Exotics and structured products

4 lessons · 4 equations. Each lesson below gives its formulas and key rules; open the lesson for the full explanation.

Digitals and barriers: why the hedging is the hard part

Digital call from a call spread

D(K)\approx\frac{C(K-\varepsilon)-C(K+\varepsilon)}{2\varepsilon}

A narrow call spread approximates the digital; under flat Black–Scholes volatility its price is $e^{-rT}\Phi(d_2)$ .

Remember

A digital is $-\partial C/\partial K = e^{-rT}\Phi(d_2)$ , and is replicated by a tight call spread.

Path-dependent exotics: Asians, lookbacks and autocallables

The geometric Asian has a closed form

\ln G \sim N\!\left(\ln S_0 + \tfrac{1}{2}\big(r - q - \tfrac{1}{2}\sigma^2\big)T,\ \tfrac{1}{3}\sigma^2T\right)

With continuous averaging under geometric Brownian motion, the log of the geometric average is normal with a third of the variance and half the drift of the terminal log price. So a geometric Asian is a Black-type option on an adjusted forward with volatility $\sigma/\sqrt3$ — and the natural control variate for pricing the arithmetic one.

Remember

Asian < vanilla < lookback in value, because of variance reduction and selection.

Multi-asset exotics: baskets, best-of, spreads and quantos

Basket and spread volatility

\sigma_{\text{basket}}^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho\sigma_1\sigma_2

A basket adds the cross term; a spread subtracts it. Setting $w_2 = -1$ turns the first into the second, which is the whole distinction.

Remember

Aggregating structures are long correlation; selecting structures are short it.

Hedging exotics: static replication, and reserving for what you cannot hedge

Put–call symmetry

P(K) = \frac{K}{H}\,C\!\left(\frac{H^2}{K}\right)\ \text{ when } S = H,\ r = q = 0

Under zero carry and a symmetric volatility smile, a put struck at $K$ and a scaled call struck at the "reflected" strike $H^2/K$ are worth the same whenever the stock is at $H$ . It is the key to static hedges of barrier options: hold the calls while above the barrier, and swap them for the put, at no cost, if it is touched.

Remember

Prefer a static hedge; it needs no rebalancing and survives gaps.