OPTION CHAINS, FUNCTIONS AND STRATEGIES
Created October 14, 2024
By Andres Talero
Overview.
This article describes option premiums as functions of time to expiration and implied volatility, and relative to strike location. Measures of deltas are presented, and the depiction of volatility surfaces and skews is reviewed. Basic option strategies are discussed. The main goal of the article is to provide a practical toolkit for options analysis and use, detailed enough to aid individual investors and non-specialized practitioners, but basic enough to be implemented directly from observing option chains.
The article is organized in 3 sections as follows:
The first section deals with the anatomy of option chains and discusses basic pricing concepts such as forward prices, put call parity and risk-free funded positions in the context of option chains. It ends by describing how volatilities, time to expiration and strike location impact option premiums and time decay.
The second section discusses option functions. It begins with a description of deltas to the underlying. It then introduces the concept of strike deltas, their connection to vertical option spreads and their use as measures of marginal risk. It later overviews volatility functions and their depiction as surfaces and curves. The section ends with a discussion of skews and their impact on option premiums, as well as their origins and some market examples.
The third section discusses option strategies, building on the concepts introduced in the first two sections. It first reviews single option strategies (outright options, covered calls, protective puts). Later portions discuss funding optimization and the impact of option skews on strategies, with a focus on vertical option spreads. The end of the section overviews marginal risk and its uses in strike selection.
Section 1: Anatomy of an Option Chain.
Option premiums (put and call prices) are observable from option chains - lists of premiums for a given expiration. Premiums for different expirations of the same underlying are usually also available. Option chains display single expiration premiums in a grid, organized from lower to higher strikes. Chains are available for expirations ranging from days, weeks, months and years in a calendar set by the exchanges where they are traded. Short term options liquidity is concentrated in strikes around the current price of the underlying. Those expiring in more than a year are referred to as LEAPS.
Figure 1. Hypothetical Option Chain
Forward Prices.
Forward prices are those at which a given underlying can be bought or sold for delivery and payment in the future. They are derived from the current price of an underlying (for immediate delivery and settlement, known as “spot prices”) and the incorporation of funding considerations (dividends and financing interest rates for stocks). Funding considerations are incorporated up to the date in which the forward transaction settles. It is useful to think of forward prices as breakeven prices net of any funding costs and income generated by the asset prior to the forward settlement.
For dividend paying stocks, the difference between funding interest rates (the cost of financing a long position) and dividends (the income received) determines the location of forward prices relative to spot prices. If dividends are higher than interest rates, holding a stock is net income producing and forward prices will be below the current price. When dividends for a stock are lower than interest rates, funding the asset causes a net cost over time, and forward prices are above the current price.
Option implied forward prices are easy to observe in option chains by finding at which strike put and call premiums intersect. Recall that option premiums have two basic components: Time value and Intrinsic value. The time value of options decreases as their strike moves further away from the forward. Put premiums decrease as strikes become lower; call premiums decrease as strikes become higher. At the forward price the premiums of calls and puts are equal, and the cost of simultaneously entering a same strike short (long) call and a long (short) put position is zero. At the strike price located at the forward a net zero cost position for purchase or sale of the stock at the expiration date can be established.
Figure 3. Forward Prices in an Option Chain - equal premium for puts and calls
At the money forward strikes have the highest time value for an option chain – above the forward option time values are a decreasing function – and are observable from calls with higher strikes than the forward; below the forward time values are a decreasing function and are observable from puts with lower strikes than the forward.
Figure 4. Observing Time Values in an Option Chain: Put strikes below at the money, call strikes above at the money.
Above the forward price, puts are in the money and calls out of the money. Below the forward price the inverse is true: calls are in the money and puts are out of the money. Out of the money options have no intrinsic value and their premiums isolate the time value of a strike – which is the same for calls and puts of the same strike.
The relationship between option implied forwards and funding implied forwards can in theory be distorted by american features of listed options. In most cases such distortion is negligible for strikes located near the forward.
