Path space rejection sampling adapted for perfect ... · Path space rejection sampling adapted for perfect simulation of exotic path dependant options Fourth year dissertation Department

Path space rejection sampling adapted for perfect

simulation of exotic path dependant options

Fourth year dissertation

Department of Statistics

Master of Mathematics, Operational Research, Statistics, and Economics

Tina Goldarreh - 1422915 - MMORSE

Supervised by Dr. Murray Pollock

1

Abstract

This dissertation explores a Monte Carlo algorithm of perfect simulation for univariate

diffusions called path space rejection sampling; a multi-dimensional extension of the

well-known rejection sampling simulation method. Exotic options often lack analytical

solutions and the only method of evaluation is by using numerical approximations whose

accuracy is dependant on the chosen method. Path space rejection sampling removes all

discretisation errors and produces perfect IID samples, and by adapting it to sample from

Geometric Brownian motion rather than simple Brownian motion, this dissertation has

introduced a modification to the algorithm to make it suitable for financial applications. To

illustrate, the sampler is then used to obtain stock prices which are then manipulated to

evaluate exotic options, in particular, Asian and Look-back options.

Acknowledgments

I would like to thank Dr. Murray Pollock for his continuous support and guidance

throughout this project. His generous contributions, enthusiastic encouragements, and

indispensable criticisms were very much appreciated and absolutely vital to my success.

2

Tina Goldarreh/Path space rejection sampling adapted for exotics 3

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Monte Carlo methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Assumptions and setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Alternative numerical methods for simulating SDEs . . . . . . . . . . . . . . . . . . 10

4.1 Euler discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 Lattice models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Path space rejection sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.1 Rejection sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.2 Simulation of Brownian motion and related processes . . . . . . . . . . . . . . . 20

5.3 Black-Scholes and its properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.4 Alternatives to Geometric Brownian motion . . . . . . . . . . . . . . . . . . . . 24

5.5 Lamperti transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.6 Girsanov transformation of measures . . . . . . . . . . . . . . . . . . . . . . . . 29

5.7 Path space rejection sampling algorithm . . . . . . . . . . . . . . . . . . . . . . 30

5.7.1 Poisson processes as unbiased estimators . . . . . . . . . . . . . . . . . . 33

5.7.2 Finalized path space rejection sampling algorithm . . . . . . . . . . . . 34

5.8 Path space rejection sampling illustrated with an example . . . . . . . . . . . . 37

6 Applications to finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.1 Simulating stock prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.1.1 Path space rejection sampling with Brownian motion proposal . . . . . 42

6.1.2 Path space rejection sampling with Geometric Brownian motion proposal 44

6.2 Exotic option pricing using path space rejection sampling . . . . . . . . . . . . 47

6.2.1 Asian options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.2.2 Look-back options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7 Conclusion and further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


1. Introduction

The trading environment has changed significantly over the past 30 years, mostly due to the

increasing role and capability of computers. This evolution has changed the nature of the

markets as well as their appearance. Nowadays, traders tend to spend most of their days

behind computer screens instead of on exchange floors, in fact, most of the old exchange

floors have been turned into museums as tokens of a more antiquated era [16, 15, 4].

Computers and automation have almost completely removed the need for spot traders,

especially in highly liquid markets such as equities and FX. This has led to more time, effort,

and resources being directed towards more complex contracts.

Options grant their owner the right but not the obligation to enter into a pre-specified

financial contract [37]. Option contracts are not a novel idea and their roots can be traced

back to ancient Greece where Thales of Miletus used an option to speculate on the olive

harvest [56], essentially betting on the weather. Options are unique in their asymmetric

nature, and hence are often used as hedges to minimize the downside of trades but keep the

upside unbounded. This asymmetry in the cost and potential benefit of option contracts,

means that they are also often used as a means of speculating on the underlying asset.

Vanilla call options grant the owner the right but not the obligation to buy the underlying

asset at a pre-determined strike price; similarly, Vanilla put options grant the right to sell the

underlying at a pre-determined strike [28]. The option space has moved far beyond these

simple vanilla contracts and can now provide much more specific bets, mainly to lower the

premium cost and risk. One can now bet on a stock’s average price (Asian options) [63]; a

stock exceeding certain price limits (Barrier options) [21]; or even modify these to include a

Knock-in or Knock-out effect whereby the option becomes viable once the Knock-in limit is

reached or ineffective when it hits the Knock-out one [37]. These are just a few examples of a

much more complex class of options, namely the exotics, which are often traded over the

counter (OTC) because of their tailor-made features, rather than on exchanges.

In computational statistics, financial problems are often expressed by using the expectation

of a random variable under some test function h. For instance, the payoff of a Vanilla call

option is simply the expectation of the stock price under a test function that evaluates the


positive difference between the stock price S and the strike price K at maturity T [28]. Only

the positive difference is considered because if the strike is higher than the spot price, the

option would simply not be exercised; it would be cheaper to buy the stock in the market.

E[h(ST )] = E[(ST −K)+] (1)

Thus, trading and investing in financial securities is often done based on the future

expectation of an asset’s return, however, this expectation is difficult to quantify due to the

uncertainty of underlying stock prices. This becomes even more complex in the case of exotic

derivatives as the test functions become considerably more intractable. The improvements in

computing power have remarkably simplified and accelerated the calculations involved in

analytical derivative pricing, where such an analytical solution exists. For the cases where

such a solution does not exist, another opportunity has arisen from the insurgence of

computers: the possibility of simulation driven solutions.

Differential equations are often used to model evolution or growth in systems; adding a

stochastic element helps represent the randomness that exists in some cases [47]. A stochastic

differential equation is represented by:

dyt = b(yt)dt+ σ(yt)dWt

y0 = x

This equation represents the growth in y with respect to time t, where W denotes Brownian

motion and σ(yt) determines the magnitude of its randomness. The first term is referred to as

the drift and it helps guide the diffusion, representing the tendancy to move a certain way in

some systems. If this term is omitted, the equation is dominated by the second one which is

known as the Brownian component, representing the randomness with σ(yt) as its volatility.

SDEs are often used to model the evolution of stock prices and capture market behaviour.

In 1973, financial markets were revolutionised when Fischer Black and Myron Scholes came


up with an approach to analytically price options based on a model for stocks. Their idea was

based on finding a self-financing trading strategy that would perfectly replicate the cash flows

of the option, and using the concept of no arbitrage to conclude that the cost of implementing

such a strategy and the price of the option must be the same [37]. They used their hypothesis

of log-normality of stocks to model stock dynamics using a Geometric Brownian motion [9]:

dSt = µStdt+ σStdWt, (2)

S0 = s0,

where µ and σ are constants and t ∈ [0, T ].

There are obvious shortcomings with this model. For example, the assumption that µ and σ

are constants is completely unrealistic. In the real world, stock volatility changes constantly

and unexpectedly. Despite its imperfections, the Black-Scholes model provides an analytical

solution to pricing options, the evaluation of which were previously not possible, and hence a

reasonable preliminary model. Since then there have been many different theories, both in

industry and academia, on the best methods for predicting stock movements and hence

option prices, but they often tend to be modifications and improvements on the now

well-established Black-Scholes model. There is a class of models that allow for non-constant

volatilities called Stochastic Volatility models [36], and others that include non-constant

drifts to introduce special properties such as mean-reversion into the models [24]. However,

usually once such modifications are implemented, closed-form analytical solutions no longer

exist and numerical methods become critical to finding a solution.

Monte Carlo simulation is a numerical method of problem solving using computer algorithms,

covered in more detail in Section 2. Monte Carlo methods are particularly useful for

answering questions that can’t be answered, or are difficult to answer, using traditional

analytical methods. Such methods were first used in corporate finance by David Hertz in

1964 [35], but they were adopted in 1977 by Phelim Boyle to evaluate derivatives [10].

Despite the early adoption of these numerical methods and the traditional preferance towards

explicitly driven analytical solutions, there is no doubt that using such numerical methods is

invaluable to obtaining solutions where, as previously mentioned, such solutions do not exist.


In addition to this, Monte Carlo methods are much more efficient when dealing with

problems in higher dimensions [19].

Monte Carlo methods are often used to obtain information on a specific target distribution

using a different distribution which is easier to manipulate. There are many different methods

and algorithms used in this field, each with its own advantages and disadvantages, and

choosing the best method for each case is an art of its own. The quality and accuracy of any

simulation based solution relies heavily on its samples, which for Monte Carlo algorithms are

required to be independent and identically distributed (IID). In 2005, Roberts and Beskos [8]

introduced a method based on a popular Monte Carlo algorithm known as rejection

sampling, to obtain perfectly IID samples from a stochastic differential equation. This was

then modified and made feasible by further clarifications and developments in [5, 6, 7]. This

dissertation focuses on these developments on the Monte Carlo algorithm known as rejection

sampling, extending it to path space rejection sampling as in [57], as a method of exact

simulation of stock prices and by extension option prices, under various models.

Exotic over the counter contracts (OTC), that were once seen as impossible to price and

hedge, have become the norm and are now so popular that market participants often engage

in option arbitrage, forcing market-makers to provide tight spreads and accurate prices. The

ferocity in this race to perfection is increasing with the growth of computing power and

hence, the field of perfect simulation is becoming more and more relevant as it constructs

perfect samples and thereby, exact solutions to previously intractable problems.

This dissertation approaches the problem of exotic option pricing, by using the path space

rejection sampler to simulate stock prices and then using these perfectly IID simulations to

derive a data-driven solution for option pricing. The paper is structured as follows: Section 2

provides a brief summary of concepts validating Monte Carlo methods in general. Section 3

defines some general assumptions that are required for building the path space rejection

sampler for a financial application. Section 4 provides the motivation behind using perfect

simulation methods instead of more popular and widely-used discretisation techniques.

