3. 1 Background
A large amount of the derivatives that are traded in the financial markets are known as
American options. Aside from in rare cases, the value of an American option does not
have an exact solution. Because of this reason, the majority of American options have
to be valued numerically. It is essential that derivatives are valued correctly so that
arbitrage (risk-free profit) is not possible. Due to this, the problem of valuing American
options is crucial and an active area of research for mathematicians.
In this report we will be focusing on the problem of valuing an American put option
on a non-dividend-paying asset. We will do this by formally posing a free boundary value
problem and considering three different numerical methods that can be used to solve
it. From here we will then be able to compare and contrast the three different methods
before deciding on the most effective one for our problem of valuing an American put
option. In studying these methods it is hoped that the reader will develop a solid
understanding so that these ideas can be applied to other American options.
1.1 Vanilla Options
Many different financial assets are traded on global exchanges in the hope of making
financial gain. These assets include the likes of equities, currencies, commodities and
indices. As an alternative to directly trading assets, investors often choose to trade
derivatives. Derivatives are contracts whose value depends on an underlying asset. An
example of a derivative could be a contract to buy a commodity, such as a barrel of oil,
for $100 in exactly one month from now.
How should we value such a contract? This question was first answered in 1973
by Black and Scholes [1]. They derived a model that values derivatives from a trading
strategy of buying and selling the underlying asset in such a way as to eliminate risk.
This Nobel Prize-winning idea allows derivatives to be a valued so that arbitrage is not
possible. The type of derivative that Black and Scholes originally focused their model
on is called an option.
An option gives us the right to buy (or sell) a certain financial asset for a fixed value
at a set time in the future. The asset that can be bought or sold using this option is
known as the underlying asset (or simply the underlying) and the fixed buy (or sell)
price in the future is called the exercise price (or strike price). A simple option such as
the one above is known as a European option. Typically, options are divided into two
categories; call options and put options. The holder of a call option has the right to buy
2
4. the underlying asset at the exercise price and the holder of a put option has the right
to sell at the exercise price.
It is now important to introduce some notation in order to take this discussion
further. Let t denote the time at which the option is written (and valued) and let T
denote the time at which the option expires. Hence, the time to expiration from the day
the option is written is given by T − t. Note that for our purposes we will measure time
in terms of years. For example, T −t = 1 means that there is one year to expiration and
T − t = 1
12 means that there is one month to expiration. Also let τ be an arbitrary time
in the interval [t, T]. We let S denote the value of the underlying asset and we will let
this be a function of time, making this explicit in our notation whenever appropriate.
We will often use the notation Sτ = S(τ). Thus, St is the value of the asset when the
option is written and ST is the value of the asset when the option expires. We let E be
the exercise price of the option and we take r to be the annual risk-free interest rate,
the rate that we would obtain from investing in a bank.
We will now consider an example to make the concept clearer. Consider shares of
Company X that are currently trading at a price of $100 each. Let us own a European
call option on a share of Company X that has a strike price of $110 and three months
until expiration. According to the above notation we have that St = 100, E = 110 and
T −t = 3
12. Put simply, by holding this option today we have the right to buy one share
of Company X in exactly three months time for the price of $110.
Say the price of the share rises to $120 over the next three months. If this were to
happen then we would have a profit-making opportunity. We could exercise our option
to buy the share for $110 and then we could immediately sell the same share in the
market for $120, giving us a $10 return. In contrast, let us consider what would happen
if the share price of Company X remained at $100 after three months. Clearly we would
not exercise our right to buy the share for $110 when we could buy the exact same share
for $100 from the market. We can see from this example that a rational investor clearly
would not exercise a European call option if at expiration the value of the underlying
asset is below the exercise price. It would be natural to ask how we could value this
option so that arbitrage is not possible.
Both European call options and European put options are examples of what are
known as vanilla options. The name is an ice cream analogy, derived from the relative
simplicity of these types of options. Options that are more complicated than vanilla
are then known as exotic options. Typically we find that vanilla options are straight
forward to value, given certain assumptions on the underlying asset value.
3
5. Figure 1: Two examples of Brownian motion evolving over time, both with zero drift.
The first with a volatility of 1 and the second with a volatility of 2.
We now want to be able to model the value of an asset and so we introduce two new
parameters µ and σ. The parameter µ is known as the drift of the asset value. The drift
is a measure of the average change of the asset value and determines the future expected
asset value in the absence of volatility. This means that if the asset has a positive drift
then we would expect the asset value to rise over time and a negative drift would lead
us to expect a fall in the asset value over time. The parameter σ is the volatility of the
asset value. A low value for σ would lead the asset value to barely deviate from the
future expected value, as determined by the drift. However, for a higher value of σ we
would find that the asset value deviates from the future expected value much more.
It is easy to distinguish between different volatilities in the plots of asset values,
and we can do this with the help of Brownian motion. Brownian motion is a stochastic
process commonly used in modelling asset values and is discussed in great detail by
M¨orters and Peres [2]. Consider Figure 1, in which both plots showcase Brownian
motion evolving over time with µ = 0. The differences come from the volatility, as the
top plot has σ = 1 and the bottom plot has σ = 2. Also note that due to the drift being
zero, neither of the two simulations strays far from the starting value. In fact, if we also
had σ = 0 then the simulation would be represented by a perfectly horizontal line. It is
harder to see drift in graphical form as it is often masked by the volatility, but the drift
determines the slope of the graph.
We are now able to build a mathematical model for the value of an asset. Consider
a time period dt in which the asset value changes from S to S + dS. We wish to model
the corresponding return on the asset over this time period, which is (S+dS)−S
S = dS
S .
4
6. We decompose this model into two parts; the deterministic part and the probabilistic
part. We know that µ is a measure of the average rate of growth of the asset value.
This means that the deterministic part contributes µdt to the return dS
S over the time
period dt. We have that the probabilistic part contributes σdX, where the term dX is
the Wiener process. This means that dX is a random variable drawn from the normal
distribution with mean 0 and variance dt. That is, the probabilistic part is the volatility
σ multiplied by a randomly generated number from the normal distribution. Combining
the two parts together, we obtain the stochastic differential equation
dS
S
= µdt + σdX ⇒ dS = µSdt + σSdX, (1.1)
that we will use to simulate asset values. This is the most widely accepted model to
value assets such as equities, currencies, commodities and indices. The deterministic
term is analogous to the return on money invested in a bank (receiving interest at a
risk-free rate of µ) and the probabilistic term represents random changes in the asset
value caused by external factors such as unexpected news. A more in-depth explanation
of the derivation of this model is written by Hull [3].
Before we continue, we must introduce two more financial concepts. The first is the
time value of money, which says that $1 today is worth more than $1 a year from now.
This is the case as $1 today can earn interest and rise above $1 over the course of a year.
This means that in order to accurately compare monetary values at different times, we
must discount the values to the same date. We will assume that all interest is com-
pounded continuously. More on the time value of money and continuously compounded
interest is written by Brown and Kopp [4].
The second financial concept that we must discuss is the no-arbitrage principle, which
we explain with an example. Consider two different investments today. If the discounted
future payoffs of these two investments are equal, then the investments must have the
same value today. If this is not the case then arbitrage is possible. This means that
under the no-arbitrage principle, options must be valued as their discounted expected
payoff.
With a model in place for the underlying asset we are now able to value the European
options that we mentioned previously. Denote the value of an option by V (S, t). Then
a European option, on an underlying asset simulated by equation (1.1), has a value
V (S, t) that satisfies the Black-Scholes equation
∂V
∂t
+
1
2
σ2
S2 ∂2V
∂S2
+ rS
∂V
∂S
− rV = 0, (1.2)
5
7. with final and boundary conditions dependent on the choice of the option (call or put).
We find that the Black-Scholes equation, with certain alterations, can model most deriva-
tives and so option valuation problems often come down to solving it. European call
options and European put options satisfy equation (1.2) in its current form.
It is worth noting that the option value V is technically a function of more than just
S and t. We could correctly write that V = V (S, t, σ, µ, E, T, r), if we wished. However,
here we simply use V = V (S, t) due to the conciseness and also our awareness that the
Black-Scholes equation only contains derivatives with respect to S and t. We actually
find that the option value V is completely independent of the drift parameter µ.
As previously mentioned, the conditions of equation (1.2) change depending on the
option being considered. Recall that ST is the value of the underlying asset at the time
of expiration. Then the final conditions (also known as the payoff functions) are
European call option: V (ST , T) = max(ST − E, 0),
European put option: V (ST , T) = max(E − ST , 0).
To explain why, consider a European call option. At expiration assume we have ST > E.
We would exercise our right to buy the underlying asset for E before selling it for ST ,
giving us a payoff of ST − E > 0. Now assume that at expiration we have ST < E. If
we exercise the option and follow the same strategy as before then the payoff would be
negative. Instead we would not exercise, letting the option expire with a payoff of 0.
Combining these two scenarios gives us the payoff as max(ST − E, 0). We can follow a
similar argument for a European put option.
The boundary conditions are
European call option:
V (0, t) = 0
V (S, t) ≈ S as S → ∞,
European put option:
V (0, t) = e−r(T−t)E
V (S, t) ≈ 0 as S → ∞.
Just as we did before, consider a European call option. By equation (1.1), if S = 0
then dS = 0. Hence, the value of S never changes. This means S = 0 at expiration
and so the option will give us a payoff of 0. If we consider S → ∞, then it is clear
that S will dominate E and we will exercise the option. This leaves the payoff as
being approximately S. We can also follow a similar argument to derive the boundary
conditions for a European put option.
If we let C(S, t) be the value of a European call option (a straight substitution
C = V , purely for clarity), then solving equation (1.2) subject to the appropriate final
6
8. and boundary conditions above gives us
C(S, t) = SN(d1) − Ee−r(T−t)
N(d2), (1.3)
where
d1 =
log( S
E ) + (r + 1
2σ2)(T − t)
σ
√
T − t
, d2 =
log( S
E ) + (r − 1
2σ2)(T − t)
σ
√
T − t
,
and N(x) = 1√
2π
x
−∞ e−1
2
s2
ds is the cumulative distribution function for the standard-
ized Normal distribution.
Equation (1.3) is known as the Black-Scholes formula for a European call option.