Put call parity.
The funding driven relationship between premiums for puts and calls, across all strikes in an option chain. For European options the payout of a long call position (net of funding) can be replicated by combining a long stock and a long-put position. The payout net of funding of a put can be replicated by combining a short stock and a long call position. Prices of put and calls for the same expiration are consistent with stock prices, funding rates and with each other.
Deviations from put call parity allow riskless positions to be taken to lock in a gain above the funding (risk free) rate. The difference in premium between same strike and expiration european puts and calls is determined by their intrinsic value relative to the forward.
Put call parity, riskless bonds and early exercise.
All strikes in an option chain are connected to funding costs via put call parity. For any strike, a combination of a long position in the stock, a short european call and a long european put produces a riskless position, with a future value equal to the strike. Such position is riskless at the time of expiration, so the net cost of the three positions must converge to the price of a riskless bond, where the strike is the future value. In this context dividends are similar to coupons, and positions in fully hedged non dividend stocks are similar to zero coupon bonds.
Most listed stock options are american, allowing for early exercise. Conceptually, put call parity is a european option construct, which does not hold if early exercise is allowed. Relative to put call parity, american options may distort the equivalence of riskless combinations of stocks and options since early exercise can alter the time to expiration. Distortions in chains resulting from early exercise features are most relevant for deep in the money puts or high dividend stocks.
High strike long term put options may become optimal to exercise if the time value of the option becomes lower than the time value of the payout (invested to the expiration time horizon). Expected dividend payments can also impact american calls. A dividend payment implies an expected drop in the stock price. For an in the money call, early exercise may be optimal if the dividend exceeds the combined time value of the option premium and the cost of funding the position to expiration.
Volatility and option premiums.
Option premiums are non- negative; as they constitute a right, not an obligation for an option holder. Option premiums increase with volatility – a measure of dispersion. For the same underlying, strike and expiration, option premiums will be higher if the underlying volatility is higher. For a given underlying, implied volatilities reflect the expected dispersion of price changes in the underlying (their average move or standard deviation). Implied volatilities incorporate information from both historical volatilities (observed from past price movements) and expected volatilities (determined by the views market participants regarding future expected volatilities). Supply and demand of options – particularly when unbalanced - may also impact option premiums, which is in turn reflected in their implied volatility.
Options premiums as a function of time.
Implied volatility is a measure closely related in its construction to a standard deviation. Recall from statistics results that variances (squared standard deviations) are additive in time, so standard deviations are additive in the square root of time. By the same token, for a given constant implied volatility level, at the money option premiums grow approximately with the square root of time. The approximation is easy to observe from option chains with different expirations and the same underlying by comparing at the money forward premiums.
An unprecise yet useful back of the envelope approximation can be used to relate at the money option premiums, time to expiration and volatility:
At the money premiums, expressed as a percentage of underlying strike price are close to 0.4 times the volatility times the square root of time. An equivalent approximation is that volatility is about 2.5 times the at the money percentage premium divided by the square root of time.
The approximations are useful for eye-balling implied volatilities from option chains without the need of a complete model, for visually inspecting model results and for articulating approximations whenever working with a volatility estimate or from an observed premium.
As a function of time, the time value of out of the money options is less predictable than for at the money options. It is sensitive to the distance of the strike to the forward, the time to expiration of the option and to skews (discussed in later sections). Out of the money calls tend to command higher premiums than similarly distant from the at the money forward strike puts – which is most noticeable if the underlying is volatile, and less noticeable or not true if the underlying is an index or negatively skewed.
As strikes move further out of the money, the relationship for the same strike between premiums and time to expiration (and the time decay of options) gets closer to being linear in time – and for way out of the money options it can be quicker than time (so that at a far distance from the at the money strike a 1 year out of the money option commands less than half the premium of the same strike 2 year option).
Section 2. Option Functions
Deltas to the underlying and strike deltas.
