Optimal Strike Prices under Uncertainty

• Optimal strike prices under uncertainty are defined as the key variable that balances risk and reward in contingent claims by solving constrained optimization problems under stochastic market conditions.
• Methodologies such as stochastic control, first-order condition analysis, and simulation-based studies are employed to derive strike levels and optimize risk-neutral profitability.
• Practical implementations involve model calibration, scenario analysis, and numerical techniques to adjust for market biases, ensuring effective hedging and risk management.

Optimal strike prices under uncertainty refer to the solution of a constrained optimization problem that selects strike levels for contingent claims—most commonly options or contracts for difference (CfDs)—in a context where the underlying price process, relevant reference indices, or payoff regimes are themselves uncertain. The strike price is typically the key tunable variable that governs both the risk–reward distribution of the contract and its hedging effectiveness. Depending on the setting, optimality may be defined by maximizing expected profit, minimizing risk, equating expected payoff to cost (risk-neutrality), or satisfying risk constraints under a scenario distribution. The determination of optimal strikes is highly sensitive to model structure, stochastic process assumptions, payoff nonlinearity, and market design.

1. Stochastic Control Formulation in Option Portfolios

Optimal strike selection in multi-option portfolios is naturally formulated as a stochastic optimal control problem with stopping. In the case of Iron Condor strategies, four strikes $\{k_1, k_2, k_3, k_4\}$ and an optional stopping time $\tau$ are chosen at inception to maximize a trade-off between normalized profit and risk under price process uncertainty. The state variable is the underlying price $S_t$, modeled (for tractability) as a bounded martingale within $[K_2, K_3]$: $dS_t = \sqrt{V_t}S_t dW_t$, where $V_t$ may be constant or stochastic. Portfolio value is a function of current marks of the four options, and risk is quantified via normalized downward shock losses. The control is thus $u = (k_i, \tau)$, and the objective is

$$\max_{k_1, k_2, k_3, k_4,\, \tau} \mathbb{E}\bigl[\,\Phi(P_\tau, S_\tau, \tau)\, \bigr]$$

subject to ordering of strikes and time constraints (Huang et al., 6 Jan 2025).

2. Submartingale Property and Optimal Stopping

If the underlying price remains confined to an interval where all options are out of the money, the discounted value process of the Iron Condor is a submartingale. In this regime, it is precisely optimal never to stop early; the optimal $\tau^*$ coincides with expiration $T$. This mathematical property is derived by examining the option value dynamics: time decay (Theta) is asymmetric at the wings, and the aggregation of Theta terms yields a nonnegative drift. Formally, the submartingale property is demonstrated via

$$\mathbb{E}[P_{t+\Delta t} \mid \mathcal{F}_t] \geq P_t$$

for all $t$, and standard optimal stopping arguments dictate $\tau^*=T$ in this setting. In contrast, for supermartingales, immediate liquidation is optimal, and for martingales any stopping policy is locally optimal (Huang et al., 6 Jan 2025).

3. First-Order Characterization of the Optimal Strikes

No closed-form solution generally exists for the four optimal strikes of an Iron Condor in stochastic models. However, first-order conditions can be written:

$$\frac{\partial}{\partial k_i} \mathbb{E}_\mathbb{Q}[\text{payoff}_T(k)] = \lambda \frac{\partial}{\partial k_i} \text{Risk}(k), \qquad i=1,\dots,4$$

where $\lambda$ is a risk-aversion parameter and the Greeks of the European put and call legs enter via the derivatives. In practical applications (e.g., under Black-Scholes or rough stochastic volatility), these conditions are solved via numerical root-finding or stochastic search. The solution structure is approximately:

$$k_2^* \simeq S_0 \exp(-\alpha^* \sigma_e \sqrt{T}), \quad k_3^* \simeq S_0 \exp(+\alpha^* \sigma_e \sqrt{T}), \quad k_1^* = k_2^* - \Delta^*, \quad k_4^* = k_3^* + \Delta^*$$

with $\alpha^*$ and $\Delta^*$ depending on market parameters and risk preferences (Huang et al., 6 Jan 2025).

4. Empirical and Simulation-Based Strike Optimization

Simulation studies under rough volatility models (e.g., Rough Heston) provide insight into strike optimization under real-world uncertainty. Results show that:

• Left-biased Condors (more OTM puts) optimize long-run profitability and risk in bullish markets, with optimal stopping at expiration.
• Right-biased Condors require aggressive early exit ($\tau^* \approx 0.5T$–$0.65T$) to control left-tail risk from market drops.
• Deeper OTM strikes (higher moneyness $x$) increase both expected profit and downside risk; risk can be trimmed by stopping at $\tau^* \sim 0.6T$–$0.75T$.
• Wider spreads ($\hat{h}$) linearly increase both max profit and max loss; again, early exit can mitigate extremes.

