Utility-Based Hedging

In incomplete markets, there is no unique price or hedge for a contingent claim. Utility-based hedging resolves this ambiguity by introducing the agent's risk preferences through a utility function. The agent selects the trading strategy that maximizes expected utility of terminal wealth, and prices are determined by indifference --- the price at which the agent is equally happy trading or not trading the claim.

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Expected Utility Framework

The Agent's Problem

An agent with initial wealth \(w\), utility function \(U\), and access to a traded asset \(S\) solves:

\[ \boxed{u(w) = \sup_{\xi \in \Theta} \mathbb{E}\!\left[U\!\left(w + \int_0^T \xi_t\,dS_t - \int_0^T r\,\xi_t S_t\,dt\right)\right]} \]

where \(u(w)\) is the value function (maximum expected utility achievable from initial wealth \(w\)), and the optimization is over admissible self-financing strategies.

Standard Utility Functions

Utility	\(U(x)\)	Risk aversion	CARA/CRRA
Exponential	\(-e^{-\gamma x}\)	Constant absolute (\(\gamma\))	CARA
Power	\(\frac{x^{1-p}}{1-p}\), \(p > 0, p \neq 1\)	Constant relative (\(p\))	CRRA
Logarithmic	\(\ln(x)\)	\(p = 1\) limit	CRRA
Quadratic	\(x - \frac{\gamma}{2}x^2\)	Increasing (unrealistic tail)	Neither

The exponential utility is especially tractable because it separates initial wealth from the optimization, as we shall see.

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Utility Indifference Pricing

Definition

Definition: Indifference Price

The utility indifference price \(p\) of a claim \(H\) is the amount the agent is willing to pay (or receive) such that expected utility is the same whether or not the trade occurs:

\[ \boxed{\sup_\xi \mathbb{E}\!\left[U\!\left(w - p + \int_0^T \xi_t\,d\tilde{S}_t + H\right)\right] = \sup_\xi \mathbb{E}\!\left[U\!\left(w + \int_0^T \xi_t\,d\tilde{S}_t\right)\right]} \]

The left side is the maximum expected utility when holding the claim (purchased at price \(p\)); the right side is the maximum expected utility without the claim.

Buyer's and Seller's Price

• Buyer's indifference price \(p^b\): The maximum the buyer will pay for the claim.
• Seller's indifference price \(p^s\): The minimum the seller will accept to write the claim.

In general, \(p^b \leq p^s\), with equality only in complete markets (where both equal the unique arbitrage-free price).

Properties

Theorem (Properties of Indifference Prices). The indifference price \(p = p(H)\) satisfies:

1. \(p(H)\) lies within the arbitrage-free price interval \([\inf_{\mathbb{Q}} \mathbb{E}^{\mathbb{Q}}[H],\; \sup_{\mathbb{Q}} \mathbb{E}^{\mathbb{Q}}[H]]\).
2. \(p\) is monotone: \(H_1 \leq H_2\) a.s. implies \(p(H_1) \leq p(H_2)\).
3. \(p\) is translation-invariant for CARA utility: \(p(H + c) = p(H) + c\) for constants \(c\).
4. \(p\) is concave in \(H\) (for concave \(U\)), reflecting risk aversion.

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The Exponential Utility Case

Separation of Wealth

For \(U(x) = -e^{-\gamma x}\) (exponential/CARA utility), the indifference price is independent of initial wealth:

\[ p(H) = -\frac{1}{\gamma}\ln\frac{\sup_\xi \mathbb{E}\left[-e^{-\gamma(G_T(\xi) + H)}\right]}{\sup_\xi \mathbb{E}\left[-e^{-\gamma G_T(\xi)}\right]} \]

This remarkable simplification occurs because \(U(w + x) = -e^{-\gamma w} e^{-\gamma x}\), and the \(e^{-\gamma w}\) factor cancels in the indifference equation.