Now in order to value a European call option it is merely a case of substituting the
known values of S (in particular, S = St), E, T, t, σ and r into the formula. Note that
µ does not appear in equation (1.3) at all.
The derivation for the Black-Scholes equation, equation (1.2), along with the deriva-
tion of the Black-Scholes formula for a European call option, equation (1.3), are both
explained by Wilmott [5].
Let P(S, t) be the value of a European put option. Now consider a European call
option and a European put option both on the same underlying asset, with the same
exercise price, time to expiration and risk-free interest rate. Then these two options are
related by the put-call parity formula
C(S, t) − P(S, t) = S − Ee−r(T−t)
.
This is derived by Gray and Malone [6]. Substituting in the expression for C(S, t), given
by equation (1.3), we find that the Black-Scholes formula for a European put option is
P(S, t) = −SN(−d1) + Ee−r(T−t)
N(−d2), (1.4)
with d1, d2 and N(x) defined as before.
Now that we have seen the relative ease of valuing these two vanilla options we can
use our experience to help us with the problem of valuing exotic options.
1.2 Exotic Options
So far we have only considered vanilla options, which are relatively simple to understand
and simple to value. Exotic options on the other hand can be more difficult to understand
and certainly more difficult to value. We managed to find a way to analytically value
European call options and European put options, but we will soon see that this is an
impossible feat for some exotic options. In order to further our understanding of exotic
7
9. options we would like to be able to classify the different types. We will classify exotic
options by considering the six different features listed below:
1. Time dependence.
2. Cash flows.
3. Path dependence.
4. Dimensionality.
5. Order.
6. Embedded decisions.
Time dependence concerns when the option in question can be exercised. Can it
be exercised at any time? Or only on certain dates? Cash flows are a big aspect
to consider when working with options on dividend-paying assets as we may need to
implement jump conditions into any model we consider. Path dependence is the degree
to which the final payoff of the option depends on past values of the underlying asset.
Dimensionality regards the number of independent variables. For example, an option
may have more than one underlying asset – which increases the dimension. Order is an
important feature of compound options. For example, a higher order option will have
a payoff that depends on the value of another option, a value which may have been
computed using the value of yet another different option. Embedded decisions are one
of the most important factors to consider with American options, where the decision is
about whether or not to exercise the option early or to continue to hold it.
We have learned about the features that make exotic options more complex and
interesting than vanilla options. Therefore, we are now ready to focus on our exotic op-
tions of choice: American options. We will look to deeply understand what an American
option is, before considering the differences between valuing a European option and an
American option.
2 American Options
An American option has the same features as a European option, except that it allows
the holder to exercise the option at any time up to and including expiration. This is
what sets American options apart from European options, as we recall that European
options can only be exercised at the expiration date. American options can be divided
8
10. into calls and puts just like with European options. This means that the payoff functions
for European options are the same in the American context and that they also apply
for whenever we choose to exercise an American option. If we let the interval [t, T] be
the lifetime of the option, then if τ is the optimal time to exercise (more on this later)
such that t ≤ τ ≤ T, then the payoffs are
American call option: V (Sτ , τ) = max(Sτ − E, 0),
American put option: V (Sτ , τ) = max(E − Sτ , 0).
In order to better understand these options it makes great sense to classify them using
the features we previously learned. American options have very little time dependence
as they can be exercised at any time (this means that we have no restrictions if we need
to discretize time). We will only consider American options with no cash flows, this
means that we will only consider options on non-dividend-paying assets. There is very
little path dependence as the payoff depends only on the value of the underlying asset
at the time that the option is exercised. In fact, the only path dependence comes from
considering whether or not the option has previously been exercised, as once exercised
the option can not be exercised again. The only independent variables here are the
underlying asset value S and the time t, just as with the European options we considered.
We will only be considering American options of first order. As we mentioned earlier,
embedded decisions are the most important feature of American options. We have the
right to exercise early, but we need to decide whether early exercise is optimal or not at
each time τ [t, T].
We know that American options and Europeans options are the same except for the
additional ability of being able to exercise American options before expiration. Now we
might question how this would effect the option value. We can conclude that the holder
of an American option has more rights than the holder of a European option. If the
ability to exercise early is useful then we would expect American options to be worth
more than European options. On the contrary, if we find that the right to exercise early
is useless then we would expect American options and European options to have the
same value.
This implies that for a comparable American option and European option (same
underlying asset, exercise price, etc.), the American option would have a higher or equal
value. But we already know the value of a European call option and a European put
option at time τ (in other words, the payoff at time τ) and so we can combine this
with what we have just learned about American options to derive inequalities for their
value. If we denote C as the value of an American call option then for all τ such that
9
11. t ≤ τ ≤ T, we know
C(Sτ , τ) ≥ max(Sτ − E, 0).
If we now denote P to be the value of an American put option then for all τ such that
t ≤ τ ≤ T, we know
P(Sτ , τ) ≥ max(E − Sτ , 0).
Now in order to discuss American call options and American put options in greater
detail we must consider them separately. We will first discuss American call options
before moving on to discussing American put options.
2.1 American Call Options
Let us now consider how to value an American call option. Due to the additional rights
of an American call option it might appear that one would be worth more than an
equivalent European call option. It actually turns out that it is not optimal to exercise
an American call option early. This means that a European call option has the exact
same value as an American call option. We will now prove this claim.
Proof. Suppose that the present value of the underlying asset is Sτ and that we own an
American call option to buy one more share of this asset at the exercise price E, and also
that this option expires after an additional time T − τ (meaning that the option expires
at time T). If we exercise the option early at this arbitrary time τ, then we will receive
the amount Sτ −E. However, consider what would transpire if, instead of exercising the
option, we sell the asset short now and then purchase the asset back at time T, either by
paying the market value at time T or by exercising our option and paying E, whichever
is less expensive. Under this strategy, we will initially receive Sτ and we will then have
to pay the minimum of the market value or the exercise price after an additional time
T −τ. At time τ prices this means that this strategy pays out Sτ −e−r(T−τ) min(ST , E).
But Sτ − e−r(T−τ) min(ST , E) > Sτ − E as clearly E > e−r(T−τ) min(ST , E). Thus, it
is clearly preferable to exercise the American call option at expiration T rather than at
an arbitrary time τ < T.
This proof follows a similar argument to that of a proof by Ross [7]. Ross also offers
an alternative derivation of the Black-Scholes formula for a European call option that
does not require knowledge of Stochastic Calculus (the Wilmott derivation [5] does).
Hence, we have proved that an American call option is worth the same amount as
a European call option. But we have already showed that we can value a European
10
12. call option by using the Black-Scholes formula we derived. Therefore, the value of an
American call option is given by equation (1.3).
2.2 American Put Options
Unlike with American call options, it can indeed be rational to exercise an American
put option early. Thus, the American put option valuation problem is very interesting
and much more complicated than anything we have seen so far.
The biggest difference between valuing an American put option and a European
put option is that for the American case we have to determine its value by computing
whether or not it should be exercised early for every time in the interval [t, T]. In fact,
at each time τ [t, T] there is a particular value of the underlying asset, that we will
denote Sb(τ), such that if Sτ < Sb(τ) then it is optimal to exercise the option and if
Sb(τ) < Sτ then it is optimal to continue to hold the option.
This means that the problem of valuing an American put option is actually a free
boundary value problem. Determining the unknown boundary S = Sb(τ) is a necessary
part of the valuation. We will refer to the curve S = Sb(τ), for t ≤ τ ≤ T, as the
optimal exercise boundary and we will discuss it in further detail later on.
We can also deduce that this free boundary value problem is uniquely specified by
four constraints. These constraints are listed below:
1. The option value P(S, t) must be greater than or equal to the payoff function.
2. The Black-Scholes equation is replaced by an inequality.
3. The option value P(S, t) must be a continuous function of S.
4. The option delta ∂P
∂S must be continuous.
We have already explained the first constraint. In particular, for an American put
option we must have that the option value P satisfies P(Sτ , τ) ≥ max(E − Sτ , 0) for all
τ [t, T]. We previously mentioned that it may be optimal to exercise the option early. If
this was always the case then the inequality would become P(Sτ , τ) > max(E − Sτ , 0).
We require the possibility of equality as not all American put options can be rationally
exercised early.
For the second constraint we must bear in mind that Wilmott’s derivation [5] of
the original Black-Scholes equation uses a no-arbitrage argument. This argument is
only partially valid in the American put scenario. We can set up a portfolio as we
would for the original Black-Scholes derivation as well as taking the same delta ∆ =
11
13. ∂P
∂S (see Wilmott). The difference now for an American put option is that it is not
necessarily possible for the option to be held both long and short, due to the times
when early exercise is optimal. Because of this, the no-arbitrage argument used for the
original derivation no longer leads to a unique value for the return on the portfolio in
the American put scenario, instead we get an inequality. Thus, for an American put
option equation (1.2) is replaced by
∂P
∂t
+
1
2
σ2
S2 ∂2P
∂S2
+ rS
∂P
∂S
− rP ≤ 0. (2.1)
When it is optimal to hold the American put option for the entirety of its lifetime then
equation (1.2) must be satisfied (hence, the equality part of equation (2.1)). However, if
it is ever optimal to exercise the option early then equation (1.2) is replaced by inequality
(2.1) (in particular with < instead of ≤). We will verify that this is indeed the case.
To show this we find that for each time τ we must divide the S-axis into two distinct
regions. We know that if the underlying asset value lies in the interval 0 < S < Sb(τ)
then it is optimal to exercise the option early and the option value is given by P = E−S.
Substituting P = E − S into the left-hand side of equation (1.2) gives us
∂P
∂t
+
1
2
σ2
S2 ∂2P
∂S2
+ rS
∂P
∂S
− rP = −rE < 0.
Now if the underlying asset value lies in the interval Sb(τ) < S < ∞ then it is not optimal
to exercise the option early and the option value P satisfies the standard Black-Scholes
equation
∂P
∂t
+
1
2
σ2
S2 ∂2P
∂S2
+ rS
∂P
∂S
− rP = 0.
Combining the above two expressions gives us inequality (2.1) as the second constraint
of the free boundary value problem.
