Tian-Shyr Dai
Department of Computer Science and Information Engineering
National Taiwan University
1 Introduction 1
1.1 Setting the G round . . . 1
1.2 Options . . . 1
1.3 Asian Options . . . 2
1.4 The Lattice Approach . . . 4
1.5 Pricing Asian Options with the Lattice Approach and Its Problems . 4 1.6 The Contributions of this Dissertation . . . 4
1.6.1 The Subexponential-Time Lattice Algorithm . . . 5
1.6.2 The Range-Bound Algorithms . . . 7
1.7 Structures of this Dissertation . . . 7
2 Preliminaries 9 2.1 Basic Assumptions . . . 9
2.2 Option Basics . . . 10
2.2.1 Definitions of Options . . . 10
2.2.2 Who Needs Options . . . 12
2.2.3 Hedging and Hedgers . . . 13
2.2.4 Pricing an Option with Arbitrage-Free Base Pricing Theory . 14 2.3 Asian Options . . . 17
2.3.1 Definitions . . . 17
2.3.2 Pricing Asian Option by Applying Risk Neutral Variation . . 17
2.3.3 Advantages of Asian Options . . . 18
2.4 Review of Literature . . . 18
2.4.1 Approximation Analytical Formulae . . . 19
2.4.2 Monte Carlo Simulation . . . 20
2.5 Lattice and Related PDE Approach . . . 21
2.5.1 The Structure of CRR Binomial Lattice . . . 22
2.5.2 How to Construct a Lattice . . . 23
2.5.3 Pricing an Ordinary Option with the Lattice Approach . . . . 23
2.5.4 The PDE Approach . . . 25
2.6 Pricing the Asian option with Lattice Approach and Its Problems . . 25
2.6.1 Approximation Algorithms and Its Problems . . . 27
3 The Integral Trinomial Lattice 30
3.1 A Simple Intuition . . . 30
3.2 The Multiresolution Lattice Model . . . 31
3.3 An Overview of the Newly Proposed Lattice . . . 34
3.3.1 The Structure of a Trinomial lattice . . . 36
3.4 Lattice Construction . . . 37
3.5 Proof . . . 39
3.5.1 Validity of the Lattice . . . 40
3.5.2 Running-Time Analysis . . . 44
4 The Range Bound Algorithms 45 4.1 The Range Bound Paradigm . . . 45
4.2 An Overview of the Proposed Range Bound Algorithms . . . 46
4.3 Preliminaries for European-Style Asian Options . . . 48
4.4 The Design and Error Analysis of the Basic Range-Bound Algorithm 49 4.4.1 Description of the Algorithms . . . 49
4.4.2 An Optimal Choice of the Number of Buckets . . . 52
4.4.3 Error Analysis . . . 53
4.5 Tighter Range Bounds . . . 54
4.5.1 A Tighter Upper-Bound Algorithm . . . 54
4.5.2 A Tighter Lower-Bound Algorithm . . . 55
4.6 Numerical Results for European-Style Asian Options . . . 57
4.7 Algorithms for American-Style Asian Options . . . 61
4.7.1 Useful Terminology . . . 62
4.7.2 Full-Range Algorithms . . . 63
4.7.3 The First Upper-Bound Algorithm . . . 64
4.7.4 The Second Upper-Bound Algorithm . . . 65
4.7.5 A Lower-Bound Algorithm . . . 68
4.8 The Range-Bound Proofs . . . 69
4.9 Numerical Results for American-Style Asian Options . . . 72
5Conclusions 77
1.1 A 3-Time-Step CRR Binomial Lattice. . . 3
1.2 The Combinatorial Explosion. . . 5
1.3 A 3-Time-Step Trinomial Lattice with Integral Asset Prices. . . 6
1.4 The Range-bound Algorithm. . . 7
1.5 Convergence of the Range-Bound Algorithm. . . 8
2.1 Profit/loss of Options. . . 12
2.2 Exchange Rate Risk. . . 13
2.3 Pricing the Currency Option by Taking Expectation. . . 16
2.4 The CRR Binomial Lattice. . . 22
2.5 Pricing an Ordinary Option on a CRR Lattice. . . 24
2.6 Prefix Sums and the Prefix-Sum Range. . . 26
2.7 Pricing an Asian Call Option on a CRR Lattice. . . 27
2.8 The Approximation in Hull-White Paradigm. . . 28
3.1 G rowth Rate Comparison. . . 31
3.2 A Sample Multiresolution Lattice. . . 32
3.3 Convergence Behavior of the Multiresolution Lattice . . . 33
3.4 Numerical Delta. . . 36
3.5 Convergence of Different Lattice Models. . . 37
3.6 A 2-Time-Step Trinomial Lattice. . . 38
3.7 A 2-Time-Step Trinomial Lattice over KS0-log-prices. . . 39
3.8 Branching Probabilities for the Node with Price Si,j. . . 42
4.1 Bucketing. . . 50
4.2 Rounding Down the Prefix Sums. . . 51
4.3 Splitting a Path. . . 55
4.4 Prefix Sum Averaging with nUnifCvg. . . 57
4.5 Comparing UnifUp, UnifAvg, and nUnifUp. . . 58
4.6 Comparing nUnifCvg and nUnifSpl against UnifUp, UnifDown, and UnifAvg. . . 62
4.7 Pricing Errors of nUnifCvg:nUnifSpl and nUnifDown:nUnifUp. . . . 63
4.8 The number of buckets and accuracy. . . 65
