18. Option Pricing without Perfect Replication

18.1 On the Edges of Arbitrage

If options really were redundant (perfectly replicable), they would not be traded as separate assets. In reality many situations break the law-of-one-price argument of Black-Scholes: jumps (Poisson jumps), stochastic interest rates/volatility (no security to hedge them), non-traded underlyings. All introduce basis risk. But we do not want to retreat to the consumption model or other "absolute" methods — we still take the prices of many assets (especially the hedging assets) as given. The idea: form an "approximate hedge" portfolio close to the focus payoff, reducing the uncertainty to pricing the residual; and since the residual is small, we can say a lot about the option price under much weaker discount-factor restrictions than absolute models. Many authors simply assume a market price of risk — but this leaves the questions: how sensitive are the results to that assumption, and what are reasonable values? The good-deal bounds systematically search over all assignments of the residual's market price of risk, constrain the total price of risk to a reasonable value, and impose no-arbitrage, to find upper and lower price bounds.

18.2 One-Period Good-Deal Bounds

This is a standard minimization with two inequality constraints, solved by trying constraint combinations in order of ease. (1) Volatility binds, positivity slack. Orthogonally decompose the payoff \(x^c=\hat x^c+w\), with \(\hat x^c=\operatorname{proj}(x^c|X)\) the approximate hedge (price known) and \(w\) the residual. As in Figure 18.1: all \(m\) pricing \(x\) lie in the plane through \(x^*\), and sweeping along it gives any price to the residual (the arbitrage bounds are the \(m>0\) triangular region); the second moment defines distance, so the \(m\) with \(E(m^2)\le A^2\) lie in a sphere about the origin (the circle). To max/minimize the residual price \(E(mw)\), \(m\) should point toward/away from \(w\) as much as possible; any movement \(\varepsilon\) orthogonal to \(w\) only raises volatility without changing the residual price. So the lower-bound discount factor is \(\underline m=x^*-\underline v\,w\), \(\underline v=\sqrt{(A^2-E(x^{*2}))/E(w^2)}\), with bound \(\underline C=E(x^*x^c)-\underline v\,E(w^2)\) (the first term the approximate-hedge value, the second the lowest residual price consistent with the volatility constraint). Check \(\underline m\ge0\): if so, this is the good-deal bound.

Figure 18.1　Construction of the discount factor for the one-period good-deal bound when positivity is slack. The \(m\) pricing the basis assets lie in the plane through \(x^*\); among those inside the \(E(m^2)\le A^2\) circle, the ones pointing most toward/away from the residual \(w\) give the upper/lower bound \(x^*\pm\underline v\,w\).

(2) Positivity binds, volatility slack. These are the classic arbitrage bounds (Ch 17), generally a linear program. One must still check whether the volatility constraint can be met at the arbitrage bound. (3) Both bind. With Lagrange multipliers, the discount factor is a truncated linear combination of the payoffs \(m=\max(-(x^c+\lambda'x)/\delta,\,0)\) (in finance terms, a call option with zero strike). Plugging back to solve for \(\lambda,\delta\) requires a nonlinear system (numerically hard); Hansen-Heaton-Luttmer (1995) recast it as an easier maximization (interchanging min/max).

Figure 18.2　Good-deal option price bounds vs. stock price. Three months to expiration, strike \(K=100\), no intermediate trading, discount-factor volatility bound \(h=1.0\) (twice the market Sharpe ratio). The good-deal upper bound is far tighter than the arbitrage upper bound (\(C\le S\) is too high to fit); the lower bound coincides with the arbitrage lower bound outside ~90–110 (positivity binds) and is tighter inside (both bind). The benchmark is the arbitrage bounds, not Black-Scholes (since there is no intermediate trading).

18.3 Multiple Periods and Continuous Time

The essence of option pricing is (even imperfect) dynamic hedging, so good-deal bounds must work in multiple periods. The key: the bounds are recursive. Today's bound is the lowest price of tomorrow's bound — the two-period \(\min E_0(m_1m_2x^c_2)\) is equivalent to first solving \(\underline C_1=\min E_1(m_2x^c_2)\), then \(\underline C_0=\min E_0(m_1\underline C_1)\) (since one must minimize \(E_1(m_2x^c)\) in each time-1 state). This recursion holds only with \(m>0\) — if \(m_1<0\) were possible, you might want to maximize in some states.

