Hedging in Incomplete Markets¶
When markets are incomplete --- due to unhedgeable risk factors, transaction costs, or trading constraints --- perfect replication is impossible. This section surveys the principal approaches to hedging under incompleteness: superreplication (eliminate all risk at maximum cost), quantile hedging (replicate with high probability), and shortfall risk minimization (control the severity of hedging failures). These methods span a spectrum from conservative to risk-tolerant, each offering a different resolution of the price-risk tradeoff.
Sources of Market Incompleteness¶
Before developing the hedging strategies, it is useful to catalog the common sources of incompleteness:
| Source | Example | Unhedgeable risk |
|---|---|---|
| Untraded risk factors | Stochastic volatility, jump intensity | Volatility, jump risk |
| Basis risk | Hedging oil with gasoline futures | Residual correlation gap |
| Transaction costs | Bid-ask spreads, market impact | Rebalancing friction |
| Trading constraints | Short-selling restrictions, margin | Constrained strategies |
| Discrete trading | Finite rebalancing dates | Discretization error |
Each source creates a gap between the set of attainable payoffs and the full space \(L^2\), leading to non-unique prices and imperfect hedges.
Superreplication¶
Definition and Theorem¶
Superreplication is the most conservative approach: find the cheapest self-financing strategy that dominates the claim in every scenario.
Definition: Superreplication Price
The superreplication price of a claim \(H\) is:
Theorem (Superreplication Duality). Under standard no-arbitrage conditions:
where \(\mathcal{M}\) is the set of equivalent martingale measures and \(\tilde{H} = e^{-rT}H\) is the discounted claim.
Proof Sketch
The inequality \(\pi^{\sup} \geq \sup_{\mathbb{Q}} \mathbb{E}^{\mathbb{Q}}[\tilde{H}]\) follows because any superreplicating strategy satisfies \(c + G_T(\xi) \geq H\), so \(c \geq \mathbb{E}^{\mathbb{Q}}[\tilde{H}]\) for every \(\mathbb{Q} \in \mathcal{M}\) (since \(\mathbb{E}^{\mathbb{Q}}[G_T(\xi)] = 0\) under any martingale measure). The reverse inequality uses Hahn-Banach separation: if \(c < \sup_{\mathbb{Q}} \mathbb{E}^{\mathbb{Q}}[\tilde{H}]\), then no superreplicating strategy exists at cost \(c\), which can be shown by constructing a separating martingale measure. \(\square\)
The Superreplication Problem¶
In practice, the superreplication price is often too expensive. For example, in a stochastic volatility model:
- A European call's superreplication price equals the price under the volatility that maximizes the option value (the worst-case vol).
- If \(\sigma \in [\sigma_{\min}, \sigma_{\max}]\), the superreplication price of a call is \(\text{BS}(S, K, T, \sigma_{\max})\).
This can be significantly higher than any reasonable market price, making superreplication impractical for most applications.
Cost of Safety
Superreplication eliminates all risk but at the cost of a potentially large premium above the "fair" price. For claims with high sensitivity to unhedgeable factors, the superreplication price may exceed the claim's intrinsic value, making it economically unviable.
Quantile Hedging¶
Motivation¶
Instead of replicating in all scenarios, quantile hedging (Follmer and Leukert, 1999) seeks the cheapest strategy that replicates with a specified probability \(1 - \varepsilon\).
Formal Problem¶
Definition: Quantile Hedging
Given a budget \(c < \pi^{\sup}(H)\) and a confidence level \(1 - \varepsilon\), the quantile hedging problem is:
Equivalently: find the cheapest strategy that succeeds with probability at least \(1 - \varepsilon\):
Solution Structure¶
Theorem (Follmer-Leukert). The quantile hedging problem reduces to superreplicating a modified claim \(H \cdot \mathbf{1}_A\) where \(A \subset \Omega\) is a suitably chosen "success set":
The optimal success set \(A^*\) is determined by the Neyman-Pearson lemma applied to the likelihood ratios \(d\mathbb{Q}^*/d\mathbb{P}\) of the least favorable martingale measure.
Interpretation¶
Quantile hedging sacrifices replication on a set of "bad" scenarios with total probability \(\varepsilon\). On the success set \(A^*\), the hedge perfectly replicates the claim. On the failure set \(A^{*c}\), the hedge may produce a loss.
The Neyman-Pearson structure implies that the scenarios sacrificed are those where the cost of hedging is highest relative to their probability --- the hedge "gives up" on the most expensive-to-hedge paths first.