4.9 Interpolation of option values in nUnifSplA in backward induction. . 66 4.10 Determination of the exercise boundary. . . 68 4.11 Pricing errors of nUnifCvgA:nUnifSplA and nUnifCvgA:nUnifSplA2. 73
2.1 Stress Tests. . . 19
3.1 Reduction of the Number of Prefix Sums. . . 33
3.2 Comparing Multiresolution Approach against Other Analytical Ap-proaches. . . 34
3.3 American-Style Asian Options. . . 35
4.1 Monte Carlo Simulation, nUnifCvg, UnifAvg, and Multiresolution Lat-tice. . . 59
4.2 Comparing nUnifCvg against Various Algorithms. . . 60
4.3 Comparing nUnifCvg against the Analytical Approaches. . . 61
4.4 Convergence of nUnifCvg:nUnifSpl. . . 64
4.5 Comprehensive Tests for nUnifSplA, nUnifSplA2, and nUnifCvgA. . . 74
4.6 Convergence of nUnifCvgA:nUnifSplA2. . . 75
4.7 Convergence of the Full-Range nUnifCvg:nUnifSpl. . . 76
Path-dependent options are options whose payoff depends nontrivially on the price history of an asset. They play an important role in financial markets. Unfortunately, pricing path-dependent options could be difficult in terms of speed and/or accuracy. The Asian option is one of the most prominent examples. The Asian option is an option whose payoff depends on the arithmetic average price of the asset. How to price such a derivative efficiently and accurately has been a long-standing research and practical problem. Up to now, there is still no simple exact closed form for pricing Asian options. Numerous approximation methods are suggested in the academic literature. However, most of the existing methods are either inefficient or inaccurate or both.
Asian options can be priced on the lattice. A lattice divides the time interval between the option initial date and the maturity date into n equal time steps. The pricing results converge to the true option value as n → ∞. Unfortunately, only exponential-time algorithms are currently available if such options are to be priced on a lattice without approximations. Although efficient approximation methods are available, most of them lack convergence guarantees or error controls. A pricing algorithm is said to be exact if no approximations are used in backward induction.
This dissertation addresses the Asian option pricing problem with the lattice ap-proach. Two different methods are proposed to meet the efficiency and accuracy requirements. First, a new trinomial lattice for pricing Asian options is suggested. This lattice is designed so the computational time can be dramatically reduced. The resulting exact pricing algorithm is proven to be the first exact lattice algorithm to break the exponential-time barrier. Second, a polynomial time approximation algo-rithm is developed. This algoalgo-rithm computes the upper and the lower bounds of the option value of the exact pricing algorithm. When the number of time steps of the lattice becomes larger, this approximation algorithm is proven to converge to the true option value for pricing European-style Asian options. Extensive experiments also reveal that the algorithm works well for American-style Asian options.
Introduction
1.1
Setting the Ground
In recent decades, financial derivatives have played an increasingly important role in the world of finance. A derivative is a financial instrument whose payoff depends on the value of other more basic underlying economical variables, like stock indexes and interest rates. Basically, derivatives can be categorized into four groups: futures, forwards, options and swaps. Standardized futures and options are traded actively in the exchanges. Forwards, swaps, and many nonstandardized derivatives are traded in the rapidly growing over-the-counter market. These complex contracts are usually hard to manage, so they give rise to new problems in designing new contracts, pric-ing these contracts and hedgpric-ing them. To deal with this problem, a new principle, named financial engineering, is founded. This new principle involves the design, man-agement, and implementation of financial instruments through which we can meet the requirements of risk managements. Knowledge from the finance, mathematics, and computer science are combined to face the new challenges from financial engineer-ing. Recently, financial engineering has become the hottest topic in both finance and applied mathematics.
