When considered in combination with the non-transferability of the options (see 8.2.3.A
above), this means that exercising the options is the only way to remove exposure to
fluctuations in value, which lowers the value of the options.
8.2.3.F Dilution
effects
When third parties write traded share options, the writer delivers shares to the option
holder when the options are exercised, so that the exercise of the traded share options
has no dilutive effect. By contrast, if an entity writes share options to employees and,
when those share options are exercised, issues new shares (or uses shares previously
repurchased and held in treasury) to settle the awards, there is a dilutive effect as a result
of the equity-settled awards. As the shares will be issued at the exercise price rather
than the current market price at the date of exercise, this actual or potential dilution
may reduce the share price, so that the option holder does not make as large a gain as
would arise on the exercise of similar traded options which do not dilute the share price.
8.3
Selection of an option-pricing model
Where, as will almost always be the case, there are no traded options over the entity’s
equity instruments that mirror the terms of share options granted to employees, IFRS 2
requires the fair value of options granted to be estimated using an option-pricing model.
The entity must consider all factors that would be considered by knowledgeable, willing
market participants in selecting a model. [IFRS 2.B4-5].
The IASB decided that it was not necessary or appropriate to prescribe the precise
formula or model to be used for option valuation. It notes that there is no particular
option pricing model that is regarded as theoretically superior to the others, and there
is the risk that any model specified might be superseded by improved methodologies in
the future. [IFRS 2.BC131].
The three most common option-pricing methodologies for valuing employee options are:
• the Black-Scholes-Merton formula (see 8.3.1 below);
• the binomial model (see 8.3.2 below); and
• the Monte Carlo Simulation (see 8.3.3 below).
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It is important to understand all the terms and conditions of a share-based payment
arrangement, as this will influence the choice of the most appropriate option
pricing model.
IFRS 2 names the Black-Scholes-Merton formula and the binomial model as examples
of acceptable models to use when estimating fair value, [IFRS 2.BC152], while noting that
there are certain circumstances in which the Black-Scholes-Merton formula may not
be the most appropriate model (see 8.3.1 below). Moreover, there may be instances
where, due to the particular terms and conditions of the share-based payment
arrangement, neither of these models is appropriate, and another methodology is more
appropriate to achieving the intentions of IFRS 2. A model commonly used for valuing
more complex awards is Monte Carlo Simulation (often combined with the Black-
Scholes-Merton formula or the binomial model). This can deal with the complexities of
a plan such as one with a market condition based on relative total shareholder return
(TSR), which 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.
8.3.1
The Black-Scholes-Merton formula
The Black-Scholes-Merton methodology is commonly used for assessing the value of a
freely-traded put or call option and allows for the incorporation of static dividends on
shares. The assumptions underlying the Black-Scholes-Merton formula are as follows:
• the option can be exercised only on the expiry date (i.e. it is a European option);
• there are no taxes or transaction costs and no margin requirements;
• the volatility of the underlying asset is constant and is defined as the standard
deviation of the continuously compounded rates of return on the share over a
specified period;
• the risk-free interest rate is constant over time;
• short selling is permitted;
• there are no risk-free arbitrage opportunities;
• there are log normal returns (i.e. the continuously compounded rate of return is
normally distributed); and
• security trading is continuous.
The main limitation of the Black-Scholes-Merton methodology is that it only calculates
the option price at one point in time. It does not consider the steps along the way when
there could be a possibility of early exercise of an American option (although as
discussed at 8.4 below this can be partially mitigated by using an assumed expected term
as an input to the calculation).
The Black-Scholes-Merton formula is an example of a closed-form model, which is a
valuation model that uses an equation to produce an estimated fair value. The formula
is as shown in Figure 30.4 below.