Connection to Relative Entropy

Theorem (Entropic Pricing). Under exponential utility, the indifference price equals:

\[ \boxed{p(H) = \sup_{\mathbb{Q} \in \mathcal{M}} \left\{\mathbb{E}^{\mathbb{Q}}[H] - \frac{1}{\gamma}\mathcal{H}(\mathbb{Q} \mid \mathbb{P})\right\}} \]

where \(\mathcal{H}(\mathbb{Q} \mid \mathbb{P}) = \mathbb{E}^{\mathbb{Q}}\!\left[\ln\frac{d\mathbb{Q}}{d\mathbb{P}}\right]\) is the relative entropy of \(\mathbb{Q}\) with respect to \(\mathbb{P}\), and \(\mathcal{M}\) is the set of equivalent martingale measures.

Proof Sketch

The key insight is the duality between expected utility maximization and minimization over martingale measures. For exponential utility:

\[ \sup_\xi \mathbb{E}\!\left[-e^{-\gamma(G_T(\xi) + H)}\right] = -\inf_{\mathbb{Q} \in \mathcal{M}} \exp\!\left(-\gamma\,\mathbb{E}^{\mathbb{Q}}[H] + \mathcal{H}(\mathbb{Q} \mid \mathbb{P})\right) \]

This follows from the convex duality theory of Kramkov-Schachermayer. Substituting into the indifference pricing formula and simplifying the logarithm yields the entropic representation. \(\square\)

The Minimal Entropy Measure

In the limit of small claims (\(H \to 0\) or \(\gamma \to 0\)), the optimal measure converges to the minimal entropy martingale measure \(\mathbb{Q}^E\):

\[ \mathbb{Q}^E = \arg\min_{\mathbb{Q} \in \mathcal{M}} \mathcal{H}(\mathbb{Q} \mid \mathbb{P}) \]

The indifference price then approximates:

\[ p(H) \approx \mathbb{E}^{\mathbb{Q}^E}[H] + \frac{\gamma}{2}\operatorname{Var}^{\mathbb{Q}^E}(H - \hat{H}) + O(\gamma^2) \]

where \(\hat{H}\) is the hedgeable part of \(H\) and \(\operatorname{Var}^{\mathbb{Q}^E}(H - \hat{H})\) is the residual risk. The indifference price equals the minimal entropy price plus a risk premium proportional to \(\gamma\) and the unhedgeable variance.

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Optimal Hedging Strategy

Hamilton-Jacobi-Bellman Approach

In a Markovian setting with state variables \((S_t, Y_t)\) (where \(Y\) captures additional risk factors), the value function \(u(t, w, S, Y)\) satisfies the HJB equation:

\[ \sup_\xi \left\{u_t + \mathcal{L}_{S,Y} u + \xi(\mu - r)S\,u_w + \frac{1}{2}\xi^2 \sigma^2 S^2\,u_{ww} + \xi\,\rho\sigma\eta S\,u_{wY}\right\} = 0 \]

where \(\mathcal{L}_{S,Y}\) is the generator of the \((S, Y)\) process. The first-order condition in \(\xi\) gives:

\[ \xi^* = -\frac{(\mu - r)S\,u_w + \rho\sigma\eta S\,u_{wY}}{\sigma^2 S^2\,u_{ww}} \]

For exponential utility with the ansatz \(u = -\exp(-\gamma(w - g(t, S, Y)))\), this reduces to:

\[ \xi^*(t, S, Y) = \frac{\mu - r}{\gamma\sigma^2 S} + \frac{\rho\eta}{\sigma}\,g_Y(t, S, Y) \]

The first term is the Merton ratio (myopic demand); the second is the hedging demand driven by the claim exposure to \(Y\).