1.2
Options
An option is a kind of financial derivatives that gives the owner the right to buy or sell another financial asset. There are two basic types of options: the call options and and the put options. A call option allows the holder to buy the underlying asset with a predetermined price at or before a certain date. On the other hand, a put option gives the holder the right to sell the underlying asset. The predetermined price mentioned above is called the strike price. The certain date that a option owner is allowed to exercise the option at or before it is known as the maturity date. The financial asset that the option holder can buy or sell with the exercise price is called
the underlying asset. Exercising an option denotes that the holder exercises the right to buy or sell the underlying asset. An American-style option can be exercised any time before maturity while a European-style option can only be exercised at maturity. More details about options are introduced in Section 2.2.
With the rapid growth and the deregulation of financial markets, nonstandardized options are created by financial institutions to fit their clients needs. These complex options are usually traded in the rapidly growing over-the-counter markets. Most of these options’ values depend nontrivially on the price history of other financial assets. We call these options path-dependent. Path-dependent options are now playing im-portant roles in financial markets. Some path-dependent derivatives such as barrier options can be efficiently priced [34]. Others, however, are known to be difficult to price in terms of speed and/or accuracy [35]. Pricing these options accurately and efficiently is an important problem in financial field.
1.3
Asian Options
An Asian option is an option whose payoff depends on the arithmetic average price of the underlying asset. Take a European-style Asian call as an example. Assume that the average price of the underlying asset between the option initial date and the maturity date is A. Then the option holder has the right to buy (or sell) the underlying asset with price A. This contract is useful for hedging transactions whose cost is related to the average price of the underlying asset (such as crude oil). Its price is also less subject to price manipulation; hence variants of Asian options are popular especially in thinly-traded markets. How to price an Asian option accurately and efficiently is important in both financial and academic fields.
The Asian option is one of the most representative example of the options that are hard to be priced in terms of speed and/or accuracy. Up to now, there is still no simple closed form for pricing Asian options. Numerous approximation methods are suggested in academic literatures. However, most of the existing methods are either inefficient or inaccurate or both [20, 21]. Generally speaking, these approximation methods can be grouped into three different categories: approximation analytical formulae, (quasi-) Monte Carlo simulations, and the lattice (and the related PDE) approach.
The analytical formulae approach denotes that the option is priced by (semi-) closed form formulae. Usually these formulae are derived by solving (stochastic) par-tial differenpar-tial equations or by applying some probability tools. The major problem of this approach is that the partial differential equations for pricing the Asian op-tion cannot be solved. Some ad-hoc approximaop-tion methods are suggested in the literature, but most of them fail in extreme cases [21]. Besides, the American-style Asian option cannot be priced by this approach easily. Related works can be found in [1, 6, 22, 23, 32, 37, 41].
0 S u S0 d S0 u P d P 2 0u S 3 0u S 0 S 2 Sd 3 0d S 0 1 2 3 u S0 d S0
Figure 1.1: A 3-Time-Step CRR Binomial Lattice.
The initial underlying asset’s price is S0. Let u and d denote the upward and the
downward multiplication factors, respectively. Let Pu and Pd denote the upward and
the downward branching probabilities, respectively.
The Monte Carlo approach divides the time interval between the option initial date and the maturity date into several time steps. Then it simulates some (discrete-time) price paths of the underlying asset. A price path consists of a series of prices that corresponds to the underlying asset’s price at each time step. The option value for each price path can be computed separately. The pricing result is obtained by averaging the option value of the simulated price paths. The major drawback of this approach is inefficiency – huge amounts of price paths should be simulated to obtain a satisfying answer. The pricing result is only probabilistic. In addition, the American-style Asian option cannot be priced by this approach easily. Related work can be found in [9, 10, 11, 28, 31, 33].
The lattice (and the related PDE) approach is more general than the first two since most methods from the first two categories suffer from the inability to price American-style Asian options without bias. Under this consideration, two proposed pricing methods in this dissertation follow the lattice approach. In the following sections, the lattice model will be introduced first. After exploring the problems of the existing lattice pricing methods, a simple intuition is given to show how the two proposed methods alleviate these problems.
1.4
The Lattice Approach
A lattice consists of nodes and edges connecting them. It simulates the (discrete-time) price process of the underlying asset from the option’s initial date to the maturity date. Assume that an option initiates at year 0 and matures at year T . A lattice divides the time interval [0, T ] into n equal time steps. Then the length of each time step is T /n. The price of the underlying asset at discrete time step l (corresponding to year lT /n) can be observed on the lattice. Take the well-known Cox-Ross-Rubinstein (CRR) binomial lattice [15] as an example. A 3-time-step CRR binomial lattice is illustrated in Fig. 1.1. At each time step, the underlying asset’s price S can ei-ther become Su—the up move—with probability Pu or Sd—the down move—with
probability Pd ≡ 1 − Pu. Note that the price process of an asset simulated by the
lattice should be guaranteed to converge to the continuous-time underlying asset’s price process as n → ∞. Thus the option value priced on the lattice also converges to continuous-time option value (call it the true option value) [19]. More detailed knowledge about the lattice construction is surveyed in Section 2.5.2.