The third constraint follows from no-arbitrage. Consider a discontinuity in the option
value P(S, t) as a function of S and also that this discontinuity persists for more than
an infinitesimal time. Then the option can make a risk-free profit if the asset value ever
reaches the value at which the discontinuity occurs. Say that for an underlying asset
value Sd, the option value has a discontinuity and simultaneously takes the values $10
and $20. If the underlying asset value ever reaches Sd then we could buy the option for
$10 and instantly sell it for $20, giving a risk-free profit. Hence, the option value P(S, t)
must be a continuous function of S.
A derivation of the fourth constraint requires knowledge of stochastic control and
optimal stopping problems and so is beyond the scope of this report. Therefore, for our
purposes we will assume that the option delta must be continuous.
12
14. Now that we have discussed the constraints of the American put option valuation
problem, we need to consider the boundary and final conditions. The boundary and
final conditions that applied to a European put option apply here too. Note that we
still require a final condition in case the option is not exercised early, and that this final
condition is merely the payoff function at expiration T. Hence, we have that the final
condition is
P(ST , T) = max(E − ST , 0),
and that the boundary conditions are
P(0, t) = e−r(T−t)E
P(S, t) ≈ 0 as S → ∞.
But we also need extra boundary conditions in order to determine the free boundary
of the problem, that is, to determine the optimal exercise boundary. Just as with the
fourth constraint of the problem, determining the boundary conditions at S = Sb(τ)
requires knowledge of stochastic control and optimal stopping problems and so we will
only give an informal derivation. We will do this by comparing (at arbitrary time τ) the
slope of the payoff function max(E − Sτ , 0) with the slope of the option value P(Sτ , τ).
Recall that for an American put option P(Sτ , τ) ≥ max(E − Sτ , 0) and so these slopes
may be different.
We start by assuming that Sb(τ) < E. This means that the slope of the payoff
function, max(E − Sτ , 0), at S = Sb(τ) is −1. Why is this the case? If Sτ < E then
max(E − Sτ , 0) = E − Sτ , and differentiating this with respect to S gives us a gradient
of −1. Hence, the slope of the payoff function is −1 for all Sτ < E, including at Sb(τ).
Bearing this in mind, we have three possibilities for the option delta ∂P
∂S at S = Sb(τ)
and they are:
ˆ ∂P
∂S < −1.
ˆ ∂P
∂S > −1.
ˆ ∂P
∂S = −1.
We will show that ∂P
∂S = −1 by proving that the first two possibilities are incorrect.
Suppose that ∂P
∂S < −1. We find that as Sτ increases from Sb(τ), P(Sτ , τ) drops below
the payoff function max(E − Sτ , 0). This is due to P(Sτ , τ) having a slope of less than
−1 whereas the payoff function has a slope of exactly −1. But this contradicts the
inequality P(Sτ , τ) ≥ max(E − Sτ , 0) that we had earlier decided must hold. Thus, this
case is not possible.
13
15. Now suppose that ∂P
∂S > −1. We will now show that an option with this delta is
incorrectly valued. The choice of Sb(τ) influences the value of P(Sτ , τ) for all values of Sτ
larger than Sb(τ) (for values of Sτ smaller than Sb(τ) we always have P(Sτ , τ) = E −Sτ
as it is optimal to exercise). Therefore, if ∂P
∂S > −1 at S = Sb(τ), then the value of
P(Sτ , τ) near S = Sb(τ) can be increased by choosing a smaller value for Sb(τ): the
exercise value then moves up the payoff curve and ∂P
∂S decreases. This means that the
option is incorrectly valued and so this case is also not possible.
This leaves us with just one possibility for the option delta here, giving us the free
boundary condition
∂P
∂S
= −1 at S = Sb(τ).
Of course we also have the condition that
P(Sb(τ), τ) = max(E − Sb(τ), 0),
due to the possibility that we could exercise at S = Sb(τ). Therefore, the additional
boundary conditions at S = Sb(τ) are that P(S, t) and ∂P
∂S are continuous (recalled from
the four constraints) and in particular that
∂P
∂S
(Sb(τ), τ) = −1 and P(Sb(τ), τ) = max(E − Sb(τ), 0).
Essentially, this is one boundary condition to locate the optimal exercise boundary and
another to determine the option value on it.
We have now formalised the problem of valuing an American put option by listing
the constraints and conditions. Finding the optimal exercise boundary has arisen as a
prominent factor in solving this problem. Before moving any further it makes sense to
spend some time on understanding this boundary.
2.3 The Optimal Exercise Boundary
We saw earlier that for an American put option, at each time τ there is a particular
value of the underlying asset, denoted Sb(τ), such that if Sτ < Sb(τ) then it is optimal
to exercise the option and if Sb(τ) < Sτ then it is optimal to hold the option. The
boundary S = Sb(τ), defined for the duration of the option’s life t ≤ τ ≤ T, is known
as the optimal exercise boundary and determining it is part of the American put option
valuation problem.
Put simply, the optimal exercise boundary is a curve that separates those values of
Sτ where it is better to exercise the American put option from those where it is better to
14
16. Figure 2: The optimal exercise boundary.
hold it. As it currently stands we know nothing about the actual shape of the boundary,
such as where it starts or where it ends, but we expect it to look somewhat like Figure 2.
The interpretation is that if the underlying asset value passes below the optimal
exercise boundary then it would be optimal to exercise the option. We are interested in
finding out more about the boundary not just for solving the option valuation problem
but also for its own standalone uses. The holder of a particular American put option
would find it incredibly useful to know the associated optimal exercise boundary, so as
to know if and when the option should be exercised. Without knowledge of this it would
be extremely difficult to use an American put option effectively.
What can we deduce about the optimal exercise boundary? We start by considering
the characteristics of the boundary at its extremes; far away from the expiration date, as
well as close to the expiration date. Our arguments will be based upon the assumption
that St ≤ E. If this assumption is not satisfied then instant exercise leads to a positive
risk-free payoff.
We can consider what the boundary looks like far away from the expiration date
T by letting T → ∞. This way, no matter which time we choose to consider, we will
always be far away from expiration. Because of this, if we consider the case T → ∞,
then for any τ < T we have that
Sb(τ) =
2rE
2r + σ2
, (2.2)
and this is proved by Musiela and Rutkowski [8]. This is clearly a bound for the optimal
exercise boundary as far away as possible from expiration. In other words it is a bound
at time t, when the option is written. We now reason that it is a lower bound. Far from
15
17. expiration it is difficult to predict the final payoff, but as we get closer to expiration T
we have a better chance of estimating what this payoff will be. Due to this, if expiration
is far away (T → ∞) then we would only exercise the option if the underlying asset
value is very low. If not there is less incentive to exercise early as the payoff would
not be particularly high but we would also be forfeiting the opportunity to exercise for
a bigger payoff sometime closer to expiration. Hence, we will take the value given by
equation (2.2) as a lower bound for the optimal exercise boundary at time t.
We can also find an upper bound for the optimal exercise boundary at time t. We
reason that the upper bound is E − P, where E is the exercise price and P is the (not
necessarily known) value of the American put option on the valuation date. Say that in
the same instant that the option is written (at time t), that the value of the underlying
asset changes from St to E −P. Assuming that we are holding the American put option,
we can buy the underlying asset in the market for E − P before exercising the option to
instantly sell the underlying asset for E. This gives us a payoff of E −(E −P) = P, but
note that in exercising the option we are forfeiting its value P, so the profit we would
receive at t is actually 0. By similar reasoning we can see that if the underlying asset
value falls below E − P at time t then exercising the option at t would generate positive
return but if the underlying asset value is above E − P then exercising would yield a
loss. We note that this is only a bound. Just because we can make a profit if the asset
value is below E − P at t does not mean that we should. It could be that it is more
profitable to exercise the option later on.
Now we can bound the optimal exercise boundary at time t from above and below.
When the option is written at time t we must have that
2rE
2r + σ2
≤ Sb(t) ≤ E − P.
Now we will try to better understand the optimal exercise boundary at expiration.
At expiration we must have have that Sb(T) = E. Assuming that the option has not
already been exercised, if ST > E then the payoff is E − ST < 0 and so it is clearly not
optimal to exercise here. Now if ST < E then the payoff is E − ST > 0, indicating that
it is optimal to exercise in this scenario (rather than let the option expire worthless).
Therefore, at expiration the optimal exercise boundary must be E.
We can also derive bounds for the entire optimal exercise boundary. For all τ such
that t ≤ τ ≤ T, we must have that Sb(τ) ≤ E. If this were not the case then it would
be optimal to exercise for the payoff E − Sb(τ) < 0, for some τ, which is clearly a
contradiction. We also have that, for all τ such that t ≤ τ ≤ T, 2rE
2r+σ2 ≤ Sb(τ). The
16
18. Figure 3: The function b∗(τ) plotted against τ, with E = 100, σ = 0.2 and T = 4
12. The
plot appears to approach an infinite gradient as τ → 4
12 ≈ 0.33.
reasoning for this is the same as for when we reasoned that 2rE
2r+σ2 is a lower bound for
the optimal exercise boundary at time t.
Now we can say that, for all τ such that t ≤ τ ≤ T, Sb(τ) must satisfy the following
inequality:
2rE
2r + σ2
≤ Sb(τ) ≤ E.
We can now bound the entire optimal exercise boundary from above and below,
including a stricter bound at time t and an exact value at time T. We can now start to
talk a bit about the actual shape of the boundary curve. In particular we find something
close to asymptotic behaviour in the boundary, just before expiration occurs. We can
say more: near expiration time T the optimal exercise boundary is approximately given
by
b∗
(τ) = E 1 − σ (T − τ) log(
1
T − τ
) ,
and this is proved by Lamberton [9]. We can get an idea of what this looks like by
plotting the graph of b∗(τ) against τ as we have done in Figure 3. For this particular
graph of b∗(τ) we have chosen the arbitrary parameters E = 100, σ = 0.2 and T = 4
12.
It is worth noting that for different parameter choices b∗(τ) may look different to Figure
3 (except near expiration). Thus, the important part of the graph to consider here is
the section near T = 4
12, which gives us the shape of the optimal exercise boundary close
to expiration.
It appears that the boundary gets steeper the closer we get to expiration before
finally approaching an infinite gradient. In fact, if we differentiate b∗(τ) with respect
17
19. Figure 4: What we know about the optimal exercise boundary, Sb(τ), in the time interval
[t, T].
to τ then we can see that as τ → T the gradient of b∗(τ) does indeed tend to infinity.