Basis risk and real options. Price a European call on a non-traded event \(V\) correlated with a traded asset \(S\) (real options, non-financial options). Model \(dS/S=\mu_S dt+\sigma_S dz\), \(dV/V=\mu_V dt+\sigma_{Vz}dz+\sigma_{Vw}dw\); the \(dw\) risk cannot be hedged by \(S\), so its market price of risk matters. The discount factor is \(\frac{d\Lambda}{\Lambda}=\frac{d\Lambda^*}{\Lambda^*}\pm\sqrt{A^2-h_S^2}\,dw\) (a loading on the orthogonal shock \(dw\) just enough to meet the volatility constraint; \(\pm\) gives the upper/lower bound). \(S,V,\Lambda\) are jointly lognormal, and the integral gives a Black-Scholes formula with an extra \(\eta\) term \(C=V_0e^{\eta T}\Phi(d+\tfrac12\sigma_V\sqrt T)-Ke^{-rT}\Phi(d-\tfrac12\sigma_V\sqrt T)\): the lower the correlation \(\rho\) and the larger the volatility bound \(A\) relative to the asset Sharpe ratios, the wider the bounds.

Market-price-of-risk language. Continuous-time problems are often stated via the "market price of risk" \(\lambda\) (the instantaneous Sharpe ratio an asset must earn if it loads on a shock) rather than the discount factor: \(E_t(dP/P)-r^f dt=\lambda\sigma\). In this language, the stock's price of risk \(h_S\) is observable and irrelevant when pricing by arbitrage (so it is absent from the BS formula); the problem reduces to choosing the market price of \(dw\) risk (not observable from a traded asset) to min/maximize the option price, subject to a total price \(\sqrt{h_S^2+\lambda^2}\le A\). Systematic continuous-time treatment: one again obtains a differential characterization — orthogonalize the discount factor as \(\frac{d\Lambda}{\Lambda}=\frac{d\Lambda^*}{\Lambda^*}-v\,dw\), where \(v\) is the vector of market prices of the \(dw\) shocks, constrained by \(vv'\le A^2-\tilde\mu_S'\Sigma_S^{-1}\tilde\mu_S\); minimizing means "at each instant, assign market prices of risk to the \(dw\) shocks to minimize the focus value." One can derive a partial differential equation for the good-deal bound (like the BS PDE, solved back from expiration); only when there is a single \(dw\) shock can \(v\) be determined directly from the volatility constraint and the bound found by forward integration without a PDE (the non-traded-underlying example above).

18.4 Other Restrictions and Approaches

Summary

In practice perfect replication often fails (jumps, stochastic volatility/rates, non-traded underlyings), leaving basis risk between the option and its hedge. The good-deal bounds address this via the discount-factor approach: find option-price bounds over all \(m\) that price the basis assets, are positive, and have limited volatility (Sharpe ratio) — far tighter than arbitrage bounds. The key is to impose positivity and volatility together (either alone is insufficient); the bounds are recursive and so extend to multiple periods/continuous time (a PDE or a Black-Scholes formula with an extra \(\eta\) term). They are the dual of the HJ bound with positivity and can be intersected with monotonicity, gain-loss, and other restrictions — the power of "stacking discount-factor restrictions" over portfolio methods. The next chapter applies this machinery (continuous time, chaining, prices of risk) to the term structure of interest rates.

Problems

1. Prove (18.32): \(\max_{R^e}\dfrac{[R^e]_+}{[R^e]_-}=\min_{m:0=E(mR^e)}\dfrac{\sup(m)}{\inf(m)}\). Start with a finite state space.
2. Binomial model. A stock at \(S\) will rise to \(uS\) with probability \(\pi_u\) or fall to \(dS\) with probability \(\pi_d\), paying no dividends, with a constant gross rate \(R^f\). (a) Find a discount factor pricing the stock and bond (its value in each state); (b) use it to price a call one step before expiration, as an expectation under risk-neutral probabilities; (c) do the same two steps before; (d) rederive via the Cox-Ross-Rubinstein (1979) hedge-portfolio method (choosing stock-bond weights to synthesize the option exactly).