Share-based
payment
2621
Figure 30.4:
The Black-Scholes-Merton formula
c = S0e–qTN(d1) – Ke–rTN(d2)
Where:
ln(S0/K) + (r–q+σ2/2)T
d
1 =
σ √T
d2 = d1 – σ √T
c =
price of a written call
S0 =
price of the underlying share
N =
the cumulative probability distribution function for a standardised normal distribution
q =
dividend yield (continuously compounded)
K =
call option exercise price
r =
the continuously compounded risk-free rate
σ =
annualised volatility of the underlying share
T =
time to expiry (in years)
Note:
‘e’ represents the mathematical constant, the base of the natural logarithm (2.718282...), and
‘ln’ is the natural logarithm of the indicated value
Whilst the Black-Scholes-Merton formula is complex, its application in practice is
relatively easy. It can be programmed into a spreadsheet, and numerous programs and
calculators exist that use it to calculate the fair value of an option. As a result, the formula
is used widely by finance professionals to value a large variety of options. However, a
number of the assumptions underlying the formula may be better suited to valuing
short-term, exchange-traded share options rather than employee share options.
The attributes of employee share options that render the Black-Scholes-Merton
formula less effective as a valuation technique include:
• Long term to expiry
The formula assumes that volatility, interest rates and dividends are constant over the
life of the option. While this may be appropriate when valuing short-term options,
the assumption of constant values is less appropriate when valuing long-term options.
• Non-transferability and early exercise
The formula assumes a fixed maturity/exercise date. While IFRS 2 provides for the use
of an ‘expected term’ in place of the contractual life to reflect the possibility of early
exercise resulting from the non-transferability of employee share options or other
reasons (see 8.5 below), this may
not adequately describe early exercise behaviour.
• Vesting conditions and non-vesting conditions
The formula does not take into account any market-based vesting conditions or
non-vesting conditions.
• Blackout periods
As the formula assumes exercise on a fixed date, and does not allow earlier exercise,
it does not take into consideration any blackout periods (see 8.2.3.D above).
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In summary, application of the Black-Scholes-Merton formula is relatively simple, in
part because many of the complicating factors associated with the valuation of
employee share options cannot be incorporated into it directly and, therefore, must be
derived outside of the formula (e.g. the input of an expected term).
IFRS 2 states that the Black-Scholes-Merton formula may not be appropriate for long-
lived options which can be exercised before the end of their life and which are subject to
variation in the various inputs to the model over the life of the option. However, IFRS 2
suggests that the Black-Scholes-Merton formula may give materially correct results for
options with shorter lives and with a relatively short exercise period. [IFRS 2.B5].
The development of appropriate assumptions for use in the Black-Scholes-Merton
formula is discussed at 8.5 below.
As noted above, the Black-Scholes-Merton formula is an example of a closed form
model that is not generally appropriate for awards that include market performance
conditions. However, in certain circumstances it may be possible to use closed form
solutions other than the Black-Scholes-Merton formula to value options where, for
example, the share price has to reach a specified level for the options to vest. These
other solutions are beyond the scope of this chapter.
8.3.2
The binomial model
The binomial model is one of a subset of valuation models known as lattice models, which
adopt a flexible, iterative approach to valuation that can capture the unique aspects of
employee share options. A binomial model produces an estimated fair value based on the
assumed changes in prices of a financial instrument over successive periods of time. In
each time period, the model assumes that at least two price movements are possible. The
lattice represents the evolution of the value of either a financial instrument or a market
variable for the purpose of valuing a financial instrument.
The concepts that underpin lattice models and the Black-Scholes-Merton formula are
the same, but the key difference between a lattice model and a closed-form model is
that a lattice model is more flexible. The valuations obtained using the Black-Scholes-
Merton formula and a lattice model will be very similar if the lattice model uses identical
assumptions to the Black-Scholes-Merton calculation (e.g. constant volatility, constant
dividend yields, constant risk-free rate, the same expected life). However, a lattice
model can explicitly use dynamic assumptions regarding the term structure of volatility,
dividend yields, and interest rates.
Further, a lattice model can incorporate assumptions about how the likelihood of early
exercise of an employee share option may increase as the intrinsic value of that option
increases, or how employees may have a high tendency to exercise options with
significant intrinsic value shortly after vesting.
In addition, a lattice model can incorporate market conditions that may be part of the
design of an option, such as a requirement that an option is only exercisable if the
underlying share price reaches a certain level (sometimes referred to as ‘target share
price’ awards). The Black-Scholes-Merton formula is not generally appropriate for
awards that have a market-based performance condition because it cannot handle that
additional complexity.
