A Framework for Optimal Strike Selection in SPX Put Spread Collars

The efficacy of an SPX put spread collar is critically dependent on the selection of its strike prices. The strikes for the long put, short put, and short call determine the level of downside protection, the point at which upside potential is capped, and the overall cost of the hedge. A systematic approach to strike selection is therefore essential for constructing a collar that aligns with a portfolio manager's specific risk management objectives. This article presents a framework for optimal strike selection, considering the trade-offs between protection, cost, and upside potential.

The Trade-Off Triangle: Protection, Cost, and Upside

Strike selection in a collar strategy can be conceptualized as a "trade-off triangle" with three competing objectives:

1. Maximizing Downside Protection: This implies selecting a long put strike (K_p1) that is as close as possible to the current price of the underlying SPX. A higher long put strike provides a greater level of protection, but also increases the cost of the hedge.

2. Minimizing Cost: The cost of the collar is a function of the premiums of the three options. To minimize the cost, a manager can either lower the strike of the long put, raise the strike of the short put, or lower the strike of the short call. Each of these adjustments, however, will have an impact on the other two sides of the triangle.

3. Maximizing Upside Potential: This involves selecting a short call strike (K_c) that is as high as possible. A higher short call strike allows for greater participation in an SPX rally, but also reduces the premium collected, thereby increasing the net cost of the collar.

A Systematic Approach to Strike Selection

A systematic approach to strike selection involves defining a set of rules based on objective criteria, such as moneyness, probability, and the VIX. Moneyness refers to the relationship between the strike price of an option and the current price of the underlying asset. It is often expressed as a percentage of the underlying price.

One common approach is to select the strikes based on a predetermined level of "moneyness." For example, a manager might decide to always buy a put spread that is 5% out-of-the-money (OTM) and sell a call that is 5% OTM. This approach provides a consistent level of protection and upside potential relative to the current market level.

The Role of Probability

Another approach is to select the strikes based on the probability of the underlying finishing within a certain range at expiration. The probability of an option finishing in-the-money can be estimated using the option's delta. For example, a 30-delta call option has an approximate 30% probability of finishing in-the-money.

A manager could, for instance, decide to sell a call with a delta of 0.25, which would have an approximate 25% probability of being exercised. This approach allows the manager to explicitly define the level of upside they are willing to forgo.

The probability of the underlying finishing between two points (e.g., between the short put and long put strikes) can be calculated using the following formula:

Probability(K1 < S_T < K2) = N(d2(K1)) - N(d2(K2))

Probability(K1 < S_T < K2) = N(d2(K1)) - N(d2(K2))

Where:

• N() is the cumulative standard normal distribution function
• d2 is the d2 from the Black-Scholes model, calculated for each strike K1 and K2

Comparison of Strike Selection Strategies

The following table compares three different strike selection strategies for a 3-month SPX collar, with the SPX at 4,500 and the VIX at 20.

Note: The above data is hypothetical and for illustrative purposes only. Premiums are calculated using a simplified Black-Scholes model.

Analysis of Strike Selection Strategies

The "5% OTM" strategy provides a balanced approach, with a small credit received for establishing the position and a symmetrical risk/reward profile. The "10% OTM" strategy provides a larger credit, but at the cost of a wider range of unprotected downside and a higher cap on the upside. The "25 Delta Call" strategy results in a net debit, but provides a higher level of downside protection and a greater potential for upside participation.

The optimal strike selection strategy will depend on the portfolio manager's specific objectives and market outlook. If the primary objective is to protect against a catastrophic crash, a wider put spread with a lower net cost may be appropriate. If the objective is to protect against a more moderate downturn while still participating in a potential rally, a tighter collar with a higher net cost may be preferable.

In conclusion, the selection of strike prices is a important element of a systematic SPX collar hedging strategy. By understanding the trade-offs between protection, cost, and upside potential, and by using a systematic framework based on moneyness and probability, a portfolio manager can construct a collar that is tailored to their specific risk management needs and that can be consistently applied over time. ""