Option valuation models

Option valuation models

Option valuation models use mathematical techniques to identify a range of possible future share prices at the exercise date. From these possible future share prices, the pay-off of an option can be calculated. These intrinsic values at exercise are then probability-weighted and discounted to their present value to estimate the fair value of the option at the grant date.

This narrative is part of the IFRS 2 series, look here.

Model selection

There are three main models used to value options:

• closed-form models: e.g. the BSM model;
• lattice models; and
• simulation models: e.g. Monte Carlo models.

These models generally result in very similar values if the same assumptions are used. However, certain models may be more restrictive than others – e.g. in terms of the different pay-offs that can be considered or assumptions that can be incorporated.

For example, a BSM model incorporates early exercise behaviour by using an expected term assumption that is shorter than the contractual life, whereas a lattice model or Monte Carlo model can incorporate more complex early exercise behaviour.

Simple model explanation

The approach followed in, for example, a lattice model illustrates the principles used in an option valuation model in a simplified manner.

Share price distribution Calculate a distribution of possible future share prices from grant date to the exercise date. The share price distribution depends on: the opening share price the risk-free rate volatility time dividends. Option value 2. Calculate the option pay-off at the exercise date (t) for each share price node – i.e. Max (0, S – X). The pay-off depends on the share price at the equivalent node in the share price tree and the exercise price – i.e. the intrinsic value. When the share price is below the exercise price, the intrinsic value at the exercise date is zero. 3. Present value, probability weight the future intrinsic values.

There are clearly many other possible future share price paths beyond the two stage up (u), down (d) shown above. Other paths have been excluded for simplicity. The formulae for u and d are provided in A2.100. Each different price point in a lattice model is referred to as a node.

Lattice models can also consider early exercise of an option at intermediate nodes so that the value of the option is not necessarily just the discounted, probability-weighted terminal intrinsic values. Such early exercise might be assumed based on the ratio of the share price at a node to the exercise price (S/X), post-vesting termination rates, payment of dividends etc.

Share price modelling does not depend on estimates of future share prices from management and/or valuation advisers. The future share prices and associated probabilities are derived from the model’s assumptions – i.e. the current share price, the risk-free rate, volatility, time and dividends.

Model selection

Model selection will depend, in part, on the complexity of an award (e.g. an award with a market condition cannot be valued using BSM) and the ability to estimate more reliable predictors of early exercise behaviour than time. Therefore, although a lattice or Monte Carlo model is able to incorporate more complex early exercise behaviour, an entity will be able to take full advantage of such features only if reliable early exercise parameters are available.

Black-Scholes-Merton

The BSM model is a closed-form equation that values plain vanilla options – i.e. options with the standard pay-off formula of the greater of zero and the share price at the end of an option’s term less the option’s exercise price.

The key advantage of the BSM model is its simplicity: it is readily available over the internet and relatively straightforward to implement, even by those with limited understanding of the underlying mathematics, once a reliable model is obtained.

However, it cannot be employed to value options with complex pay-offs – e.g. an award with a market condition. Moreover, because the BSM model is based on expected share prices at the end of an option’s term and it cannot consider early exercise, it may not be appropriate for options when early exercise is likely – e.g. for shares that are dividend-paying.

The formula for the BSM is as follows:

In the formula above:

• S is the share price;
• N() is the cumulative distribution function of the standard normal distribution;
• X is the exercise price;
• T is the expected term;
• q is the continuously compounded dividend yield;
• r is the continuously compounded risk-free rate; and
• σ is the volatility in the log-returns of the underlying share.

Lattice

A lattice model builds a lattice of future share prices from which option values can be calculated. Examples of lattice models include binomial and trinomial models. In a binomial model, shares can move to one of two possible outcomes from any given point, whereas in a trinomial model, share prices can move to one of three possible outcomes from any given point (hereafter also referred to as ‘nodes’).

Advantages of a lattice model include the following.