But recall that despite the infinite gradient, the optimal exercise boundary touches E
at expiration time T.
Given the assumption that the underlying asset value is a continuous function of
time, we would expect the optimal exercise boundary to also be a continuous function
of time. An heuristic proof of this is given by Kwok [10]. We can also say that not
only is Sb(τ) non-decreasing, it is actually strictly increasing. We can see that this
would be the case by reasoning that as the option gets closer to expiration there are
less opportunities of making a future profit, which leads to a higher (and more easily
achievable) value for Sb(τ).
Combining the above gives us Figure 4. We have the dashed lines at S = 2rE
2r+σ2 and
S = E to indicate the bounds for the entire optimal exercise boundary. We also have the
point marked at time t that is found somewhere between S = 2rE
2r+σ2 and S = E−P. Near
expiration T, the boundary looks like Figure 3 and we also have that Sb(T) = E. Figure
4 looks accurate in between t and T as we know that the optimal exercise boundary is
continuous and strictly increasing. Note that we do not know whether or not Sb(τ) is
smooth.
When we posed the problem of valuing an American put option we found that we
also had to locate the optimal exercise boundary. We now know a great deal about this
boundary and so we are in a great position to proceed with our discussion. We have
worked hard to lay the foundations for the valuation problem and so the next logical
18
20. step is to attempt to solve it.
3 Numerical Methods to Value American Put Options
Unfortunately a closed form solution does not currently exist for the problem of valuing
an American put option. As we cannot value an American put option analytically we
must use numerical methods. We will be considering binomial, finite difference and
Monte Carlo methods. This will include analysing how these methods can be applied
to valuing American put options and how they can be used to determine the optimal
exercise boundary. We will look to convince ourselves that the methods that we cover
are accurate and reliable.
We will also be comparing the different methods against one another, across the
likes of accuracy, ease of use and efficiency. It is important to consider multiple meth-
ods as different methods have varying strengths and weaknesses. This is a big factor
when valuing more complex American options. We may find that all three methods are
successful for the standard put option but only one is accurate for options of a higher
order, for example. Considering three different methods will set us up extremely well
for future research on American options.
3.1 Binomial Methods
We start by considering a binomial method for valuing an American put option. Bi-
nomial methods work by discretizing the evolution of the underlying asset’s value. We
set up a binomial tree with n time steps between the valuation date of the option at
time t and the expiration date of the option at time T. Then starting from the value of
the underlying asset at time t we work towards T computing the underlying asset value
at each node, where each node in the binomial tree represents a possible value of the
underlying asset at a given point in time. We wish to do this in such a way that we
have as many nodes as possible in the time interval [t, T].
Once we have the value of the underlying asset at each node we can begin to value
the option. We begin at the final time step (at time T) by valuing the option at each
node, using the payoff function with the relevant underlying asset value. Then we can
use an iterative relationship, with the option values we just calculated at the final time
step, to compute the option values at the penultimate time step. We then continue to
use this iterative relationship to work backwards in time all the way to time t, giving us
the option value at the valuation date.
19
21. We will assume that at each node, over a time step of size δt, that the value of the
underlying asset S will either rise to uS or it will fall to vS, where 0 < v < 1 < u. We
will also take the further assumption that u = 1
v , which reduces the number of nodes at
each time step and speeds up computation. Let us also say that the underlying asset
value will rise to uS with probability p , which also means that the underlying asset value
will fall to vS with probability 1 − p . Here p denotes the risk-neutral probability that
the underlying asset value will rise, rather than the true probability of this occurrence.
An explanation of risk-neutrality and its uses in option valuation is given by Cox and
Rubinstein [11].
In short, we use the risk-neutral probability rather than the true probability as we
assume that all investors do not require a premium to encourage them to take risks. A
consequence of this assumption is that the average return on an asset must be equal to
the risk-free interest rate r. Thus, we replace the drift µ (the expected return of the
asset) with r.
We can choose to define the parameters u, v and p in various ways but given the
earlier assumption that u = 1
v , it can be reasoned that we should take
u =
1
2
(e−rδt
+ e(r+σ2)δt
) +
1
2
(e−rδt + e(r+σ2)δt)2 − 4 (3.1)
as the definition of u. Another common choice is to take u = 1 + σ
√
δt, but this is
actually just a Taylor series approximation of equation (3.1). With u now defined, the
condition v = 1
u gives us our definition of v. Finally we take
p =
erδt − v
u − v
(3.2)
as the definition of p . Reasons for choosing equations (3.1) and (3.2), along with their
derivations are given by Wilmott [12].
Note that all of the parameters involved in equations (3.1) and (3.2) are known.
Parameters r and σ are given in the problem and we can compute δt as it is defined by
δt = T−t
n , where T −t is the time to expiration and n is the chosen number of time steps.
With this information in mind, we are now at a point where we are able to produce an
entire binomial tree of the underlying asset’s value at each node.
We can now compute the option value at each node at time step n (the final time
step) using the option’s payoff function. In particular, for the American put option
problem we will be using the payoff function V (ST , T) = max(E − ST , 0). We define a
coordinate system for the binomial tree as in Figure 5. That is, the (i, j)-th node is i
time steps away from time t and is j nodes away from the bottom node of the i-th time
20
22. Figure 5: Our chosen coordinate system for the binomial tree.
step. Further, we will denote Si,j as the underlying asset value and Vi,j as the option
value at the (i, j)-th node. Thus, the value of the option at its valuation date is given
by V0,0.
Now for the American put option problem we can rewrite the payoff function as
Vn,j = max(E − Sn,j, 0). This will be the formula used to compute the option value for
each node at expiration. Now we need an iterative relationship in order for us to work
from the right of the binomial tree to the left. Under a no-arbitrage argument the value
of a European option at time step i is equal to the expected value of its payoff at time
step i + 1, discounted at the risk-free interest rate. This means that for a European
option (call or put) we can use the relationship
Vi,j = e−rδt
p Vi+1,j+1 + e−rδt
(1 − p )Vi+1,j. (3.3)
This relationship is a weighted average of the two option values at the (future) nodes
that are connected to the node that we wish to consider. The weights come from the
(risk-neutral) probabilities of moving up or down. The e−rδt term is the discount factor
that we need to apply at each time step, as we work from the right of the tree to the
left. Note that after all n time steps we will have discounted the value from time T by
e−nrδt = e−r(T−t), giving us the value of the option at time t prices, as desired.
Equation (3.3) does not suffice for an American put option as we must account for
the possibility of early exercise. At a given node it is simple to decide whether or not
21
23. it is optimal to exercise the option. We do this by calculating the value of the option
at the node using equation (3.3) and also by calculating the payoff we would receive
from exercising at this node. Then we can compare the two values, with the highest
value deriving from the optimal action. We can easily implement early exercise into our
method by replacing equation (3.3) with
Vi,j = max(e−rδt
p Vi+1,j+1 + e−rδt
(1 − p )Vi+1,j, E − Si,j). (3.4)
Now we have a binomial method ready to go, we can explain it more clearly by
showing an example. For our example, we will take the underlying asset to be worth
S = 100 at time t, the exercise price to be E = 100, the risk-free interest rate to be
r = 0.1 and the volatility to be σ = 0.2. Let there also be 4 months until expiration,
that is T − t = 4
12 years to expiration. If we take n = 4 time steps then we get the size
of a time step as δt = T−t
n = 1
12. In other words, we are taking time steps of 1 month.
If we substitute these values into equation (3.1) then we receive u = 1.0604. Recall
that v = 1
u , which gives us v = 0.9431. Now substituting the above values into equation
(3.2) gives us p = 0.5567.
At time t we have that S = 100, so in our binomial tree notation we have that
S0,0 = 100. Recall that an upward move means we multiply the underlying asset value by
u and a downward move means we multiply the underlying asset value by v. We can then
compute that S1,0 = vS0,0 = 94.31 (to 2 decimal places) and that S1,1 = uS0,0 = 106.04.
We carry on computing the values of Si,j this way until we reach the final nodes S4,0,
S4,1, S4,2, S4,3 and S4,4, completing the entire binomial tree as in Figure 6.
Now that we have the value of the underlying asset at each node we can begin to
value the option. We start from the right of the binomial tree, computing the payoff
function before working to the left of the tree using equation (3.4). Here the payoff
function would be Vn,j = max(100 − Sn,j, 0). As an example let us take the bottom
node on the far right, that is node (4, 0). Here we have that S4,0 = 79.10, so this means
that the option is worth V4,0 = max(100 − 79.10, 0) = 20.90 at this node. We then
repeat this on the nodes above (4, 0) to find the values of V4,1, V4,2, V4,3 and V4,4.
Having computed the value of the American put option at each of the nodes at
expiration, we must work our way leftward down the binomial tree using equation (3.4).
We will show how to calculate V0,0 from V1,0 and V1,1. We can calculate that V1,0 = 6.24
and V1,1 = 1.10, which we then substitute into equation (3.4) along with E = 100,
22
24. Figure 6: The binomial tree of the underlying asset value.
r = 0.1, δt = 1
12, p = 0.5567 and S0,0 = 100 to give us
V0,0 = max(e−rδt
p V1,1 + e−rδt
(1 − p )V1,0, E − S0,0)
= max(0.9917 × 0.5567 × 1.10 + 0.9917 × 0.4433 × 6.24, 100 − 100)
= max(3.35, 0)
= 3.35.
Note that at this particular node we have
3.35 = e−rδt
p Vi+1,j+1 + e−rδt
(1 − p )Vi+1,j > E − Si,j = 0,
and so for this node it would be optimal to hold the option rather than exercise it.
We will now show an example of a node where it is optimal to exercise early. Let
us consider node (3, 0). We have that V4,0 = 20.90 and V4,1 = 11.06, so using equation
(3.4) as above gives us
V3,0 = max(e−rδt
p V4,1 + e−rδt
(1 − p )V4,0, E − S3,0)
= max(0.9917 × 0.5567 × 11.06 + 0.9917 × 0.4433 × 20.90, 100 − 83.88)
= max(15.29, 16.12)
= 16.12.