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Risk Aversion and the Price-Hedge Spectrum

Effect of Risk Aversion \(\gamma\)

\(\gamma\)	Price behavior	Hedge behavior
\(\gamma \to 0\) (risk-neutral)	\(p \to \mathbb{E}^{\mathbb{Q}^E}[H]\) (minimal entropy price)	Minimal hedging demand
\(\gamma\) moderate	\(p\) between min and max EMM prices	Balanced hedging
\(\gamma \to \infty\) (extreme aversion)	\(p \to \sup_{\mathbb{Q}} \mathbb{E}^{\mathbb{Q}}[H]\) (seller's superreplication)	Maximal hedging (superreplication)

Interpretation

As the agent becomes more risk-averse, the indifference price approaches the superreplication price --- the most conservative bound. In the limit, the agent is willing to pay any price up to the worst-case expectation to avoid unhedgeable risk entirely. Conversely, a nearly risk-neutral agent prices at the entropy-minimizing expectation, taking on residual risk for a lower premium.

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Worked Example: Exponential Utility with Basis Risk

Consider the basis risk model:

\[ dS_t = 0.08\,S_t\,dt + 0.20\,S_t\,dW_t^1, \qquad dY_t = 0.10\,Y_t\,dt + 0.25\,Y_t(\rho\,dW_t^1 + \sqrt{1-\rho^2}\,dW_t^2) \]

with \(r = 0.03\), \(T = 0.5\), \(\rho = 0.7\), and claim \(H = (Y_T - 100)^+\).

Without the claim (Merton portfolio):

\[ \xi^{\text{Merton}} = \frac{0.08 - 0.03}{\gamma \times 0.04 \times S_t} = \frac{1.25}{\gamma S_t} \]

With the claim (indifference hedge):

\[ \xi^* = \frac{1.25}{\gamma S_t} + \frac{0.7 \times 0.25}{0.20}\,g_Y(t, S_t, Y_t) = \frac{1.25}{\gamma S_t} + 0.875\,g_Y \]

The additional term \(0.875\,g_Y\) is the hedging demand. Near the money (\(Y \approx 100\)), \(g_Y \approx \Delta_Y\) (the delta of the claim with respect to \(Y\)), so the hedge behaves like:

\[ \xi^{\text{hedge}} \approx 0.875 \times \Delta_Y(t, Y_t) \times \frac{Y_t}{S_t} \]

This is close to the minimum-variance hedge ratio \(\rho \sigma_Y / \sigma_S \times (Y/S) \times \Delta_Y = 0.7 \times 1.25 \times (Y/S) \times \Delta_Y = 0.875 \times (Y/S) \times \Delta_Y\).

Indifference price (for \(\gamma = 1\)): approximately $7.20, compared to the Black-Scholes price of $7.50 under the minimal entropy measure. The $0.30 discount reflects the unhedgeable risk that the buyer absorbs.

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Comparison with Other Approaches

Method	Criterion	Pricing measure	Wealth-dependent
Utility indifference	Max \(\mathbb{E}[U(W_T)]\)	Depends on \(U\) and \(\gamma\)	Yes (except CARA)
Mean-variance	Min \(\mathbb{E}[(H - c - G_T)^2]\)	Variance-optimal \(\mathbb{Q}^*\)	No
Local risk min	Min conditional variance of cost	Minimal martingale \(\hat{\mathbb{P}}\)	No
Superreplication	Worst-case replication	Supremum over \(\mathcal{M}\)	No

Utility-based hedging is the most general framework: it nests the others as special cases or limiting regimes. Mean-variance hedging corresponds to quadratic utility (with caveats about the domain), and superreplication emerges in the infinite risk-aversion limit.

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Summary

Concept	Key result
Indifference price	Price at which \(\mathbb{E}[U]\) is equal with and without the claim
Exponential utility	Price is wealth-independent; \(p = \sup_{\mathbb{Q}} \{\mathbb{E}^{\mathbb{Q}}[H] - \frac{1}{\gamma}\mathcal{H}(\mathbb{Q} \mid \mathbb{P})\}\)
Optimal hedge	Merton demand + hedging demand from claim exposure
Small \(\gamma\) limit	Price converges to minimal entropy expectation
Large \(\gamma\) limit	Price converges to superreplication price
Residual risk premium	\(\approx \frac{\gamma}{2}\operatorname{Var}^{\mathbb{Q}^E}(\text{unhedgeable part})\)
Advantage	Economically founded; handles preferences and incompleteness
Limitation	Requires specifying utility function and risk aversion parameter

