UNIVERSITY OF CALIFORNIA

Los Angeles

A Monte Carlo Approach to Pricing

An Exotic Currency Derivative Structure

A thesis submitted in partial satisfaction

of the requirements for the degree of Master of Science

in Statistics

by

Ivan O. Asensio

2004

The thesis of Ivan O. Asensio is approved.

Ker-Chau Li

Rick Schoenberg

Ying Wu, Committee Chair

University of California, Los Angeles

2004

This thesis is dedicated to my parents, Edgard and Ramona Asensio. I would like to

express all the gratitude in the world for their sacrifice and hard work throughout my

years, for the impeccable blueprint they have laid before me for living a good life and

achieving success at home and at work, and especially for always greeting me with

warmth, laughter and smiles.

TABLE OF CONTENTS

I. Introduction

II. The Process

III. The Methodology

IV. The Structure Defined

V. The Structure Analytically

VI. Monte Carlo Process

VII. Simulation Procedure

VIII. Precision

IX. Results

X. Conclusion

ABSTRACT OF THE THESIS

A Monte Carlo Approach to Pricing

a Complex Currency Derivative Structure

by

Ivan O. Asensio

Master of Science in Statistics

University of California, Los Angeles, 2004

Professor Ying Wu, Chair

This paper presents a non-analytic simulation framework for pricing a complex foreign currency

derivative structure. The paper will establish that currency prices move according to a stochastic

process, and more specifically that currency returns follow an Ito process with a specified

expected mean and variance. Monte Carlo simulation is used for generating random realizations

of the currency, creating an entire distribution of potential paths. The price of the structure is

estimated by taking an arithmetic average of the discounted payoffs for the structure at each of

the paths. Results of this study will be validated using the Black-Scholes option pricing model.

INTRODUCTION ~

This paper presents a non-analytic simulation technique for pricing an exotic foreign

currency derivative structure known as a forward extra. The original work on deriving

financial option prices through simulation was proposed by Phelim Boyle in 1977 1 This

method uses the fact that the distribution of asset values at option expiration can be

determined by the process which characterizes the evolution in the future value of the

asset itself. If this process can in fact be specified, then it can be simulated on a computer

‘many’ times, and thus a distribution of terminal asset values can be created for the

option. The probability weighted average terminal value then would yield the option

price. The original work by Boyle was proposed for stocks. His framework can similarly

be applied to other assets including currencies, short-term interest rates, and commodity

prices provided the process which describes the evolution of the instrument is suitably

described. Most recently his approach is being applied to the simulation of property

values in the US. 2

Results of this paper will be validated with the use of the analytic solution furnished by

the Black-Scholes option-pricing model developed in 1973. It was the first general

equilibrium solution provided for the valuation of options, and one that revolutionized

financial markets promoting the proliferation of option creation, usage, and trading across

a variety of assets. Mark Garman and Steven Kohlhagen3 ten years later developed the

particular extension of the model that applies to currency options. They cleverly

concluded that a dividend on a stock during the option coverage period was analogous to

the interest rate differential paid or received in currency transactions by holding the risk-

free rate in one country versus another. Inherent in all foreign exchange forward

transactions is the concept of holding one currency versus another. In example, suppose

an entity exchanges euros for US dollars for delivery something in the future. The entity

is selling euros and buying dollars and in doing so equivalently borrowing euros and

investing dollars at the respective risk-free rates in each country. Because interest rates

differ across borders, there is generally a discount or a premium that applies to such

currency trades. It is this sum that serves as the “dividend”.

The motivation for this study is to establish an intuitive framework for pricing complex

structures with non-linear payouts, as an alternative to Black-Scholes. Although more

computationally expensive, simulation techniques offer greater flexibility for relaxing

any of the various assumptions made under Black-Scholes. The model assumes for

instance that both interest rates and volatility are static. Any participant in global

financial markets can assert this is in fact not the case. These variables are dynamic in

nature. They can be stochastic or deterministic. If stochastic, assumptions about cross-

correlations would need to be made in order to generate suitable random realizations. If

deterministic, the pricing framework must be flexible enough to allow for proper

calibration. Dynamic interest rate and volatility assumptions will not be addressed in this

paper. The focus here will be to establish the framework for replicating Black-Scholes

results through simulation.