1.5Pricing Asian Options with the Lattice
Ap-proach and Its Problems
The difficulty with the lattice method in the case of Asian options lies in its expo-nential nature: Since the price of the underlying asset at each time step influences the option’s payoff, it seems that 2npaths have to be individually evaluated for an n-time-step binomial lattice (see Fig. 1.2). This makes the pricing problem intractable even with a small n. Many proposed approaches solve this combinatorial explo-sion by employing approximation [4, 13, 20, 26, 29, 43]. The resulting algorithms become more efficient, but most of them lack error controls [20]. Thus an efficient and accurate pricing algorithm is needed.
Some shorthand that will be used frequently later are introduced below. A pricing algorithm is said to be exact if no approximations are used in backward induction. Besides, a polynomial-time algorithm means that the running time is polynomial in n. Similar convention is adopted for exponential-time and subexponential-time algorithms.
1.6
The Contributions of this Dissertation
This dissertation provides two different lattice methods to address the pricing prob-lem. First, a new trinomial lattice for pricing Asian options is proposed. The resulting exact pricing algorithm is proved to break the exponential time barrier. This algo-rithm is hence more efficient than any existing exact pricing algoalgo-rithm. Second, an
S0
n
2n
Figure 1.2: The Combinatorial Explosion.
There are 2n paths (one of which follows the darkened edges in the plot) in this binomial lattice.
approximation algorithm based on the CRR lattice model is derived. This approxi-mation algorithm computes the upper and the lower bounds (call it range bound) that bracket the option value of the exact pricing algorithm. When pricing European-style Asian options, this range-bound algorithm is guaranteed to converge to the true option value. Extensive experiments also reveal that the algorithm works well for American-style Asian options.
1.6.1
The Subexponential-Time Lattice Algorithm
We next give the intuition about how the lattice looks like and why the computational time of the resulting exact algorithm is dramatically reduced. Imagine a new lattice composed of integral asset prices. The asset price sum of any arbitrary price path on this lattice must be an integer. Thus all possible asset price sums must be integers between the maximum and the minimum asset price sums, which can be easily calcu-lated. Take a hypothetical 3-time-step lattice in Fig. 1.3 as an example. The asset’s prices are printed on the nodes. Consider the price paths that reach the shaded node with an asset price of 4. The paths with the maximum price sum and the minimum
price sum are (8, 12, 8, 4) and (8, 4, 2, 4), respectively. The maximum and the mini-mum price sums are thus 8 + 12 + 8 + 4 = 32 and 8 + 4 + 2 + 4 = 18, respectively. The possible asset price sums at that node must be some of the 15 integers between 18 and 32, inclusively. Recall that the value of an Asian option depends on the average price of the asset. The number of possible asset’s price sums at an arbitrary node N equals the number of possible option values there. An exact pricing algorithm suffers from computational intractability since the number of possible asset’s price sums is exponential in n. If it can be shown that the total number of possible asset’s price sums on our new lattice is dramatically reduced, then an efficient exact algorithm can be successfully constructed. Indeed, the proposed new exact algorithm is much more efficient as it is subexponential in n. A simple numerical example on a 160-time step lattice helps us to realize the drastic difference: While the total number of pos-sible asset’s price sums to the middle node at the maturity date is the astronomical 8.429 × 1074, the total number of price sums at the same node in the proposed new lattice is only 57887. 8 12 4 8 18 12 8 4 2 24 18 12 8 4 2 1
Figure 1.3: A 3-Time-Step Trinomial Lattice with Integral Asset Prices. All paths reaching the shaded node have integral price sums. The maximum price sum at the shaded node is achieved by the upper path in thickened lines, whereas the minimum price sum at the shaded node is achieved by the lower path in thickened lines.
Although the performance of the proposed exact pricing algorithm is improved significantly, it is still not a polynomial-time algorithm. The efficiency problem re-mains open. The computational performance can be further improved by employing some approximations, but the accuracy problem should be considered simultaneously. The next approach is proposed to meet these requirements.
1.6.2
The Range-Bound Algorithms
Efficient approximation pricing algorithms that provide pricing error control by pro-ducing provable range bounds are introduced. On an n-time-step lattice model, a range-bound approximation algorithm can produce upper and lower bounds that bracket the option value of the exact pricing algorithm (call it the desired option value) as shown in Fig. 1.4. The desired option value becomes practically available if the upper bound and the lower bound are essentially identical. Note that the difference between the upper bound and the lower bound, call it e, gives an upper limit of the pricing error between exact and approximation pricing algorithms. The desired option value is known to converge to the true option value as n → ∞. Thus the approximation algorithm is guaranteed to converge to the true option value if
e → 0 as n → ∞. This relationship is illustrated in Fig. 1.5. In this dissertation,
a range-bound algorithm for pricing European-style Asian options is developed, and this algorithm is proved to converge to the true option value. Extensive experiments also reveal that the algorithm works well for American-style Asian options.