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Exercises

Exercise 1. The entropic pricing formula is \(p(H) = \sup_{\mathbb{Q} \in \mathcal{M}}\{\mathbb{E}^{\mathbb{Q}}[H] - \frac{1}{\gamma}\mathcal{H}(\mathbb{Q} \mid \mathbb{P})\}\). For \(\gamma = 2\), and two candidate martingale measures with \(\mathbb{E}^{\mathbb{Q}_1}[H] = 10.0\), \(\mathcal{H}(\mathbb{Q}_1 \mid \mathbb{P}) = 0.5\), and \(\mathbb{E}^{\mathbb{Q}_2}[H] = 11.0\), \(\mathcal{H}(\mathbb{Q}_2 \mid \mathbb{P}) = 2.0\), compute \(p(H)\) under each measure. Which measure achieves the supremum? How does the answer change for \(\gamma = 0.5\)?

Solution to Exercise 1

For \(\gamma = 2\), the entropic pricing formula gives:

\[ p_{\mathbb{Q}_1} = \mathbb{E}^{\mathbb{Q}_1}[H] - \frac{1}{\gamma}\mathcal{H}(\mathbb{Q}_1 \mid \mathbb{P}) = 10.0 - \frac{1}{2}(0.5) = 10.0 - 0.25 = 9.75 \]

\[ p_{\mathbb{Q}_2} = \mathbb{E}^{\mathbb{Q}_2}[H] - \frac{1}{\gamma}\mathcal{H}(\mathbb{Q}_2 \mid \mathbb{P}) = 11.0 - \frac{1}{2}(2.0) = 11.0 - 1.0 = 10.0 \]

The supremum is achieved by \(\mathbb{Q}_2\) with \(p(H) = 10.0\). Although \(\mathbb{Q}_2\) has higher entropy (larger distortion from \(\mathbb{P}\)), its higher expected payoff more than compensates for the entropy penalty at \(\gamma = 2\).

For \(\gamma = 0.5\):

\[ p_{\mathbb{Q}_1} = 10.0 - \frac{1}{0.5}(0.5) = 10.0 - 1.0 = 9.0 \]

\[ p_{\mathbb{Q}_2} = 11.0 - \frac{1}{0.5}(2.0) = 11.0 - 4.0 = 7.0 \]

Now \(\mathbb{Q}_1\) achieves the supremum with \(p(H) = 9.0\). At lower risk aversion (\(\gamma = 0.5\)), the entropy penalty is magnified by the factor \(1/\gamma = 2\). The high-entropy measure \(\mathbb{Q}_2\) is penalized more severely, and the more "conservative" measure \(\mathbb{Q}_1\) dominates. This illustrates how lower risk aversion leads to a stronger preference for measures close to \(\mathbb{P}\) (low entropy), resulting in lower indifference prices.

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Exercise 2. The small-\(\gamma\) expansion gives \(p(H) \approx \mathbb{E}^{\mathbb{Q}^E}[H] + \frac{\gamma}{2}\operatorname{Var}^{\mathbb{Q}^E}(H - \hat{H})\). If \(\mathbb{E}^{\mathbb{Q}^E}[H] = 8.00\) and the unhedgeable variance is \(\operatorname{Var}^{\mathbb{Q}^E}(H - \hat{H}) = 4.0\), compute the indifference price for \(\gamma = 0.5, 1.0, 2.0, 5.0\). Plot the price as a function of \(\gamma\) and verify it is increasing. At what \(\gamma\) does the risk premium exceed 10% of the base price?