THE PROCESS ~

Currency price movements can be modeled as a continuous-variable, continuous-time

stochastic processes. The ‘continuous variable’ assumption implies that the asset can take

on any future value, changes are not restricted to pre-determined increments. The

‘continuous-time’ assumption implies that price changes can take place at any time, in

other words implying continuous trading. Although the assumption is also suitable for

other assets including equities, commodities, and certain short-term interest rates,

participants in foreign exchange markets can attest to the fact that in no other market is

this assumption more appropriate, as currencies trade, largely unregulated, 24 hours a

day, 365 days a year. By comparison, stocks and bond prices changes are restricted to

discrete values (32nds or 64ths out of a dollar). And these changes occur only when markets

are open.

The process described can first be characterized as Markov. By definition then, only the

present value of the currency is considered for describing the evolution of the variable

through time. The past history of the variable and the way in which the present has

emerged from the past would thus be irrelevant. Mechanically, the Markov property

implies that the variance of currency price changes in successive periods is additive.

Suppose USD/JPY is currently 100.00 and the change in value for the following year is

expected to be _(0,10), where _ (µ ,σ) denotes a probability distribution that is normally

distributed with mean µ and standard deviation σ.The change in two years then is the

sum of two normal distributions, the change is 5 years the sum of 5 normal distributions,

each with mean of 0 and standard deviation of 10. The probability distributions are

independent, as they are Markov. It follows then that the mean of the change during two

or five years would be therefore zero, and the variance 20.0 and 50.0, respectively.

Otherwise stated, the change in the variable over 2 years is _(0, √20.0) and _(0, √50.0)

over 5. Similarly we can characterize a 3-month move according to _(0, √2.5).

Further clarification is needed in asserting that currency prices do not exactly follow a

normal distribution. Black and Scholes4 established, with empirical backing, that asset

prices more accurately follow a lognormal distribution. It is price changes or ‘returns’

that are normal. By carrying out a simple transformation of prices, we can further

characterize the evolution of normally distributed returns as a Wiener process. This is a

special type of Markov process in which the mean is zero and the variance rate is 1.0 per

unit of time, so that if the value of the currency is x0 at time zero, at time T it is normal

with mean return x0 and standard deviation of √T.

Two properties hold. The change ∆z during a small period of time ∆t is

∆z = _ √∆t

where _ is a random drawing from a standardized normal distribution _(0,1). And again,

because the process is Markov, the values of ∆z for any two different short intervals of

time ∆t are independent.

A generalized Weiner process enables us to further describe our process in that it allows

us to specify a drift of ‘a’ per unit of time and a variance rate of ‘b2’ per unit time, where

a and b are constants. The value of the currency would be x0 at time zero, and normally

distributed with a mean of x0 + aT and a standard deviation of b√T at time T. The drift

component is especially relevant and necessary in financial markets as it allows us to

synthesize spot prices with forward or future prices. In other words, suppose the

EUR/USD rate for delivery of dollars for euros today is 1.2000, the rate today for

delivery sometime in the future will be higher or lower depending on the differential of

risk-free rates in each currency regime. The drift assumption asserts that just as today’s

equilibrium rate of exchange is the current spot rate, tomorrow’s equilibrium rate for

delivery can be found along the forward curve.

The final step in our development for the process we will use to describe currency price

changes establishes it as an Ito process. This outlines a generalized Weiner process where

the parameters a and b are actually functions of the value of the underlying variable, x,

and time, t. The process, widely used in the valuation of financial derivatives, involves

two parameters, µ and σ. The parameter, µ, is the expected return for the currency while

the parameter driven again by the spot-forward relationship; σ, is the expected volatility

for the underlying currency pair. This important result was discovered by K. Ito in 19515.

Algebraically, an Ito process can be written:

dx = a(x,t) dt + b(x,t) dz

Both the expected drift and variance rate of an Ito process are liable to change over time.

In a small time interval between t and t + Δt, the variable changes from x to x + Δx

where:

Δx = a(x,t) Δt + b(x,t)ε√ Δt

This relationship works best for very small periods of time. For such it is a good

approximation of a normally distributed x, but its change over longer periods of time is

much more likely to be non-normal. In addition, it assumes that the drift and variance rate

of x remain constant, an assumption we will preserve in our simulation, also an

assumption preserved under Black Scholes. It is however this assumption that can be

more easily relaxed under a Monte Carlo framework.