• Because it shows share price movements over an option’s term, early exercise parameters can be built into the model. For example, if there is evidence that option holders exercise their options when an entity’s share price is twice the amount of an option’s exercise price, then a lattice model can be built that monitors points or nodes in a share price lattice at which the share price is twice the amount of the exercise price and assumes exercise at that point.
• More complex pay-offs can be measured. For example, if an option has a capped return, then this could be incorporated into a pay-off formula. The standard pay-off formula for a plain vanilla option at exercise is its intrinsic value, which can never be negative. In the case of a capped option, the pay-off formula would be the lower of the intrinsic value or the cap.
• A lattice model may be used to measure an award with a market condition when the market condition is measured on the exercise date. For an award with a market condition that is measured on the exercise date, the pay-off formula at the terminal nodes would check if the share price was greater than that required under the market condition, in which case intrinsic value would be assumed; otherwise the pay-off would be zero. However, if a market condition is measured at a date other than the exercise date, which is normally the case, then a Monte Carlo model may be more appropriate. The reason for this is that there are a number of routes through a lattice by which a market condition may be met and by which a terminal node is reached and it is difficult to monitor which route may have been followed. This is illustrated below.

The pay-off under a share option with no market condition would be the same for different share price paths that end at the same terminal node. If an award has a market condition that is measured before the end of the expected term, then both the share price path taken and the terminal node reached will be important. For example, assume an option award with a market condition that the share price must exceed 12 after three years. Without a market condition, the intrinsic value at exercise would be the same for paths A and B. However, only path A meets the market condition so even though path B ends in the money, the pay-off to the recipient under B would be 0 because the market condition is not met. The difficulty of mapping share price paths through a lattice model generally makes Monte Carlo models easier to use for market conditions.

Implementing a lattice model

A lattice model can be built in a spreadsheet, once certain core formulae are known. The ability to build a lattice model allows a valuer the flexibility of modelling awards with complex pay-offs not modelled easily using closed-form models such as the BSM.

The steps in valuing an option using a lattice model are as follows.

1. Calculate the distribution of possible future share prices from grant date to the end of the option term (moving left to right). The share price distribution depends on the opening share price (S), the risk-free rate, volatility, time and dividends.
2. Calculate the option pay-off at the end of the option term (t) for each share price node. The pay-off depends on the share price at the equivalent node in the share price tree and the exercise price (X) – i.e. the intrinsic value. If the share price is below the exercise price, then the intrinsic value at the exercise date is zero.
3. Calculate the discounted, probability-weighted intrinsic values at the terminal nodes, working backwards in the lattice (moving right to left).
4. Consider early exercise of the option at intermediate nodes. Such early exercise might be assumed based on the ratio of the share price to the exercise price [S/X], post-vesting termination rates, dividends etc.

Calculating the share price distribution

From the opening share price (S), the share price is assumed to be able to increase to a value (Su) or decrease to a value (Sd; ‘u’ is referred to as the ‘up factor’ and ‘d’ is referred to as the ‘down factor’).

From each of these nodes, the share price can then increase by a factor (u) or decrease by a factor (d). The probability of an increase is (p) and the probability of a decrease is (1-p); these probabilities are not necessarily equal.

This can be seen as follows.

5. These parameters are constant over the contractual life of the option in most lattice models.

Because there are only two possible moves, the probability of either an increase or a decrease must be equal to one. This can be seen as follows: the probability of an up and down move is p + (1 – p) = 1 (the ps cancel out).

This can be implemented as follows.

Notes

1. Su = 10 x 1.3269.

2. Sd = 10 x 0.75364.

3. Suu = 10 x 1.3269 x 1.3269.

4. Sud = 10 x 1.3269 x 0.75364 or Sdu = 10 x 0.75364 x 1.3269.

Notes

1. p x p – i.e. 0.4557 x 0.4557.

2. p x (1 – p) + (1 – p) x p – i.e. 0.4557 x (1 – 0.4557) + (1 – 0.4557) x 0.4557.