Here we have
15.29 = e−rδt
p Vi+1,j+1 + e−rδt
(1 − p )Vi+1,j < E − Si,j = 16.12,
23
25. Figure 7: The binomial tree of the American put option value.
and so at this node it is optimal to exercise the American put option early. Repeating
similar calculations for every single node gives us Figure 7. For clarity, the option values
at nodes where it is optimal to exercise the option early are boxed.
Continuing all of the way to the left of the binomial tree gives us the value of the
American put option as V (S, t) = V0,0 = 3.35. This may not be the true value of the
option however. We have discretized the evolution of the underlying asset value using
just 4 time steps. But of course options in the real world are not treated in this way, their
value and the value of their underlying assets vary continuously. We can approximate
continuity by allowing the size of our time steps, δt, to approach zero. Why is this
the case? Think of the binomial tree as a mesh across the whole domain of possible
times and underlying asset values. All possibilities that are not covered by a node are
dismissed and so to improve the accuracy of our binomial method we want as many
nodes as possible covering the domain. With a time step size approaching zero we can
think of our mesh tightening to cover the entire domain.
We can let δt → 0 by allowing n → ∞. This means that the more time steps we
take over the interval [t, T], the more accurate we expect the binomial method to be.
Above we found an approximation to the option value using only 4 time steps and this
was still time consuming. We have reached the limit of what we can reasonably achieve
without the use of a computer. It is possible to write a program in Matlab that will
follow the same algorithm that we outlined above, calculating the underlying asset and
24
26. the option value at each node. We do this and call the Matlab program Binomial.
Running Binomial for the same option and with n = 4 time steps, the exact same
calculation as done above, gives us V (S, t) = 3.35 (to 2 decimal places). This of course
is the exact same value we received from running the algorithm by hand. We can now
use this program to run the algorithm with more time steps to improve the accuracy of
our simulation. If we use 10 time steps the program values the option as 3.3879, 100
time steps give 3.4087, 1,000 time steps give 3.4106 and finally 10,000 time steps give
us 3.4107. It is encouraging that these values of V (S, t) appear to be stabilising and
converging.
With numerical solutions we have the issue of convincing ourselves that our method
is indeed giving us the correct value. We have already seen one way of doing this, in the
form of using the Matlab code to make the exact same calculations that we ourselves
made by hand. This should give us some confidence that our Matlab code is accurate.
We can also use the fact that, everything else being equal, an American put option is
worth at least as much as a European put option. Valuing a European put option with
the same parameters as for the American equivalent we have just considered gives us a
value of 2.9588, using equation (1.4). We have that 3.4107 ≥ 2.9588, which adds to our
code’s credibility.
We can go further than this by attempting to use our Matlab code to value something
that we actually know the analytical solution of. In particular, let us adapt our code
and use it to value a European call option which we can also value using equation (1.3).
To do this, all we need to alter in the code is the payoff function. Whenever the
payoff of a put option appears in our code we must change it to the payoff of a call
option. Note that this is the only change that we need to make to the code. We do not
have to remove the possibility of early exercise as we recall that it is never optimal to
exercise an American call option early. Now if we use equation (1.3) to value a European
call option with S = 30, E = 34, T − t = 3
12, r = 0.08 and σ = 0.2 (chosen arbitrarily),
we find that V (S, t) = 0.2383. This is the analytical solution – the value that should
be the output of our Matlab code for large n. In fact if we run the adapted code with
n = 1, 000 time steps we get that exact same value. We can be confident in our binomial
method.
It is also possible to adapt the code of Binomial so that it can find and plot the
optimal exercise boundary of an American put option. This works at each time step
by determining the first node (working downwards from the top of the tree) where it is
optimal to exercise and then noting the associated underlying asset value. In particular,
25
27. Figure 8: The optimal exercise boundary computed with 4 time steps.
the programme starts at the top node of each time step and works downward checking
whether or not it is optimal to exercise early. If we assume that at the i-th time step
the first node where early exercise is optimal is at node (i, j), then we take (iδt, Si,j) as
a point on the optimal exercise boundary.
Recall the example we presented for applying the binomial method to value an
American put option. Referring back to Figure 7 we saw that at some nodes it was
optimal to exercise our option early rather than to continue holding it. If we find that
at a certain node it is optimal to exercise, then at the node below it will also be optimal
to exercise. This is because at the lower node the underlying asset will be worth less
and so the payoff of the option will be even higher than that of the above node. We can
see an example of this in Figure 7. Recall that we boxed the option value at the nodes
where it is optimal to exercise the option early. We can see that 5.69 is boxed and, as
we would expect, 16.12 is boxed too.
Now we can begin to picture a curve on our binomial tree, separating the boxed nodes
from all of the other nodes. This curve is then our approximation to the optimal exercise
boundary. Rather than plotting the option values at these nodes we use the associated
underlying asset values as we recall that the optimal exercise boundary separates those
underlying asset values where it is optimal to exercise from those where it is not. Then
it is simple for the program to plot these underlying asset values for each time step. We
use Binomial to do this for our 4 time step example in order to find Figure 8.
Let us discuss why Figure 8 is correct. Considering Figure 7, we see that it is not
optimal to exercise at time steps 0 or 1 (none of the values are boxed). Therefore,
recalling that one time step is 1
12, the first two points of the optimal exercise boundary
26
28. Figure 9: The optimal exercise boundary computed with 10,000 time steps.
are (0 × 1
12, 0) and (1 × 1
12, 0). At time step 2 we see that it is optimal to exercise at
only node (2, 0). At this node the underlying asset is worth 88.94, and we know that
if the underlying asset is worth less than this then it is still optimal to exercise as the
payoff will be even higher. Therefore, another point for the boundary is (2 × 1
12, 88.94).
At time step 3 we have more than one node where it is optimal to exercise but we are
interested in the one where the underlying asset value is highest. This of course is at
node (3, 1) where the underlying asset is worth 94.31. This gives us the final point of
the boundary as (3 × 1
12, 94.31). Why is the fourth and final time step missing from this
graph? The final time step is at expiration where it is no longer early and hence, early
exercise is not possible at this time step.
The above reasoning gives us confidence that our program is giving us an accurate
approximation to the optimal exercise boundary for the given number of time steps. We
will be even more confident if Binomial, with large n, outputs a graph that resembles the
curve given in Figure 4. We run Binomial with n = 10, 000 time steps to find Figure 9.
This gives us a much more accurate looking approximation to the optimal exercise
boundary. The only problem with this approximation is that it gives a vertical line near
the beginning of the time interval and we know from our earlier discussions that this is
incorrect. By the nature of the binomial method it is always going to take a number of
time steps for the underlying asset to first get to a level where it is optimal to exercise
and so our approximation to the boundary will always be zero at the first time steps. As
we take smaller and smaller time steps we will see this vertical line tend more and more
towards the Sτ -axis. It is also possible to use extrapolation techniques to approximate
the boundary to the left of this vertical line. Extrapolation is explained very well by
27
29. Sidi [13].
We can also see that this optimal exercise boundary appears to satisfy the conditions
we stated previously. The entire optimal exercise boundary appears to be bounded by
2rE
2r+σ2 = 83.33 and E = 100. Using extrapolation we can expect Sb(t) ≈ 89 and we
should have that this lies above 2rE
2r+σ2 = 83.33 and below E − P = 100 − 3.41 = 96.59,
which it clearly does. We can also see that Sb(T) = 100 = E. Also we have that the
boundary is strictly increasing and near expiration it resembles the function b∗(τ) that
we defined earlier.
We now know in great detail how to use the binomial method to value an American
put option and to determine the optimal exercise boundary. We have also seen that
this method appears to be accurate, but we must not stop here as we reasoned earlier
that we must consider multiple methods. Once we have discussed some more methods
we will have a solid basis for comparison and we will return to our discussion of the
binomial method. Our next step is to find a generalisation of the binomial method.
3.2 Finite Difference Methods
Now we will consider the use of finite difference methods. Finite difference methods can
be used to value options by approximating the Black-Scholes equation using discrete-
time difference equations. These discrete-time difference equations can then be solved
iteratively to find a numerical solution to the original continuous-time Black-Scholes
equation, giving us the value of the option. The most popular finite difference methods
used in computational finance are the Explicit Euler, Implicit Euler, and Crank-Nicolson
methods. Explicit Euler suffers from stability issues under certain conditions whereas
Crank-Nicolson is a combination of the two Euler methods. Therefore, we will focus on
the Implicit Euler method.
A finite difference is an expression of the form
f(x + b) − f(x + a),
where f is an arbitrary function and a and b are arbitrary constants. A finite difference
equation is a natural way to approximate a derivative. Recall that the definition of the
derivative of a function f at a point x is given by
f (x) = lim
h→0
f(x + h) − f(x)
h
.
Thus, we know that the finite difference f(x+h)−f(x), divided by small h, is an accurate
approximation of the derivative f (x). Due to this we can use finite difference equations
to approximate and replace the derivatives appearing in a differential equation.
28
30. Figure 10: The 2-dimensional lattice of discretized time and underlying asset value, with
our chosen coordinate system.
We will value options by expressing the derivatives in the Black-Scholes equation by
discrete-time difference equations, just as we reasoned above. To do this we will model
the evolution of the option value by a 2-dimensional lattice, in a similar way to how we
used binomial trees previously. The best way to comprehend the 2-dimensional lattice is
to consider a grid such as in Figure 10. One dimension of the lattice is time, which runs
from the valuation date t to the expiration date T and the other dimension is the value
of the underlying asset which runs from 0 to an adequately high value. Finite difference
methods then work by creating a mathematical relationship which links together every
node on the lattice. In fact, finite difference methods are just a generalization of the
binomial method we saw previously.
We utilize the known boundary and final conditions to compute the values at some
of the nodes before using an iterative relationship to find the values at the remaining
nodes. We start by using the payoff function to compute the option value of each node
at expiration. Then the option values at the other nodes can either be computed from
the boundary conditions or recursively from the values at expiration, working backwards
in time to t. It is the recursive step where we use a discretized form of the Black-Scholes
equation in such a way that the option value at an arbitrary node is given by a function
of the option value at adjacent nodes. The value of the option at a chosen time and
underlying asset value is then either given by a particular node or can be found by
29
31. interpolating between nodes.