Example¶
Consider a European call in a stochastic volatility model with budget equal to 90% of the superreplication price. Quantile hedging:
- Perfectly replicates the call on paths where volatility stays moderate.
- Fails to replicate on paths where volatility reaches extreme levels (the most expensive scenarios).
- Achieves, say, \(\mathbb{P}(\text{success}) = 0.97\) --- the hedge works in 97% of scenarios.
Shortfall Risk Minimization¶
Beyond Success Probability¶
Quantile hedging controls the probability of failure but ignores the severity of failure. A strategy that fails with probability 3% but loses $1 million when it fails is very different from one that fails with probability 3% but loses only $100.
Shortfall risk minimization addresses this by incorporating a loss function that penalizes the magnitude of the shortfall.
Formal Problem¶
Definition: Shortfall Risk Minimization
Given a loss function \(\ell : [0, \infty) \to [0, \infty)\) (convex, increasing, \(\ell(0) = 0\)), the shortfall risk of a strategy \((c, \xi)\) is:
The shortfall risk minimization problem is:
Common Loss Functions¶
| Loss function | \(\ell(x)\) | Interpretation |
|---|---|---|
| Indicator | \(\mathbf{1}_{\{x > 0\}}\) | Quantile hedging (probability of loss) |
| Linear | \(x\) | Expected shortfall |
| Quadratic | \(x^2\) | Mean-squared shortfall |
| Power | \(x^p\), \(p > 1\) | Penalizes large shortfalls more heavily |
Solution Structure¶
Theorem (Follmer-Leukert, 2000). For convex loss functions, the optimal shortfall-minimizing strategy superreplicates a modified claim:
where \(\varphi^* \in [0, 1]\) is a "randomized test" determined by the Neyman-Pearson lemma applied to the modified problem. The function \(\varphi^*\) assigns weights to scenarios based on the likelihood ratio and the loss function.
For the linear loss \(\ell(x) = x\) (expected shortfall):
The optimal strategy minimizes the expected amount by which the hedge falls short. This is a natural criterion for risk management, as it directly controls the average loss in failure scenarios.
Comparison of Approaches¶
The Hedging Spectrum¶
The various approaches can be ordered by their conservatism:
| Approach | Cost | Risk tolerance | Loss control |
|---|---|---|---|
| Superreplication | Highest | None (zero risk) | Perfect |
| Quantile hedging | High | Small probability of total failure | Probability only |
| Expected shortfall min | Moderate | Accepts some loss | Average loss |
| Mean-variance | Moderate | Symmetric \(L^2\) loss | Variance |
| Utility-based | Depends on \(\gamma\) | Preference-based | Utility-weighted |
Price Ordering¶
For a given claim \(H\):
The superreplication price is the upper bound; the infimum over martingale measures is the lower bound. All other prices lie in between, depending on the risk criterion.
Practical Considerations¶
Choosing an Approach¶
The choice depends on the context:
- Regulatory capital: Superreplication or quantile hedging at high confidence (\(\varepsilon = 0.01\)) may be required.
- Trading desk P&L: Mean-variance or shortfall minimization balances cost and risk.
- Insurance/pension: Utility-based approaches with appropriate risk aversion.
- Market making: Delta hedging with transaction cost optimization (practical rebalancing).
Computational Methods¶
| Approach | Computation | Difficulty |
|---|---|---|
| Superreplication | Solve worst-case PDE or linear program | Moderate |
| Quantile hedging | Neyman-Pearson + superreplication of modified claim | Hard |
| Shortfall minimization | Modified Neyman-Pearson + optimal transport | Hard |
| Mean-variance | Regression / backward induction | Moderate |
| Utility-based | HJB PDE or Monte Carlo | Moderate to hard |
Worked Example: Stochastic Volatility¶
Consider a European call (\(K = 100\), \(T = 1\)) under a model where volatility can take values \(\sigma \in [0.15, 0.35]\).
Prices under different criteria (with \(S_0 = 100\), \(r = 0.03\)):
| Approach | Price | Comment |
|---|---|---|
| Superreplication | $14.80 | \(\text{BS}(\sigma = 0.35)\) |
| Sub-replication | $6.20 | \(\text{BS}(\sigma = 0.15)\) |
| Quantile hedge (\(\varepsilon = 0.05\)) | $12.50 | Gives up 5% worst paths |
| Expected shortfall min | $11.00 | Controls average loss |
| Mean-variance | $10.30 | Under variance-optimal \(\mathbb{Q}^*\) |
| Utility indifference (\(\gamma = 1\)) | $10.80 | Exponential utility |
The $8.60 gap between superreplication and sub-replication reflects the degree of incompleteness. All hedging approaches produce prices within this interval, with more conservative approaches closer to the upper bound.