Option deltas are mathematical expressions relating changes in the price of the underlying to marginal changes in the premium of a specific option. They are important in the hedging of options – as they are used to determine hedging ratios (units of the underlying to be bought or sold to offset the change in option premiums that result from moves in the underlying).
For puts and calls, deltas absolute values range from zero (way out of the money, to 1 (way in the money). Deltas are positive for calls and negative for puts – but are usually referred to by their absolute value. At the money deltas are close to 50% and get progressively smaller as options move out of the money, and higher as they move in the money - progressively approaching 100%. Deltas of puts and calls with the same strike exhibit symmetry and add up to 100%.
Deltas of options are determined by the sum of sensitivity to moves of the underlying of the time value and the intrinsic value. Changes in intrinsic value of an in the money option are partially offset by changes in the time value, as whenever the underlying moves further in the money it moves away from the strike, if it moves out of the money it gets closer to the strike.
As the underlying trades away from the strike in any direction the time value decreases. The change in intrinsic value is determined by the type of option and the direction of the move; it is either linear if the option is in the money and zero if it is out of the money.
The time value for a given strike is highest when the underlying is trading at the same price as the strike (on a forward basis). The change in time value is the same for either type of option and is determined by the strike and its position to where the underlying is trading. Changes in time values are highest when the underlying and the strike are near, and lower if they are far from each other.
Changes in delta are not constant as the sensitivity of options to the underlying changes as the underlying price changes. The variation in sensitivity of an option to the underlying is called gamma, which corresponds to the change in delta due to changes in the price of the underlying. Gamma is highest for at the money short term options, and lower for further out of the money and long term options.
Strike Deltas.
For purposes of this article, a strike delta is defined as the difference in premium for two adjacent strikes of the same expiration, divided by the difference in strikes of the two options in the spread. Each pair of options used to calculate a strike delta is also a vertical spread (a call spread or a put spread). Strike Deltas are similar to option deltas calculated from sensitivities to the underlying. They measure how option premiums change because of changes in strikes (not the underlying).
A critical distinction between deltas to the underlying and strike deltas is that strike deltas are not model based constructs, but observations derived from available market option premiums. When observing an option chain, calculating strike deltas does not require the use of an option pricing model, as strike deltas are observed directly by comparing market premium differences. The direct connection between strike deltas and vertical option spreads (call spreads and put spreads) facilitates using strike deltas to conceptualize and implement trading strategies.
Strike deltas also measure the marginal cost from selecting one strike versus selecting the immediately adjacent one. They capture the rate of change in premiums as strikes are set further away from the at the money forward. To preclude riskless opportunities for profit (arbitrage opportunities), the rate of decrease must be smaller as the distance of strikes from the forward increases – otherwise an out of the money vertical spread could be hedged with a closer to at the money spread.
Recall that at the money forward strike call and put premiums are equal. However, the strike delta is not necessarily equal for at the money forward strike put spreads and call spreads– only the premiums are equal at the at the money forward strike. As discussed in later sections, skews impact the rate at which option premiums change and impact strike deltas.
Figure 10. Observing strike deltas for calls in an option chain.
Implied Volatility, Skews and the Volatility Surface.
The use of standard option pricing models serves as a quoting convention when both dealers and other market participants use the same model. Other than volatility, other parameters used in a Black and Scholes formula are set independently of premiums and easy to observe in the market. Implied volatilities are calculated from observing option premiums (as traded in the market) and backing out the standard model implied volatility.
When backing out volatilities from observed option premiums, the implied volatility observed for each strike usually varies. Implied volatilities in option premiums may vary both for different strikes with the same expiration, and for similar strikes with different expirations.
Volatility functions are full sets of specific implied volatilities for each strike and expiration combination. Plotting all expirations and strikes together generates a volatility surface; plotting implied volatilities for a single expiration generates a volatility curve. Volatility surfaces are defined by 3 axes: volatility, time to expiration and strikes, or moneyness if strikes are expressed as ratios between the strike and the underlying price.
