• The option payoffs at the final time node (time 5 above) must be calculated. This is the share price less
the exercise price, or zero if the payoff is negative.
• Then the option values must be calculated at the previous time point (time 4 above). This is done by
calculating the expected value of the option for the branch paths available to the particular node being
valued discounted at the risk-free rate. For example, for S4,4 in the chart above the option value is the
probability of going to node S5,5 multiplied by the option value at that node plus the probability of going
to node S5,4 multiplied by the option value at that node, all discounted at the risk-free rate):
=
e–r.dt{p.111.82 + (1 – p)34.82}= 0.95{0.4068 × 111.82 + 0.5932 × 34.82} = 62.92
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This would be the value at node S4,4 if the option were European and could not be exercised earlier.
As the binomial model can allow for early exercise, the option value at node S4,4 is the greater of the
option value just calculated and the intrinsic value of the option which is calculated the same way as
the end option payoff. In this case, as the intrinsic value is $63.89 ($73.89 – $10.00), the node takes
the value of $63.89.
• The previous steps are then repeated throughout the entire lattice (i.e. for all nodes at time 4, then all
nodes at time 3, etc.) until finally the option value is determined at time 0 – this being the binomial
option value of $4.42.
• Additionally, if there is a vesting period during which the options cannot be exercised, the model can be
adjusted so as not to incorporate the early exercise condition stipulated in the previous point and allow
for this only after the option has vested and has the ability to be exercised before expiry.
One of the advantages of a lattice model is its ability to depict a large number of possible
future paths of share prices over the life of the option. In Example 30.33 above, the
specification of an interval of 12 months between nodes provides an inappropriately
narrow description of future price paths. The shorter the interval of time between each
node, the more accurate will be the description of future share price movements.
Additions which can be made to a binomial model (or any type of lattice model) include
the use of assumptions that are not fixed over the life of the option. Binomial trees may
allow for conditions dependent on price and/or time, but in general do not support
price-path dependent conditions and modifications to volatility. This may affect the
structure of a tree making it difficult to recombine. In such cases, additional
recombination techniques should be implemented, possibly with the use of a trinomial
tree (i.e. one with three possible outcomes at each node).
For the first three assumptions above, the varying assumptions simply replace the value
in the fixed assumption model. For instance, in Example 30.33 above r = 0.05; in a time-
dependent version this could be 0.045 at time 1, 0.048 at time 2 and so on, depending
on the length of time from the valuation date to the individual nodes.
However, for a more complicated addition such as assumed withdrawal rates, the equation:
= e–r.dt {p.111.82 + (1 – p)34.82}
may be replaced with
= (1 – g) × e–r.dt {p.111.82 + (1 – p)34.82} + g × max (intrinsic value, 0)
where ‘g’ is the rate of employee departure, on the assumption that, on departure, the
option is either forfeited or exercised. As with the other time- and price-dependent
assumptions, the rate of departure could also be made time- or price-dependent (i.e. the
rate of departure could be assumed to increase as the share price increases, or increase
as time passes, and so forth).
8.3.2.A
Lattice models – number of time steps
When performing a lattice valuation, a decision must be taken as to how many time
steps to use in the valuation (i.e. how much time passes between each node). Generally,
the greater the number of time steps, the more accurate the final value. However, as
more time steps are added, the incremental increase in accuracy declines. To illustrate
the increases in accuracy, consider the diagram below, which values the option in
2626 Chapter 30
Example 30.33 above as a European option. In this case, the binomial model has not
been enhanced to allow for early exercise (i.e. the ability to exercise prior to expiry).
Option value
$5.00
$4.80
$4.60
$4.40
$4.20
$4.00
$3.80
$3.60
1
3
5
7
9
11
13
15
17
19
35
65
100
Time steps
Whilst the binomial model is very flexible and can deal with much more complex
assumptions than the Black-Scholes-Merton formula, there are certain complexities
it cannot handle, which can best be accomplished by Monte Carlo Simulation –
see 8.3.3 below.
The development of appropriate assumptions for use in a binomial model is discussed
at 8.5 below.
In addition to the binomial model, other lattice models such as trinomial models or finite
difference algorithms may be used. Discussion of these models is beyond the scope of
this chapter.
