PRICING AMERICAN OPTIONS USING MONTE CARLOSIMULATION

NAIRN MCWILLIAMS

Abstract. Option contracts are used by all major financial institutionsand investors either to speculate on stock market trends or to controltheir level of risk from other investments. American options form themajority of those traded today. Yet pricing such options, even in thestandard case of a lognormal process for the underlying asset, is still anarea of active research.

Contents

1. Introduction 12. Simulating a stock price path 33. Using Monte Carlo to price options 44. Pricing American options 75. Using stopping times to obtain a martingale 86. Finding the optimal stopping time 86.1. Using the binomial tree model 96.2. Making use of the Black-Scholes model 116.3. Using simulation to obtain an exercise boundary 127. Conclusion 17References 17

1. Introduction

An options contract on an underlying asset, such as a stock, allows thecontract holder the right to purchase or sell the asset within a predeterminedtime period. The end of this period is known as the maturity date and,within this period, the holder of the option can purchase or sell the asset foran agreed amount - the exercise price. Options differ from other derivativesecurities, such as forward contracts, in that the holder has the right toexercise the option at some point in the future, but not the obligation. Ifexercising the option is unfavourable, it does not have to be exercised.

There are two basic types of options available. A call option allows theholder the right to purchase an asset at the exercise price by the maturitydate. The owner of a put option can sell for the agreed amount within theagreed time period. American options can be exercised at any time up tothe maturity date. European options can only be exercised on the maturitydate itself.

Date: July 15, 2005.

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2 NAIRN MCWILLIAMS

Figure 1. Payoff from the European call option.

Example 1. Suppose an investor buys a European call option on a non-dividend paying stock1 and the exercise price is £5. The maturity date onthe option is 1 year later.

From now on, let t represent time in years, with T representing maturity.Let K be the exercise price and St the value of the asset at time t.

If, at maturity, the asset price exceeds the exercise price, ST > K = £5,the investor would exercise the option and buy the stock. They can thenimmediately sell the asset, realising a payoff of

PT = ST −K > 0

If the value at maturity is less than the exercise price, the stock could bebought for less elsewhere, so the option would not be exercised. The totalpayoff to the investor is

PT = (ST −K)+ := max(ST −K, 0) :=

ST −K ST −K > 00 otherwise

The present value of the payoff corresponds to the value of the payoffat the time the contract is sold. If r denotes the risk-free interest rate2

compounded continuously, the present value of the payoff is

e−rT PT = e−rT (ST −K)+

Figure 1 shows how the payoff varies with stock price at maturity. In thecase of the option being a European put, the present value of the payoff ise−rT (K − ST )+.

In the above scenario the investor can guarantee not to lose anything fromholding the option. In the worst case the option is not exercised and theinvestor earns (and loses) nothing. If the stock price at maturity exceedsthe exercise price, the investor makes a profit despite no initial investment.

1In this project, we shall assume all assets encountered do not pay any dividends.2The risk-free rate is usually obtained from the treasury rate of short-dated government

bonds in the required currency.

PRICING AMERICAN OPTIONS USING MONTE CARLO SIMULATION 3

The notion of risk-less profit is known as arbitrage. To prevent arbitrage,the investor pays a premium to buy the option, regardless of whether it isexercised. The value of the premium is the present value of the expectedpayoff from the option.3 The total payoff to the investor is PT − h, where hdenotes the option premium.

Example 2. Let Xt denote the present value of money worth Xt at time t.In the case of a European call option, the premium is

h = E[PT ] := E[e−rT (ST −K)+]

The present value of the expected payoff to the investor is then

E[PT − h] = E[PT − E[PT ]] = E[PT ]− E[E[PT ]] = E[PT ]− E[PT ] = 0

Since there is no expected return (or loss) there is no arbitrage. With aEuropean put option, the premium is h = E[e−rT (K −ST )+], and the sameresult follows.

The aim of this project is to attempt to find the premium for other op-tions - notably American options - by simulating the price of the asset overa specified time period. Monte Carlo methods can be used to obtain anestimate of the option price.