Figure 11 – Volatility Surface.
Volatility functions plotted for single expirations result in a curve, a single expiration slice of the volatility surface. Volatility curves are defined by two axes (volatility and strikes). The volatility curve may be flat (similar volatility for all strikes); skewed (biased) to higher volatilities for either low or high strikes (negatively or positively skewed) – or a smile (with higher implied volatilities for out of the money options on both higher and lower strikes).
Implied volatilities for the same underlying have a term structure which can be depicted as a curve – most commonly for at the money (vs the forward) options. Depending on the underlying, it can be upward sloping, in part because of interest rate volatility of longer time to expiration strikes. The term structure of volatilities can also exhibit some backwardation if mean reversion is a factor.
An option chain: the premium function for a single expiration.
Volatility curves are useful conceptual constructs, but do not represent directly tradeable instruments. A volatility curve for a single expiration is consistent with a corresponding premium function. An option chain displays the premiums that option market participants can obtain in the market, usually listed in an exchange or provided by an options broker.
Skews and their origin.
A standard Black and Scholes type of model assumes constant marginal percentage changes drive the behavior of the underlying in a manner consistent with a single level flat volatility curve. Market makers - key providers of liquidity in option markets - incorporate in their option pricing models features differentiating the behavior of each underlying as a variable – which may not correspond directly to a standard model, or use more elaborate versions of it, or incorporate other peculiarities of the underlying asset or its derivatives market.
Nonstandard features captured in option premiums may include stock momentum, different type of underlying distributions, unequal risk aversion in different directions, interrelations between stock prices and volatilities, increased correlation in the tails of indexes and impacts from imbalanced options supply and demand. From the lens of the standard Black and Scholes model, characteristics and asymmetries incorporated in option pricing create skews, which are implied volatility biases consistent with market observed option premiums.
Market examples of skews.
For index options, economic and market risks and higher volatility tend to be pronounced in the direction of lower strikes. Correlations among index constituents also tend to increase when economic fundamentals deteriorate, negating some of the diversification effects of index portfolios and resulting in increased index volatility when index constituents sell-off. As a result, compared to high strikes of the same expiration, low strike index options usually have higher implied volatilities.
The opposite effect is observed in option markets for which the underlying is momentum based – where higher strikes (in the same direction as the momentum) tend to be priced at higher implied volatilities. By the same token, options where the underlying is a heaven asset tend to exhibit skews towards high strikes.
The overdriving observation regarding skews, is that when the risk and expected speed of move of an underlying is asymmetric in different directions (above or below at the money strikes) the risk asymmetry tends to be captured in higher option premiums in the direction of the skew, which is reflected in directional skews of the option volatility function.
Follow this link for a Cboe article on index skews.
Section 3. Option strategies.
This section illustrates how the characteristics of option functions described in earlier sections of this post can be applied when articulating option strategies. The focus is on strategies using a single option and on vertical spreads.
Single Option Strategies.
Analyzing premiums and time values when creating single option strategies is straightforward as they can be directly observed from option chains. The most common strategies are outright options (calls, puts), and single option overlays (Covered calls, protective puts).
For any strike the time value component is the out of the money option (calls for high strikes and puts for low strikes). As mentioned earlier, the evolution over time of at the money premiums tends to follow the square root of time (assuming no movement in the underlying). The time decay components of out of the money options is less predictable and is affected by the time to expiration, the distance to the strike and skews.
Relative to the time decay of at the money options, the time decay of out of the money options is closer to linear – and faster than linear for way out of the money options. The expected time decay of an option (assuming the same underlying price) can be observed from option chains by comparing the out of the money premium of different expirations for the same strike.
Comparing premiums from chains across expirations is a static analysis with limitations and should be thought as an expected average base case. Future option premiums are unpredictable, because they depend on the future value of the underlying. Market volatilities and skews also evolve dynamically through time and affect premiums.