3. (1 – p) x (1 – p) – i.e. (1 – 0.4557) x (1 – 0.4557).

4. The option value at this node is equal to the probability-weighted present value of the succeeding nodes – i.e. ((21 x 45.47%) + (7.61 x (1 – 45.47%))) x exp (-0.05 x 2.96%).

The following points are relevant in relation to the diagram.

• A very small number of nodes has been used to simplify presentation of the analysis – i.e. only four time periods. Based on an assumed time period of two years, share prices are calculated only every six months. Significantly more calculated share prices would be used in an actual valuation (e.g. 100 sub-periods).
• Even with only four time periods, the resulting lattice-derived value is within 5.6 percent of the value reached under the BSM model. Increasing the number of nodes would cause the lattice value to converge to the BSM value, if the same assumptions are used.
• More complex lattice models could consider early exercise in the periods before the end of the contractual life. Therefore, the lattice model can be amended to include more complex assumptions than BSM.

Monte Carlo

A Monte Carlo model simulates future share prices. The share prices can be simulated at the end of the expected term or, if the share price path is needed from the grant date through to the end of the expected term, more frequent estimation of share prices is possible.

Once the share price at the end of the option’s term has been simulated, the pay-off to the option can be calculated based the option’s pay-off function. As with a lattice model, because of the transparency of the simulated share prices, it is possible to model complex pay-offs using Monte Carlo simulation. The average pay-off is then discounted to the present value using the risk-free rate.

More complex modelling may be required when it is necessary for the Monte Carlo model to consider early exercise of the option. In these circumstances, techniques such as a regression-based least squared method may be required. Detailed discussion of such techniques is beyond the scope of this narrative.

Monte Carlo simulation is computationally intensive because a high number of simulations have to be run to ensure a sufficiently large sample. Generally, tens of thousands of simulations of share prices are run. If it is necessary to model share prices over the expected term, then this will increase the computational intensity.

However, it is not generally necessary to run share price simulations over each trading day before the end of the expected term.

For example, if there is a fixed date on which achievement of the market condition is measured and the end of the expected term is later, then two simulations can be performed for every share price path: one to the measurement date of the market condition, and the second to the end of the expected term.

More frequent simulations may be required when early exercise parameters are applied over an award’s contractual life.

The results of a Monte Carlo simulation can be tested by assessing whether the average simulated future share price is broadly equal to the current share price ‘increased’ at the risk-free rate less the dividend yield over the expected term.

The formula in a Monte Carlo simulation for each simulated share price is as follows.

In this formula:

• rf is the risk-free rate (continuously compounded);
• q is the dividend yield (continuously compounded);
• σ is volatility;
• t is the start of the time period;
• ∆t is the length of the time period over which the share price is simulated, expressed in years; and
• z is a random number.

It can be seen from the above that the formula contains two elements: (1) the share’s drift, based on the risk-free rate less the dividend yield, which is the same in all simulations; and (2) a variable element based on the application of a random number to the share’s volatility that changes in each simulation as a new random number is generated.

Comparing option valuation models

Normally, the BSM model is applicable only for European-style options and the lattice model can be applied to both European and American options. Although the Monte Carlo is normally used to value European options, it can be modified to estimate the value of American options.

However, the use of an expected term assumption, under which an option is assumed to be exercised at the end of the expected term, which is shorter than the contractual life, essentially converts an American or Bermudian option into a European option from a financial modelling perspective because early exercise is reflected in the input assumption into the option pricing model rather than in the model itself.

A comparison of different option valuation models is set out below.

Complex capital structures

Complexity arises in valuing shares or share options when the entity has a complex capital structure – i.e. when the entity that grants shares or share options in a share-based payment award has different forms of capital (e.g. ordinary and preferred shares, and/or convertible instruments).

In these circumstances, the valuation of individual elements of the capital structure in order to value shares or share options granted is difficult because of the interacting rights of different securities. Any valuation of individual elements of the capital structure should consider their specific contractual terms.