Section 5 begins by explaining the basics of the well-known rejection sampling method, and

then extending and explaining the ideas and key concepts required to extend it to become


the path space rejection sampler. Section 6 then applies the derived method to finance by

simulating stock prices using different proposals and then evaluating some exotic path

dependant options. Finally, Section 7 concludes the paper by summarizing the arguements

and suggests some avenues that can be used to improve and build on it in the future.

2. Monte Carlo methods

Monte Carlo methods have been used for years in many different industries to simulate

systems with significant uncertainty. In astrophysics, Monte Carlo methods are used to model

the evolution of the galaxy [49]. In quantum mechanics, they are used to solve the

many-body problem [44, 20]. In engineering, they are used to solve the Boltzmann equation

for finite Knudsen number fluid flows [22]. In biology, they are used for studying systems such

as genomes by Bayesian inference in phylogeny [53]. In game development, they are used in

artificial intelligence based programmes to model the opponent’s moves [13]. Lastly, as

previously mentioned, in finance, they are used to evaluate investments and derivatives [11].

Due to their efficiency and simplicity, Monte Carlo methods are very popular as numerical

methods and hence are widely used and have countless applications.

Monte Carlo methods are validated by two very strong results from probability theory, the

Central Limit Theorem [32] and the Strong Law of Large Numbers [34]. By sampling

y1, y2, ..., yn ∼ f and applying the Strong Law of Large Numbers, a consistent estimator can

be approximated [57]:

limn→∞1

n

n∑i=0

h(yi) =

∫ ∞0

h(yi) = Ef [h(y)]

Then, assuming the test function of the samples have finite variance σ2, the Cental Limit

Theorem can be used to show that the errors converge in distribution to a normal variable

with variance σ2:


limn→∞√n

(Ef [h(y)]− 1

n

n∑i−0

h(yi)

)D= ε where ε ∼ N(0, σ2)

These results combined state that over a large number of samples, Monte Carlo estimators

converge to the true values. This implies that by using a large number of independent and

identically distributed (IID) random simulations, the samples converge to the true

distribution. Therefore, the main problem with option pricing using Monte Carlo methods

boils down to obtaining such IID samples. This is why this dissertation focuses on path space

rejection sampling that makes simulating such IID samples possible.

3. Assumptions and setup

There are some general assumptions that are either required by various models or are merely

made for simplicity; further assumptions will be undertaken and similarly justified wherever

required. The first implicit assumption is that of continuity of time, i.e. t ∈ [0,∞). This is

obvious and is reiterated mostly for clarity and completeness.

Assumption 1. Stock-prices are always non-negative, St ≥ 0, ∀t ∈ [0,∞).

An application specific assumption is required to ensure the stock prices always remain

non-negative. This is because once the stock price reaches zero, the company is essentially

bankrupt and hence the stock price can not go down any further. This assumption guarantees

realistic results and ensures that the true properties of stocks are not violated.

Assumption 2. The drift b(yt) and volatility σ(yt) of the SDEs used in this dissertation are

sufficiently regular.

An additional assumption is undertaken here to guarantee the existence of a unique,

non-explosive, weak solution to SDEs as described in Section 1.


Remark. Many financial applications assume that log return of stock prices is an

infinitesimal random walk with drift, i.e. a Geometric Brownian motion with constant drift

and volatility as in Equation 2: dSt = µStdt + σStdWt. This is the main assumption

underlying the Black-Scholes model of stocks [9]. This dissertation allows for simulation of

stock prices under custom made models that better represent market conditions, without

undertaking such restrictive assumptions.

4. Alternative numerical methods for simulating SDEs

Continuous-time problems are often solved by dividing each time interval into N equal

subintervals and finding solutions to the analogous N step discrete-time problems as N tends

to infinity. Such discretisation methods can be used to solve simulation-based problems as

well, and indeed have been used for many years, with various degrees of convergence and

success. The simplest such method is the Euler discretisation scheme which makes use of the

left-hand side rule for integrals to transform a continuous diffusion problem to the discrete

case by considering a fine mesh in finite space. Another approach that is more popular in

finance due to its path-dependant properties, in particular whilst dealing with interest rate

products, is evaluating derivatives using Lattice models. These models are built on the same

basic principles as that of a Binomial tree, as they discretize the number of possibilities at

each step and achieve the properties of a continuous model by including many steps in the

approximation.

4.1. Euler discretisation

Euler Discretisation can be applied to Black-Scholes (Equation 2), covered in detail in 5.3, by

dividing the time interval [0,T] into N equal discrete subintervals. The equality condition is

not necessary but has been assumed for simplicity, however, this is not a restricting condition

as any result for equally spaced intervals can be easily extended to the general case. So let

these equal subintervals be denoted by ti such that t0 = 0 and tn = T . Then the distance

between ti and ti+1 can be represented by dt. To discretize a stochastic process using the

Euler scheme, dSt is integrated from t to t + dt and then since the value of St at time t is


known, the left-point rule is used [60]:

St+dt = St +

∫ t+dt

tµ(Su)du+

∫ t+dt

tσ(Su)dWu

≈ St + µ(St)

∫ t+dt

tdu+ σ(St)

∫ t+dt

tdWu

≈ St + µ(St)dt+ σ(St)(Wt+dt −Wt)

= St + µ(St)dt+ σ(St)√dtZ

Implementing such an algorithm is simple as the Brownian motion component, which accounts

for the randomness of the equation, can be simulated using normally distributed random

variables due to the normality of its increments. Paths using such a discretisation method can

be simulated as follows:

Euler discretisation scheme - Algorithm 1

1) Set ∆ = 1/N , Y = N + 1 , S0 = starting value = x2) Sim W ∼ N(0, 1)

3) Update Si+1 using Si+1 = Si + µ∆ + σ√

∆W4) Return path S = (S0, ..., ST )


−4 −2 0 2 4 6 8

0.0

0.1

0.2

0.3

0.4

Density of Euler Discretised Samples with N = 1000 Possible Stock Path with Euler Discretisation (N = 1000)

time

Sto

ck P

rice

0.0 0.2 0.4 0.6 0.8 1.0

1.0

1.5

2.0

2.5

3.0

−4 −2 0 2 4 6 8

0.0

0.1

0.2

0.3

0.4

Density of Euler Discretised Samples with N = 3 Possible Stock Path with Euler Discretisation (N = 3)

time

Sto

ck P

rice

0.0 0.2 0.4 0.6 0.8 1.0

0.8

1.0

1.2

Fig 1: Two cases of Euler Discretisation with N=3 and N=1000 and their respective densities

Intuitively, a finer mesh (larger N) results in a more accurate approximation, since the

continuous case is achieved by extending N to infinity, and Figure 1 supports this. It is

important to note that there is a trade off between computational cost and the accuracy of

this algorithm and extending N to infinity is impossible. The error induced by using such a

discretisation scheme is most noticable in the case with small N as portrayed in Figure 2.

time

Sto

ck P

rice

0.0 0.2 0.4 0.6 0.8 1.0

−0.

50.

00.

51.

0

Fig 2: Euler Discretisation with N=3


Figure 2 shows another possible path for the stock price using Euler discretisation with N=3,

where the simulated stock price is negative which is not a possibility in the real markets,

violating Assumption 1. There are more sophisticated methods available such as the Milstein

scheme [14], however, such errors persist. This graph clearly portrays the problem with such

discretisation methods, and beckons a more accurate method of simulation. The need for a

simulation method without such systematic errors, is the motivation behind this paper, the

main focus of which is perfect simulation using the path space rejection sampler.

4.2. Lattice models

A Binomial Lattice model re-iterates a two-dimensional probability tree at each lattice node

for the required number of time steps M ≥ 1 over [0, T ] [2]. As with any Binomial model,

there are two subsequent states available at every node: u > 1 where stock price goes up and

d < 1 where stock price goes down with P(u) = p and P(d) = 1 − P(u) = 1 − p = q, where

p ∈ [0, 1]. Let the mth time step be tm where m = 0, 1, ...,M , and similarly to the Euler

scheme in Section 4.1, let dt = (T − t0)/M denote the size of each time step. Then the stock

price at node (m,n) is:

Smn = dm−nunS0

Thus, given S0, the stock price S at any time step can be found. Then using risk-neutral

pricing [63] and the concept of no arbitrage, the returns of the stock must equate to those of

the risk-free rate r which is taken to be constant for simplicity. Then introducing a volatility

parameter σ and noting that the equation above must hold for all S,

puS + (1− p)dS = erdtS

pu2 + (1− p)d = e(2r+σ2)dt

Remark. As dt tends to 0, σ converges to the log-normal volatility in the Black-Scholes

model [2] represented in Equation 2.


For example, consider the case where p = q = 0.5, and S0 = 100, then by setting u = 1.2 and

d = 0.8, the stock’s evolution in 5 steps is as represented in Figure 3.

100

80

120

64

96

144

51.2

76.8

115.2

172.8

40.96

61.44

92.16

138.24

207.36

32.77

49.15

73.73

110.59

165.89

248.83

Fig 3: Evolution of a stock’s price using the Binomial Lattice model

Binomial Lattice models were first introduced to finance in 1979 as a method of option

pricing [17]. Since then, many variations of this model have been developed; some with direct

extensions such as including more states at each time step like the Trinomial model [10], and

some with more functionality, such as the Edgeworth Lattice model that captures the

kurtosis and skew of the underlying asset [61].

These models were some of the first simulation based solutions used in finance and they made

pricing of some exotic path dependant options possible for the first time [58]. However,

despite their initial popularity in finance, such models were nearly all replaced by Monte

Carlo methods due to efficiency. To obtain a single price in a Lattice model, all branches

must be fully calculated and as the number of states and time steps increases, such a

calculation becomes increasingly inefficient and slow. Similarly to the Euler scheme, this

approach incurs a discretisation error, which persists even with infinitesimal time steps,

causing inaccuracies. This again beckons a more accurate and efficient approach which the

path space rejection sampler provides.


5. Path space rejection sampling

This section covers the theoretical concepts used to develop the path space rejection sampler.

