Static Replication of Option Payoffs

Static replication represents a terminal payoff using cash, a forward, and a strip of vanilla options held to maturity. It is powerful because market option prices reveal risk-neutral distributional information without requiring a full dynamic model. It is also conditional: the clean identity assumes a continuum of strikes, frictionless trading, and a payoff defined at one terminal date. Real chains are sparse, tails are unquoted, and path dependence cannot generally be removed with a single static strip.

The payoff decomposition

For a twice-differentiable payoff g(S_T) and reference strike K0,

g(S_T) = g(K0) + g'(K0)(S_T-K0)
       + integral_0^K0 g''(K)(K-S_T)+ dK
       + integral_K0^infinity g''(K)(S_T-K)+ dK

The first two terms are cash and a forward. The lower integral is puts and the upper integral is calls. The second derivative determines option weights: convex payoffs require long options, while concave payoffs require short options.

Payoff	g''(K) implication	Replication intuition
Call	point mass at strike	one call
Digital	derivative of point mass	tight call spread
Log contract	inverse-square weighting	many OTM options
Capped payoff	local positive/negative curvature	vertical-spread strip

Variance as the central example

The log contract yields the variance swap replication formula. Under continuous diffusion and proper forward normalization, fair variance is proportional to OTM option prices weighted by 1/K^2:

K_var = (2*exp(rT)/T) *
 [integral_0^F P(K)/K^2 dK + integral_F^infinity C(K)/K^2 dK]

The left tail receives enormous weight. This is why a variance strike calculation is often more sensitive to extrapolated puts than to the many liquid ATM quotes. It also explains why a numerical integration can look precise while carrying material model uncertainty.

Discrete strike implementation

Listed options turn the integral into a weighted sum. Use true strike spacing, OTM option selection around the forward, discounting, and an explicit tail rule.

def static_strip_weights(strikes, curvature, widths):
    # weights are for option prices at the same maturity
    return 0.5 * curvature(strikes) * widths

def trapezoid_widths(k):
    widths = k * 0.0
    widths[1:-1] = (k[2:] - k[:-2]) / 2
    widths[0], widths[-1] = k[1]-k[0], k[-1]-k[-2]
    return widths

The code does not decide extrapolation. That decision is economic: zero price beyond the last quote, a parametric wing, and a conservative bid-side bound produce different hedge costs. Report the sensitivity rather than hiding it in a default.

Static is not always dynamic

Put-call parity lets cash, a call, and a put synthesize a forward at maturity. But barriers, American exercise, dividends, and path-dependent payoffs normally require dynamic hedging or additional instruments. Even a European payoff can have dynamic hedge risk before maturity: a static terminal replication may exist, while its mark-to-market is sensitive to an evolving surface and funding.

For example, a digital option can be approximated by a narrow call spread:

digital(K) ≈ [C(K-epsilon) - C(K+epsilon)] / (2*epsilon)

Reducing epsilon improves terminal payoff accuracy but makes the package sensitive to bid/ask, strike granularity, and pin risk. “Exact” replication exists only when the market supplies arbitrarily close, liquid strikes.

Pricing versus hedging

Replication establishes an arbitrage relation only if each component is tradable at the assumed price. Use executable bids to value a short liability hedge and asks for a long hedge. Include commissions, margin, borrow, funding, and roll costs. If the strip needs periodic rolling, its realized result is no longer static.

The choice of forward matters as much as option data. Equity forwards require dividends and borrow; index forwards require the correct futures basis; FX requires both rates. A wrong forward selects the wrong OTM option around K=F and distorts every integrated weight.

Key takeaways

• Smooth terminal payoffs decompose into cash, forwards, puts, and calls weighted by curvature.
• The variance-swap identity exposes severe inverse-square left-tail sensitivity.
• Discrete strikes and tail extrapolation are first-order practical assumptions.
• Static terminal replication does not eliminate pre-expiry surface, funding, or liquidity risk.
• Treat executable quotes and forward conventions as part of the replication, not data cleanup.