8.3.3
Monte Carlo Simulation
In order to value options with market-based performance targets where the market
value of the entity’s equity is an input to the determination of whether, or to what
extent, an award has vested, the option methodology applied must be supplemented
with techniques such as Monte Carlo Simulation.
TSR compares the return on a fixed sum invested in the entity to the return on the
same amount invested in a peer group of entities. Typically, the entity is then
ranked in the peer group and the number of share-based awards that vest depends
on the ranking. For example, no award might vest for a low ranking, the full award
might vest for a higher ranking, and a pro-rated level of award might vest for a
median ranking.
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The following table gives an example of a possible vesting pattern for such a scheme,
with a peer group of 100 entities.
Ranking in peer group
Percentage vesting
Below
50
0%
50
50%
51-74
50% plus an additional 2% for
each increase of 1 in the ranking
75 or higher 100%
Figure 30.5 below summarises the Monte Carlo approach.
Figure 30.5:
Monte Carlo Simulation approach for share-based payment
transactions
Performance projections
Option valuation
Value option using
Binomial option
pricing model
Yes
Simulate
Has the
performance:
performance
Update average
Repeat (up to desired
– Company
hurdle been
option valuations
number of simulations)
– Index
achieved?
No
Value of
option = 0
The valuation could be performed using either:
• a binomial valuation or the Black-Scholes-Merton formula, dependent on the
results of the Monte Carlo Simulation; or
• the Monte Carlo Simulation on its own.
The framework for calculating future share prices uses essentially the same underlying
assumptions as lie behind Black-Scholes-Merton and binomial models – namely a risk-
neutral world and a log normal distribution of share prices.
For a given simulation, the risk-neutral returns of the entity and those of the peer group
or index are projected until the performance target is achieved and the option vests. At
this point, the option transforms into a ‘vanilla’ equity call option that may be valued
using an option pricing model. This value is then discounted back to the grant date so
as to give the value of the option for a single simulation.
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When the performance target is not achieved and the option does not vest, a zero value
is recorded. This process is repeated thousands or millions of times. The average option
value obtained across all simulations provides an estimate of the value of the option,
allowing for the impact of the performance target.
8.4
Adapting option-pricing models for share-based payment
transactions
Since the option-pricing models discussed in 8.3 above were developed to value freely-
traded options, a number of adjustments are required in order to account for the
restrictions usually attached to share-based payment transactions, particularly those
with employees. The restrictions not accounted for in these models include:
• non-transferability (see 8.4.1 below); and
• vesting conditions, including performance targets, and non-vesting conditions that
affect the value for the purposes of IFRS 2 (see 8.4.2 below).
8.4.1 Non-transferability
As noted at 8.2.3.A above, employee options and other share-based awards are almost
invariably non-transferable, except (in some cases) to the employee’s estate in the event
of death in service. Non-transferability often results in an option being exercised early
(i.e. before the end of its contractual life), as this is the only way for the employee to
realise its value in cash. Therefore, by imposing the restriction of non-transferability,
the entity may cause the effective life of the option to be shorter than its contractual
life, resulting in a loss of time value to the holder. [IFRS 2.BC153-169].
One aspect of time value is the value of the right to defer payment of the exercise price
until the end of the option term. When the option is exercised early because of non-
transferability, the entity receives the exercise price much earlier than it otherwise
would. Therefore, as noted by IFRS 2, the effective time value granted by the entity to
the option holder is less than that indicated by the contractual life of the option.
IFRS 2 requires the effect of early exercise as a result of non-transferability and other
factors to be reflected either by modelling early exercise in a binomial or similar model
or by using expected life rather than contractual life as an input into the option-pricing
model. This is discussed further at 8.5.1 below.
Reducing the time to expiry effectively reduces the value of the option. This is a
simplified way of reducing the value of the employee stock option to reflect the fact
that employees are unable to sell their vested options, rather than applying an arbitrary
discount to take account of non-transferability.
8.4.2
Treatment of vesting and non-vesting conditions
Many share-based payment awards to employees have vesting and non-vesting
conditions attached to them which must be satisfied before the award can be exercised.