It explains the link between these ideas and provides a hollistic overview. Since the path

space rejection sampler is an extension of the well-known rejection sampler, this section

begins by reviewing the key concepts behind rejection sampling and introducing the theory

required for the extension. The section ends by providing the full algorithm, amalgamating

the ideas discussed.

5.1. Rejection sampling

Simulating from densities without any closed form solution, or densities with computationally

costly solutions can be very difficult and cumbersome: such densities are often collectively

referred to as inaccessible densities. Fortunately, Monte Carlo methods provide multiple ways

to deal with such problems. One such method is by sampling using an accessible proposal

density instead, then retaining the suitable samples and rejecting the unsuitable ones.

Let f represent the inaccessible target density and g the accessible proposal. The choice of

proposal is imperative as the proposal must have thicker tails than the target density to fully

cover it. Note that, by definition, the area under all densities is equal to one. As the proposal

needs to fully cover the target density it must be scaled by a factor M as represented in

Figure 4.


Target f

Proposal M.g

Fig 4: Graphical representation of rejection sampling

Uniform samples y are then taken under M.g(x), where x is sampled from g, and all samples

that lie below f are accepted, and all others are rejected. The result is a list of

two-dimensional cartesian variables (x, y), and by dropping the y-coordinates, a list of

x-coordinates is obtained which is distributed according to the law of f . The rejection

sampling algorithm follows the steps described above as follows:

Rejection sampling [41] - Algorithm 2

1) Sim X ∼ g2) Sim Y ∼ Uni[0,M.g(X)]3) If Y ≤ f(X) accept the sample, else reject and go back to 1)4) Return X

Using geometrical intuition and Figure 4, the acceptance probability in such a sampler is:

p(X) := P(Acceptance) =Areaf

AreaM.g=

f(X)

M.g(X)

From this acceptance probability, it is clear that the choice of M is imperative to the

efficiency of this algorithm: the smaller M is, the closer the acceptance probability will be to


one, and thus more samples are accepted, and vice versa. Hence the rejection sampling

algorithm can be rewritten in terms of this acceptance probability:

Rejection sampling [41] - Algorithm 3

1) Sim X ∼ g2) Accept Y with probability f(X)

M.g(X)and return X, else reject

and go back to 1)

This paper provides a simple proof of rejection sampling that relies on probabilities under

different measures; it shows how starting with a sample following the proposal density g, a

sample with density f can be derived. There are of course alternative more rigorous proofs

available [57].

Proof. [41] Let A be an arbitrary set. Then,

Px∼g (x ∈ A and x is accepted) = Eg[1(x∈A), y ≤

f(x)

M.g(x)

]=

∫Ag(x) · f(x)

M.g(x)dx =

∫A f(x)dx

M

Px∼g (x is accepted) = Px∼g (x ∈ Ω and x is accepted) =

∫Ω f(x)dx

M

Then using Bayes’ Rule:

Px∼g (x ∈ A|x is accepted) =Px∼g (x ∈ A and x is accepted)

Px∼g (x is accepted)=

∫A f(x)dx

M∫Ω f(x)dx

M

=

∫A f(x)dx∫Ω f(x)dx

=

∫A f(x)dx

1

=

∫Af(x)dx = Pf (1(x∈A))

It is worth noting that rejection sampling’s acceptance probability is proportional to the

Radon-Nikodym derivative. This is particularly useful for applying the algorithm in higher

dimensions and thus will be used regularly in the extension to path space rejection sampling.


p(x) :=1

M· dQdW

(x) =1

M· dfdg

(x), M := supx∈W

df

dg(x)

It is possible to obtain samples using rejection sampling even when the full distribution of the

target is unknown, the target only needs to be identified up to a constant of proportionality.

Proof. Let f(x) = c · π(x), then

p(x) :=1

M· dQdW

(x) =1

M· dfdg

(x) =1

M· dc · πdg

(x)

=c

M· dπdg

(x) =1

M ′· dπdg

(x)

M ′ := c · supx∈W

dπ

dg(x), M ′ := c ·M

With the establishment of the basic framework of rejection sampling, a natural question

arises: how long would the algorithm have to run until a single sample is accepted?

To answer this question, it is enough to notice that the acceptance-rejection process is

essentially a random variable following a Geometric distribution with parameter 1/M. This

implies that the amount of time expected to pass before the first acceptance is the

expectation of the Geometric distribution, which is M. Thus, as previously mentioned,

choosing the smallest M possible is vital to the efficiency of this algorithm, both in terms of

time and computational power required to successfully obtain the required number of

samples.

The rejection sampling algorithm is expected to complete in finite time by the argument

above, however, computational power and resources are scarce and sometimes finiteness is

not enough. For instance, assume 10,000 samples are required. Since each run of the

algorithm is independent, the most intuitive approach is to run the algorithm 10,000 times,

however, this can be very time consuming. It is tempting to introduce a time constraint, one

hour for example, at the end of which the algorithm is to stop and return the samples

obtained. Such an approach is deeply problematic as it introduces stoppping time bias and


deteriorates the overall integrity of the sampling process, and thus must be avoided at all

times [5]. A possible solution to such an issue is to obtain samples in smaller subgroups as

the samples are independent of one another. For example, to obtain 10,000 samples, one can

take 1000 samples ten times and aggregate the results.

Remark. Rejection sampling is a special case of a Monte Carlo method called importance

sampling which uses weights to get information from every single sample instead of throwing

some away. Since it uses all simulated samples, importance sampling is sometimes seen as a

more efficient simulation method. Despite its advantages, using weights of zero and 1, i.e.

rejection sampling, is much simpler and the efficiency problem can be resolved by using

appropriate proposal densities and scaling factors.

The extension of rejection sampling to path space rejection sampling occurs by allowing the

target and proposal to be path measures of diffusions, essentially moving from the

one-dimensional case to the multi-dimensional one. This is a very useful concept because it

allows sampling from stochastic differential equations with non-constant drift, using

stochastic differential equations with constant drift such as the Black-Scholes one. So the

target Q, the proposal W, and the acceptance probability p(S) can be chosen as follows:

Q : dSt = µ(St)Stdt+ σStdWt

W : dSt = µStdt+ σStdWt

p(S) :=1

M· dQdW

(S), M := supS∈W

dQ

dW(S)

Targets of this form, i.e. Stochastic Differential Equations with non-constant drift, are

intractable and do not have closed-form solutions, but they are very useful and can introduce

useful properties into financial models as explained in Sections 5.3 and 5.4. As mentioned, the

lack of an analytical solution means the only possible method of obtaining stock prices from

such a model is by using numerical methods.


Since path space rejection sampling mainly deals with Stochastic Differential Equations and

replicates their randomness through Brownian motion, these concepts and their properties

are briefly covered in the following sections.

5.2. Simulation of Brownian motion and related processes

Brownian motion and Brownian bridge concepts are defined here, due to their vital role in

understanding SDEs and the ideas behind the construction of the path space rejection

sampler. Brownian motion was initially developed by a physicist named Robert Brown as a

way to model the random movement of diffusion of particles [36]. Since then, it has been

applied to many different fields including finance to describe the randomness of specific

components.

Definition 1. A d-dimensional Brownian motion [59] (Wt)t≥0 is a stochastic process

indexed by [0,∞) taking values in Rd such that:

(B0) Initialisation: W0(w) = 0 ∀w ∈ Ω

(B1) Independent Increments: Wtn − Wtn−1 , ...,Wt1 − Wt0 are independent ∀n ≥ 0 ,

0 = t0 ≤ t1 < t2 < ... < tn <∞

(B2) Stationary Increments: Wt −Ws ∼Wt+h −Ws+h ∀0 ≤ s < t, h ≥ −s

(B3) Normal Increments: Wt −Ws ∼ N(0, t− s)

(B4) Continous Paths: Brownian paths Wt(w) are continuous ∀w ∈ Ω

The following algorithm allows for simulation of Brownian motion points at a predetermined

set of times. Using the stationarity, independence and normality of the increments of Brownian

motion allows for simulation of each point individually.

Brownian motion at t1, t1, ..., tn [57] - Algorithm 4

1) For each i in 1 to n , sim Wti ∼ N(Wti−1 , ti − ti−1)


One simple modification of this concept is that of a Brownian bridge, which is a Brownian

motion whose endpoint is apriori known. This is a very useful notion as it models the random

behaviour of a particle between two points.

Definition 2. A Brownian bridge [18] βt is a continuous stochastic processs in [0, T ] defined

for t ∈ [0, T ] by,

βt := Wt −t

TWT

Given a set of points, simulating a Brownian bridge requires first identifying the known

points around the point sought. Once the general location is obtained by finding the closest

points (one on either side in the two-dimensional case), the required point can be simulated

using the same logic as in Algorithm 4. Let LS and RS represent the point directly to the left

and right of the point considered respectively, as below:

Brownian bridge at t1, t2, ..., tn given the process at 0, r1, ..., rm, T [57] - Algorithm 5

1) Set K := (0, S0), (ri, Sri)mi=1, (T, ST )

2) For each i in 1 to n,2.1) Set LS := supK : K ≤ ri and RS := infK : K ≥ ri2.2) Sim Wri ∼ N(WLS +

(ri−LS)(WRS−WLS )

RS−LS, (RS−ri)(ri−LS)

RS−LS)

Figure 5 is a sample produced using Algorithm 5; it is clear that the process goes through the

apriori known dark blue points, and that it moves randomly between these points as

required.


0 1 2 3 4 5

−0.

50.

00.

51.

01.

52.

02.