Hedging strategies:
- Superreplication: Hedge with \(\Delta(\sigma_{\max}) = \Delta(0.35)\), always over-hedging.
- Mean-variance: Hedge with \(\Delta(\hat{\sigma})\) where \(\hat{\sigma}\) is an intermediate volatility, accepting residual risk.
- Quantile hedging: Hedge with the superreplicating delta on "normal" paths; no hedge on extreme paths.
The Risk-Cost Frontier¶
The fundamental message of hedging in incomplete markets is that there exists a risk-cost frontier: one cannot simultaneously minimize both cost and risk. Each approach represents a different point on this frontier:
- Moving toward zero risk (superreplication) increases cost.
- Reducing cost below the superreplication price necessarily introduces some form of residual risk.
- The optimal point depends on the agent's risk tolerance, regulatory requirements, and market conditions.
The various approaches studied --- mean-variance, local risk minimization, utility-based, quantile, and shortfall --- provide a rich toolkit for navigating this frontier.
Summary¶
| Approach | Criterion | Key result |
|---|---|---|
| Superreplication | \(c + G_T(\xi) \geq H\) a.s. | \(\pi^{\sup} = \sup_{\mathbb{Q}} \mathbb{E}^{\mathbb{Q}}[\tilde{H}]\) |
| Quantile hedging | \(\max \mathbb{P}(\text{success})\) for given budget | Neyman-Pearson on modified claim |
| Shortfall risk min | \(\min \mathbb{E}[\ell((H - c - G_T)^+)]\) | Generalized Neyman-Pearson |
| Mean-variance | \(\min \mathbb{E}[(H - c - G_T)^2]\) | \(L^2\) projection / FS decomposition |
| Utility-based | \(\max \mathbb{E}[U(W_T)]\) | Entropy / HJB duality |
| Price ordering | \(\pi^{\sup} \geq \pi_\varepsilon \geq p_{\text{indiff}} \geq \inf \mathbb{E}^{\mathbb{Q}}[\tilde{H}]\) | |
| Key insight | Incompleteness creates a risk-cost frontier; no approach eliminates the tradeoff |
Exercises¶
Exercise 1. The superreplication price of a European call under volatility uncertainty \(\sigma \in [0.15, 0.35]\) is \(\text{BS}(S, K, T, \sigma_{\max})\). For \(S = K = 100\), \(T = 1\), \(r = 0.03\), compute the superreplication price and the sub-replication price \(\text{BS}(\sigma_{\min})\). What is the width of the no-arbitrage price interval? Express this width as a fraction of the midpoint price.
Solution to Exercise 1
Superreplication price (using \(\sigma_{\max} = 0.35\), \(S = K = 100\), \(T = 1\), \(r = 0.03\)):
Using \(N(0.2607) \approx 0.6029\) and \(N(-0.0893) \approx 0.4644\):
Sub-replication price (using \(\sigma_{\min} = 0.15\)):
Using \(N(0.2750) \approx 0.6084\) and \(N(0.1250) \approx 0.5498\):
(Note: the exact values depend on the normal CDF precision. The worked example in the text uses \(\$14.80\) and \(\$6.20\), which may reflect slightly different parameter conventions or rounding.)
Width of the no-arbitrage interval:
As a fraction of the midpoint price:
The no-arbitrage interval is remarkably wide --- nearly \(68\%\) of the midpoint price. This reflects the significant degree of incompleteness: with volatility ranging from \(0.15\) to \(0.35\), the uncertainty in the option's fair value is enormous. Any hedging approach must operate within this interval, and the choice of method determines where in the interval the price falls.
Exercise 2. Quantile hedging with budget \(c = 0.90 \times \pi^{\sup}(H)\) replicates the claim on a success set \(A^*\) chosen by Neyman-Pearson. If the superreplication price from Exercise 1 is \(\pi^{\sup}\), compute the budget \(c\). The claim fails to replicate on paths where realized volatility exceeds some threshold \(\sigma^*\). Assuming that \(\mathbb{P}(\sigma > \sigma^*)\) can be made as small as 5%, explain qualitatively which paths are "sacrificed" and why the Neyman-Pearson structure selects the most expensive-to-hedge scenarios first.
Solution to Exercise 2
The budget is:
(using the worked example value \(\pi^{\sup} = \$14.80\)).