We mentioned that the value of the underlying asset runs from 0 to an adequately
high value. Denote this adequately high value by Smax. Note that in practice it is
possible (albeit unlikely if Smax is chosen befittingly) that the underlying asset value
may exceed Smax sometime in the time interval [t, T] and that ideally we would choose
Smax = ∞. However, this clearly is not possible for our discretized problem. We must
choose our own value of Smax, acknowledging that the higher the value of Smax the better
we expect our approximation to be and also that we will expect more computation. We
also need to choose the number of discretizations of the asset value (call this M) and
the number of discretizations of time (call this N). The values of M and N dictate the
size of the steps between different nodes in a similar way that the number of time steps
n did for the binomial method. Now note that we have
MδS = Smax and Nδt = T,
where δS is the step size between successive underlying asset values and δt is the size of
each time step. As with Smax, we would expect that picking large values of M and N
would increase accuracy but also increase the number of computations we require.
Now define Vi,j = V (iδt, jδS) as the value of the option at the node (i, j), that is
at the i-th time step and the j-th underlying asset value step. We now want to use the
boundary and final conditions to compute some of the nodes to give us a starting point
before we can make use of any iterative relationship. Recall that the final condition for
an American put option is V (ST , T) = max(E − ST , 0). We can use this condition for
all of the nodes at expiration time T = Nδt: the nodes VN,j for all j {0, 1, 2, ..., M}.
Hence, in terms of the lattice notation we get that VN,j = max(E − (jδS), 0) for all
j {0, 1, 2, ..., M}.
Now we consider the boundary conditions, starting with V (0, t) = e−r(T−t)E which
becomes Vi,0 = e−r[(N−i)δt]E for all i {0, 1, 2, ..., N} under the lattice notation. We can
see that this is the case as at time t, T −t is the remaining time to expiration and at the
equivalent discrete time iδt (for any i), (N − i)δt is the remaining time to expiration.
The remaining boundary condition to consider is V (S, t) ≈ 0 as S → ∞. In the
lattice notation this becomes Vi,M = 0 for all i {0, 1, 2, ..., N}. We see that this is the
case because V (S, t) ≈ 0 as S → ∞ is equivalent to V (Smax, t) ≈ 0, since Smax is
our highest possible value. Recalling that MδS = Smax, the condition becomes exactly
Vi,M = 0 for i {0, 1, 2, ..., N} in the discrete lattice notation. Summarising the conditions
30
32. for our discrete problem we have
VN,j = max(E − (jδS), 0), j = 0, 1, . . . , M
Vi,0 = e−r(N−i)δtE, i = 0, 1, . . . , N
Vi,M = 0, i = 0, 1, . . . , N.
(3.5)
From the final condition we know the option values at expiration and so we need to
solve the Black-Scholes equation backwards, working from the expiration date T to the
valuation date t. There are different ways to use finite differences to solve the Black-
Scholes equation, but as we mentioned earlier we will be using the Implicit Euler method.
Explicit methods calculate the state of a system at a later (or earlier) time by solving an
equation using the state of the system at the current time, whereas implicit methods do
this by solving an equation involving both the current state of the system and the later
(or earlier) one. In general, implicit methods are more difficult to implement but offer
greater stability than explicit methods. More is said on the difference between explicit
and implicit finite difference methods by Lyuu [14].
If we recall the Black-Scholes equation, equation (1.2), we can seen that there are
three derivatives that we need to approximate by finite difference equations. One possi-
ble discretization of the Black-Scholes equation, for use with the Implicit Euler method,
is
Vi,j − Vi−1,j
δt
+
1
2
σ2
(jδS)2 Vi−1,j+1 − 2Vi−1,j + Vi−1,j−1
(δS)2
+r(jδS)
Vi−1,j+1 − Vi−1,j−1
2δS
−rVi−1,j = 0.
We find that this can be rewritten as
Vi,j = AjVi−1,j−1 + BjVi−1,j + CjVi−1,j+1, (3.6)
with
Aj =
1
2
δt(rj − σ2
j2
), Bj = 1 + (σ2
j2
+ r)δt, Cj = −
1
2
δt(rj + σ2
j2
). (3.7)
Note that there are multiple ways to approximate the derivatives in the Black-Scholes
equation. For example, with a first derivative we could have chosen a forward, backward
or even a central difference. A great deal more is written on this area by Brandimarte
[15]. Brandimarte also takes Lyuu’s comparison [14] of explicit and implicit finite dif-
ferences further in the direction of option valuation.
Here we have used a backward difference for the time derivative, a central difference
for the first derivative with respect to the underlying asset value and the standard
31
33. (central) difference for the double derivative with respect to the underlying asset value.
We must choose a backward difference for the time derivative as we are solving backwards
in time from expiration, whereas we have used central differences for the other two
derivatives as they are generally the most accurate.
As we are working backwards in time, equation (3.6) gives us three unknown values
(at the (i − 1)-th time step) linked to one known value (at the i-th time step). Recall
that the number of discretizations of the asset value is M and that the number of
discretizations of time is N. We will see that for each time step i we have M−1 equations
in M − 1 unknowns, plus two more values derived from the boundary conditions. The
values VN,0, VN,1, ..., VN,M in the final time step N are derived from the final condition
given in equation (3.5) and we use these as our starting point. To solve this problem we
work backwards in time solving a sequence of systems of linear equations for each time
step i = N − 1, ..., 0. The system of linear equations for time step i is then given by
B1 C1 0
A2 B2 C2
A3 B3 C3
...
...
...
AM−2 BM−2 CM−2
0 AM−1 BM−1 CM−1
Vi−1,1
Vi−1,2
Vi−1,3
...
Vi−1,M−2
Vi−1,M−1
=
Vi,1
Vi,2
Vi,3
...
Vi,M−2
Vi,M−1
+
A1Vi−1,0
0
0
...
0
CM−1Vi−1,M
, (3.8)
where the matrix in the system is tridiagonal.
At each time step i we know Vi,1, ..., Vi,M−1 from computing the previous time step
(as we are working backwards in time) and all of the Aj, Bj, Cj can be computed directly
from equation (3.7) for any j. Finally, we can see that Vi−1,0 and Vi−1,M can be computed
from the boundary conditions given in equation (3.5). Then all that remains is to solve
this system in order to find Vi−1,1, ..., Vi−1,M−1.
Up until now we have kept our discussion of the Implicit Euler method general, that
is, we have not concentrated on American options. In explicit methods the extension to
American options is simple and shares similarities to that of the binomial method. In
32
34. particular, at each node we would compute the option value as derived from the iterative
relationship but we would also compare this value to the one that we would receive from
exercising at this node. Doing this, the option value at a given node would be given by
a relationship analogous to equation (3.4).
However, this is more difficult for implicit methods. We must solve a linear system,
equation (3.8), to progress from one time step to the next and we do not know the value
of the option at a given node until we get to the next time step. We can see this from
equation (3.6). We know Vi,j and we need to know two of the option values at time step
i − 1 in order to compute the third one (all of the option values are implicitly related to
one another). But in solving the system directly we do not get to implement the ability
to exercise early until each of the options at the nodes have already been valued. If the
option values have been computed from one another before early exercise is implemented
then the option values are clearly incorrect. To bypass this issue we must solve each
linear system iteratively rather than using a direct method.
The iterative method we will use is known as Successive Over Relaxation (SOR)
iteration. We will use SOR iteration to solve the linear system given in equation (3.8)
for each time step i. Note that equation (3.8) can be written in the form Ax = b, where
A is a k × k matrix and x, b are vectors of size k. Applying SOR iteration to equations
of the form Ax = b is described by Burden and Faires [16].
SOR iteration projects the option values so that we can check if early exercise is
optimal. This works by starting with an initial approximation of x(0) and iterating to
find the approximate solution x(k). The elements of the vector x(k) are computed from
the elements of x(k−1) and so we can compute each x
(k)
i,j ≈ Vi,j one at time, implementing
early exercise.
Now we must apply SOR iteration to each time step. Having done this we will know
the correct value of Vi,j at each node of the 2-dimensional lattice. All that is left is to
find the value of the option at a chosen time and underlying asset value. As we stated
earlier, this is either given by the option value Vi,j at a particular node or can be found
by interpolating between nodes.
A Matlab code that implements the Implicit Euler method with SOR iteration is
given by Richardson [17]. We call this code ImplicitEuler. We find that ImplicitEuler
is already built to interpolate for the option value at our chosen values of S and t. Now
we need to verify that this code is indeed giving us accurate option values, just as we
did with the binomial method.
We omit a comparison of the output of the code to the output of computing an
33
35. example by hand, as solving each system of linear equations is time consuming. If we
choose to value a European put option with S = 36, E = 40, T − t = 3
12, σ = 0.25
and r = 0.08, then by using equation (1.4) we get that P(S, t) = 3.8978. Now if we
use ImplicitEuler, with M = 100, N = 400 and Smax = 200, to value the equivalent
American put option then we get 4.1425. We recall that the American put option should
be worth at least as much as the European put option and we get that 4.1425 ≥ 3.8978,
which satisfies this.
Just as we did for the binomial method, we want to try changing the code slightly so
that we can use it to value an option that we can already value analytically. The easiest
way to do this with ImplicitEuler is to remove the ability to exercise early so that it
values a European put option. Doing this with M = 100, N = 400 and Smax = 200 gives
us 3.8759, which is very close to the analytical solution that we previously computed.
Note that we can make this approximation even closer to the analytical solution by
changing the parameters M, N and Smax. Conducting these tests gives us confidence
that the Matlab code ImplicitEuler is accurate.
It is also possible to compute the optimal exercise boundary using finite difference
methods. This works for each discrete time iδt by setting Sb(iδt) to be the last contact
point with the payoff. This is analogous to how we computed S = Sb(τ) with the
binomial method, working through all of the nodes at a particular time step to find the
last node where it is optimal to exercise.
Now that we have found a generalisation of the binomial method, it would be logical
to pursue a method that uses a completely different approach. Studying a completely
different method arms us with more ideas for when we look to study American options
with higher complexity. One alternative approach is to consider a probabilistic method.
3.3 Monte Carlo Methods
Unlike the previous methods we have considered, Monte Carlo methods are probabilistic.