Upper bound
e Lower bound
Desired option value
Figure 1.4: The Range-bound Algorithm.
The difference between the upper bound and the lower bound pricing results e de-notes the upper limit of the pricing error between exact and approximation pricing algorithms.
1.7
Structures of this Dissertation
This dissertation is organized as follows. Some background knowledge, including re-quired financial knowledge, mathematical tools, and the survey on related literatures, is reviewed in Chapter 2. The first subexponential exact pricing algorithm and the lattice it based on is introduced in Chapter 3. Chapter 4 introduces the convergence approximation algorithm for pricing European-style Asian options. The improvement of the accuracy for the approximation algorithm for pricing American-style Asian op-tions is also introduced in this chapter. Finally, Chapter 5 concludes the paper.
Approximation Algorithm Exact Algorithm (Desired Option Value) True Option Value e Computationally Intractable Efficient but Error
Control Required
Unsolved
∞
→
n
Figure 1.5: Convergence of the Range-Bound Algorithm.
The true option value is unsolved. An exact lattice pricing algorithm is computation-ally intractable. Thus an approximation pricing algorithm with good error control is attractive. The pricing error between the approximation pricing algorithm and the exact pricing algorithm is bounded by e. Thus the approximation algorithm is guaranteed to converge to the true option value if e → 0 as n → ∞.
Preliminaries
Some required background knowledge is introduced in this chapter. Basic assumptions on financial markets and the related mathematical concepts are introduced first. A simple survey on options and their advantages is given next. The modern arbitrage-based pricing theory is then introduced to describe how the Asian option can be priced. Academic literature related to the pricing of Asian options is then introduced. These academic works are grouped into three categories: approximation analytical formulae, (quasi-) Monte Carlo simulations, and the lattice (and the related PDE) approach. The drawbacks of these academic works and the improvements on them are also discussed.
2.1
Basic Assumptions
Some economic assumptions that are adopted in this dissertation are illustrated as follows.
1. The mean and the volatility of the underlying asset’s price, and the risk-free interest rate are assumed to be fixed constants.
2. The short selling of financial assets with full use of proceeds is permitted. 3. All assets are perfectly divisible.
4. There are no transactions costs or taxes.
5. No stocks pay dividends during the life of the option. 6. There are no risk-less arbitrage opportunities.
7. Asset trading is continuous.
8. These is no liquidity problem. That is, you can always trade at the market price.
A stochastic process is a variable that changes over time in an uncertain way. Eco-nomical variables, like the asset’s price and the exchange rates, are usually modelled as stochastic processes in academic models. The randomness of these processes are usually governed by some fundamental stochastic processes like the Brownian mo-tion. Define {Bt} as a Brownian motion where Bs denotes the process value at time
s. Then we have the following properties [30]:
1. Normal increments: Bt− Bs has normal distribution with mean 0 and variance
t − s.
2. Independent of increments: Bt− Bs is independent of the past, that is, of Bu,
where 0 ≤ u ≤ s.
3. Continuity of paths: Bt, t ≥ 0 are continuous functions of t.
In this dissertation, a financial asset’s price is assumed to follow a log-normal stochastic process. To be more specific, consider a time interval starting at year 0 and ending at year T . Define S(t) as the price of a financial asset at year t. The price process follows the continuous-time diffusion process as follows:
S(t + dt) = S(t)exp[(r − 0.5σ2)dt + σdBt], (2.1)
where r is the risk-free interest rate per annum, and σ is the annual volatility. The continuous log-normal stochastic price process can be approximated by a discrete-time model like the lattice model. A lattice model partitions the time between year 0 and year T into n equal time steps. The length of each time step ∆t is equal to T /n. For convenience, all the time notations are expressed in terms of the number of time steps unless stated otherwise. Let Si denote the asset’s price at (discrete)
time i. Then Si corresponds to S(i∆t) in the continuous-time model. For the pricing
purpose, the price process simulated by a lattice model is required to converge to continuous log-normal stochastic process as n → ∞. The sufficient conditions to achieve this is done by calibrating the drift term (r − 0.5σ2) and the volatility term (σ) of the underlying asset’s price process (in Eq. 2.1) [19]. More details about the lattice constructions are introduced in Subsection 2.5.2.
2.2
Option Basics
Fundamental knowledge about options is introduced here. This includes the types of options, the payoff of each type of option, and how to price them.