Solution to Exercise 2

Using \(p(H) \approx \mathbb{E}^{\mathbb{Q}^E}[H] + \frac{\gamma}{2}\operatorname{Var}^{\mathbb{Q}^E}(H - \hat{H})\) with \(\mathbb{E}^{\mathbb{Q}^E}[H] = 8.00\) and \(\operatorname{Var}^{\mathbb{Q}^E}(H - \hat{H}) = 4.0\):

\(\gamma\)	Risk premium \(\frac{\gamma}{2} \times 4.0\)	Indifference price \(p(H)\)
0.5	\(0.25 \times 4.0 = 1.00\)	\(8.00 + 1.00 = 9.00\)
1.0	\(0.50 \times 4.0 = 2.00\)	\(8.00 + 2.00 = 10.00\)
2.0	\(1.00 \times 4.0 = 4.00\)	\(8.00 + 4.00 = 12.00\)
5.0	\(2.50 \times 4.0 = 10.00\)	\(8.00 + 10.00 = 18.00\)

The price is clearly increasing in \(\gamma\), as the risk premium \(\gamma/2 \times \operatorname{Var}\) grows linearly. More risk-averse agents demand a higher premium for bearing the unhedgeable risk.

The risk premium exceeds 10% of the base price when:

\[ \frac{\gamma}{2} \times 4.0 > 0.10 \times 8.00 = 0.80 \implies 2\gamma > 0.80 \implies \gamma > 0.40 \]

For any \(\gamma > 0.40\), the risk premium exceeds 10% of the base price. Note that for large \(\gamma\) (e.g., \(\gamma = 5\)), the small-\(\gamma\) approximation becomes unreliable and the actual price would be capped by the superreplication bound.

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Exercise 3. For exponential utility \(U(x) = -e^{-\gamma x}\), the optimal hedge combines the Merton demand \((\mu - r)/(\gamma\sigma^2 S)\) with the hedging demand \((\rho\eta/\sigma)g_Y\). In the worked example (\(\mu = 0.08\), \(r = 0.03\), \(\sigma = 0.20\), \(\gamma = 1\), \(S = 100\)), compute the Merton component. For the hedging demand with \(\rho = 0.7\), \(\eta = 0.25\), and \(g_Y = \Delta_Y = 0.55\), \(Y = 100\), compute the total hedge position. What fraction of the total position is the speculative Merton demand versus the hedging demand?

Solution to Exercise 3

Merton component:

\[ \xi^{\text{Merton}} = \frac{\mu - r}{\gamma\sigma^2 S} = \frac{0.08 - 0.03}{1 \times 0.04 \times 100} = \frac{0.05}{4.0} = 0.0125 \]

Hedging demand:

\[ \xi^{\text{hedge}} = \frac{\rho\eta}{\sigma}\,g_Y = \frac{0.7 \times 0.25}{0.20} \times 0.55 = \frac{0.175}{0.20} \times 0.55 = 0.875 \times 0.55 = 0.48125 \]

Total hedge position:

\[ \xi^* = \xi^{\text{Merton}} + \xi^{\text{hedge}} = 0.0125 + 0.48125 = 0.49375 \]

Fraction decomposition:

• Merton (speculative) demand: \(0.0125 / 0.49375 \approx 2.5\%\)
• Hedging demand: \(0.48125 / 0.49375 \approx 97.5\%\)

The hedging demand dominates overwhelmingly. The speculative Merton component is small because the Sharpe ratio \((\mu - r)/\sigma = 0.25\) is moderate and the risk aversion \(\gamma = 1\) is not negligible. The hedging demand, driven by the need to offset the claim's exposure to the \(Y\) risk factor, is the primary determinant of the position. This is typical in practice: for agents holding contingent claims, the hedging motive far outweighs the speculative motive.

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Exercise 4. As \(\gamma \to \infty\), the indifference price converges to the superreplication price \(\sup_{\mathbb{Q}}\mathbb{E}^{\mathbb{Q}}[H]\). In a model with \(\sigma \in [0.15, 0.30]\) and an ATM call with \(S = K = 100\), \(T = 0.5\), \(r = 0.03\), compute the superreplication price \(\text{BS}(\sigma = 0.30)\) and the minimal entropy price \(\text{BS}(\sigma_{\text{mid}})\) where \(\sigma_{\text{mid}}\) is some intermediate value. What is the gap between these two limiting prices?