5

Times

Val

ues

Fig 5: Brownian bridge with process known at 0, 1, 2, 3, 4, 5

5.3. Black-Scholes and its properties

The popular Black-Scholes model is based on another subclass of stochastic differential

equations with Geometric Brownian motion [47]. It provides a solution to evaluating option

prices based on stock volatility, interest rate, and price at a previous point in time through

dynamic calculation of stock prices. The Black-Scholes assumption requires both the standard

deviation σ(yt) and the drift b(yt) to be constants, so the equation can be rewritten as:

dSt = µStdt+ σStdWt

The explicit solution to this equation is found by using Ito’s Formula on log(St):

St = S0 · exp

σWt + (r − σ2

2)t

Since the main aim of this paper is to provide a simulation based solution, the explicit solution

is included only here for completeness and to exhibit that the Black-Scholes equation does

indeed have an analytical solution, unlike those with non-constant drift. It is important to note


however, that this explicit solution could provide useful properties for checking the validity of

the simulated samples.

Fig 6: Geometric Brownian motion with µ = 0.05, and σ = 0.3 [29]

The Black-Scholes model is very popular in finance and most of the more advanced models

used in the industry are adaptations and modifications of this model. One special property of

this model which makes it especially convenient for financial applications is that it essentially

guarantees non-negativity of stock prices (Assumption 1) as shown in Figure 6. This is

provided by St in the drift and the Brownian components: as the stock price approaches zero,

i.e. as St gets smaller, the volatility of the Brownian component gets smaller, lowering the

randomness of the function, increasing the influence of the drift. Since µSt ≥ 0 , the stock

price will increase again reversing the effect described, increasing the randomness again.

Whilst the Black-Scholes model provides a good starting point for modelling stock prices, the

assumptions of a scalar drift and volatility are not very realistic and are often modified in

practice. Once such modifications are applied, however, a problem arises: closed-form

solutions are no longer available and as mentioned, this is where Monte Carlo simulation

based approaches become invaluable.


5.4. Alternatives to Geometric Brownian motion

Slight variations in the coefficients of an SDE can result in some very interesting properties,

which can be used to model different processes. One example is a well-known subclass, called

Ornstein-Uhlenbeck processes [24] which are used in multiple financial models such as the

Vasicek [65] model of stock prices:

dSt = θ(µ− xt)dt+ σdWt

0 2 4 6 8 10

−2

−1

01

2

Times

Val

ues

Fig 7: Example of a mean reverting process

This process has constant volatility and a very special property, that of mean-reversion.

Stocks modelled based on this process exhibit mean-reversion as in Figure 7, i.e. if the stock

price is above its mean, then it will fall and return to its mean and if it is below its mean,

then it will rise and return to its mean. This means that in the long run, stock prices tend to

fluctuate around their mean. Similarly to σ scaling the Brownian component, θ provides a

rate for the mean-reversion. This is a property often observed in financial markets; in fact,

many trading strategies are build around this concept of reversion to the long term mean

[3, 33].


For simplicity, Monte Carlo algorithms often use an unbiased estimator with the same

acceptance probability to obtain samples. This is a clever trick that permits quicker and more

efficient sampling. This is similar to obtaining samples from a coin by throwing a die instead

and marking heads for evens and tails for odds. Over a large number of throws, both

experiments result in an acceptance probability of 0.5 and hence the two experiments are

indistinguishable by Strong Law Large Numbers as described in Section 2.

This trick allows for exploration of much more realistic and interesting financial models. For

example, one popular belief about movements of stock prices is centered around an idea

called predilection. This belief proposes that stock prices tend to gravitate towards integer

values and it can be modelled by including a periodic drift, like that of a sinusoidal function:

Q : dSt =1

πsin(St)dt+ dWt (3)

S0 = s0

Such a target models a financial belief but also has useful properties that can be used to

evaluate the results obtained from the path space rejection sampler, and thus, this

modification to the target will be explored in this paper: the methodology will be established

using this particular target and financial securities will be evaluated using this model.

Despite this, the theory is clearly robust and can be applied to any other target that satisfies

the required assumptions.

It is worth noting that a limitation of path space rejection sampling is that rejection

sampling needs tractable proposals whose diffusion paths are absolutely continuous with the

target SDE [27]. However, if the target has unit volatility and drifts of the gradient form,

then this condition is satisfied. The required conditions on the drift are satisfied by

Assumption 3, and the unit volatility requirement can be achieved using a Lamperti

Transformation, both of which are discussed in the next section.


5.5. Lamperti transformation

Just as Fischer Black and Myron Scholes revolutionised finance, the Japanese mathematician

Kiyosi Ito reformed the field of stochastic calculus by developing methods to compute

stochastic integrals in continuous time [40]. Despite the popularity of Ito’s formula, for

simplicity, Ito’s lemma is stated here as it fits better with the Lamperti transformation and

avoids the complications introduced by using the full version of Ito’s formula.

Theorem 1. (Ito’s Lemma):[54][40] Let St be an Ito process given by,

dSt = b(St)dt+ σ(St)dWt

Let ψ(St) ∈ C2[0,∞]×R. Then Zt = ψ(St) is again an Ito process and:

dZt =dψ

dt(St)dt+

dψ

dx(St)dSt +

1

2

d2ψ

dx2(St)(dSt)

2

A Lamperti transformation [45] can be used to convert diffusions with volatility larger than 1

for use in the methodology which requires unit volatility. To do this Assumptions 3 and 4,

ensuring continuity and boundedness of the growth of the diffusion, are required.

Assumption 3. [57] The drift coefficient b ∈ C1. The volatility coefficient σ ∈ C2 and σ > 0.

Assumption 4. [57] ∃K > 0 such that |b(x)|2 + |σ(x)|2 ≤ K(1 + |x|2) ∀x ∈ R.

If these assumptions are satisfied the Lamperti transformation can be stated as in Theorem 2.


Theorem 2. (Lamperti Transformation): [38] Let St be an Ito process as defined in Ito’s

lemma and define,

ψ(St) =

∫1

σ(St)ds|s=St

If ψ is one to one from the state space of St onto R for every t ∈ [0,∞), then choose Zt = ψ(St).

Otherwise, if σ(St, t) ≥ 0,∀St choose,

Zt = ψ(St) =

∫ s

ξ

1

σ(u)du|s=St

where ξ is some point inside the state space of St. Then Zt has unit diffusion and is governed

by the SDE:

dZt =

(dψ

dt(Zt) +

f(ψ−1(Zt))

σ(ψ−1(Zt))− 1

2σs(ψ

−1(Zt))

)dt+ dWt

Simplifying the notation helps clarify the proof, let f = f(St) , σ = σ(St) , ψ = ψ(St).

Proof. [48][52] Let ψu = dψdu and ψuu = d2ψ

du2 .

Then dZt from Ito’s lemma (Theorem 1) can be simplified:

dZt = ψtdt+ ψs(fdt+ σdWt) +1

2ψss(fdt+ σdWt)

2

= (ψt + ψsf)dt+ ψsσdwt +1

2ψssσ

2dt

= (ψt + ψf +1

2ψssσ

2)dt+ ψsσdWt

From this it is clear that the level dependent diffusion σ(St) can be removed from the SDE

above by modifying the transformation ψ:

ψ(St) =

∫1

σ(u)du|u=St

ψx(St) =1

σ(St)


Taking derivatives with respect to time and S gives:

ψss(St) = −σs(St)σ(St)2

ψt(St) =d

dt

∫1

σ(u)du|u=St

Then substituting into the expression derived for dZt gives:

dZt =

(d

dt

∫1

σ(u)du|u=St +

f

σ− 1

2

σsσ2σ2

)dt+ dWt

Then by cancelling the denominator and numerator in the last term, substituting in ψ and

noting that St = ψ(−1)(Zt) the proof is completed:

dZt =

(dψ

dt(Zt) +

f(St)

σ(St)− 1

2σs(St)

)dt+ dWt

=

(dψ

dt(Zt) +

f(ψ−1(Zt))

σ(ψ−1(Zt))− 1

2σs(ψ

−1(Zt))

)dt+ dWt

The resulting variable Zt has unit volatility and a new drift α, i.e. dZt = α(yt)dt + dWt. The

sampling process can now be done using this transformed variable Zt which has unit volatility

as required. It is imperative to implement the inverse transformation St = ψ−1(Zt) once the

samples are obtained, otherwise the samples obtained are not from the true target diffusion.

Remark. Lamperti transformation is generally only suitable for one-dimensional diffusions

[27].

This transformation is very powerful as it removes the restrictions on the targets that can be

explored with the path space rejection sampler. The next step in completing the extension

from rejection sampling to path space rejection sampling is obtaining and simplifying the

required acceptance probability, this is done in the next section using a well-known result

from stochastic calculus: Girsanov’s theorem for Brownian motion [36].


5.6. Girsanov transformation of measures

Recall that the acceptance probability in rejection sampling is proportional to the

Radon-Nikodym derivative. As this can be a complicated expression in the multi-dimensional

case, it is useful to systematically simplify it prior to implementing the algorithm. Girsanov’s

theorem and transformation of measures are often used in finance especially to move between

traditional and risk neutral measures for asset pricing [63].

The Girsanov transformation of measures [54] combined wtih the differentiability of α (Note

that Assumption 3 states the drift must be differentiable, and that in the case where a

Lamperti transform was not required α = b) implies that the Radon-Nikodym derivative can

be rewritten and that the Ito integral can be eliminated by applying Ito’s lemma to A(Wt)

for A(u) :=∫ u

0 α(y)dy, u ∈ R [5],

dQ

dW(x) = exp

∫ T

0α(Wt)dWt −

1

2

∫ T

0α2(Wt)dt

= exp

A(WT )−A(x)−

∫ T

0

1

2(α2(Wt) + α′(Wt))dt

= exp

A(WT )−A(x)−

∫ T

0φ(Wt)dt

,

where φ = 12(α′ + α2). Here, the methodology involved in building the path space rejection

sampler diverges based on whether or not this function φ can be bounded. For the

unbounded case Bessel bridges [5, 50] are used instead of Brownian bridges to construct

paths as in [5]. However, this paper focuses on the simpler bounded case and hence an extra

assumption is undertaken here to ensure φ is bounded.

Assumption 5. The function φ is compact, and thus bounded above by Ux ∈ R and below by

Lx ∈ R.