2.2.1
Definitions of Options
An option is a right to buy or sell the a specific asset at (or within) a certain date with a predetermined price. This specific asset is called the underlying asset. Generally
speaking, options can be classified into two groups: call options, and put options. A call option gives the holder the right to buy a financial asset with a specific price at (or within) some certain time, while a put option gives the holder the right to sell it. The price for the holder to buy or sell the asset is called the strike price (denoted as
X). The date the option is expired is called the maturity date (denoted as T ).
The options can also be classified based on the time in which they can be exercised. An American-style option can be exercised at any time up to the maturity date; while a European-style option can only be exercised at the maturity date. Since an American option gives all the advantages that a European option possesses plus the extra advantage of early exercise, the value of an American option is at least as great as that of a European one, other conditions being equal.
There are two sides to every option contract. On the one side is the investor who take the long position (i.e., he buys the option), while on the other side is the investor who takes the short position (i.e., he sells the option). An option holder is given the right of gaining benefit without any obligation. An option will be exercised only when it is the best choice for the holder to gain maximum benefit. Take a European-style option as an example. Recall that a European option can only be exercised at the maturity date. The payoff for the long position at the maturity date is max(0, S(T ) − X) for call options; max(0, X − S(T )) for put options. On the other hand, the loss for a short position in call options can be expressed as
− max(0, S(T ) − X) = min(0, X − S(T )),
while the profit for a short position in put options is
− max(0, X − S(T )) = min(0, S(T ) − X).
Fig. 2.1 illustrates profit/loss of a European option graphically.
The payoff for an American-style option is more complex since the option holder can exercise the option early before the maturity date. Define τ as the time the option is exercised, then the payoff to exercise an option at year τ is max(S(τ ) − X, 0) and max(X − S(τ ), 0) for call options and put options, respectively. Similarly, the loss for a short position in call options is expressed as
− max(S(τ) − X, 0) = min(X − S(τ), 0),
while the loss for a short position in put options is
− max(X − S(τ), 0) = min(S(τ) − X, 0).
An option can be priced in a discrete time model. The payoff for a European-style option at the maturity (time n) is
Payoff =
max(Sn− X, 0), for a call option
max(X − Sn, 0), for a put option .
Payoff Payoff Payoff X X X X (a) (b) (d) (c) Put Call Payoff ) (T S ) (T S ) (T S ) (T S
Figure 2.1: Profit/loss of Options. (a) Long a call. (b) Short a call. (c) Long a put. (d) Short a put.
The payoff to exercise an American-style option at time i (i ≤ n)is Payoff =
max(Si− X, 0), for a call option
max(X − Si, 0), for a put option ,
(2.3)
where i denotes the time step when the option is exercised.
2.2.2
Who Needs Options
Basically speaking, options attract three different types of traders: speculators, hedgers, and arbitragers. A speculator tries to take a position to gain more benefits in the market by forecasting the future. He longs (or shorts) an option if he believes it is beneficial. A hedger is the one who tries to avoid risk by buying or selling the options. A simple example is given in the next section to show how a hedger hedges the risk. An arbitrager is the one who can gain risk-less profit if the option value is not “fair.” A trading strategy used to gain risk-less profits by taking advantages of mispriced options is called arbitrage. Theoretically, there is a fair price for each option in the market. The market price of an option should be equal to its fair price; otherwise the
arbitragers can take advantage of it. How to price the option by arbitrage-free based pricing theory is described in Subsection 2.2.4.
2.2.3
Hedging and Hedgers
ˆˈ ˆˉ ˆˇ 7
10
5
.
3
×
710
6
.
3
×
710
4
.
3
×
˧̂˷˴̌ ˢ́˸ʳˠ̂́̇˻˟˴̇˸̅ ˖˴̆˸ʳ˄ ˖˴̆˸ʳ˅Figure 2.2: Exchange Rate Risk.
A simple one-step model illustrates the evolution of the exchange rate. The exchange rate (TWD/USD) is 35 today. The exchange rate may go up to 36 (in case 1) or go down to 34 (in case 2) one month later. The first column for each node denotes the spot exchange rate, and the second column denotes the cost to buy 1 million USDs.
A simple example on hedging the foreign exchange rate risk is given as follows. The risk-free rate is set to 0 for simplicity. Assume that there is a foreign trading company XY Z in Taiwan. It is required to pay 1 million USDs next month.1 Assume that the exchange rate (TWD/USD) today is 35. Obviously, XYZ can buy 1 million USDs with 35 million TWDs today, keep these USDs for a month, and then pay back the debt next month.