The issues that arise in these circumstances can be illustrated using a simplified example. Company P is acquired by a private equity fund for 100. The transaction is funded using 65 of third party debt at 5 percent and 35 from the private equity fund.

Thirty of the 35 is attributed, without doing a formal valuation, to preferred shares with a coupon of 6 percent, recognised at par. The residual amount of 5 is attributed to ordinary equity. P then issues ordinary shares or options on ordinary shares to management based on the residual amount attributed to the ordinary shares in the transaction.

Considered from a commercial perspective, the preferred shares have limited upside potential (6 percent per annum), whereas they carry significant downside risk because there is only a very limited equity ‘cushion’ in the capital structure. In reality, the preferred coupon is probably below market rates for instruments with similar risk profiles and as such the value of the preferred shares is economically below their par value.

If, in the example above, the value of P increases by 35 percent at the end of one year, then the third party debt will have earned interest of 3.25, the preferred shares will have earned a coupon of 1.8 and the balance of the increase in value of 29.95 will have been to the benefit of the ordinary shareholders, a rate of return of almost 600 percent.

If, however, the value of the entity decreased by 35 percent at the end of one year, then the ordinary shareholders and the preferred shareholders will have been wiped out.

In this example, although the ordinary shareholder bears significant risk, they are compensated for this risk through very large upside. The preferred shareholders also bear significant risk without the benefit of significant upside to compensate. This is illustrated below.

Although this example highlights that the preferred shares may not be worth their par value when they are issued, the private equity fund is likely to focus on the value of its aggregate position and may not differentiate between the value of its ordinary shareholding and its preferred shareholding in P.

From this viewpoint, when the investment is made, undervaluation of the ordinary shares would compensate for the extent to which the preferred shares are overvalued.

Although the private equity fund in these circumstances focuses on its aggregate position and may be unconcerned about undervaluation of the ordinary shares, such undervaluation would lead to understatement of the value of the ordinary shares or options granted to management and the related share-based payment cost.

There are several techniques that may be used to value complex capital structures. One of the most comprehensive descriptions of possible approaches in this area is the practice aid produced by the American Institute of Certified Public Accountants, Valuation Of Privately Held Company Equity Securities Issued As Compensation, which, although it is not authoritative and not developed for application to IFRS 2, provides a useful discussion of methods that may be used to value share-based payments in these circumstances.

In particular, it sets out approaches that can be followed to value complex capital structures, once the value of the aggregate entity has been estimated using standard valuation techniques such as market multiples, DCF etc. These enterprise value allocation techniques include the following.

• The current value method: This method assumes that the business is sold at the valuation date for the enterprise value, which is distributed among the elements of the capital structure, based on their contractual rights. The weakness of this method is that it ignores the potential upside inherent in the instruments. Therefore, the current value method would value the ordinary shares at 5 in the example above.
• Probability-weighted expected return (PWER) approach: The PWER approach looks at different scenarios under which the investment would be expected to be realised – e.g. IPO, trade sale etc. This approach estimates the probability of each such scenario and calculates the pay-off to different elements of the capital structure under each scenario. The present value of the probability-weighted pay-offs of each instrument under all scenarios represents the value of the elements of the capital structure. The difficulty in applying the PWER method is the estimation of the different possible outcomes and the probabilities associated with such outcomes. The PWER approach may be especially useful close to an expected exit event – e.g. during the preparation for an upcoming IPO.
• The option pricing method: This method uses option mathematics to value different elements of the capital structure. Under this approach, various levels of security holders in the entity are seen as holding a series of options.

Under the option pricing method, using the same example, the debt holders are seen as owning P but having written an option to the preferred shareholders to buy P for an exercise price equal to the third party debt including accumulated interest.