We will close this section by again reiterating that the application of the previously

developed process rests on the transformation of prices into returns. Currency prices or

rates do not follow an Ito process, currency returns however do.

METHODOLOGY ~

Monte Carlo simulation involves generating random numbers on a computer to construct

entire distributions of particular real-world phenomena. A casino for instance can

replicate an entire night’s play on their blackjack tables in order to estimate not only their

expected take, but also shed a light on the potential for unexpected losses or ‘player

luck’. Monte Carlo offers great power and flexibility, and in our age of advanced

computing power it has become realistic to implement the methodology across a wide

range of scientific and engineering problems without must sacrifice.

Certain mathematical problems lend themselves well to closed form solutions or

approximations. The Black Scholes model and the umpteenth others that followed from it

offer prices ‘in closed form’ for a wide range of financial options6. Although more

expeditious and less computationally demanding, such models often times are subject to

potentially rigid assumptions. Monte Carlo is commonly used to break, relax, or change

such assumptions.

The process works as follows. Many possible inputs are generated randomly for the

problem under study and after recording the results for each possible input, statistical

methods are used to deduce a solution based on the distribution of outcomes. The key to

using Monte Carlo methods is the ability to describe the system of inputs in terms of

random variables. The preceding section established an Ito process for describing the

evolution of currency price changes through time.

A ‘large number’ of potential currency paths are generated, ideally representing all

potential outcomes for a predetermined future period. The number of replications should

be as large as feasible, in other words it should be large enough to prevent biases that

may arise due to odd combinations of random number draws. For each path the payoff of

the derivative is calculated and discounted at the risk-free rate. The arithmetic average of

the discounted payoffs is the theoretical price of the option.

THE STRUCTURE DEFINED ~

A ‘forward extra’ structure is a foreign currency derivative tool that is used by primarily

for risk management purposes. The complex structure brings together the favorable

features of a forward contract with the flexibility of options. It would allow a corporate

hedger to lock in a predetermined worst-case rate of conversation for a particular

currency, while also offering a limited amount of upside potential. The reason forward

extras are so popular among participants looking to mitigate foreign currency conversion

risk is that the structure can be entered into at no upfront cost. It is generally the norm for

derivatives that offer flexibility and upside potential (i.e. options) to bring along an initial

price tag. Typically the premium paid for option protection can be in the neighborhood of

1 to 5% of the notional amount being protected.

The example developed will involve a US company that sells goods into the Euro zone.

The goods are sold denominated in the local currency, the Euro. Because the company

ultimately cares about the number of dollars it collects for its goods, they are inevitably

concerned about fluctuations of the EUR/USD exchange rate. A stronger Euro and thus

weaker Dollar would be favorable to the company, as it would bring more dollars upon

conversion. The reverse is also true. A weaker Euro and thus stronger dollar would

reduce the bottom-line profit margin for the goods after repatriation occurs. Further

suppose the company is planning to complete their sales in the region and bring back the

funds in six month’s time. As far as mitigating conversion risk during this period is

concerned, the company has alternatives. The EUR/USD spot exchange rate at the

beginning of the period is 1.2500.

At one end of the spectrum of risk management tools available is a forward contract.

Forwards would enable a corporate hedger to lock in a rate of conversion today, for

execution sometime in the future. The rate, as compared to current spot, will be different

however. Forward contracts or rates are priced such that they incorporate the interest rate

differential between two underlying currencies for whatever tenor chosen. In our case

since Euro zone interest rates are slightly lower than those in the US, selling (buying)

Euro forward would yield a more (less) favorable forward contract rate than current spot.

This is because holding (lending) the higher yielding currency versus the lower yielding

one would produce an interest rate differential net gain (cost). There is no economic

benefit to dealing in forward markets; the spot-forward differential is simply a function of

‘time value of money’. Thus with the current spot rate at 1.2500, the six-month forward

rate would be 1.2530. The company in our example can lock in a rate of 1.2530 for

repatriation of Euros, to occur in six months’ time. This strategy eliminates any

uncertainty about the rate at which the funds will be converted. The forward however

provides no flexibility as the rate is locked in.

