SσeffT,σeff≈σ3C_{\text{ATM}} \approx 0.4\,S\sigma_{\text{eff}}\sqrt{T}, \qquad \sigma_{\text{eff}} \approx \frac{\sigma}{\sqrt{3}}
  2. Substitute
    σeff=0.201.732=0.1155\sigma_{\text{eff}} = \frac{0.20}{1.732} = 0.1155
  3. Solve
    C≈0.4×100×0.1155×1C \approx 0.4 \times 100 \times 0.1155 \times 1
  4. ≈4.62\approx 4.62
  5. Answer
    about $4.6, a little over half the vanilla\text{about } \$4.6, \text{ a little over half the vanilla}

Sanity check. The ratio is 1/3≈0.581/\sqrt{3} \approx 0.58, which is the whole content of the answer: averaging cuts the effective volatility by that factor, and an at-the-money option is close to linear in volatility.

Proposition 2.3

Barriers and monitoring

A continuously monitored barrier can be breached at any instant; one monitored on daily closes can be crossed intraday and recovered. So discrete monitoring makes a knock-out *more* valuable and a knock-in *less*, and the size of the effect scales with σΔt\sigma\sqrt{\Delta t} — there is a standard correction that shifts the barrier away from the spot by about 0.58σΔt0.58\sigma\sqrt{\Delta t}.

Holds when

  • In–out parity holds for either convention, as long as both legs use the same one.
  • The correction constant is −ζ(1/2)/2π≈0.5826-\zeta(1/2)/\sqrt{2\pi} \approx 0.5826 — close to the 1/3≈0.5771/\sqrt{3} \approx 0.577 above, but a different number arising for unrelated reasons.

Why barriers are hard and Asians are easy. Both are path-dependent, and they are opposite in difficulty. Averaging *smooths*: the payoff depends gently on a hundred observations, so no single day matters much and the Greeks are tame. A barrier *concentrates*: the entire value of the option turns on whether one number was ever touched, so the delta near the barrier is enormous and flips sign as it is approached. Path dependence is not one thing — what matters is whether the path enters the payoff smoothly or through a discontinuity.

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