2. Simulating a stock price path

The project requires simulating asset prices over the time interval [0, T ].To do this, we require the following.

Definition 1. Let Ω = ω1, ω2, . . . denote the sample space. The σ-algebraF is the set of all observable events for a single trial, with the followingproperties.

• ∅,Ω ∈ F , where ∅ is the empty set.• If A ∈ F , then Ω \A ∈ F .• If A1, A2, . . . ∈ F , then

⋃∞i=1 Ai ∈ F .

P is the probability measure on F , where P(ω) ∈ [0, 1],∀ω ∈ Ω and P(Ω) = 1.The probability space (Ω,F , P) is equipped with a filtration - a collection ofσ-algebras Ft0≤t≤T , where

F0 ⊆ Fs ⊂ Ft ⊆ FT

for 0 ≤ s < t ≤ T . We assume F0 = ∅,Ω and FT = F = P(Ω).4

Definition 2. A stochastic process Xt0≤t≤T is a collection of randomvariables on (Ω,F). For each fixed ω ∈ Ω, Xt(ω) is the sample path of Xt

associated with ω.

The following gives an example of a stochastic process.

Definition 3. Let (Ω,F , P) be a probability space. A one-dimensionalBrownian motion is a continuous stochastic process Wt0≤t≤T such that

3At the time the option is exercised. In the case of American options, this is notnecessarily maturity time.

4The power set P(Ω) is the set of all possible subsets of Ω. If Ω has n elements, thenits power set contains 2n distinct subsets.

4 NAIRN MCWILLIAMS

• W0 = 0 a.s.5

• For 0 ≤ s < t ≤ T , Wt −Ws is independent of Fs and is normallydistributed with mean 0 and variance t− s.6

• For 0 = t0 < t1 < . . . < tm = T , the increments Wtj − Wtj−1 areindependent and their distribution depends only on the differencetj − tj−1. Wt is said to have independent and stationary incre-ments.

Definition 4. The stochastic integral is defined by the random variable

XT :=∫ T

0g(t) dWt

that isdXT = g(t) dWt

Here, dWt can be thought of as an N(0, dt) random variable.

We assume the stock price St follows a geometric Brownian motion, givenby the stochastic differential equation (SDE)

dSt = rSt dt + σSt dWt

where r denotes the risk-free rate and σ is the volatility, such that σ√

δt isthe standard deviation of stock price return over a time period of lengthδt. By using Ito’s Lemma (see Hull [2], p226-228), we obtain the equationbelow. This will be used to describe a simulated stock price path.

St = S0e

(r−σ2

2

)t+σWt = S0e

(r−σ2

2

)t+σ

√tZ (2.1)

Here, Z is a standard normal random variable.7 Since Wt is normallydistributed, it follows that log St

S0∼ N

((r − σ2

2

)t, σ2t

). Hence, St follows

a lognormal distribution. We have

E[St] = S0ert Var[St] = S2

0 e2rt(eσ2t − 1

)Example 3. Figure 2 shows a simulated path that follows equation 2.1. Theinitial stock price S0 = 50, interest r = 0.05, volatility σ = 0.35. Maturityis T = 1 year from now.

3. Using Monte Carlo to price options

We will simulate n stock price paths St(ω1), . . . , St(ωn) over the timeperiod [0, T ]. We can obtain the payoff h = h(ST (ωi)) of a particular optionwhen the stock price is associated with the sample point ωi ∈ Ω. By findingthe sample mean of the discounted payoffs we obtain

1n

n∑i=1

h(ST (ωi)) → E[h(ST ) |S0] =: h0

5A property holds almost surely if it holds everywhere except on a set whose probabilityis zero.

6We will write X ∼ N(µ, σ2) to denote the random variable X as normally distributed,with mean µ and variance σ2.

7That is, Z ∼ N(0, 1).

PRICING AMERICAN OPTIONS USING MONTE CARLO SIMULATION 5

Figure 2. Simulated stock price path.

as n → ∞. This allows us to obtain an estimate of the expected payoff attime 0 - an estimate of the option price.