Having obtained a SDE with unit volatility and drift α and a new simplified acceptance

probability dQdW as described above, the rejection sampling algorithm can be modified to

obtain the path space rejection sampler.


5.7. Path space rejection sampling algorithm

Sampling full diffusion paths is very difficult as these are infinite-dimensional objects. To

implement rejection sampling for such targets the rejection sampling algorithm in Section 5.1

needs to be modified significantly. This section focuses on these modifications and the

rationale behind the implementation of these changes.

There are multiple problems with implementing the standard rejection sampling algorithm

here. First, it is impossible to simulate entire trajectories of infinite-dimensional paths.

Second, obtaining a bounding constant M is not possible as the acceptance probability in its

current form is unbounded.

The second problem is most significant, since the only condition in building a rejection

sampler is finding a bounding constant M . However, the expression derived above for the

Radon-Nikodym derivative is a useful tool for understanding and fixing this issue. Recall

from Section 5.6,

dQ

dW(x) = exp

A(WT )−A(x)−

∫ T

0φ(Wt)dt

,

where φ =1

2(α′ + α2).

From this expression, it is clear that whilst the negative exponentials can be bounded, the

exp(A(Wt)) is unbounded. Thus, despite finding the source of the problem, finding a bounding

constant M is still not possible. It is hence justified that the first modification to the algorithm

involves modifying the proposal to obtain a bounded acceptance probability. To do this, the

notion of Biased Brownian motion [8] is introduced: this is essentially a Brownian motion whose

end point as well as its start point is known.


Definition 3. Biased Brownian Motion [57] is the process ZtD= (Wt|W0 = x,WT := y ∼ h)

with measure Zx,y0,T , where x, y ∈ R, t ∈ [0, T ] and h is defined as,

h(y;x, T ) :=1

c(x, T )exp

A(y)− (y − x)2

2T

The function h is a simulatable tilted Gaussian and therefore the endpoint can be simulated

using standard rejection sampling methods described in Section 5.1. Using such a proposal

the Radon-Nikodym derivative can be simplified as well:

dQ

dZ∝ exp

−∫ T

0φ(xs)ds

,

where φ = α2+α′

2 . This acceptance probability is proportional to a negative exponential and

thus it is always bounded above by 1 and a suitable scaling constant M is attainable. In fact,

an explicit formula for M can be derived by finding the lower bound of this fraction. Let,

infu∈[0,T ]φ(xu) ≥ Φ

Then,

M := exp(−ΦT )

is always a viable choice.

Now the first problem can be approached, that of the infinite-dimensionality of Brownian

motion paths. To resolve this problem a particularly useful property of diffusions can be

used, one which states that each diffusion path can be recognized and identified using

finitely-many points. These points are called skeletal points and the rough path determined

by them is referred to as the skeleton of the diffusion, denoted in this dissertation by K to

avoid confusion with the stock price S.


Definition 4. A Skeleton [57] K is a finite dimensional representation of a diffusion

sample path (S ∼ Qx0,T ), that can be simulated without any approximation error by means of

a proposal sample path drawn from an equivalent proposal measure (Wx0,T ) and accepted with

probability proportional todQx0,TdWx

0,T(S), which is sufficient to restore the sample path at any

finite collection of time points exactly with finite computation where S|K ∼ Wx0,T |K. A

skeleton typically comprises information regarding the sample path at a finite collection of

time points and path space information which ensures the sample path is almost surely

contained to some compact interval.

To avoid simulating entire infinite-dimensional trajectories an additional step is added to the

algorithm to allow for simulation of a finite-dimensional skeleton. This is then followed by the

simulation of the missing points to obtain the full trajectory. Simulating the skeleton instead

of the entire path by proposing from a biased Brownian motion measure Zx0,xT0,T is

implemented into the algorithm as follows.

PSRS 1

1) Sim x[0,T ] ∼ Zx,xT0,T

1.1) Sim xT ∼ h1.2) Sim xfin ∼Wx,xT

0,T

1.3) Sim xcts ∼Wx,xT0,T |x

fin

2) Accept with probability 1M

dQdZ

(x) else reject and go back to 13) Return x|(accept) ∼ Q

Although these improvements have fixed some of the issues rising from the path space

extension, the algorithm is still far from computational implementability. The

acceptance-rejection processs requires the evaluation of an infinite-dimensional object under a

continuous function φ (continuity of φ is a result of linearity of differentiable and ergo

continuous functions).

To resolve this, recall the arguement earlier in Section 5 regarding throwing a die and flipping

a coin. As discussed, over a large number of samples both experiments result in the same


outcome by the Strong Law of Large Numbers. This clearly demonstrates the imperative role

and significance of induced randomness regardless of the initiating process. This is the vital

principle behind the well-known statistical concept of unbiased estimators. To implement the

algorithm an unbiased estimator in the form of a Poisson process is introduced into the path

space rejection sampler.

5.7.1. Poisson processes as unbiased estimators

Poisson processes are used to model the number of times a specific event happens in a

continuous time interval. This section provides the general idea behind these complex

processes and will only cover the essentials required for their simulation. In the context of the

path space rejection sampler, Poisson processes play a small part as an unbiased estimator to

simplify the algorithm. Note that, as this is their only role in this methodology they can be

replaced by any other process that allows for the same unbiased estimation. For example, a

Binomial process can be used instead in a manner parallel to the construction of the

Bernoulli factory [46].

Definition 5. A Poisson Process [23] with parameter λ is a counting process (strictly

increasing) (Nt)t≥0 with the following properties:

1. N0 = 0

2. For all t > 0, Nt has a Poisson distribution with parameter λt

3. (Stationary increments) For all s, t > 0, Nt+s − Ns has the same distribution as Nt. That

is P(Nt+s −Ns = k) = P(Nt = k) = e−λt(λt)k

k! , for k = 0, 1, ...

4. (Independent increments) For 0 ≤ q < r ≤ s < t, Nt −Ns and Nr −Nq are independent.

Poisson processes can be either homogeneous or nonhomogeneous. Definition 5 above refers to

a homogeneous Poisson process with parameter λ, as Nt ∼ Poi(λt). Nonhomogeneous Poisson

processes, on the other hand, have a time dependent intensity λ(t). Note that although time

dependence is the most popular type, the intensity can also be space dependent or both time

and space dependent [39]. The noteworthy feature of nonhomogenous Poisson processes is that

their intensity is variable.


Remark. Homogeneous Poisson processes are a special case of the nonhomogeneous Poisson

processes with λ(t) = λ.

Since Poisson processes are used as unbiased estimators in the path space rejection sampler,

the following algorithms that simulate such processes are essential.

Time homogeneous Poisson process [57] - Algorithm 6

1) Sim n ∼ Poi(λt)2) If n 6= 0, Sim iid u1, ..., un ∼ U[0, T ]3) If n 6= 0, set q1, ..., qn to be the order statistics of the set u1, ..., un

Simulating a nonhomogeneous Poisson process is done in a very similar manner, in fact it can

be done by conducting the algorithm above followed by a Poisson thinning process. The

thinning process omits some of the points simulated in the algorithm above, hence creating a

Poisson process that is not spread out in a homogeneous manner.

Time nonhomogeneous Poisson process [57] - Algorithm 7

1) Sim proposal event times (p1, ..., pn) of a time homogeneous Poisson process with intensity λ′

using Algorithm 32) If n 6= 0, then set j = 0 and i in 1 to n,2.1) With probability λ(pi)/λ

′ set j = j + 1 and qj = pi

5.7.2. Finalized path space rejection sampling algorithm

Recall that the goal here is to obtain an unbiased estimator that allows for finite-dimensional

calculations. To this end, a Poisson process is used to obtain the number of skeletal points

required in each time interval to decide whether or not the path identified by that skeleton

should be accepted or rejected. Let κ represent this number, then only κ many Brownian

motion points are required to compute the acceptance probability of the path. To be able to


use this Poisson estimator the acceptance probability p(S) needs to be expressed in terms of

κ. To achieve this goal a Taylor series expansion is conducted on P (S) as in [57] and [7]:

P (S) =1

M

dQ

dZ= eΦT exp

−∫ T

0φ(xs)ds

= e−(Lx−Φ)T e−(Ux−Lx)T exp

∫ T

0Ux − φ(xs)ds

= e−(Lx−Φ)T

[ ∞∑i=0

e(Ux−Lx)T [(Ux − Lx)T ]i

i!

∫ T

0

Ux − φ(xs)

(Ux − Lx)Tds

i],

where the last step is done by using the fact that the Radon-Nikodym derivative for a

homogenous Poisson process with rate λ on [0, T ]× [0, 1] with respect to φ [7] is,

exp−λTλκT κ

κ!

(exp(−T )

T κ

κ!

)−1

= exp T (1− λ)λκ,

again as in [57] and [7], define a measure M such that κ has distribution Poi((Ux − Lx)T ),

and Uκ such that (ξ1, ..., ξκ) are independent and identically distributed U[0, T ].

P (S) = e−(Lx−Φ)TEM

[(∫ T

0

Ux − φ(xs)

(Ux − Lx)Tds

)κ|xfin

]

= e−(Lx−Φ)TEM

[EUκ

[κ∏i=0

(Ux − φ(xξi))

Ux − Lx|xfin

]|xfin

]

This representation of the acceptance probability requires only finitely-many points and

hence it is an implementable, finite step computation. The PSRS 1 Algorithm can now be

modified by including extra steps to sample this auxiliary Poisson variable κ and changing

the acceptance probability accordingly. Also to make the entire algorithm implementable and

finite, note that step 1.3 in PSRS 1 is not actually required for the acceptance rejection

process, it is only included to obtain a complete path rather than the skeleton. So if the aim

is to obtain a skeleton it can be completely omitted. Thus, the PSRS algorithm can be

finalized as follows.