The preferred shareholders’ holding is seen as equal to the value of the option written by the debt holders (which has an exercise price equal to the debtor holders’ claims on the business, both principal and interest) net of the option that the preferred shareholders have written to the ordinary shareholders (which has an exercise price equal to the par value of the preferred shares plus the preferred dividends plus the amount payable to the debt holders).

In essence, P can be seen as a cascading chain in which it is owned in the first instance by the entity that has the most senior security – i.e. the debt holders – but in which the upside can be secured by those further down in security in the capital structure by paying off those elements of the capital structure higher up in security. The amount to be paid off is treated as the exercise price of the option. This can be illustrated using the BSM as follows.

This table can be interpreted as follows.

• The debt holders have the first claim on the value of P – i.e. they have the right to be repaid before other providers of capital. Once they have been repaid, any residual value is available to the other sources of finance. The first right to enterprise value can be modelled as a call option with an exercise price of zero. This option is referred to as Option 1 in the table and has a value of 100, equal to the entity’s current enterprise value.
• Once they have been repaid, the debt holders do not have any further right to the remaining enterprise value of P. This can be shown to be equivalent to the debt holders writing (or giving) the next highest ranking source of capital, in this case the preferred shareholders, a call option on the enterprise value with an exercise price equal to the future amount due on the debt (71.7). Essentially, the preferred shareholders have a choice to repay the debt to secure the residual future enterprise value. If the amount due on the debt is greater than the enterprise value, then the option will not be ‘exercised’ and the debt holders secure all of the enterprise value at that point in time. However, if the enterprise value at that point in time is greater than the outstanding debt, then holders of the preferred securities will ‘exercise the option’ by paying off the debt and securing the remaining enterprise value. This is referred to as Option 2 in the table and has a value of 35.
• The preferred holders’ claim on the business is also fixed. Therefore, if they are paid off in full, then any residual value is available to the ordinary shareholders. The ordinary shareholders have a residual claim on the business – i.e. they have the right to receive the upside on the business once the debt holders and preferred shareholders are paid off. This can be viewed as a call option written by the preferred shareholders giving the ordinary shareholders the right to acquire the enterprise value with an exercise price equal to the required pay-off to the debt and preferred shareholders. In certain scenarios, the enterprise value will be below the debt and preferred shareholders’ claims and the ordinary shares would have no value. This is equivalent to an option being out-of-the-money. In other scenarios, the enterprise value may exceed the debt and preferred shareholders’ claims and the ordinary shares would have value. This is equivalent to the option being in-the-money. This is shown as Option 3 in the table and has a value of 17.24.
• The value of each security is equal to the value of the implicit options that they hold, net of the implicit options that they have written. The table above shows that the values of debt, preferred and ordinary shares are 65, 17.8 and 17.2, respectively. The difference between the fair values identified above and the values attributed on a par value basis reflects the option characteristics of the securities. The par value of the securities was based on their rights assuming an immediate wind-up, which is equivalent to the current value method. This approach is equivalent to the intrinsic value of the securities. The option pricing method reflects both intrinsic and time value.
• The amount of the share-based payment is the difference between the value of the ordinary shares and the amount paid by management. For example, if the option pricing method was used to value ordinary shares granted to or acquired by management, with a price paid per share of 5, and the value per share under this approach is 17.2, then the price paid of 5 would result in a share-based payment of 12.2. This differs from the result under the current value method, under which the value of the shares and the amount paid are both 5, resulting in a share-based payment of zero.
• The valuation depends on similar assumptions to any option valuation, including time, volatility, risk-free rate etc. The time assumption, in particular, is a matter of judgement because there may be no contractual restriction on the life of an individual security. Factors to be considered include the expected holding period of investors – e.g. private equity or venture capital investors may target an exit event in three to seven years – the ability of investors to delay a liquidity event if circumstances are unfavourable and the willingness of debt holders to continue to finance the entity, which is especially important when the approach is applied to heavily indebted entities.

The table uses the BSM model to value the embedded options. These could also be valued using a lattice model, which is useful particularly for securities with more complex rights.

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