Example 4 (Using Monte Carlo to price path-dependent options). In orderto simulate a path we shall consider the price of an asset on a finite set ofm + 1 evenly-spaced dates 0 = t0, t1, . . . , tm = T , where tj = jδt = j T

m isthe time of the jth observation. We shall use the following equation.

Stj = Stj−1e

(r−σ2

2

)δt+σ

√δtZj Zj ∼ N(0, 1) j = 1, . . . ,m (3.1)

From now on, r denotes the risk-neutral rate of interest.8 σ representsvolatility, as before. We shall assume the options are European in thatthey can only be exercised at maturity time. The payoffs for each optionare given in table 1. S, Smax and Smin denote the average, maximumand minimum values of a particular stock price path over the set of timest0, t1, . . . , tm, respectively. By letting m → ∞ (thus, letting δt → 0), thepath is observed more frequently, leading to the appropriate values over thecontinuous interval [0, T ].

Option type PayoffAsian call (S −K)+

Asian put (K − S)+

Lookback call (Smax −K)+

Lookback put (K − Smin)+

Floating lookback call (ST − Smin)+

Floating lookback put (Smax − ST )+

Table 1. Payoffs from path-dependent options.

8In the risk-neutral world, investors are indifferent to risk. They will not pay to avoidrisk and will not actively take any risks. The expected return on any investment is therisk-free rate

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(1) For i = 1, . . . , n(a) For j = 1, . . . ,m

(i) Obtain the stock price at time tj by using equation 3.1.(ii) Update the minimum and maximum values obtained so

far if necessary.(b) We’ve reached maturity time for this particular path - obtain

the value for ST = Stm(ωi) and the average valueS = 1

m+1

∑mj=0 Stj (ωi).

(c) Let hi be the present value of the payoff from the selected optionwhen path i is followed.

(2) Obtain the sample mean h = 1n

∑ni=1 hi.

(3) Obtain the sample variance s2 = 1n−1

∑ni=1(hi − h)2.

(4) Return h as the estimate of the option price, along with a 95%confidence interval h± 1.96 s√

n.

Figure 3. Algorithm for pricing path-dependent options.

Figure 4. Estimating the price of an Asian call option.

We will use Java to simulate n stock price paths, each over t0, t1, . . . , tm,where m denotes the number of time intervals in (0, T ]. Figure 3 provides a’pseudo-code’ representation of the algorithm used to price path-dependentoptions.

Figure 4 shows an estimate of the price of an Asian call option whenS0 = 50, K = 56, r = 0.05, σ = 0.35 and T = 1.9 The time between ob-servations is δt = T

m = 110000 = 0.0001 years, making t0, t1, . . . , tm almost

continuous. From figure 4 the option price is estimated as 2.310. Increasingthe number of simulated paths should provide a smaller confidence interval.

9It is possible to specify time in days or months, as well as years. The program willconvert time to the appropriate value for T in years. However, we assume that there are252 trading days in a year and not 365.

PRICING AMERICAN OPTIONS USING MONTE CARLO SIMULATION 7

4. Pricing American options

American options differ from European options in that they can be ex-ercised at any time over [0, T ], not just at maturity. While the price of aEuropean option depends only on the expected payoff at maturity, pricingan American option depends on expected payoff over the entire interval.

In this project, we consider m+1 < ∞ equally-spaced dates t0, t1, . . . , tm,t0 = 0, tm = T , where exercise is possible. Increasing m makes exercisepossible at almost any point over [0, T ].10

We assume that the current asset price St is Ft-measurable. That is, itis formed on the basis of prices over the period [0, t]. This follows sinceinvestors know present and previous values, but not future ones. St, andfunctions dependent on it, are said to be adapted to Ft0≤t≤T .

Some useful properties, related to the conditional expectation of suchadapted sequences - with their proofs - are discussed in Bingham and Kiesel [4],p51-53. A particularly important property is given below.