PSRS 2 [57] - Final Version

1) Sim y := xT ∼ h2) Sim skeleton points xfin = (xξ1 , ..., xξκ)2.1) Sim κ ∼ Poi((Ux − Lx)T ) and iid skeleton times ξ1, ..., ξκ ∼ U [0, T ]2.2) Sim sample paths at skeleton times xξ1 , ..., xξκ ∼W

x0,y0,T

3) With probability∏κi=0

(Ux−φ(xξi))

Ux−Lx accept the entire path, else reject and go back to 1

This is the final version of the path space rejection sampler. It is fully implementable as it

only requires a finite amount of computation and by using this algorithm perfectly simulated

samples from the true transition density of the target can be obtained. Of course this is not a

perfect substitute for full paths, however, given the computational restrictions at this point in

time a skeleton will suffice. This is because given the skeleton, the path between any two

skeletal points can be reconstructed using a Brownian bridge as described in Section 5.2 and

hence despite not immediately obtaining the full trajectory from the path space rejection

sampler, the full trajectory is indeed accessible through the construction of Brownian bridges.

Finally, note that the intensity of the auxiliary variable κ is time dependent and hence using

longer time intervals will likely result in a larger κ increasing the execution time of the

algorithm. To obtain samples for larger time intervals efficiently independence of Brownian

motion increments and the Markov property can be used to expedite the simulation process.

Note that the samples obtained are Markovian and thus by dividing large time intervals into

i ∈ 1, ..., n smaller ones and running the algorithm iteratively such that the start point of

each interval i ∈ 2, ..., n, xi0,= xi−1T/i . This is because for any such filtration of the sample

space the acceptance probability can be decomposed as

p(S) =dQx0

0,T

dWx00,T

(S) ∝ exp

−∫ T

0φ(Su)du

= exp

−∫ t1

0φ(Su)du

· exp

−∫ t2

t1

φ(Su)du

· ... · exp

−∫ T=tn

tn−1

φ(Su)du


Now that the sampler and methodology and concepts used to obtain it have been fully

developed and explained, path space rejection sampling is illustrated using an example that

is very similar to the target defined in Equation 3 but with more distinguishable properties.

5.8. Path space rejection sampling illustrated with an example

For illustration purposes, consider a modified target:

Q′ : dSt = sin(St)dt+ dWt

W : dSt = dWt

Then to conduct the first step in the algorithm, h(y;x0, T ) needs to be calculated to allow for

simulation of the Biased Brownian motion. Recall from Section 5.1 that it is enough to know

the target up to a constant of proportionality. Since the endpoint is to be simulated using

rejection sampling, the normalizing constant can be omitted to simplify h(y;x0, T ).

h(y;x0, T ) =1

c(x0, T )exp

A(y)− (y − x0)2

2T

∝ exp

A(y)− (y − x0)2

2T

Recall from Section 5.6 that A(y) =∫ y

0 α(u)du and in this case as the SDE has unit volatility,

the drift α = sin(St).

A(y) =

∫ y

0sin(u)du = −cos(u)

]y0

= 1− cos(y)


Substituting this into the expression of h(y;x0, T )

h(y;x0, T ) ∝ exp

A(y)− (y − x0)2

2T

∝ exp

1− cos(y)− (y − x0)2

2T

∝ exp

−cos(y)− (y − x0)2

2T

Any Monte Carlo method can be used to obtain this endpoint y, however, as rejection

sampling has already been covered in Section 5.1 this is the method chosen in this report. So

the next step is to find the bounding constant M . Looking at the target in this case it seems

reasonable to use a normal proposal with mean x0 and variance T

f(y) = h(y;x0, T ) ≤M.g(y)

exp

−cos(y)− (y − x0)2

2T

≤Mexp

−(y − x0)2

2T

exp −cos(y) ≤M

cos(y) ∈ [−1, 1] , exp−cos(y) ∈ [e−1, e1] and so choosing M = e1 will suffice.

Once the endpoint y is simulated the only other requirement to implement the path space

rejection sampler for this target is to find the bounds of φ and thus, the next step is to

compute φ.

φ(x) =1

2(α2 + α′)

=1

2

((sin(x))2 + cos(x)

)

Again, note that cos(x) ∈ [−1, 1] and sin(x) ∈ [−1, 1], so sin2(x) ∈ [0, 1], so Lx = −0.5 and

Ux = 1 can serve as the required bounds. Adjusting the algorithm for this particular eaxmple

results in the pseudocode below.


Implemented PSRS - dSt = sin(St)dt+ dWt

1) Sim y := xT ∼ h(y;x, T ) ∝ exp−cos(y)− (y−x0)2

2T using rejection sampling

1.1) Sim u ∼ N(x0, T )

1.2) Sim q ∼ U [0,− e1(u−x0)2

2T]

1.3) If q ≤ exp−cos(q)− (q−x0)2

2T

2) Sim skeleton points xfin = (xξ1 , ..., xξκ)2.1) Sim κ ∼ Poi(1− (−0.5))T ) = Poi(1.5T ) and iid skeleton times ξ1, ..., ξκ ∼ U[0, T ]

2.2) Sim sample paths at skeleton times xξ1 , ..., xξκ ∼Wx0,y0,T

3) With probability∏κi=0

(Ux−φ(xξi))

Ux−Lx =∏κi=0(1− 1

2((sin(xξi))

2 + cos(xξi))))/1.5 accept theentire path, else reject and go back to 1

0 1 2 3 4 5

−6

−4

−2

02

46

Time

X

Fig 8: PSRS samples with target dSt = sin(St)dt+ dWt and BM proposal, starting at 0 anda time interval of [0, 5]

Figure 8 shows some samples obtained by using the pseudocode derived. The graph is

somewhat symmetrical as expected since the drift is sinosoidal. Notice also that despite the

paths’ randomness and movement within the space the endpoints seem to be concentrated

around two particular modes of attraction. Figure 9 explains the intuition behind where the

endpoint concentrations occur. Consider the case where the starting point x0 = 2. In this

case, there is positive initial drift represented by a green arrow pushing the sample to the

right until it reaches the first orange line at π. At π the sin graph is at half cycle and changes

sign so now the drift is negative and the sample is pushed back to π. As sin is periodic the

same phenomenon occurs half way into every cycle hence concentrations occur at odd

multiples of π, resulting in the existence of these hubs of attraction.


−10 −5 0 5 10

−1.

0−

0.5

0.0

0.5

1.0

Time

Drif

t

Fig 9: Relationship between sin(x) and endpoint concentration

Using the properties of a sinosidal drift to gain an understanding of the shape of the target

distribution is very useful as intuitively illustrated above. Further, the conclusions drawn

regarding the behaviour of the endpoints can be verified using the density of the endpoints as

in Figure 10 which portrays the density of the endpoints of 10,000 samples, with orange lines

representing π and −π.

−10 −5 0 5 10

0.00

0.05

0.10

0.15

0.20

Den

sity

Fig 10: Density of the endpoints of paths obtained using path space rejection sampling


The endpoints are heavily concentrated around π and −π as discussed, confirming that the

samples are drawn from the true density of the diffusion. Notice how Figure 10 and Figure 8

are fully compatible with each other and both exhibit the same properties, showing the

consistency between the intuitive and quantitative explanations.

Now that the samples have been deemed suitable by using intuition and logic, a more robust

method for checking them is required. Recall the Euler discretisation scheme from Section

4.1, despite the discrepencies displayed for small N, for large enough N and by obtaining

sufficienly many samples n, the density of the Euler discretisation scheme is fairly accurate.

Thus, it can be used to ensure the path space rejection sampler is returning the correct set of

paths. Figure 11 provides a comparison between the densities of samples obtained from the

two methods and it can be seen that they are nearly indistinguishable. Since the samples

from both methods coincide it can be concluded that the samples are viable.

−10 −5 0 5 10

0.00

0.05

0.10

0.15

0.20

Den

sity

PSRS SamplesEuler SamplesBM Proposal

Fig 11: For T = 5, x0 = 0, N = 1000, n = 10, 000, densities of BM proposals, the samplesobtained from path space rejection sampling, and the samples from Euler discretisation

6. Applications to finance

This section applies the path space rejection sampler to finance by perfectly simulating stock

prices under two proposals and then using the resulting paths to evaluate exotic options.


6.1. Simulating stock prices

6.1.1. Path space rejection sampling with Brownian motion proposal

As previously mentioned, the predilection of stocks can be modelled using a SDE with

periodic drift as in Equation 3. Initially let the proposal be Brownian motion, then:

Q : dSt =1

πsin(St)dt+ dWt

W : dSt = dWt

Then as in Section 5.8, the first step is to find the biased Brownian motion:

h(y;x0, T ) ∝ exp

A(y)− (y − x0)2

2T

Recall from Section 5.6 that A(y) =∫ y

0 α(u)du and in this case as the SDE has unit volatility,

the drift α = 1π sin(St),

A(y) =

∫ y

0

1

πsin(u)du = − 1

πcos(u)

]y0

=1

π(1− cos(y))

Substituting this into the expression of h(y;x0, T ) as previously,

h(y;x0, T ) ∝ exp

A(y)− (y − x0)2

2T

∝ exp

1

π(1− cos(y))− (y − x0)2

2T

∝ exp

− 1

πcos(y)− (y − x0)2

2T

Then to construct the rejection sampler for the endpoint, the bounding constant M is

required:


f(y) = h(y;x0, T ) ≤M.g(y)

exp

− 1

πcos(y)− (y − x0)2

2T

≤Mexp

−(y − x0)2

2T

exp

− 1

πcos(y)

≤ exp −cos(y) ≤M

thus it can be seen that the same bounding constant M = e1 as that of Section 5.8 can be

used. It is worth noting however, that as mentioned in Section 5.1, smaller values of M will

provide a more efficient algorithm.