In our context, they work by using random number generation to simulate the evolution
of asset values. This is often done multiple times before finding the average of these
simulations. We assume that asset values move according to the stochastic differential
equation dS = µSdt + σSdX, as we mentioned previously. Random number generation
is a large topic and it is discussed by Press et al. [18], along with some of its applications.
A simple example of how Monte Carlo methods work, outside of the realm of finance,
is to estimate the integral of a function f over the unit interval [0, 1]. Let α denote the
34
36. integral
α =
1
0
f(x)dx.
Now suppose that we can randomly generate points U1, U2, ..., Un independently and
uniformly from the interval [0, 1]. Then, as we know f, we can evaluate the function
values at these n points before finding the average. This gives us the Monte Carlo
estimate
ˆαn =
1
n
n
i=1
f(Ui).
Therefore, if f is integrable over [0, 1] then, by the Strong Law of Large Numbers,
ˆαn → α as n → ∞ with probability 1. This means that we can use randomly generated
numbers to approximate the value of an integral. The Strong Law of Large Numbers is
important for Monte Carlo methods and is described by Ross [19].
To value a European put option we could simulate the value of the underlying asset
at expiration before using this value in the discounted payoff function. As this is a
probabilistic simulation we would expect different results each time. To combat this, we
can repeat this process n times before computing the mean of the n option values. By the
Strong Law of Large Numbers we expect that for large n, this value will approximately
be the true value of the option.
Recall that we model the underlying asset’s value using the stochastic differential
equation dS = µSdt + σSdX. We assume risk-neutrality here and, as we discussed
previously, in this scenario we replace µ with r. Now also recall that dX is a Wiener
process, a random variable drawn from a normal distribution with mean 0 and variance
dt. Bearing this in mind we can let dX = φ(dt)
1
2 , where φ is a random variable drawn
from a normal distribution with mean 0 and variance 1. Now our underlying asset value
is modelled by
dS = rSdt + σSφ(dt)
1
2 . (3.9)
Equation (3.9) is a lognormal random walk. A lognormal random walk is ideal for
modelling asset values as the random walk can never go negative and it can also never
reach infinity in finite time. Physically this makes sense as we know that the prices
of equities, currencies, commodities and indices can never go negative or reach infinity.
We can actually see that this model has these properties by solving equation (3.9). But
first, let us state a result from Ito’s lemma.
Assume that the underlying asset value S satisfies the stochastic differential equation
dS = a(S)dt + b(S)dX,
35
37. where a(S) and b(S) are both arbitrary functions of S and dX is the Wiener process.
Now let F be another arbitrary function of S. Then by Ito’s lemma, the function F
satisfies the stochastic differential equation
dF =
dF
dS
dS +
1
2
[b(S)]2 d2F
dS2
dt.
Let us take the function F(S) = log(S). Now if we apply this result from Ito’s lemma
on the function F, knowing that S satisfies equation (3.9) (recalling that dX = φ(dt)
1
2 ),
then we get
dF =
dF
dS
dS +
1
2
σ2
S2 d2F
dS2
dt
= (
1
S
)dS +
1
2
σ2
S2
(
−1
S2
)dt
=
1
S
(rSdt + σSφ(dt)
1
2 ) −
1
2
σ2
dt
= (r −
1
2
σ2
)dt + σφ(dt)
1
2 .
Note that we used b(S) = σS and equation (3.9) itself to do this. Ito’s lemma and
stochastic calculus are explained in great depth by Calin [20]. Note that the stochastic
calculus outlined by Calin is necessary background information for Wilmott’s derivation
[5] of the Black-Scholes equation.
As dt is the size of the time increment we are considering, we will take dt = T − t.
This now means that dF (the change in F over time interval dt) is also given by
dF = F(ST ) − F(St) = log(ST ) − log(St) = log
ST
St
.
Now combining the above gives us
ST = Ste(r−1
2
σ2)(T−t)+σ
√
T−t φ
. (3.10)
Equation (3.10) confirms that, assuming St > 0, for all future T we have ST > 0.
That is, when modelled by equation (3.9), the underlying asset value can never go
negative. Note that the exponential function ex cannot reach ∞ in finite time. Thus,
when modelled by equation (3.9), the underlying asset value can never reach ∞ in finite
time also.
We can discretize both equation (3.9) and equation (3.10) to simulate the underlying
asset value over the period T − t:
Sn+1 = Sn + rSnδt + σSn
√
δtφ, (3.11)
36
38. Figure 11: An asset value simulated by a lognormal random walk with S = 100, T = 4
12,
r = 0.1, σ = 0.2 and 1,000 time steps.
Sn+1 = Sne(r−1
2
σ2)δt+σ
√
δt φ
, (3.12)
where δt = T−t
N−1 and N = number of time steps.
For example, we may decide that we are going to simulate the asset value over the
period T − t using 100 time steps. Say the current asset value is S = 100. Then we
take S1 = 100 and substitute in the known values of r and σ into either of the above
equations. For φ we substitute in a randomly generated number taken from the normal
distribution with mean 0 and variance 1. The output we get from doing this gives us
S2, which we now substitute into the right-hand side of either equation with another
randomly generated φ. We continue this process until we get to S100, the simulated
value of the asset at expiration time T.
We write a Matlab program called AssetSimulation that values assets in this way,
using equation (3.12). In Figure 11 we can see an example of AssetSimulation used
with the parameters S = 100, T = 4
12, r = 0.1, σ = 0.2 and 1,000 time steps. Note
that Figure 11 looks similar to the graph of an equity price, just as we might find in a
financial publication.
Now we can value a European put option by first using AssetSimulation to approxi-
mate ST . We can then substitute this approximation into the discounted payoff function
e−r(T−t)
max(E − ST , 0)
to give us one approximation to the option value. Then as we reasoned previously, we
can do this n times before computing the mean of the n option values. For large n we
expect this mean value to be an accurate approximation of the true option value.
37
39. Now we are ready to use Monte Carlo methods to value an American put option.
The obvious obstacle here is the embedded decision of when (and if) to utilise early
exercise. This means that simply simulating ST and computing the discounted payoff of
the option does not suffice here. We know that at every time before expiration we must
consider whether it is optimal or not to exercise the option. This free boundary value
problem is difficult to consider when using Monte Carlo methods.
Before continuing, we need to formulate the problem we are trying to solve in a way
that is appropriate for use with Monte Carlo methods. We know that for an American
put option, the payoff for exercising at time τ is max(E − Sτ , 0). Now let T be the set
of all times before expiration T at which we can exercise the option. Then, assuming it
is optimal to exercise early, we can find the value of the option by computing
sup
τ T
{e−rτ
max(E − Sτ , 0)}.
We can see that this is the case as we are considering all of the possible values that
could occur from early exercise and then we are taking the largest of these discounted
payoffs as the option value.
Also, if we are able to find the optimal exercise boundary, Sb(τ), for t ≤ τ ≤ T then
we can also find the value of the option by computing
τ∗
= inf{τ ≥ t : Sτ ≤ Sb(τ)}. (3.13)
Here τ∗ is the first time that the asset value drops below the optimal exercise boundary.
By the definition of the boundary it would be optimal to exercise at time τ∗, giving
e−rτ∗
max(E − Sτ∗ , 0) as the option value. The Least Squares method is one such
Monte Carlo method that can be used to value American put options using this kind of
reasoning. The Least Squares method was first introduced by Longstaff and Schwartz
[21] and we will now attempt to generalise it.
We start by taking the time interval [t, T] and discretizing it. We will take the
discretization t = t0, t1, . . . , tn = T. Also assume that the step size is equal between
consecutive time steps, in other words δt = tj −tj−1 for all j = 1, . . . , n. The next task
is to simulate the asset value over the n time steps just as we did previously, using either
equation (3.11) or equation (3.12). From doing this, we know the underlying asset value
at t0, t1, . . . , tn – in other words, we know St0 , St1 , . . . , Stn .
Now consider the final time step tn. Assume that the option has not been exercised
prior to tn. We compute the payoff of the option here, which is max(E − Stn , 0). Now
we introduce a matrix that keeps track of our cash flow at each time step. So far we
38
40. would have
t1 t2 . . . tn−1 tn
. . . max(E − Stn , 0)
.
Our objective is to fill in the empty entries of our cash flow matrix. The entry we
currently do have at tn may change as we assumed, possibly wrongly, that the option
would not have been exercised prior to the final time step.
Next we must decide whether it is optimal to exercise early at time step tn−1 or
if it is optimal to wait until the next time step, tn. We start by assuming that the
option has not been exercised before time step tn−1. If the option’s payoff is 0 here,
then the tn−1 entry of the cash flow matrix is 0. If the option’s payoff is positive here,
then let X denote the asset value at tn−1, which is X = Stn−1 . Now also let Y denote
the discounted (from tn to tn−1) payoff function at time step tn. In other words, let
Y = e−rδt max(E − Stn , 0).
We want to estimate the expected cashflow from not exercising the option, condi-
tional on the asset value Stn−1 . We do this by regressing Y on a function of X. Regres-
sion, particularly in the context of valuing American options, is discussed by Glasserman
[22]. There are many choices for the regression basis functions but for implementation in
Least Squares, weighted Laguerre polynomials are commonly used. The n-th Laguerre
polynomial Ln is defined by
Ln(x) =
ex
n!
dn
dxn
(e−x
xn
).
A discussion of different regression basis functions that could be used for Least Squares,
including Laguerre polynomials, is given by Moreno and Navas [23]. From regressing
Y according to these functions we get the conditional expectation E[Y |X], which is a
function in terms of X. Longstaff and Schwartz [21] also give a numerical example of
this regression, with simple basis functions.
The next task is to compare the value of exercising at time step tn−1 with the value
from not exercising. We have that the value of exercising at time step tn−1 is given
by max(E − Stn−1 , 0) and also that the value from not exercising is given by E[Y |X],
evaluated at the value of X we previously defined. The decision here, upon whether
exercising the option is optimal or not is then given by the higher value. We can then
use this information to update our cash flow matrix. If it is optimal to exercise here
then the tn−1 entry is given by E − Stn−1 and the tn entry becomes 0 (there can never
be positive cash flow at a time step after the option has been exercised). If it is not
optimal to exercise here then the tn−1 entry is 0 (we receive no income here) and the tn
39
41. entry remains unchanged. This gives us our updated cash flow matrix:
t1 t2 . . . tn−1 tn
. . . 0 or E − Stn−1 0 or max(E − Stn , 0)
.