Definition 5. Let Yt be adapted to Ft.• Yt is said to be a martingale if ∀s ∈ [t, T ], E[Ys | Ft] = Yt.• Yt is a supermartingale if ∀s ∈ [t, T ], E[Ys | Ft] ≤ Yt.• Yt is a submartingale if ∀s ∈ [t, T ], E[Ys | Ft] ≥ Yt.

Let Zt = Z(St) denote the payoff from exercising the American optionat time t (if it hasn’t been exercised already). In the case of a put option,Zt = (K − St)+. Let ht = h(St) be the price of the option at time t. Weshall first consider the price at maturity tm = T . Since this is the finalopportunity to exercise, the payoff at this step is

hT = ZT

At the previous time step tm−1, the option holder can either exercise atthis point or wait until tm, the value being the expected value, at time tm−1,of the payoff at time tm. Since the option holder wishes to maximise theirpayoff, the option price at time tm−1 is

htm−1 = max(Ztm−1 , e−rδtE[htm | Ftm−1 ])

where δt = Tm = tj − tj−1 is the length of time from one step to the next.11

Using backward induction (see Lamberton and Lapeyre [3], p11) and takingthe value at time 0 gives the following dynamic programming (DP) recur-rence.

hT = ZT

htj−1 = max(Ztj−1 , E[htj | Ftj−1 ]) j = 1, . . . ,m(4.1)

The problem is solved by finding h0 = h0. From proposition 1.3.6 ofLamberton and Lapeyre [3], we find that htjm

j=0 is a supermartingale.12

As a result

10Such options, where exercise is only possible on discretely spaced dates are known asBermudan options. By increasing m these options become American in style.

11Since all time steps are evenly-spaced.12It is the smallest supermartingale that dominates the sequence Ztj of discounted

early exercise prices. It is therefore said to be the Snell envelope of Ztj.

8 NAIRN MCWILLIAMS

E[ht] = E[ht | F0] ≤ h0 ∀t ∈ t0, t1, . . . , tmThe next objective to obtain a martingale from the sequence htj, in

order to price American options.

5. Using stopping times to obtain a martingale

Definition 6. A stopping time ν is a random variable taking values int0, t1, . . . , tm such that

ν ≤ tj := ω | ν(ω) ≤ tj ∈ Ftj j = 0, 1, . . . ,m

In the case of American options, ν can be thought of as the time whereexercise takes place. T0,T is the set of all stopping times over the interval[0, T ]. If ν = tk, we have

hνtj :=

htj 0 ≤ j < k

htk k ≤ j ≤ T

where hνtj is the present value of the option price at time tj when exercise

takes place at time ν. Let τ = infν ∈ T0,T | hν = Zν denote the smallestoptimal stopping time - the earliest time over t0, t1, . . . , tm where exerciseis preferable. From proposition 2.2.1 of Lamberton and Lapeyre [3], hτ

tjmj=0

is a martingale. Hence

h0 = E[hτT ] = E[hτ ] = E[Zτ ] = sup

ν∈T0,T

E[Zν ]

The objective is then to find the smallest optimal stopping time, whichwe can use to find the present value of the option at time τ and, thus, theAmerican option price.

6. Finding the optimal stopping time

The smallest optimal stopping time for a path St(ωi) corresponds to thepoint where the option should be exercised. It is the earliest point wherepayoff from exercising the option exceeds the value of holding until a laterdate. In order to determine the best time to exercise when given St(ωi),an optimal stopping rule needs to be found. Chapter 8 of Glasserman [1]suggests forming an optimal exercise boundary b∗t . For each simulated path,τ is the time when St crosses b∗t . That is

τ =

infν ∈ T0,T |Sν ≥ b∗ν for an American call optioninfν ∈ T0,T |Sν ≤ b∗ν for an American put option

Example 5. Figure 5 shows a possible exercise boundary for an Americanput option. The stock price is simulated up until the time t where St ≤ bt.

As before, we consider exercise on a finite set of evenly-spaced pointst0, t1, . . . , tm. The stock price is treated as a Markov process, where

E[Sq | Ft] = E[Sq |St] ∀q ∈ [t, T ]

That is, the expected stock price in the future depends only on the currentstock price, not on the history over [0, t).