The only other variables that need to be identified for the implementation of the path space

rejection sampler are now the bounds of φ:

φ(x) =1

2(α2 + α′)

=1

2

((1

πsin(x))2 +

1

πcos(x)

)

Noting cos(x) ∈ [−1, 1] and sin2(x) ∈ [0, 1], it can be concluded that choosing the bounds as

Lx = −0.5 and Ux = 1 suffices. This results in a nearly identical algorithm to that used in

Section 5.8, which is expected as the targets are nearly identical but for the addition of a

dividing scalar. The only difference is the modification in equation for φ as derived above.

Figure 12 shows some samples obtained by allowing for the suggested modification. The

graph is again symmetrical due to the sinosoidal drift. However, note that now the endpoints

and even the paths themselves tend to gravitate towards integer values, representing the

predilection effect.


0 1 2 3 4 5

−6

−4

−2

02

46

Time

X

Fig 12: PSRS samples with target dSt = 1π sin(St)dt + dWt and BM proposal, starting at 0

and a time interval of [0, 5]

These samples, though from the true distribution and hence representing predilection, are

fundementally unsuitable for a financial application as they violate Assumption 1 of

non-negativity of stock prices. This is because the measure W used as a proposal is a

Brownian motion, i.e. it is symmetric around zero. As explained in Section 5.3, Geometric

Brownian motion (GBM) which is the basis for the Black-Scholes model resolves this problem

and therefore it is a natural proposal candidate.

6.1.2. Path space rejection sampling with Geometric Brownian motion proposal

As mentioned in Section 5.3, Geometric Brownian motion has a unique property in the sense

that it guarantees Assumption 1, the non-negativity of stock prices, which is the reason why

such a proposal is a natural candidate for financial applications. Geometric Brownian motion

has other useful properties as well: expected returns of a Geometric Brownian motion are

independent of stock prices and scale with investments which is realistic [37] and allows for

relatively simple calculations.

A Geometric Brownian motion proposal is an SDE with non-unit volatility and recall that by

using a Lamperti Transformation from Section 5.5 any such SDE can be transformed into one

with unit volatility. It is intuitive that in the same fashion a Brownian motion with unit

volatility can be transformed into a Geometric Brownian motion with non-unit volatility


using the Inverse Lamperti Transformation. From Section 5.5 and letting σ(St) = σSt, then:

Lamperti Transformation: Zt = ψ(St) =

∫1

σsds|s=St =

1

σlog(St)

Inverse Lamperti Transformation: St = ψ−1(Zt) = eσZt

One approach at this stage is to return to the path space rejection sampler and recalculate

the acceptance probability based on this new proposal and repeat the simulations. However,

an easier method exists: instead of transforming the acceptance probability and the proposals,

one can transform the obtained samples from the path space rejection sampler with Brownian

motion. By conducting the inverse Lamperti Transformation above on the stock paths obtained

in Section 6.1.1, the following paths are acquired.

0.0 0.2 0.4 0.6 0.8 1.0

−1

01

23

45

Time

S

Fig 13: Samples from path space rejection sampling with target dSt = 1π sin(St)dt+ dWt and

GBM proposal, starting at 0 and a time interval of [0, 5]

Note how all paths are now positive and therefore suitable for a financial application. Figure

14 provides the density of the endpoints of the paths obtained from a Geometric Brownian

motion proposal for comparison with the endpoints of the paths obtained from a Brownian

motion proposal. From this graph, it is evident that the stock prices exhibit log-normality as

expected. This is consistent with the Black-Scholes assumption, further validating the results

for a financial application.


0 5000 10000 15000

0e+

002e

−04

4e−

046e

−04

Den

sity

Fig 14: Densities of the samples obtained using path space rejection sampling with GBMproposals

As illustrated in this section, by using path space rejection sampling, stock prices can be

perfectly simulated under the chosen financial model in a manner that is consistent with

real-life expectations. Since the samples are perfectly IID, any result driven from the path

space rejection sampling supersedes those derived from discretisation schemes or other

methods that allow for correlation in the samples. Given these simulated prices, it is possible

to evaluate path dependant exotic options that lack analytical solutions and this is the goal

of the next section.

Remark. Obtaining samples immediately from a target such as dSt = 1π sin(St)Stdt+ σStdWt

would remove the issue of negative stock prices from a Brownian motion proposal that was

resolved here by using Geometric Brownian motion instead. However, bounding such a target

is difficult. Note using drift α = 1π sin(St)St,

A(y) =

∫ y

0

u

πsin(u)du = − 1

π(sin(u)− ucos(u))

]y0

=1

π(sin(y)− ycos(y))


h(y;x0, T ) ∝ exp

A(y)− (y − x0)2

2T

∝ exp

1

π(sin(y)− ycos(y))− (y − x0)2

2T

Then to bound the target:

f(y) = h(y;x0, T ) ≤M.g(y)

exp

− 1

π(sin(y)− ycos(y))− (y − x0)2

2T

≤Mexp

−(y − x0)2

2T

exp

− 1

π(sin(y)− ycos(y))

≤M

exp

−sin(y)

π+ycos(y)

π

≤M

Then it is clear that whilst the negative exponential is easily bounded as previously illustrated,

the positive exponential, exp( yπ cos(y)), can not be bounded. To use such a target directly, one

can set an arbitrary upper bound for y, for example 100000 × S0 and bound the expression

above. Another solution is to truncate the target such that it can be bounded. However,

modifying the target as above and using a Geometric Brownian motion proposal results in

samples from virtually the same density and that is why such a decision was taken in this

dissertation.

6.2. Exotic option pricing using path space rejection sampling

The increase in computing power and automation have enhanced the development of

simulation based financial solutions. Perfect simulation algorithms such as the path space

rejection sampler were not possible in the past, however, with the current resources stock

simulation as described in Section 6.1 is readily available now using an average computer.

Therefore, taking these simulated paths as given, one can proceed to evaluate exotic option

prices under the chosen financial model dSt = 1π sin(St)dt + σdWt, reflecting predilection of

stocks. This section focuses on two path-dependant options: Asian options and Look-back

options, explaining their dynamics and use followed by their evaluation.


6.2.1. Asian options

Asian options are derivatives that use the average price of the underlying rather than the

spot price at maturity [63]. Asian options were introduced in 1987 by two Bankers Trust

fixed-income arbitrage proprietary traders as an option linked to the average price of crude

oil [25]. They named the option, the Asian Option as they were on a business trip in Asia

when they came up with the idea and pricing formula [55]. Since then, Asian options have

come to exist in most markets, including equities, as they are advantageous to European and

American options in a number of ways.

One advantage of Asian options is that they are much less prone to manipulation of the

underlying moments before maturity because they take into account the entire path rather

than just ST , stock price at maturity [43]. Another advantage of such options includes the

reduced volatility due to the averaging feature, which then results in reduced risk and hence

a lower price. This is particularly useful for firms that are interested in rewarding their

employees using stock options as it allows them to reward their employees more at a lower

cost [26]. Lastly and most obviously, these options provide better risk management than

other types for firms that have cashflows throughout a certain period, as they can be seen to

have a similar structure to a swap. In the same manner that a swap’s daily value changes

based on the relative cashflow of the floating leg compared to the fixed leg, the daily value of

the Asian option is also affected based on the stock price on that day compared to the

average up to that point in time.

There are different types of Asian options based on the averaging time period and the

method of averaging used [63]. If the chosen averaging time period is daily, for example, then

the stock price at a pre-determined time is taken every day. Similarly, if a monthly averaging

time period is chosen, the stock price is taken at a pre-determined time and day each month.

These are discrete examples, however, there is a case for using continous averaging time

periods, continuously obtaining averages over the lifetime of the contract. The averaging

methods can be summarized:


Averaging Type Formula Averaing Time Period

Arithmetic Average 1T

∫ T0Stdt Continuous

Geometric Average exp( 1T

∫ T0

ln(St)dt) Continuous

Discrete Monitoring Average 1T

∑T−1t=0 St Discrete

For simplicity, this paper will focus on the Discrete Monitoring case, however, the others can

be done in a similar manner using the paths obtained via some modifications and

adjustments to the pricer.

Consider the Daily Discrete Monitored Asian call option with a one year maturity with strike

k = 90. Let today’s stock price S0 = 100, and assume that the market behaves according to

Q : dSt = 1π sin(St)dt + dWt. It can be seen that the value of such an option denoted by

VAsian(n, T, S0, k) is

VAsian(1, 365, 100, 110) = EQ[(St − k)+

]= EQ

(( 1

365

365−1∑t=0

St

)− 90

)+

= 11.833651

where the last step is done by using the samples obtained in Section 6.1.2 in the formula

above. Note that this is the price of an Asian call on one share of this stock and options are

normally written on 100 shares [37]. Therefore, the value of a standard Asian contract as

described above is 11.833651× 100 = 1183.37.

Comparing the results obtained from this pricer with those from others, for example, the

Asian Pricer from the R fOptions package [66] or the MATLAB Asian Option Pricer [51], it

can be seen that there are some discrepencies. This is, however, to be expected for two

reasons: first, the other pricers mentioned include some error as they do not use perfect IID

samples; and secondly and more importantly, these pricers assume that stocks behave

according to the Black-Scholes model with fixed drift and volatility, whilst the path space


rejection sampler used in this report assumes non-constant drift and unit volatility. Thus,

due to the inherent difference between the two underlying models and the error included in

methods of imperfect simulation, it is reasonable that the two pricer do not yield the same

exact value.

Finally, it is worth mentioning that this report has used a relatively simple Asian option to

demonstrate the value of path space rejection sampling in finance but there are much more

complex versions available. For example, consider an average strike variation whereby the

strike price is also averaged daily based on some pre-determined set of rules. These variations

display the diversity readily available in exotic contracts: they can be modified and made

more complex based on client needs due to their OTC nature. This is another reason why

simulation based approaches are essential: developing analytical solutions for each small

variation is time consuming and includes many mathematical intricacies; whereas using

numerical solutions is becoming faster and cheaper due to the increase in automation and

computing power.

Remark. The value of an option and its price are not usually the same. To obtain the price

of an option, its value as calculated above needs to be discounted by a suitable risk-free rate.

