Section 25.3

Static Replication

Building payoffs (min, max, |S-K|, binaries) from calls, puts and bonds; binaries as limits of spreads.

Toolkit

Any piecewise-linear payoff can be built exactly from zero-coupon bonds, forward/stock positions, and calls/puts struck at each kink, since adding $(S_T-K)^+$ bends the line upward by one unit of slope at $K$ . Match value and slope at each breakpoint to find the static portfolio; the price is then the same combination of bond price, spot and option premiums, with no dynamic trading needed.

Key identities:

\max(S,K)=K+(S-K)^+,\qquad \min(S,K)=S-(S-K)^+=K-(K-S)^+,

|S-K| = (S-K)^+ + (K-S)^+ = K - S + 2(S-K)^+.

A binary (digital) call pays $1\{S_T>K\}$ ; a binary put pays $1\{S_T<K\}$ , and

\text{binary call}(K)+\text{binary put}(K) = \text{zero-coupon bond}.

Binaries are limits of tight call/put spreads: $1\{S_T>K\}=\lim_{\epsilon\to0}\frac{(S_T-K)^+-(S_T-K-\epsilon)^+}{\epsilon}$ . Any payoff $f(S_T)$ decomposes via

f(S_T)=f(K_0)+f'(K_0)(S_T-K_0)+\int_{-\infty}^{K_0} f''(k)(k-S_T)^+\,dk+\int_{K_0}^{\infty} f''(k)(S_T-k)^+\,dk,

and in particular $(K-S_T)^+=\int_0^K 1\{S_T<k\}\,dk$ , so staircases of binaries sub/super-replicate and bound vanilla prices.

Problems