Unfortunately, XYZ may not have 35 million TWDs today. It may decide to buy 1 million USDs next month. But this will introduce foreign exchange rate risk. That is, the cost to buy 1 million USDs next month is not determined today as the exchange rate is floating. A picture to describe this risk is illustrated in Fig. 2.2. Assume that the USD may appreciate to 36 TWDs (case 1) or devalue to 34 TWDs (case 2) next month. Note that XYZ has to pay one more million TWDs (= 3.6 × 107− 3.5 × 107) if the USD appreciates to 36 TWDs. Company XYZ might not put up with such a huge risk.
A currency call option can help XYZ to avoid such uncertainty. A currency option is an option whose underlying asset is a foreign currency. XYZ can buy a currency call on 1 million USDs. To hedge the exchange rate risk, the contract for this call is designed as follows: the maturity date for this option is one month from now, and
the strike price is 35 TWDs for each USD. Now the exchange rate risk is completely hedged by this call option. If the exchange rate moves up to 36 a month later, XYZ can simply exercise the option to buy a million USDs with 35 million TWDs. On the other hand, if the exchange rate moves down to 34, XYZ can simply junk the option and buy the spot from the foreign exchange market.
Obviously, this option gives XYZ the right to buy the USDs but no obligation. XYZ should be charged by the option seller for this right. It is important to find out the fair option’s price that XYZ needs to pay.
2.2.4
Pricing an Option with Arbitrage-Free Base Pricing
Theory
In financial markets, an option buyer pays so called option premium to a seller at the option initial date. This premium can be viewed as the fair price of an option. The fair option’s price can be priced by the arbitrage-free base pricing theory. In this subsection, I will show how the fair option price is obtained by replication first. Next, I will show that why the market price must be equal to the fair price by considering the behavior of arbitragers. Finally, a simple sketch is given to show that option can be priced by taking the expectation of future discounted payoff under the so-called risk-neutral probability.
Replicate an Option
An option is said to be replicated by a portfolio A if A can be constructed so that the future payoff of A is always equal to the payoff of the option. It is intuitive that the fair price of the option should be equal to the cost of constructing portfolio A since the future payoffs of A and the option are the same. Take the currency call option mentioned above as an example. The replication portfolio A is assumed to be consisted of x units of USD and y units of TWD. Assume that the USD appreciates to 36 TWDs (case 1). Since the value of the option is worth 1 million TWDs (= max(36− 35, 0) × 106) in this case, the value of A (= 36x + y) should be also equal to 1 million TWDs. On the other hand, assume that the USD devalues to 34 TWDs (case 2). The value of the option is 0 (= max(34− 35, 0) × 106). Therefore, the value of A (= 34x + y) is also equal to 0. By solving the two equations,
36x + y = 106,
34x + y = 0,
we obtain x = 5 × 105 and y = −1.7 × 107. Thus the initial cost to construct the portfolio A is equal to 5 × 105 × 35 − 1.7 × 107 = 5× 105. The fair price of the call option is 5× 105.
Arbitrage and Arbitragers
We have argued that the fair price of the option is equal to the cost of the replication portfolio. Moreover, we can also claim that the market price of the option should be equal to the fair price. Otherwise, the arbitragers can gain riskless profit by taking advantage of the mispriced options. Take the currency call option mentioned above as an example. Assume that the currency call’s market price V is larger than 5 × 105. An arbitrager can short a call and buy the portfolio A. He can earn V − 5 × 105 > 0
today. At maturity (one month later), he will neither win nor lose anything since the final payoffs of the call and A are equal. In other words, he can gain V − 5 × 105 > 0
without paying anything or suffering any risk! Economists argue that such arbitrage opportunities should not exist for long since every market participant will try to gain from this “free lunch.” Thus the market price of this currency call will quickly go back to the fair price 5× 105. On the other hand, if the option value V is lower than 5× 105, an arbitrager can construct an arbitrage strategy by longing a currency call and shorting the portfolio A. By the same argument, the market price of the option will finally go back to the fair price 5× 105. So we conclude that the market price of the option is equal to the cost of the replication portfolio under arbitrage-free considerations.
Risk Neutral Valuation
Option pricing problem can be reduced to the expectation evaluation problem under the so-called risk-neutral probability [24]. The expected return of any security is risk-free rate under risk-neutral probability. Take the currency option mentioned in Fig. 2.2 as an example. Assume that the USD may appreciate to 36 TWDs with probability P (in case 1) and devalue to 34 TWDs with probability 1 − P (see Fig. 2.3). Since the risk-free rate (r) is set to 0, the expected return of the USD should also be 0. Thus we have
36× P + 34 × (1 − P ) = 35 × er×1/12,
where 1/12 denotes the time span of one month (in years). We have P = 0.5 by solving the above equation. The payoff of the option in case 1 (marked by *) is
max(36− 35, 0) × 106 = 106, and the payoff of the option in case 2 (marked by **) is
max(34− 35, 0) × 106 = 0.