We consider three approaches to pricing American options.

PRICING AMERICAN OPTIONS USING MONTE CARLO SIMULATION 9

Figure 5. Exercise boundary for an American put option.

6.1. Using the binomial tree model. The Cox-Ross-Rubinstein methodis a discrete-time approach to pricing options. In the case European options,it is a discrete time version of the Black-Scholes model. At each time steptj , assume that the current asset price can go up by an amount u = eσ

√δt

or down by d = e−σ√

δt = 1u . The probability of an upward movement is

p := P[Stj+1 = Stju |Stj ] =erδt − d

u− d(6.1)

with the probability of downward movement being 1−p. At each time step,the stock price takes the value

Stj = S0uidj−i i = 0, 1, . . . , j

where i is the number of upward movements. Firstly, consider the value attime tm = T . From equation 4.1

hT = Z(S0uidm−i) =

(S0u

idm−i −K)+ for a call option(K − S0u

idm−i)+ for a put option

We take T to be a stopping time, regardless of the stock price. This meansthe option ends by this time, if not before.13 At time tj < T , the expectedvalue of the option at time tj+1 is

E[htj+1 |Stj = S0uidj−i] = p Z(S0u

i+1dj−i) + (1− p) Z(S0uidj−i+1)

If Ztj ≥ e−rδtE[htj+1 |Stj = S0uidj−i], then tj is an optimal stopping time.

Using equation 4.1, we have

h(S0uidm−i) = Z(S0u

idm−i)htj = max(Z(S0u

idj−i), p Z(S0ui+1dj−i) + (1− p) Z(S0u

idj−i+1))

i = 0, 1, . . . , j j = 0, 1, . . . ,m

Using backwards induction finds the option price from this model. Thepoints at which exercise is optimal can be used to form the exercise boundary

13We cannot have τ > T , as the option expires after this time.

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Figure 6. Using the binomial model to price an Americanput option.

b. Since the number of paths is finite, we can find both the price of theAmerican option and the average optimal stopping time.

Example 6. Figure 6 gives an example of using this model to price anAmerican put option. Here, S0 = 100, K = 110, r = 0.1 and σ = 0.34641.The length of the option is T = 1

3 years14 and there are m = 4 intervals over(0, T ], each one month in length. The price of such an option is 12.862 andthe average optimal stopping time is 3.25 months after the option is sold.

However, a problem with pricing options in this way follows from thisbeing a discrete model. At each time step, it only considers a fixed set ofpossible stock prices. For example, at the first time step t1, it will onlyconsider the stock price as S0u or S0d, even though it could be between(or above or below) these two values. In theory, this can be solved byconsidering a larger value of m. However, the model requires all possiblevalues at each time step t0, t1, . . . , tm.15 Using m + 1 time steps requiresstoring 1

2(m + 1)(m + 2) values in memory and means that 2m possiblepaths must be considered. Computing the average optimal stopping time inthis way becomes a problem as m becomes very large.

14In this example, the time to maturity is specified as l = 4 months instead of l = 13

years. The appropriate value for T is obtained by taking T = l12

. In the case of specifying

the option length in days, we would take T = l252

.15This is a result of trying to compute the option price using optimal stopping times. If

we didn’t need to know the average optimal stopping time, we would be able to computethe option price ’dynamically’. This means we would be able to take a much larger valueof m.

PRICING AMERICAN OPTIONS USING MONTE CARLO SIMULATION 11

(1) For i = 1, . . . , n(a) Set j = 0, where t = tj is the current time step.(b) While j ∈ 0, 1, . . . ,m and an optimal stopping time for this

path hasn’t been found(i) Compute d1 and d2 using equation 6.3, given the param-

eters above, the current stock price St = Stj and theremaining time until maturity T − t = T − tj .