Time-to-exit $\tau^*$, except for the left-biased case, generally falls within $50\%\text{–}75\%$ of the option term (Huang et al., 6 Jan 2025).

5. Optimal Strikes in Multi-Asset Option Pricing

In exotic multi-asset derivatives, optimal choice of strikes for implied volatility input is critical for accurate pricing. For exchange options under stochastic volatility, Malliavin calculus yields an optimal log-linear convention:

$$k_X^* = (1-a^*)x + a^*y, \quad k_Y^* = a^*x + (1-a^*)y$$

$$a^* = \frac{\rho_X \lambda_X - \rho_Y \lambda_Y} {\rho_X(\lambda_X - \rho \lambda_Y) - \rho_Y(\lambda_Y - \rho \lambda_X)}$$

where $x$ and $y$ are log-spot prices, $\rho_*$ are model-implied correlations, and $\lambda_*$ are volatility scale factors. The first-order condition arises from short-maturity asymptotics and is model-agnostic: only the ATM limits and skews matter, not the specific form of the stochastic volatility model. Comprehensive grid testing demonstrates that this convention achieves lower mean absolute error and flatter implied correlation than naive ATM lookup or volatility-based lookup (Alòs et al., 2018).

6. Strike Optimization in Contract-for-Difference Designs under Market Uncertainty

For CfDs in electricity markets, uncertainty in spot prices, volumes, and reference indices necessitates scenario-based strike optimization. Optimal strike $K^*$ is defined by the first-order risk-neutral condition:

$$\mathbb{E}_{s\in S}\bigg[\frac{\partial}{\partial K}\pi_{i,n,s}(K^*)\bigg] = 0$$

where $\pi$ is scenario profit, $S$ the scenario set. For production-based CfDs: $$K^*_{basic} = \frac{\mathbb{E}_s[C_{i,n,s}]}{\mathbb{E}_s[\sum_t q_{t,i,n,s}]} = c + \frac{A M}{\sum_t \mathbb{E}_s[f_{t,i,n,s}]}$$ with $f_{t,i,n,s}=q_{t,i,n,s}/Q_{i,n,s}$ (Johanndeiter et al., 19 Dec 2025).

For variants with alternative reference indices (annual average market value, revenue per MW), corresponding closed-form expressions use scenario averages and covariances. These are computed in a single pass through market scenarios, requiring no iterative solver. Optimal $K^*$ aligns expected zero-profit bidding with system incentives, with risk premia introduced via a mean–variance objective.

Empirical results show that all CfD variants achieve substantial volatility reduction in profits, with capacity-based references minimizing cost-recovery variance but at the cost of weaker price signals for dispatch and investment (Johanndeiter et al., 19 Dec 2025).

7. Practical Strike Determination—Rules of Thumb and Implementation

Implementation steps for strike optimization under uncertainty include: model calibration (volatility, correlation, cost parameters), numerical solution of the first-order system (via root finding or scenario averaging), and Monte Carlo simulation for validation. Generalized rules include:

• When the underlying is expected to be range-bound, set strikes at the no-profit bounds ($k_2=K_2, k_3=K_3$) and hold to expiration.
• For bullish markets, left-bias the strikes (deep OTM puts), set wider spreads, and hold to maturity.
• For sideways or uncertain markets, use symmetric or mildly left-biased structures with moderate moneyness and spreads, exiting near $0.6T$.
• For situations exposing downside tail risk, adopt right-biased configurations only with mandatory early exit.
• In CfD contexts, set strike by $$K^* \approx \text{expected LCOE} + (\mathbb{E}[p_{ref}] - \mathbb{E}[p_{ind}])$$ to align with risk-neutral zero-profit incentives.

All such strategies depend crucially on accurate scenario analysis and, if desired, explicit risk-adjustment (mean–variance or CVaR) in the strike selection criterion (Huang et al., 6 Jan 2025, Johanndeiter et al., 19 Dec 2025).

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Selected Key References:

• "Stochastic Optimal Control of Iron Condor Portfolios for Profitability and Risk Management" (Huang et al., 6 Jan 2025)
• "On the optimal choice of strike conventions in exchange option pricing" (Alòs et al., 2018)
• "Most certainly certain? The Impact of Contract for Difference Design on Renewables' Strike Prices and Electricity Market Risks" (Johanndeiter et al., 19 Dec 2025)