At the other end of the spectrum is an option. The concept of an option is analogous to

that of an insurance contract. The company can purchase a six-month EUR put / USD call

option which can be used to lock in the six-month forward rate of 1.2530 as the worst

possible rate of conversion for buying Dollars and selling Euros. Should the Euro weaken

to 1.1500 for instance, the option would enable the company to repatriate funds at the

more favorable locked in rate of 1.2530. However, should the Euro strengthen to a rate of

1.3500 in six months’ time, the company would carry out the conversion at the more

favorable market rate and the option would expire worthless, just as an insurance contract

would expire without monetary value should the event it was designed to protect against

fail to occur. In contrast to the forward, the option offers protection in addition to a great

deal of flexibility, for a cost. Purchasing option protection for six months would cost the

company a premium of 1 – 5% of the total amount being protected. The amount paid

depends largely on the expected volatility of the currency for the forthcoming period. The

higher the expected volatility, the greater the premium. For this analysis, we will assume

the EUR/USD exchange rate is expected to fluctuate at an annualized rate of 10%.

The complex structure serving as the subject for this paper combines features from both

ends of the spectrum. It offers the lock-in feature of a forward, while also providing the

company the limited opportunity to benefit from a favorable rate move in the Euro. And it

does so at no upfront cost.

The forward extra would offer a worst-case rate of 1.2450 for repatriation of Euros into

dollars. The rate is slightly less favorable the six-month forward of 1.2530, however in

exchange of this offset, the company can participate in favorable rate movements up to

1.3225. Should the EUR/USD exchange rate trade higher than 1.3225, the currency

exchange would occur at the locked-in rate of 1.2450.

The following exhibit illustrates the gain or loss profile at the end of the six-month period

for each strategy aforementioned. Included is also a “do-nothing” case. We will assume

the company sales for the six-month period in question total 10,000,000 EUR.

Exhibit 1

Gain/Loss At Six-Month Period End

$(1,500,000)

$(1,000,000)

$(500,000)

$-

$500,000

$1,000,000

$1,500,000

1.15

00

1.17

00

1.19

00

1.21

00

1.23

00

1.25

00

1.27

00

1.29

00

1.31

00

1.33

00

1.35

00

EUR/USD Exchange Rate

Forward

Option

Forward Extra

Do Nothing

Also a recap of the particulars of this example is below:

Exhibit 2Beginning EUR/USD rate 1.2500

EUR/USD 6-month forward rate 1.2530

EUR/USD expected volatilityper annum

10%

Forward extra forward level 1.2450

Forward extra barrier level 1.3225

THE STRUCTURE ANALYTICALLY ~

Once the process has been developed, and the structure described, we have enough to

proceed in estimating the theoretical price for the forward extra non-analytically through

simulation. Because, however, we are validating our results with those that would be

derived from the Black Scholes pricing model, a discussion on how the model would

arrive at an analytic result is also warranted.

The forward extra structure can synthetically be created by carrying out the following

two transactions involving options:

1. Purchase a EUR put / USD call option with a strike set at a less favorable rate

than the current forward. In our case the 6-month forward is 1.2530, thus the

strike for the option purchased would be 1.2450.

2. Write or sell a EUR call / USD put knock-in barrier option with the same strike

rate of 1.2450 and a barrier level of 1.3225.

Transaction number one is straightforward. Transaction two is slightly more complicated

as it involves the writing or selling of options, in addition to a special type of option

known as a knock-in barrier.

Writing or selling an option is the mirror image of purchasing an option. The roles are

essentially reversed. In writing the option, a company will receive an up-front premium

(as opposed to paying) and the risk potential is unlimited should prices move adversely

(as opposed to the risk limited to the premium paid).

Barrier options are options where the ultimate payoff depends on whether the underlying

price touches or does not touch a pre-determined level. Knock-ins are a special type of

barrier that come into existence only if a pre-determined level is reached. Technically

they do not exist when purchased, they are essentially ‘knocked into existence’ by the

contingent event. The written knock-in option in our example does not exist unless the

barrier level of 1.3225 is reached. If during the 6-month period, the EUR/USD rate stays

below this hurdle, the company will technically not possess a sold option. If, however,

the rate does touch 1.3225, the company will indeed possess a sold option position with a

strike price of 1.2450.