It must be remembered that a non-market vesting condition, while reducing the ‘true’
fair value of an award, does not directly affect its valuation for the purposes of IFRS 2
(see 6.2 above). However, non-market vesting conditions may indirectly affect the
value. For example, when an award vests on satisfaction of a particular target rather
than at a specified time, its value may vary depending on the assessment of when that
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target will be met, since that may influence the expected life of the award, which is
relevant to its fair value under IFRS 2 (see 6.2.3 and 8.2.2 above and 8.5 below).
8.4.2.A
Market-based performance conditions and non-vesting conditions
As discussed at 6.3 and 6.4 above, IFRS 2 requires market-based vesting conditions and
non-vesting conditions to be taken into account in estimating the fair value of the options
granted. Moreover, the entity is required to recognise a cost for an award with a market
condition or non-vesting condition if all the non-market vesting conditions attaching to the
award are satisfied regardless of whether the market condition or non-vesting condition is
satisfied. This means that a more sophisticated option pricing model may be required.
8.4.2.B Non-market
vesting conditions
As discussed at 6.2 above, IFRS 2 requires non-market vesting conditions to be ignored
when estimating the fair value of share-based payment transactions. Instead, such
vesting conditions are taken into account by adjusting the number of equity instruments
included in the measurement of the transaction (by estimating the extent of forfeiture
based on failure to vest) so that, ultimately, the amount recognised is based on the
number of equity instruments that eventually vest.
8.5
Selecting appropriate assumptions for option-pricing models
IFRS 2 notes that, as discussed at 8.2.2 above, option pricing models take into account,
as a minimum:
• the exercise price of the option;
• the life of the option (see 8.5.1 and 8.5.2 below);
• the current price of the underlying shares;
• the expected volatility of the share price (see 8.5.3 below);
• the dividends expected on the shares (if appropriate – see 8.5.4 below); and
• the risk-free interest rate for the life of the option (see 8.5.5 below). [IFRS 2.B6].
Of these inputs, only the exercise price and the current share price are objectively
determinable. The others are subjective, and their development will generally require
significant analysis. The discussion below addresses the development of assumptions
for use both in a Black-Scholes-Merton formula and in a lattice model.
IFRS 2 requires other factors that knowledgeable, willing market participants would
consider in setting the price to be taken into account, except for those vesting
conditions and reload features that are excluded from the measurement of fair value –
see 5 and 6 above and 8.9 below. Such factors include:
• restrictions on exercise during the vesting period or during periods where trading
by those with inside knowledge is prohibited by securities regulators; or
• the possibility of the early exercise of options (see 8.5.1 below). [IFRS 2.B7-
9].
However, the entity should not consider factors that are relevant only to an individual
employee and not to the market as a whole (such as the effect of an award of options
on the personal motivation of an individual). [IFRS 2.B10].
2630 Chapter 30
The objective of estimating the expected volatility of, and dividends on, the underlying
shares is to approximate the expectations that would be reflected in a current market
or negotiated exchange price for the option. Similarly, when estimating the effects of
early exercise of employee share options, the objective is to approximate the
expectations about employees’ exercise behaviour that would be developed by an
outside party with access to detailed information at grant date. Where (as is likely) there
is a range of reasonable expectations about future volatility, dividends and exercise
behaviour, an expected value should be calculated, by weighting each amount within
the range by its associated probability of occurrence. [IFRS 2.B11-12].
Such expectations are often based on past data. In some cases, however, such historical
information may not be relevant (e.g. where the business of the entity has changed
significantly) or even available (e.g. where the entity is unlisted or newly listed). An
entity should not base estimates of future volatility, dividends or exercise behaviour on
historical data without considering the extent to which they are likely to be reasonably
predictive of future experience. [IFRS 2.B13-15].
8.5.1
Expected term of the option
IFRS 2 allows the estimation of the fair value of an employee share award to be based
on its expected life, rather than its maximum term, as this is a reasonable means of
reducing the value of the award to reflect its non-transferability.
Option value is not a linear function of option term. Rather, value increases at a
decreasing rate as the term lengthens. For example, a two year option is worth less than
twice as much as a one year option, if all other assumptions are equal. This means that
to calculate a value for an award of options with widely different individual lives based
on a single weighted average life is likely to overstate the value of the entire award.
Accordingly, assumptions need to be made as to what exercise or termination behaviour
International GAAP® 2019: Generally Accepted Accounting Practice under International Financial Reporting Standards Page 524