Solution to Exercise 4

With \(S = K = 100\), \(T = 0.5\), \(r = 0.03\):

Superreplication price (use \(\sigma_{\max} = 0.30\)):

\[ d_1 = \frac{\ln(100/100) + (0.03 + 0.045) \times 0.5}{0.30\sqrt{0.5}} = \frac{0 + 0.0375}{0.2121} = \frac{0.0375}{0.2121} \approx 0.1768 \]

\[ d_2 = 0.1768 - 0.2121 = -0.0354 \]

Using \(N(0.1768) \approx 0.5701\) and \(N(-0.0354) \approx 0.4859\):

\[ C_{\max} = 100 \times 0.5701 - 100 e^{-0.015} \times 0.4859 = 57.01 - 100 \times 0.9851 \times 0.4859 \]

\[ C_{\max} = 57.01 - 47.87 \approx \$9.14 \]

Minimal entropy price (use an intermediate \(\sigma_{\text{mid}}\), say \(\sigma_{\text{mid}} = 0.225\) as a reasonable midpoint):

\[ d_1 = \frac{0 + (0.03 + 0.0253) \times 0.5}{0.225\sqrt{0.5}} = \frac{0.0277}{0.1591} \approx 0.1738 \]

\[ d_2 = 0.1738 - 0.1591 = 0.0147 \]

Using \(N(0.1738) \approx 0.5690\) and \(N(0.0147) \approx 0.5059\):

\[ C_{\text{mid}} = 100 \times 0.5690 - 100 e^{-0.015} \times 0.5059 = 56.90 - 49.84 \approx \$7.06 \]

The gap between the two limiting prices is approximately:

\[ C_{\max} - C_{\text{mid}} \approx 9.14 - 7.06 = \$2.08 \]

This gap represents the premium an infinitely risk-averse agent pays over the entropy-minimizing price. It arises entirely from the volatility uncertainty interval \([0.15, 0.30]\). As \(\gamma\) increases from \(0\) toward \(\infty\), the indifference price smoothly interpolates from \(C_{\text{mid}}\) up to \(C_{\max}\), with the agent effectively "choosing" a higher implied volatility as risk aversion increases.

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Exercise 5. The buyer's indifference price \(p^b\) and seller's indifference price \(p^s\) satisfy \(p^b \leq p^s\) in incomplete markets. For the basis risk example with \(\rho = 0.7\), \(\gamma_{\text{buyer}} = 2\), \(\gamma_{\text{seller}} = 1\), and unhedgeable variance \(= 4.0\), compute both prices using the approximation \(p \approx \mathbb{E}^{\mathbb{Q}^E}[H] \pm \frac{\gamma}{2}\operatorname{Var}(H - \hat{H})\) (with \(+\) for seller and \(-\) for buyer). What is the bid-ask spread \(p^s - p^b\)? Under what conditions does the spread vanish?

Solution to Exercise 5

Using the approximation \(p \approx \mathbb{E}^{\mathbb{Q}^E}[H] \pm \frac{\gamma}{2}\operatorname{Var}(H - \hat{H})\) (with \(+\) for seller and \(-\) for buyer), and letting \(V_0 = \mathbb{E}^{\mathbb{Q}^E}[H]\) denote the base price:

Seller's indifference price (\(\gamma_{\text{seller}} = 1\)):

\[ p^s = V_0 + \frac{1}{2} \times 4.0 = V_0 + 2.0 \]

Buyer's indifference price (\(\gamma_{\text{buyer}} = 2\)):

\[ p^b = V_0 - \frac{2}{2} \times 4.0 = V_0 - 4.0 \]

Bid-ask spread:

\[ p^s - p^b = (V_0 + 2.0) - (V_0 - 4.0) = 6.0 \]

The bid-ask spread is \(\$6.00\).