Covered Calls.
One of the most common option strategies, mostly used as an income generating strategy. A popular strike selection is slightly out of the money – balancing the premium received with some capital appreciation potential. Because at the money option premiums are linked to the square root of time, the choice of expiration is sometimes set shorter than the time horizon – seeking to maximize the annualized premium and assuming it is maintained on a rolling basis.
Covered calls can also be used for risk management as the premium received can offset the volatility of the underlying. Strike selection may be set lower when the strategy is used to control risk positioning.
Deep in the money LEAPS are a defensive positive carry strategy and can be used for funding optimization. In the money covered call positions held to expiration can absorb substantial market moves, offset by the time value and the distance to the low strike. Long term deep in the money covered calls behave like bonds around the initial underlying price but have higher tail risk. Note that the mark to market risk of a covered call with its strike set in the money at inception can increase substantially in a selloff even if the underlying is still trading above the strike.
Covered calls have three important drawbacks: First, they cap the potential payout of a position and thus limit the combined position appreciation potential. Second, losses in the position can far exceed the time value of the net premium received. A third and less obvious drawback (for in the money covered calls) is that the time value of the premium can increase in value substantially if the volatility increases and the underlying sells off - even if the underlying is trading above the strike. Such changes are due to changes in the time value of the option.
The mark to market risk of a covered call position can be higher than the perceived economic risk of the strike. High strike covered calls also experience a rapid increase in the option premiums if the underlying rallies – even if it is still trading below the strike. The rate of increase of the option’s time value is not constant and rises as the underlying increases in price towards the high strike, and peaks when the strike is at the money.
Protective puts.
Mostly used as a risk management tool, frequently by selecting out of the money (low) strikes to minimize option premiums. Incorporating out of the money puts constrains the payouts of the combined underlying and option position (or of a portfolio, when used systematically across all positions).
The use of protective puts for risk management has some advantages over managing risk based on a set of expected volatilities and correlations. A program of out of the money protective puts can set an exact limit (not subject to estimation error) to the maximum downside of both individual and aggregate positions, make the cost of risk to a given horizon explicit and improve portfolio convexity in market selloffs. The main drawback of protective puts is that the cost of premiums can create a drag on a position or portfolio.
Protective puts can be used as a building block when combined with other strategies. To avoid a drag in a portfolio, the premium cost of protective puts can be financed by selling options, usually out of the money calls or bear call spreads. Combining covered calls with protective puts can offer risk management precision and flexibility (from the many choices in strike location) – with potential payouts and net premiums being determined by the location of the strikes. The resulting strategy is a collar. A previous article in the blog discusses collars in more detail.
Funding optimization.
Protective puts can be inefficient from a funding perspective. The portion of value below the strike of a protective put does not constitute active risk – and net of the premium yields a return equal to the risk free rate. In line with the above put call parity discussion, the payout of the protective put strategy, net of funding is similar to the payout of a call with the same strike and expiration.
A strategy with a payout similar to a protective put is to use an outright call with the same strike – and replace the underlying asset value below the strike with another asset, which either finances the time value of the premium or has desirable characteristics such as high convexity, low or negative correlation with the directional portion of a portfolio or better compensation for tail risk.
The separation of funding and risk - illustrated above for a protective put - can be extended into different portfolio implementations. For example, portfolios can be split into active directional positions and funding assets. Bifurcation of positions for portfolio construction can also be incorporated via separate construction criteria for at the money (more active) and out of the money (tail like) risks.
Funding optimization can be aided by the use of minimum funding return criteria (for example a minimum annualized premium threshold or credit spread being applied to all positions) and marginal return criteria for at the money or near at the money risks and recovery views on a funding underlying.
Strategy variations for a single underlying.
To ponder the permutations of basic option strategies, it is useful to think of the relationship between risk and funding of option portfolios in a simplified framework resulting from put call parity. Each permutation can easily be observed from an option chain.