(ii) Compute E[ST |St] using equation 6.2.(iii) Compute the payoff, Zt from exercise at time t.(iv) If Zt ≥ E[ST |St], the boundary has been crossed. Let

τi = tj be the optimal stopping time for the path St(ωi).(v) Otherwise, set j = j + 1, obtain the stock price at the

next time step using equation 3.1 and continue.(c) If an optimal stopping time wasn’t found for this path, τi = T .(d) Set hi = Zτi to be the present value of payoff from exercise at

time τi when path St(ωi) is followed.(2) Obtain the average optimal stopping time τ = 1

n

∑ni=1 τi.

(3) Obtain the variance v2 = 1n−1

∑ni=1(τi − τ)2 for optimal stopping

time.(4) Obtain the sample mean h = 1

n

∑ni=1 hi.

(5) Obtain the sample variance s2 = 1n−1

∑ni=1(hi − h)2.

(6) Return τ as the estimate of stopping time, h as the estimate ofthe option price, along with a 95% confidence intervals τ ± 1.96 v√

n,

h± 1.96 s√n

for stopping time and option price, respectively.

Figure 7. Algorithm using the Black-Scholes model as anexercise boudnary.

6.2. Making use of the Black-Scholes model. The Black-Scholes for-mula for calculating the price of a European option at time t is

E[ST |St] =

StN(d1)−Ke−r(T−t)N(d2) for a call optionKe−r(T−t)N(−d2)− StN(−d1) for a put option

(6.2)

where

d1 =log(St

K ) + (r + σ2

2 )(T − t)σ√

T − td2 = d1 − σ

√T − t (6.3)

and N(x) = P[Z < x] is the cumulative distribution function of an N(0, 1)random variable. The Black-Scholes value of an option at time t will be usedas a second example of an exercise boundary.

For each simulated path, the optimal stopping time is the first pointτ ∈ t0, t1, . . . , tm where the value of the payoff at time tj exceeds theBlack-Scholes price, at that time, of an asset worth Stj , when K, r, σ andT are the exercise price, risk-neutral rate of interest, volatility and time ofmaturity, respectively.

Figure 7 shows the algorithm used to price American options in this way,when given S0, r, σ, T , the number of intervals, m, over (0, T ] and thenumber of paths n.

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Figure 8. Using the Black-Scholes model to estimate theprice of an American put option.

Example 7. Figure 8 gives an example of the model being used to price anAmerican put option using the same parameters as in example 6. We use aninterval length of δt = 0.0004 months, giving an almost continuous model,as opposed to the model we can use in section 6.1.

Notice that the option price from example 6 isn’t included in the con-fidence interval in example 7. One problem from using this model is thatit will usually underestimate the actual option price. The example belowdemonstrates why this is the case.

Example 8. Consider the subset of the binomial model in figure 9, whereexercise is optimal at points 1, 2, 3 and 4. At points 1 and 2, we have

htj = Ztj ≥ E[htj+1 |Stj ]

However. the Black-Scholes model assumes

htj = E[htj+1 |Stj ]

Hence, the model underestimates the option price at time tj when thestock price is at point 1 or 2. As a result, it underestimates the option priceat time tj−1 when the stock price is at point 3.16 It will continue backwardsin this way. Since the price at time zero depends on paths that go throughthe numbered points (see equation 4.1), the estimate of the option price willbe less than the actual value h0.

6.3. Using simulation to obtain an exercise boundary. The final ap-proach we will use to price American options is to obtain an estimate of theoptimal exercise boundary using a sample of n1 simulated paths. In this

16It will also underestimate at the unnumbered node above point 3.

PRICING AMERICAN OPTIONS USING MONTE CARLO SIMULATION 13

Figure 9. Binomial model showing a four early exercisepoints. Using the Black-Scholes model will underestimatethe American option price.

section, we will only consider American put options.17 Increasing n1 shouldresult in a more accurate estimate of b∗.