Now we move on to deciding whether or not it is optimal to exercise early at time
step tn−2. As done previously we assume that the option has not been exercised before
tn−2. Once again we must note whether or not the option’s value is positive here, as
if it is not then the relevant cash flow is 0. Let us assume that the option’s value is
indeed positive here. We let X be the asset value here, so X = Stn−2 . Now we let Y be
the discounted value of the payoff at whichever future time step the option is exercised
at. At tn−2 the only future time steps are tn−1 and tn. We take the non-zero entry
of the cash flow matrix at one of these future time steps as the payoff here (or 0 if
both entries are 0). We always discount to the current time step we are considering.
Hence, if the option is exercised at tn−1 then we discount by one time step, but if it
is exercised at tn then we must discount by two time steps. Therefore, we take either
Y = e−rδt max(E − Stn−1 , 0) or Y = e−2rδt max(E − Stn , 0).
Then, as before, we estimate the expected cash flow from not exercising the option,
conditional on the asset value Stn−2 , by regressing Y on Laguerre functions of X. This
gives us a new conditional expectation function E[Y |X], different to the previous one.
We then compare the value of exercising (computing the payoff function at Stn−2 ) to
the value of not exercising (evaluating E[Y |X] at X), just as we did for time step tn−1.
Then, as done previously, we update the tn−2 entry of the cash flow matrix with either
0 or E − Stn−2 . If it is optimal to exercise here then the cash flow at all future time
steps becomes 0. Either way, there will only be one non-zero entry in the second row of
the cash flow matrix (apart from if it is not optimal to exercise at any time step).
We repeat this process for each time step, working backwards from tn−3 to t1, to
give us the cash flow matrix
t1 t2 . . . tn−1 tn
0 or E − St1 0 or E − St2 . . . 0 or E − Stn−1 0 or max(E − Stn , 0)
.
Assuming that our option does not expire worthless, we will find that the second row of
our cash flow matrix has exactly one non-zero entry. This is because there is only one
time step where it is optimal to exercise and it is known as the optimal stopping time
(recall τ∗ from equation (3.13)).
If we assume that the only non-zero entry of the cash flow matrix is at time step ti,
40
42. then our cash flow matrix becomes
t1 t2 . . . ti . . . tn−1 tn
0 0 . . . E − Sti . . . 0 0
.
All that remains to do, in order to receive our approximation of the American put option
value, is to discount this payoff to time step t0. Here the approximation would be given
by
e−riδt
(E − Sti ).
Note that if it was never optimal to exercise the American put option then our approx-
imation to its value would be 0. Now it is a case of running M of these simulations to
receive M approximations, before computing their mean value. This mean value is then
an accurate approximation to the true value of the American put option for large M.
We have a Matlab code that runs the Least Squares method called LeastSquares
and it was written by Phoulady [24]. Note that this code uses equation (3.11) rather
than equation (3.12) to simulate the underlying asset value, although this can easily be
changed if desired. Now we want to check that this code is accurate, just as we did for
the previous methods. It is difficult to follow this method by hand due to the random
number generation, so we need to use other checks to verify the code.
Let us consider an American put option with S = 18, E = 20, T −t = 6
12, σ = 0.2 and
r = 0.05. Using equation (1.4) to find the analytical value of the analogous European
put option gives us P(S, t) = 1.9759. Using LeastSquares, with M = 5, 000 different
simulations and N = 50 time steps, to value the American put option gives us 2.1456.
As 2.1456 ≥ 1.9759, we have satisfied the constraint that the American put option must
be worth at least as much as the equivalent European put option.
We can also adapt the LeastSquares code so that it values a European put option, in
a way similar to that of the code AssetSimulation. If we remove all of the lines of code
regarding regression and early exercise then LeastSquares must value a European put
option. Running the adapted code for this problem, with M = 5, 000 and N = 50, we
receive the approximation P(S, t) ≈ 1.9731. This is very close to the analytical solution
P(S, t) = 1.9759 and gives us confidence that our Matlab code is accurate.
Although it would be difficult, it is certainly possible to adapt the LeastSquares
code so that it outputs the optimal exercise boundary. Let us consider one particular
simulation of the underlying asset value, where it is optimal to exercise the option early.
Say this optimal exercise occurs at time τ , where the underlying asset is worth S . Then
we take Sb(τ ) = S as an approximation of a point on the optimal exercise boundary.
If we do this for a large number of simulations then we will find many (approximate)
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43. Figure 12: Here we can see that ˜S is a better approximation to Sb(τ ) than S is.
points of the boundary. Of course, those simulations where it is not optimal to exercise
early will not contribute here.
Assume that two simulations are optimally exercised at τ but additionally assume
that both simulations have different underlying asset values at this time. That is, in the
first simulation the underlying asset is worth S at τ and in the second simulation the
underlying asset is worth ˜S at τ . If we assume that ˜S ≥ S , then we take Sb(τ ) = ˜S .
This is because if it is optimal to exercise then the underlying asset value is on or below
the optimal exercise boundary. As we have discretized the evolution of the underlying
asset value it may be that the value jumps to the other side of the boundary without
ever touching it. As can be seen in Figure 12, the higher underlying asset value will be
closer to (or is) the correct value of the optimal exercise boundary.
We have now introduced and discussed three different numerical methods for valuing
an American put option. The next step is to compare them against one another. We
wish to discover the most effective of the methods for solving our problem by considering
aspects such as ease of use and accuracy.
3.4 Comparing the Methods
We have spent a lot of time introducing different numerical methods for valuing an
American put option and now we would like to compare them. Recall that we have
considered a binomial method, a finite difference method (Implicit Euler) and a Monte
Carlo method (Least Squares). Before we go any further it is worth noting a major
similarity of the three methods.
Recall that American options are the same as European options, but with the addi-
tional ability to exercise the option at any time before expiration. Bermudan options fall
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44. somewhere in between American options and European options. A Bermudan option
is a European option, but with the additional ability to exercise the option at certain
pre-determined dates prior to expiration. An example is a Bermudan option due to
expire in three months that could be exercised at the end of month one and also at the
end of month two. In valuing this option, we would have to consider whether or not
exercising is optimal at the end of each month.
It is now clear to see that all of the methods we have considered for valuing an
American put option have actually been valuing a Bermudan put option. In all three of
our methods we have had to discretize the time interval [t, T]. Rather than being able to
exercise the option at any time, we have been considering exercising at a finite amount
of times prior to expiration. All three of the methods have worked by approximating
the American put option as a Bermudan put option with many possible exercise dates
(preferably tending to ∞). This means that we have to approximate the option before
we even apply a numerical scheme, which is a common drawback of all three of the
methods. Unfortunately this is a difficulty that cannot be avoided when considering
American options. Now that we have discussed a common caveat of all three of the
methods, we can begin to discuss their differences from a theoretical perspective.
The binomial method is by far the simplest of the three methods to implement. At
each node we can compute the value of exercising and the value of not exercising the
option directly. This means that it is clear to see where the option values at each node
are derived from. In fact, we saw that we could use this method by hand due to its
simplicity and conciseness, whereas this was not the case for Implicit Euler and Least
Squares. Another benefit of the binomial method is that it is effective for finding the
optimal exercise boundary.
A downside of the binomial method is that the only parameter we can change is
the number of time steps, n. To ensure a high level of accuracy we require n to be a
large value, which requires a lot of computational power. Another drawback is due to
the nature of the binomial tree. A 2-dimensional lattice covers a larger portion of the
underlying-time space than a tree, and so we expect that the accuracy from the tree
will be lower than from the 2-dimensional lattice (more on this in a moment).
Earlier we stated that finite difference methods are a generalization of the binomial
method. We can see this by comparing the binomial tree in Figure 5 to the 2-dimensional
lattice in Figure 10. Consider the time near the valuation date, where the binomial tree is
narrow and the 2-dimensional lattice is covering a large number of nodes. In deriving the
optimal exercise boundary using the binomial method we encountered the problem that
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45. it takes a number of time steps for the underlying asset to fall far enough so that exercise
is optimal. This is not an issue with Implicit Euler as the 2-dimensional lattice covers
many possible underlying asset values, even at the first few time steps. Therefore, we
would expect Implicit Euler to be more accurate than the binomial method for valuing
the option and finding the optimal exercise boundary. We also have the additional
flexibility of being able to change the parameters M, N and Smax, which can allow us
to optimize the trade off between accuracy and efficiency.
However, Implicit Euler is difficult to use and implement. In general, the implemen-
tation of implicit methods is difficult but this difficulty is necessary for circumventing
the stability issues of explicit methods (we mentioned this earlier). Also, the use of iter-
ative methods for solving the systems of linear equations makes the entire algorithm less
intuitive. For example, during the iterative step, it is certainly not clear at which nodes
it is optimal to exercise. SOR iteration also adds to the complexity of Implicit Euler,
as it requires the user to choose extra parameters. Up until now we have considered
these parameters as fixed, but it is entirely possible that a poor choice of these SOR
parameters may lead to an inaccurate approximation of the option value.
If we use a large number of simulations with Least Squares then, due to the Strong
Law of Large Numbers, we would expect the accuracy to be high, just as if we had used
a 2-dimensional lattice of nodes. This is because over the course of multiple simulations
we expect a large portion of the underlying-time space to be covered by these simulations
(assuming the time steps are small). The simplicity of Least Squares comes somewhere
in between that of the binomial and Implicit Euler methods. It is arguably easier to
understand and implement the regression in Least Squares than the SOR iteration in
Implicit Euler.
We have already mentioned two of the biggest issues with Least Squares. The first is
that it is probabilistic. Unless we use infinitely many simulations, running the method
more than once will give us different option values. This problem can be avoided to
an extent by running a very large number of simulations, but of course this will slow
the algorithm down. We also covered how we could use Least Squares to compute the
optimal exercise boundary and we saw that this also requires many simulations.
Now we would like to analyse the respective accuracies of the three methods. We
will do this by valuing a collection of different American put options. We will start by
picking a selection of option parameters to consider, before altering this selection one
parameter at a time. Doing this allows us to compare the option values against one
another, so that we can see if the outputs of the methods are changing as we would
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