Conceptually, the minimum risk to a given horizon is a riskless bond maturing on the expiration date; in the context of an option chain, the riskless bond is constructed from a combination of the underlying and same strike short calls and long puts.
The selection of a specific strike (the same for the long put and the short call) is a starting point that determines the overall positioning for different permutations. For long positions high strikes are bullish and low strikes are conservative. Risk can be added marginally to the riskless position by creating a distance between the starting point strikes: choosing a higher strike for the call or a lower strike for the put – at which point the portfolio becomes a collar – a wider collar has higher risk and higher potential return.
Dispensing of the short call creates a protective put – a directional long position. Dispensing of the long put creates a covered call – a bet that the asset price will not fall significantly below the strike. Portfolio leverage for any net long positions can be controlled by replacing with a call the protective put component (simultaneously selling the put and the underlying).
Short positions can be built from dispensing of the covered call (and the underlying) and preserving the put. The cost of put premiums can be improved by selling lower strike puts, which also allow recovery views to be incorporated. The difference in premiums of adjacent strikes is the marginal cost of buying or selling the options and can be used to observe strike deltas.
Option strategies and skews.
Implied volatility skews have a direct impact on premiums, which are higher in the direction of the skew. Relative to flat volatility premiums, the impact of directional skews is asymmetric – increasing premiums of options with strikes located in the direction of the skew and decreasing premiums of options with strikes away from the direction of the skew.
An important observation regarding the attractiveness of options strategies is that the impact of skews on strike deltas and vertical spreads is in the opposite direction of their impact on option premiums. As skews increase, the relative price of out of the money options in the direction of the skew increases compared to at the money options. As a result, strike deltas and vertical spreads net premiums in the direction of a skew decrease as skews increase - since they are determined by the difference of option premiums.
To summarize: In the direction of the skew, vertical option spreads with payouts in the same direction become attractive relative to outright options. If volatility is skewed (is higher) towards low strikes (a negative skew, as in equity indexes) bear put spreads become attractive. If volatility is skewed towards high strikes (a positive skew as in heaven assets and momentum stocks) bull call spreads become attractive. By the same token, outright options with strikes located in the side opposite to the skew are usually priced attractively (command lower premiums) relative to options with strikes located on the direction of the skew, which command higher premiums.
The presence of a skew is easy to spot in an option chain by observing the premiums of calls and puts located at an equal distance to the forward. A useful metric when comparing premiums of strikes located above and below the forward is the at the money premium for the same expiration – which can be used as a yardstick. For example, out of the money options with strikes located one premium up (calls) and one premium down (puts) from the forward can be observed to check for asymmetries in the distribution of time values.
At the money premiums for a given expiration are connected to volatilities in a predictable manner: the implied (standard deviation) dispersion is about 2.5 times the at the money premium. When the distance between strikes is set at a constant number of premiums away from the at the money strike, it is using a measure derived from implied volatilities and standard deviations – and directly connected to the expected dispersion of potential outcomes for the underlying. Comparing premiums in such manner helps create a simplified and consistent framework across expirations and underlying assets.
Figure 23 – Hypothetycal Option Chain – Flat volatility example.
Out of the money options have similar premiums when the volatility curve is flat – Comparing similar distance strikes from at the money options
Figure 24 – Hypothetycal option chain – Positive skew example.
High strike calls are more expensive than low strike puts when the volatility curve has a positive skew – Comparing similar distance strikes from at the money strike
Figure 25 – Hypothetycal Option Chain – Negative skew example.
Low strike puts are more expensive than high strike calls when the volatility curve has a negative skew – Comparing similar distance strikes from at the money strike
Relative value application to specific strategies.
The direction of skews influences the relative attractiveness of option spreads versus outright options in a predictive manner that can be used to guide strategy selection. The relative impact of skews on bull call spreads and bear put spreads is inverse to the impact in option premiums. A skew that makes outright options more expensive makes spreads in the same direction less expensive.