Which paths are sacrificed: The quantile hedging solution superreplicates the modified claim \(H \cdot \mathbf{1}_{A^*}\), where \(A^*\) is chosen by the Neyman-Pearson lemma. The lemma selects the success set \(A^*\) to maximize \(\mathbb{P}(A^*)\) subject to the budget constraint \(\pi^{\sup}(H \cdot \mathbf{1}_{A^*}) \leq c\).
The Neyman-Pearson structure ranks scenarios by the ratio \(d\mathbb{Q}^*/d\mathbb{P}\), which measures how "expensive" each scenario is to hedge per unit of probability. Paths with high \(d\mathbb{Q}^*/d\mathbb{P}\) are the most expensive to insure against --- these correspond to scenarios where the claim's payoff is extreme and the cost of replication is highest.
In the stochastic volatility context, paths where realized volatility is very high produce:
- Large option payoff variability, making replication costly.
- High values of the martingale measure density (the risk-neutral measure assigns higher probability to these adverse states).
The optimal strategy sacrifices these high-volatility paths first because they are the most expensive per unit of probability saved. By giving up on the \(5\%\) most costly-to-hedge scenarios (extreme volatility paths), the strategy achieves a \(10\%\) reduction in cost while only failing in a small fraction of realizations. This is efficient because these paths contribute disproportionately to the superreplication cost but occur with relatively low probability.
Exercise 3. The shortfall risk with linear loss is \(\mathcal{R}(c, \xi) = \mathbb{E}[(H - c - G_T(\xi))^+]\). For a simple two-state model where the hedge falls short by \(\$5\) with probability 0.10 and short by \(\$0\) otherwise, compute the expected shortfall. Compare this to a strategy that falls short by \(\$2\) with probability 0.20. Which strategy has lower shortfall risk? Which has lower failure probability (quantile hedging criterion)?
Solution to Exercise 3
Strategy A: Falls short by \(\$5\) with probability \(0.10\), and \(\$0\) otherwise.
Failure probability: \(0.10\).
Strategy B: Falls short by \(\$2\) with probability \(0.20\), and \(\$0\) otherwise.
Failure probability: \(0.20\).
Comparison:
| Criterion | Strategy A | Strategy B | Preferred |
|---|---|---|---|
| Expected shortfall | \(\$0.50\) | \(\$0.40\) | B |
| Failure probability | \(10\%\) | \(20\%\) | A |
Strategy B has lower shortfall risk (\(\$0.40 < \$0.50\)) because its failures are less severe, even though they occur more frequently.
Strategy A has lower failure probability (\(10\% < 20\%\)), so it would be preferred under the quantile hedging criterion.
This example illustrates precisely why shortfall risk minimization and quantile hedging can lead to different optimal strategies. Quantile hedging ignores the severity of failure and focuses only on its frequency, while expected shortfall penalizes both frequency and magnitude. A risk manager concerned about the average loss in failure scenarios should prefer Strategy B; one concerned about the probability of any failure at all should prefer Strategy A.
Exercise 4. The price ordering states \(\pi^{\sup}(H) \geq \pi_\varepsilon(H) \geq p_{\text{indiff}}(H) \geq \inf_{\mathbb{Q}}\mathbb{E}^{\mathbb{Q}}[\tilde{H}]\). Using the worked example values (\(\pi^{\sup} = \$14.80\), \(\pi_\varepsilon = \$12.50\), \(p_{\text{MV}} = \$10.30\), sub-rep \(= \$6.20\)), verify the ordering. Compute the "conservatism premium" \(\pi^{\sup} - p_{\text{MV}}\) and express it as a percentage of \(p_{\text{MV}}\). For what type of institution would the superreplication price be appropriate?
Solution to Exercise 4
Verifying the price ordering with the worked example values:
The ordering holds. (Note: the utility indifference price of \(\$10.80\) from the worked example also fits between \(\pi_\varepsilon\) and \(p_{\text{MV}}\), consistent with the general ordering.)
Conservatism premium:
The superreplication price carries a \(43.7\%\) premium over the mean-variance hedging price. This is the cost of eliminating all residual risk.
For whom is superreplication appropriate? Superreplication is most appropriate for institutions with:
- Regulatory requirements that demand zero residual risk (e.g., certain insurance solvency standards).
- Systemic importance where even small hedging failures can trigger cascading effects.
- Fiduciary obligations where the institution cannot tolerate any shortfall against promised liabilities.
- Small position sizes relative to capital, where the absolute cost premium is manageable.
For a typical trading desk, the \(43.7\%\) premium makes superreplication uneconomical. Mean-variance hedging or utility-based approaches provide better cost-risk tradeoffs for profit-seeking entities willing to absorb some residual risk.