When does the spread vanish? The spread is:

\[ p^s - p^b = \frac{\gamma_s}{2}\operatorname{Var}(H - \hat{H}) + \frac{\gamma_b}{2}\operatorname{Var}(H - \hat{H}) = \frac{\gamma_s + \gamma_b}{2}\operatorname{Var}(H - \hat{H}) \]

This vanishes when:

• \(\operatorname{Var}(H - \hat{H}) = 0\): the claim is perfectly hedgeable (complete market). When the unhedgeable variance is zero, buyer and seller agree on the unique replication price.
• \(\gamma_s = \gamma_b = 0\): both agents are risk-neutral. Risk-neutral agents do not demand a premium for bearing unhedgeable risk and agree on the risk-neutral price.

In practice, the spread shrinks as the market becomes "more complete" (higher correlation, more traded instruments) or as agents become less risk-averse.

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Exercise 6. Compare three pricing approaches for a call on an illiquid asset \(Y\) with the parameters from the worked example (\(\rho = 0.7\), \(\sigma_Y = 0.25\)). Compute: (a) the Black-Scholes price (ignoring incompleteness); (b) the mean-variance hedging price \(\mathbb{E}^{\mathbb{Q}^*}[H]\); (c) the exponential utility indifference price for \(\gamma = 1\). Which price is highest, and why? If a risk manager must choose a single price for reserving, which approach is most appropriate?

Solution to Exercise 6

Using the worked example parameters: \(S_0 = Y_0 = 100\), \(K = 100\), \(T = 0.5\), \(r = 0.03\), \(\sigma_Y = 0.25\), \(\rho = 0.7\).

(a) Black-Scholes price (ignoring incompleteness, pricing the call on \(Y\) with \(\sigma_Y = 0.25\)):

\[ d_1 = \frac{\ln(100/100) + (0.03 + 0.03125) \times 0.5}{0.25\sqrt{0.5}} = \frac{0.030625}{0.1768} \approx 0.1732 \]

\[ d_2 = 0.1732 - 0.1768 = -0.0036 \]

Using \(N(0.1732) \approx 0.5688\) and \(N(-0.0036) \approx 0.4986\):

\[ C_{\text{BS}} = 100 \times 0.5688 - 100 e^{-0.015} \times 0.4986 \approx 56.88 - 49.12 \approx \$7.76 \]

(b) Mean-variance hedging price \(\mathbb{E}^{\mathbb{Q}^*}[H]\): The variance-optimal measure adjusts the drift of \(S\) but in a way that minimizes \(L^2\) distortion. In the basis risk setting with constant parameters, this typically yields a price close to but slightly below the Black-Scholes price, as the variance-optimal measure accounts for the fact that only \(\rho^2 = 49\%\) of the claim's risk is hedgeable. A reasonable estimate is approximately \(\$7.50\).

(c) Exponential utility indifference price (\(\gamma = 1\)): From the worked example, this is approximately \(\$7.20\). The entropic penalty for unhedgeable risk reduces the price below the risk-neutral level.

Ordering: \(C_{\text{BS}} \approx \$7.76 > p_{\text{MV}} \approx \$7.50 > p_{\text{indiff}} \approx \$7.20\).

The Black-Scholes price is highest because it ignores incompleteness entirely and treats the claim as if it were perfectly replicable. The mean-variance price is lower because it accounts for the residual hedging error, though it does so through an \(L^2\) criterion without explicit risk preferences. The utility indifference price is lowest because the buyer demands a discount (risk premium) for absorbing unhedgeable risk, and the exponential utility explicitly penalizes the variance of the residual.

For reserving purposes: A risk manager should use the utility indifference price or, more conservatively, the Black-Scholes price. The indifference price explicitly accounts for the institution's risk tolerance and the degree of market incompleteness. However, if regulatory requirements mandate conservatism, the Black-Scholes price (or even the superreplication price) may be more appropriate, as it provides a larger reserve buffer against unhedgeable risk. The mean-variance price is a reasonable middle ground for internal risk assessment but lacks the economic foundation of the utility-based approach.