This approach uses the methods described in Glasserman [1], section8.2. The first step is simulate n1 stock price paths St(ω1), . . . , St(ωn1),t ∈ t0, t1, . . . , tm, using equation 3.1. Let θj denote the value of the exer-cise boundary at time tj . Since we only exercise the option at maturity (ifwe haven’t already done so) when the asset price is below the exercise price,we set θm = K. The average present value of the payoff from exercising alloptions at this time is

1n1

n1∑i=1

Z(ST (ωi)) =e−rT

n1

n1∑i=1

(K − ST (ωi))+

We now consider the value of the boundary at time tj < T . Since thepresent value of an option exercised at this point is greater than the presentvalue of an option exercised at a later date when the stock price doesn’tchange18, we have θj ≤ θj+1, ∀j = 0, 1, . . . ,m − 1. Therefore, we try θj =θj+1 as the first potential value for the exercise boundary at time tj . Forevery simulated path where Stj ≤ θj , we exercise the option at this time. Forall other paths, we exercise at the optimal stopping time, τj ∈ Ttj+1,T , overtj+1, . . . , tm. The present value of the payoff from exercising the option

17Although it is possible to price them using sections 6.1 and 6.2 above, it is neveroptimal to exercise an American call option before maturity time. One can use the methodsof section 6.1 to check this.

18Unless of course we have no interest r = 0. We always assume r > 0, however. Hence,if the stock price remains constant over [t, t′], we have eq−tSq = eq−tSt < St,∀q ∈ (t, t′].

14 NAIRN MCWILLIAMS

Figure 10. Using three paths to estimate b∗.

when given St(ωi) is

Z(Stj (ωi)) =

(K − Stj )+ Stj ≤ θj

(K − Sτj )+ otherwise (6.4)

We repeat this process using all stock prices Stj (ωi) < θj+1 as potentialvalues for θj , taking the θj that maximises

1n1

n1∑i=1

Z(Stj )

as the value of the exercise boundary at time tj . This results in the followingDP recurrence.

θm = K

θj = maxθj+1,Stj <θj+1 1

n1

∑n1i=1 Z(Stj ) j = 0, 1, . . . ,m− 1 (6.5)

Finding θ0, θ1, . . . , θm using equation 6.5 provides the estimate of the op-timal exercise boundary b∗.

Example 9. Figure 10 shows the result of simulating three paths, usingthe same values for S0, K, r, σ and T as in examples 6 and 7.19 At timetm−1, the potential values for θm−1 are K and all three simulated valuesStm−1(ω1), Stm−1(ω2) and Stm−1(ω3) (represented by the red, green and bluelines, respectively). It turns out that Stm−1(ω2) maximises the payoff oversteps tm−1, tm so it becomes the value for θm−1. For the time steps im-mediately before this point, all three paths are greater than θm−1, so theboundary retains this value. It is only when St(ω2) meets the boundaryagain that this becomes any smaller.

Once the boundary takes a value around 86.6557, it remains below allsimulated paths. This is the value of the exercise boundary at time 0.

19In practice, we would always use more than three simulated paths and have donethis for display purposes only.

PRICING AMERICAN OPTIONS USING MONTE CARLO SIMULATION 15

(1) Simulate n1 stock price paths St(ω1), . . . , St(ωn1).(2) Set θm = K.(3) For j = m− 1,m− 2, . . . , 1, 0

(a) Set θj = θj+1 and θj = θj to be the maximising value of θj sofar.

(b) Set P = 1n1

∑n1i=1 Z(Stj (ωi)), using equation 6.4 as the value for

Ztj .(c) For i = 1, . . . , n1

(i) If Stj (ωi) < θj+1

(A) Try θj = Stj (ωi).(B) Set P = 1

n1

∑n1i=1 Z(Stj (ωi)) using equation 6.4 and

this value of θj .(C) If P > P then P = P and θj = θj

(d) Set θj = θj as the exercise boundary value at time tj .(4) For i = 1, . . . ,m

(a) Set j = 0(b) While j ∈ 0, 1, . . . ,m and an optimal stopping time for this

path hasn’t been found(i) If Stj (ωi) ≤ θj then this path is below the boundary. Let

τi = tj be the optimal stopping time for this path.(ii) Otherwise, set j = j + 1, use equation 3.1 to obtain the

stock price for the next step and continue the while loop.(c) If an optimal stopping time for this path wasn’t found, τi = T .(d) Set hi = (K −Sτi(ωi))+ to be the present value from exercising

this path at time τi.(5) Obtain the average optimal stopping time τ = 1

n

∑ni=1 τi.