Exercise 5. A trader sells a European call for \(\$11.00\) (the expected shortfall minimization price from the worked example) and delta-hedges using mean-variance optimal deltas. Under the worst-case volatility scenario (\(\sigma = 0.35\)), the Black-Scholes price is \(\$14.80\). Compute the maximum shortfall \(\$14.80 - \$11.00\). Under the best-case scenario (\(\sigma = 0.15\)), the price is \(\$6.20\). Compute the best-case profit. What is the expected P&L if the trader believes \(\sigma\) is uniformly distributed on \([0.15, 0.35]\)?
Solution to Exercise 5
Maximum shortfall (worst-case, \(\sigma = 0.35\)):
Under the worst-case volatility, the claim is worth \(\$14.80\) at maturity, but the trader collected only \(\$11.00\), producing a loss of \(\$3.80\) per unit.
Best-case profit (\(\sigma = 0.15\)):
Under the lowest volatility, the claim is worth only \(\$6.20\), and the trader collected \(\$11.00\), earning \(\$4.80\) per unit.
Expected P&L under uniform \(\sigma \sim U[0.15, 0.35]\): If \(\sigma\) is uniformly distributed on \([0.15, 0.35]\), the expected claim value is approximately the Black-Scholes price at the mean volatility (by Jensen's inequality this is an approximation, since BS is convex in \(\sigma\) for calls):
(using the midpoint volatility; the actual expectation is slightly higher due to the convexity of BS in \(\sigma\), but \(\$10.30\) is the value given in the worked example for mean-variance).
The expected P&L is:
The trader expects a positive P&L of approximately \(\$0.70\) per unit. This positive expected profit arises because the expected shortfall minimization price (\(\$11.00\)) exceeds the mean-variance fair value (\(\$10.30\)) --- the additional \(\$0.70\) is the compensation the trader earns for bearing the tail risk of extreme volatility outcomes.
Exercise 6. The risk-cost frontier illustrates the fundamental tradeoff in incomplete markets. For the stochastic volatility example, plot the following five points on a risk-cost diagram: (a) superreplication: cost \(= \$14.80\), residual risk \(= 0\); (b) quantile hedging: cost \(= \$12.50\), failure probability \(= 5\%\); (c) expected shortfall min: cost \(= \$11.00\); (d) mean-variance: cost \(= \$10.30\), residual std \(\approx \$2.50\); (e) sub-replication: cost \(= \$6.20\), maximum risk. Is the frontier convex or concave? Explain why no strategy can achieve simultaneously lower cost and lower risk than any point on the frontier.
Solution to Exercise 6
The five points on the risk-cost diagram (with cost on the vertical axis and residual risk on the horizontal axis) are:
| Point | Approach | Cost | Risk measure |
|---|---|---|---|
| (a) | Superreplication | \(\$14.80\) | \(0\) (zero risk) |
| (b) | Quantile hedging | \(\$12.50\) | \(5\%\) failure prob |
| (c) | Expected shortfall min | \(\$11.00\) | Controlled avg loss |
| (d) | Mean-variance | \(\$10.30\) | \(\$2.50\) residual std |
| (e) | Sub-replication | \(\$6.20\) | Maximum risk |
The frontier traces from the top-left corner (high cost, zero risk) to the bottom-right (low cost, high risk).
Shape of the frontier: The frontier is convex (when cost is plotted against risk). This means that the marginal cost of risk reduction is increasing: eliminating the last few percent of risk is disproportionately expensive compared to the initial risk reduction. For instance:
- Moving from sub-replication to mean-variance: costs \(\$4.10\) extra to reduce risk from maximum to \(\$2.50\) std.
- Moving from mean-variance to quantile hedging: costs \(\$2.20\) extra for further risk reduction.
- Moving from quantile hedging to superreplication: costs \(\$2.30\) extra to eliminate the last \(5\%\) failure probability.
The incremental cost per unit of risk reduction grows as risk approaches zero.
Why no strategy can be below the frontier: The frontier represents the set of Pareto-optimal strategies. For any given level of risk, the frontier point achieves the minimum cost; for any given cost, it achieves the minimum risk. If a strategy existed with both lower cost and lower risk than a frontier point, one could construct a strategy that dominates the frontier point. But this would contradict the optimality of the frontier strategy, which was derived by solving a constrained optimization problem. Formally, the existence of such a dominating strategy would imply an arbitrage opportunity or a violation of the duality theorems (superreplication duality, Neyman-Pearson optimality) that underpin each approach. The frontier is therefore an efficient boundary: improvements in one dimension necessarily come at the expense of the other.