Positive skews (high strikes having higher implied volatilities and premiums) result in less expensive bull call spreads, more expensive high strike calls and less expensive out of the money puts. Attractive strategies to capture positive skews include bull call spreads, collars, low strike puts for lower volatility assets, out of the money call strikes for covered calls. The combination of high volatilities and positive skews can create relatively inexpensive in the money bull call spreads and collars with attractive payouts.
Figure 26 – Hypothetycal Option Chain – Positive skew and strategies.
Bull call spreads and low strike puts are relatively attractive. Skew directional impact is inverse in option premiums compared to option spreads.
Negative skews (low strikes having higher implied volatilities and premiums) result in inexpensive bear put spreads, inexpensive high strike calls and more expensive out of the money puts. Attractive strategies to capture negative skews include bear vertical spreads, long out of the money calls, risk reversals and if the skew is very pronounced ratios (e.g. selling an in the money covered call, buying two out of the money calls – with the strikes located at similar distance).
Figure 27 – Hypothetycal Option Chain – Negative skew and strategies.
Bear put spreads and high strike calls are more attractive. Skew directional impact is inverse in option premiums compared to option spreads.
Using skews to select vertical spreads to a given horizon helps identify attractive payouts but has limitations in the context of rebalancing. When the underlying moves the value of the spread might move less than expected, and at times the skew might increase as the underlying moves towards strikes in the direction of the skew – potentially making early rebalancing of a spread position difficult and the position less attractive from a mark to market perspective.
An example is put spreads used as a hedge for an index – when the index sells off the skew can sometimes simultaneously increase, partially offsetting the increase in intrinsic value, and rendering a poor performance of the spread as a mark to market hedge. If the position is held to expiration an attractive risk reward can be captured – but in the process the mark to market may be less attractive.
Using strike deltas in strike selection and rebalancing.
There is a tradeoff in strike selection. Further out of the money options and vertical spreads are increasingly cheaper, but their payouts also have lower probability of occurring. Out of the money options are subject to increased time decay. As option strikes come closer to at the money, options have increasingly higher probability of a positive payout but command higher premiums.
Establishing criteria to decide at which point the increasing premium cost out weights the increasing probability of receiving a payout is important for consistent position management. Strike deltas can be used to articulate trading and rebalancing criteria for options strategies. A separate article in this blog discusses two approaches to the above tradeoff, both from a perspective of marginal costs – measured by strike deltas. The idea is to observe marginal costs between strikes and use them to select strikes.
The first approach is to arbitrarily set a maximum marginal cost using a utility-based criteria derived from risk reward preferences. The second approach is to find optimal marginal costs from an optimization standpoint, in which the change in strike deltas is observed as a curve to find an optimal point, at which the rate of change in deltas decreases.
Marginal cost criteria can be included by establishing the maximum marginal price to pay when initiating an option position, as well as the marginal price at which the position is rebalanced or sold. For example, a call position strike can be chosen by purchasing an option with an initial strike delta of 50% or less. If the underlying moves favorably the option becomes further in the money as the position becomes profitable. It can be rebalanced into a higher strike if the marginal cost (the strike delta) reaches a threshold, for example when the position strike delta becomes 75%.
For an option spread used to capture directional views (e.g. a bull call spread or bear put spread) there is two strikes for which a decision is needed – a near or further in the money strike (for the option being purchased) and a far or further out of the money strike (for the option being sold).
Marginal cost criteria are easier to conceptualize and implement for an option being purchased (where the strike delta represents a cost weighted against a potential payout) than for an option being sold (where the strike delta represents the premium saved, weighted against giving up a potential expected payout in the direction of the view being captured.
A similar consideration to the option being sold in a spread position is at which point a long position in a single option that is becoming out of the money and progressively losing value should be exited or rebalanced. The optimal location of an out of the money strike being sold is less obvious, and depends on the market view being captured, the expected speed and size of moves in the underlying and the motivations for using an option strategy to capture the market view or signal.