(6) Obtain the variance v2 = 1n−1

∑ni=1(τi − τ)2 for optimal stopping

time.(7) Obtain the sample mean h = 1

n

∑ni=1 hi.

(8) Obtain the sample variance s2 = 1n−1

∑ni=1(hi − h)2.

(9) Return τ as the estimate of stopping time, h as the estimate ofthe option price, along with a 95% confidence intervals τ ± 1.96 v√

n,

h± 1.96 s√n

for stopping time and option price, respectively.

Figure 11. Algorithm that uses simulated paths to obtainthe boundary, then estimates the price as before.

Having computed an estimate of the exercise boundary, we simulate n2

new paths St(ω1), . . . , St(ωn2) and estimate the American put option priceas before. The optimal stopping time for a particular path is the timeτ ∈ t0, t1, . . . , tm when

Sτ (ωi) ≤ θτ

Figure 11 summarises this method of pricing American options.

Example 10. Figure 12 shows the result of using the above algorithm withthe same parameters as in examples 6 and 7. As in 7, we can use a small

16 NAIRN MCWILLIAMS

Figure 12. Using an estimated exercise boundary to pricean American put option.

Figure 13. Specifying too many intervals or simulatedpaths results in failure to price the option

value for δt, giving an almost continuous model. Unlike 7, by letting δt → 0(from letting n1, n2 →∞), it ought to be the case that h → h0.

However, in order to estimate the payoff boundary using this method,we need to store all simulated paths St(ω1), . . . , St(ωn1) in memory at onetime. The case of using Java to estimate the boundary using n1 paths withm+1 intervals t0, t1, . . . , tm requires storing an m+1×n1 array of double-precision decimal numbers.20 Using the same value for m as in example 7requires too much memory for even a high-end system, as figure 13 shows.

20In the above example, m = 10000 and n1 = 1000. A double in Java is 8 bytes inlength, resulting in 80,008,000 bytes ≈ 76.3 megabytes needed for the array alone.

PRICING AMERICAN OPTIONS USING MONTE CARLO SIMULATION 17

7. Conclusion

The Black-Scholes model provides a way of pricing European call andput options. No such model exists for American options, so numerical pro-cedures need to be used. In theory, we can price American options usingoptimal stopping times. In practice, finding the optimal stopping rule be-comes a challenge, itself. Whilst the binomial tree method of Cox, Ross andRubinstein provides a method for pricing American options in this way, itonly considers a discrete set of stock price values at each time period. Wecan generalise to a continuous interval of possible stock prices by increasingthe number of periods. However, doing this results in the number of possiblepaths we need to consider increasing exponentially.

Monte Carlo simulation estimates the option price by considering a sampleof possible stock price paths. We can take one sample to create an estimate ofthe optimal exercise boundary, and use it with another sample to estimatethe option price. However, the method of determining the boundary is,computationally, memory-intensive. Pricing options over a large continuousinterval means we need to consider a larger number of time steps. As aconsequence, a smaller sample must be used to determine the boundary,resulting in reduced accuracy. Whilst limited memory-usage is becoming lessof a problem over time, it is still significant in the area of pricing Americanoptions.

References

[1] Glasserman P. Monte Carlo methods in financial engineering. Springer, 2004.[2] Hull J. Options, futures, and other derivatives - fifth edition. Prentice Hall Interna-

tional, 2003.[3] Lamberton D., Lapeyre B. Introduction to stochastic calculus applied to finance.

Chapman & Hall, 1996.[4] Bingham N. H., Kiesel R. Risk-neutral valuation: pricing and hedging of financial

derivatives - firstd edition. Springer, 1998.[5] Kopka H., Daly P. W. A guide to LATEX- third edition. Addison-Wesley, 1999.