Share-based
payment
2623
Most valuation specialists believe that lattice models, through their versatility,
generally provide a more accurate estimate of the fair value of an employee share
option with market performance conditions or with the possibility of early exercise
than a value based on a closed-form Black-Scholes-Merton formula. As a general rule,
the longer the term of the option and the higher the dividend yield, the larger the
amount by which the binomial lattice model value may differ from the Black-Scholes-
Merton formula value.
To implement the binomial model, a ‘tree’ is constructed the branches (or time steps) of
which represent alternative future share price movements over the life of the option. In
each time step over the life of the option, the share price has a certain probability of
moving up or down by a certain percentage amount. It is important to emphasise the
assumption, in these models, that the valuation occurs in a risk-neutral world, where
investors are assumed to require no extra return on average for bearing risks and the
expected return on all securities is the risk-free interest rate.
To illustrate how the binomial model is used, Example 30.33 below constructs a simple
binomial lattice model with a few time steps. The valuation assumptions and principles
will not differ in essence from those in a Black-Scholes-Merton valuation except that
the binomial lattice model will allow for early exercise of the option. The relevant
difference between the two models is the specification of a very small number of time
steps, for illustrative purposes, in the binomial lattice model (see also 8.3.2.A below). We
discuss below how the model can be augmented for a more complex set of assumptions.
Example 30.33: Binomial model
A share option is issued with an exercise price of $10, being the share price on the grant date. This Example
assumes a constant volatility (50%) and risk-free rate (5% continuously compounded) although, as discussed
later, those static assumptions may not be appropriate when valuing a long-term share option. It is also
assumed that: the grantor pays dividends with a yield of 2% (continuously compounded) on its shares; the
term of the option is five years; and each branch of the tree represents a length of time of one year.
At t = 0 (the grant date), the model is started at the grant date share price ($10 in this Example). At each node
(the base of any price time step), two possible price changes (one increase and one decrease) are computed
based on the volatility of the shares. The two new share prices are computed as follows:
The up-node price utilises the following formula:
u = eσ√dt = e0.5*√1 = 1.6487
Where:
σ = annualised volatility of the underlying share
dt = period of time between nodes
The down-node is the inverse of the up-node:
1
1
d =
=
= 0.6065
u
1.6487
The probability of each upward and downward price movement occurring is calculated from:
• the probability of an upward movement in price:
e(r–q)dt – d
e(.05–.02)*1 – 0.6065
p =
=
= 0.4068
u – d
1.6487 – 0.6065
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Where:
r = continuously compounded risk-free rate
<
br /> q = dividend yield (continuously compounded)
dt = period of time between nodes
• the probability of a downward movement in price:
= 1 – p = 1 – 0.4068 = 0.5932
Using the above price multiples, the price tree can then be constructed as shown diagrammatically below –
each rising node is built by multiplying the previous price by ‘u’ and each falling node is similarly calculated
by multiplying the previous price by ‘d’.
Assumptions:
S5,5
Share price
$10
121.82
Exercise price
$10
Risk-free rate
5%
S4,4
111.82
Dividend yield
2%
Volatility
50%
73.89
Life/Term
5 years
S3,3
63.89
S5,4
44.82
44.82
S2,2
34.82
S4,3
34.82
27.18
27.18
S1,1
18.02
S3,2
17.18
S5,3
16.49
16.49
16.49
S0,0
9.04
S2,1
8.06
S4,2
4.42
6.49
Share price
10.00
10.00
10.00
Option FV
4.42
S1,0
3.67
S3,1
4.42
2.51
S5,2
6.07
6.07
6.07
1.63
S2,0
4.42
0.97
S4,1
4.42
0.00
3.68
3.68
0.38
S3,0
4.42
0.00
S5,1
2.23
2.23
0.00
S4,0
0.00
1.35
0.00
S5,0
0.82
0.00
Time/Years
0.0
1.0
2.0
3.0
4.0
5.0
To calculate the option value:
International GAAP® 2019: Generally Accepted Accounting Practice under International Financial Reporting Standards Page 523