This is due to the idea of replicating strategies [37]; this essentially states that to satisfy no

arbitrage, two portfolios with the same payoff must have the same present value. Thus, if a

replicating portfolio is obtained in such a way to have the same payoff as the option, it must

have the same present value as the option which is obtained by discounting the future value of

the option obtained from the pricer. Note, in addition to this, as market-makers, banks often

also include a bid-ask spread to ensure profits and manage the competitivity of their offer in

order to manage the corresponding risk in their book.

6.2.2. Look-back options

Look-back options are derivatives that use the most favourable price during the life time of

the contract for the owner instead of the spot price at maturity [21]. This essentially removes

the problem of market timing as it permits investors to exercise the option at the most


optimal time which is often difficult to determine due to market uncertainties [31]. In

addition to this obvious advantage of minimizing regret, Look-back options have many other

benefits.

These options give life to the investor fantasy of buying low and selling high [30] and allow

for hedging against human behavioural flaws [12]. By definition, the option guarantees the

lowest buying price and highest selling price which is partly the reason for its high price. The

second advantage mentioned is more delicate: investors often hesitate trading in times of high

volatility, mostly in fear that the market could move against them or more in their favour.

This causes a state of paranoia and paralysis during periods of high volatility, reducing

liquidity and further increasing the volatility level. Use of such options, removes the human

flaws and grants more stability in the market [62], albeit at the cost of higher than average

premiums.

Similarly to the case of Asian options, there are different types of Look-back options as well.

Look-back options vary based on whether the strike is fixed or floating [21]. Fixed strike

Look-backs pay the difference between the chosen strike and the highest stock level in the

case of a Look-back call, and the difference between the strike and the lowest stock level in

the case of a Look-back put. Similarly, Floating strike Look-backs pay the difference between

the lowest stock level and the spot price at maturity in the case of the Look-back call, and

the difference between the highest stock level and spot at maturity in the case of a Look-back

put. Again for simplicity, this paper focuses on the Fixed strike case but the Floating strike

case is easy to replicate. Since the path space rejection sampler simulates the endpoint of

each path in its first step, the extension to the Floating case is simply done by substituting

the chosen strike with the endpoint which is always perfectly simulated by construction of

the sampler.

The payoff of a fixed strike Look-back call option is very similar to the payoff of a Vanilla call

(Equation 1) with the exception of the maximum spot price throughout the life time of the

contract instead of the spot at maturity:


VLB = EQ[(max(St)− k)+

]

Thus, to evaluate such an option, the maximum of each path needs to be found. One method

of obtaining this maximum is by computing the minimum of its reflection [57]. Then let the

required maximum be denoted by s := supSq : q ∈ [0, T ] = infSrefq : q ∈ [0, T ] and the

time at which it is attained be t := supq ∈ [0, T ] : Sq = s where S ∼ Qx0,y0,T is a path space

rejection sampled simulated stock path, then s = max(St) = min(Sreft ) = St. Then the join

distribution of s and t is given by [42]:

P(s ∈ ds, t ∈ dq|S0 = x0, ST = y)

∝ (w − x0)(w − y)√(T − q)3(q − 0)3

· exp

−(w − x0)2

2(q − 0)− (w − y)2

2(T − q)

ds · dq

∝ (w − x0)(w − y)√(T − q)3q3

· exp

−(w − x0)2

2q− (w − y)2

2(T − q)

ds · dq

Then a sample (t, s) can be drawn from this joint density as in [57] by using u1, u2 ∼ U[0, 1]

and setting,

s = x0 −1

2

[√(y − x0)2 − 2T log(u1)− (y − x0)

]

t =T

1 + V

where: V =

v1,where v1 ∼ IGau( y−sx0−s ,

(y−s)2

T ), if u2 <x0−s

x0+y−2s ,

1v2,where v2 ∼ IGau(x0−s

y−s ,(x0−s)2

T ), if u2 ≥ x0−sx0+y−2s ,

where IGau(µ, λ) denotes the Inverse Gaussian distribution with mean µ and shape

parameter λ. To computationally obtain such samples upper a2 and lower a1 bounds of the

maximum must be set then s can be obtained as in Algorithm 8 [57].


Simulation of (t, s) ∈ [a1, a2], where a1 < a2 ≤ x0 & y conditional on S0 = x0 and ST = y - Algorithm 8

1) Sim u1 ∼ U[M(a1),M(a2)] where M(a) := exp(−2(a− x0)(a− y)/T ) and U2 ∼ U[0, 1]

2) Set s = x0 − 12[√

(y − x0)2 − 2T log(u1)− (y − x0)]

3) If u2 ≤ x0−sx0+y−2s

, then V ∼ IGau( y−sx0−s

, (y−s)2T

), else 1V∼ IGau(x0−s

y−s ,(x0−s)2

T)

4) Set t = T1+V

Note that Algorithm 8 simulates s and t separately and since the Look-back option only

requires the maximum and not the time of its occurance, this algorihm can be further

simplified.

Simulation of s ∈ [a1, a2], where a1 < a2 ≤ x0 & y conditional on S0 = x0 and ST = y - Algorithm 9

1) Sim u1 ∼ U[M(a1),M(a2)] where M(a) := exp(−2(a− x0)(a− y)/T ) and U2 ∼ U[0, 1]

2) Set s = x0 − 12[√

(y − x0)2 − 2T log(u1)− (y − x0)]

Therefore, by reflecting the paths obtained in Section 6.1.2 and implementing Algorithm 9,

the maximum stock price during the life time of the option can be obtained. Evaluating the

option is now feasible.

Consider a 3 month Fixed Look-back call option with S0 = k = 100. Then paths can be

simulated using path space rejection sampling and s found by using Algorithm 19, and then

the value of the option is

VLB = EQ[(s− k)+

]= EQ

[(s− 100)+

]= 20.5122

Notice that similarly to the case of the Asian option the value obtained differs slightly from

those obtained by using other methods [21, 64]. This is again likely due to errors induced by


other methods and more importantly the different underlying models. The advantages and

risk managing properties of these options compared to the Asian ones are reflected in their

higher value as shown above. These options have recently become very popular with volatile

cryptocurrency markets and thus there has been an increase in research around cheaper

variations such as the Amnesiac Lookback Option [12]. This option reduces the premium of a

standard Look-back option by selectively choosing monitoring windows based on expected

periods of high volatility.

7. Conclusion and further research

The increase in computing power has brought about a new age of technology and automation

to finance freeing up scarce resources for more complex exotic contracts. The rise in

computational mathematics and simulation driven solutions has made evaluation of

complicated contracts, that were once seen as impossible to price, possible.

There are many different models and algorithms for simulation of stock prices each

exceptional and revolutionary at introduction but most consistently including systematic

errors. As explained, most discretisation schemes incur some error resulting in inaccuracies.

This has motivated the field of perfect simulation, obtaining perfect IID samples as those

achieved by using the path space rejection sampler. Despite its univariate restriction on

diffusions, as shown in this dissertation, the sampler can be used to simulate samples from a

whole host of diffusions by using the Lamperti transformation. The transformation can also

be used to transform Brownian motion proposed samples into Geometric Brownian motion

ones, therefore, ensuring the samples are suitable for finance and that they do not violate any

real market conditions. These perfectly simulated samples can then be used to evaluate

exotic options that lack analytical solutions under various models, as illustrated in this

dissertation with Asian and Look-back options.

There are some obvious ways by which the results here can be extended. The first mode of

modification is the use of more interesting and realistic targets, for example models

exhibiting mean reversion, or those that include parameters to account for the effect of trader


interactions. Another modification can be conducted by using different proposals thereby

restricting the samples without affecting the underlying model as was done here by using

Geometric Brownian motion instead of standard Brownian motion proposals. Also, as

mentioned, this dissertation focused on the case where the unbiased estimatior φ is bounded.

Therefore, another method of extending the results here is exploring targets for which this

unbiased estimator is unbounded and using the techniques used for Look-back options to

obtain minimums or maximums of the paths to build Bessel bridges and obtain samples.

Path space rejection sampling as explained in this dissertation, can be applied to many other

fields such as physics and engineering, where other Monte Carlo algorithms are currently

used, improving accuracy and removing the error induced by dependence of samples.

Finally, perfect simulation methods as developed here can be used to evaluate other exotics.

For example, Knock-in and Knock-out options mentioned in Section 1 are evaluated by

obtaining hitting times of a boundary at the knock-in/knock-out level and then

correspondingly cutting off or initiating the evaluation. This is very similar to barrier options

whereby the payoff is calculated at the hitting time. All these variations, however, require a

method for evaluating the hitting time of a particular barrier. Path space rejection sampling’s

samples under Brownian motion can be used to evaluate this hitting time as any other

Brownian path would be, but they are inherently unsuitable for a financial application as

explained in Section 6.1.1. Path space rejection sampling’s samples obtained via a Lamperti

transformation, on the other hand, are completely unsuitable for such manipulation despite

being apt for finance. This is because to obtain the hitting time under the Geometric

Brownian motion measure the barrier must be transformed using the Lamperti

Transformation into the Brownian motion measure first. Such a transformation results in a

non-linear barrier and hence obtaining hitting times is no longer feasible. This beckons a

different transformation or the use of other Monte Carlo methods for evaluation of such

barrier related options. Thus evaluating other exotics or modifying the transformation is

another possible extension.

Overall, path space rejection sampling allows for perfect simulation of IID samples with no

error and as such is a superiour Monte Carlo method to those currently used in many

industries, such as finance. As a numerical method it grants solutions to problems that are


unsolvable using traditional analytical methods, for example, the evaluation of exotic path

dependant options under complex models as explored here. The path space rejection sampler

is a flexible algorithm as it allows for modifications both through the target and the proposal,

and hence is open to many applications. As the technology progresses and robotics and

automation become the norm such perfect simulation methods are absolutely indispensible as

they can completely remove errors but also be easily integrated into existing systems.


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