Created synthetically, the purchase of the plain vanilla option and the sale of the knock-in

barrier option will result in a net premium of zero for the company. The structure is a

hybrid; it provides the no-cost feature of a forward contract, while providing a limited

degree of upside potential.

MONTE CARLO PROCESS ~

We have established that currency price changes follow an Ito process. Suppose S is the

underlying currency price series. The process is characterized as

dS = µS dt + _S dz

where dz is a Wiener process, µ is the expected return for the currency, and _ is the

anticipated volatility of the currency. To simulate the path followed by S, we divide time

into N short intervals of length ∆t and approximate price changes as

S(t + ∆t) – S(t) = µS(t) ∆t + _S(t) _ √∆t

Where S(t) denotes the value of S at time t, _ is a random sample from a normal

distribution with a mean zero and a standard deviation of one. This effectively enables

each value of S at each successive interval to be calculated from the value of the interval

before it.

One simulation trial involves constructing a complete path for S using N random samples

from a normal distribution. This is done ‘many’ times, for each time, the payoff of the

option, which is dependent on the value of S, is calculated. The mean of the payoffs is

computed and discounted at the risk-free rate of the company’s functional currency, in

this case USD. This discounted mean represents the theoretical option price.

As asserted in the previous section, the structure can be synthetically constructed by

conducting two transactions: the purchase of a EUR put / USD call option struck at

1.2450 and the sale of a EUR call / USD put knock-in also struck at 1.2450, with a barrier

at 1.3225. Our simulation procedure then will occur in two stages.

SIMULATION PROCEDURE ~

The first stage will require price calibration for the purchased EUR/USD option struck at

1.2450. For this, we observe S at then end of each simulated path. If S > 1.2450, then the

option will expire worthless for purposes of this replication. The payoff in this case is

zero. If S < 1.2450 at the end of the six-month period, the option expires in the money,

and has a value on that date equal to difference between 1.2450 and S. The payoff in this

case is calculated according to a notional holding of 10,000,000 EUR. Once again, an

average of the payoffs is computed and discounted back to the present day (ie. six

months) using the US risk-free rate. This yields the price of the first piece of the

structure. Using Black Scholes, we arrive at a premium of 2.61% of the USD notional.

We will approximate this through simulation.

The second stage is a bit more involved. Recall the forward extra structure has an initial

value of zero. Essentially this means the price of the purchased option (2.61%) must

equal the price of the sold option. For the sold knock-in barrier then, since we already

know the value it must converge to (2.61%), and we know that the strike must also equal

to the strike of the purchased piece (1.2450), we must carry out a convergence process in

order to arrive at the particular barrier level that would yield this pre-determined price.

Once again, the barrier represents quite a significant element regarding profitability. It

determines the level of upside participation for the structure. The company may repatriate

the Euros at favorable exchange rate up, but not equaling or exceeding the barrier rate.

Once the barrier is reached, repatriation will occur at the predetermined level of 1.2450,

which again is slightly less favorable than the original 6-month forward rate of 1.2530.

Valuation of the first piece simply required we look at the value of S at the end of each 6-

month path. For valuation of the second piece, however, we must look at every value

along each path, since, we must evaluate whether the barrier was reached or not. If it was

in fact reached, then the payoff of the sold option is computed according to the

methodology previously described for the purchased option. If the barrier is not crossed,

the option is not “knocked-in” and thus does not come into existence. The payoff in this

case would be zero.

An iterative process is then carried out in order to arrive at the particular barrier level that

will converge to the price of 2.61%. The bisection method is used to achieve

convergence.

PRECISION ~

The more simulations M generated, the more stable the results. Just as we can compute

the average of the discounted payoffs, similarly we can compute the standard deviation of

those payoffs and use this metric to calibrate the degree of precision achieved, for a given

number of trials. So if the mean is _ and the standard deviation is _, the standard error of

the price estimate is

_ _√M

and thus a 99% confidence interval for the price, p, of the currency option is given by

_ – 2.576 _ < p < _ + 2.576 _ √M √M

The uncertainty about the value of the derivative is thus inversely proportional to the

square root of the number of trials. So to double the accuracy of the simulation, the

number of trials must be increased by a multiple of 4. Similarly increasing the number of

trials by a factor of 10,000 would increase the accuracy by a factor of 100.

Before deciding on the number of simulations needed to achieve stable results, we will

also implement a variance reduction approach known as antithetic variable technique.

This will increase the accuracy, for every given number of trials. It involves calculating

two values of the derivative per simulation. The first value, p1, is calculated the usual way

as described in the previous section. The second, p2, is calculated by changing the sign of

all random samples from standard normal distributions. So if _ is used to calculate p1, -_

is used to calculate p2. The sample value for each trial is then calculated by taking the

average of p1 and p2., denoted by p.

The derivative value is the discounted average of the p’s. So if _ is the standard deviation

of the p’s, and again M is the number of simulations, the standard error of the estimate is

_ _√M

and this is generally much less than the standard error calculated using 2M trials.

We will carry out the Monte Carlo simulation using 5000 trials. This will enable us to

achieve a suitable degree of precision, while still keeping computation time to a

reasonable level.

RESULTS ~

The Black Scholes price for the purchased EUR put / USD call option struck at 1.2450 is

2.61% and thus this is also the price of the sold EUR call / USD put knock-in, also struck

at 1.2450, with a barrier at 1.3225. The following table contains the simulated results for

the structure. The process was carried out 25 times, each time 5000 trials.

Exhibit 3

Simulation No. Option Prices Barrier Level

1 2.66% 1.32092 2.69% 1.32283 2.63% 1.32094 2.56% 1.32285 2.58% 1.32186 2.64% 1.32287 2.69% 1.32188 2.57% 1.32099 2.68% 1.3228

10 2.62% 1.320911 2.72% 1.322812 2.70% 1.322813 2.55% 1.320914 2.69% 1.322815 2.65% 1.322816 2.63% 1.320917 2.61% 1.320918 2.62% 1.321819 2.60% 1.322820 2.53% 1.322821 2.67% 1.322822 2.55% 1.322823 2.58% 1.321824 2.53% 1.322825 2.64% 1.3209

Average 2.62% 1.3220 Std Error 0.06% 0.0009

Max 2.72% 1.3228 Min 2.53% 1.3209

With only 5000 trials we were able to bring down the standard error to 6 hundredths of a

percent for the price of the options, and 9 ten thousands. If additional precision is needed,

more simulations can be added or more trials per simulation. Along similar lines,

bisection convergence can be taken to the hundred thousandths place.

CONCLUSION ~

The Black Scholes family of models provides pricing in closed-from to a host of different

types of options and related derivatives. Simplifying assumptions are often required

depending on the particular version of the model used, often they are considered

weaknesses, but they are also strengths as they provide insight into the workings of the

particular model. The Garman-Kohlhagen version of Black Scholes used to price options

on foreign currency rates assumes that the volatility of the currency pair stays constant,

and it assumes that the prevailing interest rates in each country also remain constant. This

is obviously not the case, as both volatilities and interest rates change with time. Relaxing

these assumptions, provided you could characterize the process for the evolution of both

volatilities and interest rates would in theory yield more accurate results.

This paper presents a simulation framework for replicating Black Scholes prices non-

analytically. Simulation techniques, although computationally more expensive, are more

flexible and thus better suited to handling dynamic volatility and interest rate assumptions

than closed-form approaches. The framework presented here is stable and suitably

replicates prices computed analytically. It can be further developed to incorporate

dynamic volatility and interest rate processes, even such that these processes occur in co-

movement with one another.

1 Boyle, Phelim, P., 1977, Options: A Monte Carlo Approach, Journal of Financial Economics 4 (May1977), pages 323-338.2 According to a model developed and licensed by the company Economy.com and widely used inmortgage markets in conjunction with other bank proprietary models for expected default and loss severityfor residential and commercial mortgage products.3 Garman, Mark, and Stephen Kohlgahen, 1983, Foreign Currency Option Values, Journal of InternationalMoney and Finance 2 (December 1983), pages 231-237.4 F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy,81 (May-June 1973), 637-659.5 K. Ito, On Stochastic Differential Equations, Memoirs, American Mathematical Society 4 (1951), pages1-51.6 From Black-Scholes to Black Holes, New Frontiers in Options, Risk Books (1992), Various authors.