The option value is evaluated by taking expectation of the discounted future payoff under risk neutral probability as follows:
106× P + 0 × (1 − P )
er×1/12 = 10
Note that the option value evaluated by using replication is the same as the value evaluated by taking expectation, but the latter method is simpler.
˃ˁˈ
˃ˁˈ
510
5
×
610
0
35
36
34
ʽ
ʽʽ
Figure 2.3: Pricing the Currency Option by Taking Expectation.
The (risk-neutral) probability for each case is marked directly on the branch. The first column of each node denotes the spot exchange rate and the second column denotes the option value at each node.
The above result can be formalized as follows: In a continuous time model, the value of a European-style option can be expressed as
Option Value =
e−rTE[max(S(T ) − X, 0)], for a call option
e−rTE[max(X − S(T ), 0)], for a put option . (2.4) The value of an American-style option can be expressed as
Option Value =
E[e−rτmax(S(τ ) − X, 0)], for a call option
E[e−rτmax(X − S(τ ), 0)], for a put option ,
where τ denotes the time when the option is optimally exercised. In a discrete time model, the European-style option’s value is obtained by changing S(T ) in Eq. (2.4) into Sn. The American-style option’s value is
Option Value =
E[e−ri∆tmax(Si− X, 0)], for a call option
E[e−ri∆tmax(X − Si, 0)], for a put option ,
2.3
Asian Options
2.3.1
Definitions
An Asian option is an option whose payoff depends on the average price of the under-lying asset during a specific period. Define the average price of the underunder-lying asset from year 0 to year t as
A(t) ≡
t
0 S(u) du
t . (2.5)
Then the payoff for a European-style Asian option at the maturity date is
Payoff =
max(A(T ) − X, 0), for a call option max(X − A(T ), 0), for a put option . The payoff for exercising an American-style Asian option at year τ is
Payoff =
max(A(τ ) − X, 0), for a call option max(X − A(τ ), 0), for a put option .
In a discrete time model, the average price of the underlying asset is redefined as
Aj ≡
j
i=0Si
j + 1 . (2.6)
Thus the payoff for a European-style Asian option is
Payoff =
max(An− X, 0), for a call option
max(X − An, 0), for a put option .
(2.7)
The payoff for exercising an American-style Asian option at time i is Payoff =
max(Ai− X, 0), for a call option
max(X − Ai, 0), for a put option .
(2.8)
2.3.2
Pricing Asian Option by Applying Risk Neutral
Vari-ation
The value of an Asian option can be evaluated by taking expectation of the future discounted payoff as we do in Eq. (2.4). For a European-style Asian option, the option value is
Option Value =
e−rTE[max(A(T ) − X, 0)], for a call option
e−rTE[max(X − A(T ), 0)], for a put option . The option value for an American-style Asian option is
Option Value =
E[e−rτmax(A(τ ) − X, 0)], for a call option
In a discrete time model, the value of a European-style Asian option is evaluated as
Option Value =
e−rTE[max(An− X, 0)], for a call option
e−rTE[max(X − An, 0)], for a put option ,
(2.9)
while the value for an American-style Asian option is
Option Value =
E[e−ri∆tmax(Ai− X, 0)], for a call option
E[r−ri∆tmax(X − Ai, 0)], for a put option .
(2.10)
2.3.3
Advantages of Asian Options
Since the payoff of an Asian option depends on the average price of the underlying asset, it is useful for hedging transactions whose cost is related to the average price of the underlying asset. Take the foreign exchange rate risk case discussed above as an example. Assume that the company XYZ needs to pay 106 USDs per month for the next six months. XYZ may buy six currency call options maturing at the next month, two months later, . . ., and six months later, respectively. On the other hand, XYZ may hedge the exchange rate risk by buying six Asian call options that matures six month later. It can be observed in the markets that an Asian option is usually much cheaper than an otherwise identical ordinary option. Thus XYZ can reduce the hedge cost significantly by buying six Asian call options instead of six ordinary call options.
Besides, the price of an Asian option is also less subject to price manipulation due to the following reasons: The payoff of a European-style ordinary option is determined by the underlying asset’s value at the maturity date, while the payoff of a European-style Asian option is determined by the average price of the underlying asset between the option initial date and the maturity date. And it is easier to control an asset’s price at a specific time point than to manipulate the whole price path. Preventing price manipulation is an attractive property of an option especially in thinly-traded markets.
In practice, as varieties of asian option are widely traded in today’s financial markets, how to price them accurately and efficiently is important in both financial and academic fields.
2.4
Review of Literature
The major problem in pricing Asian option is that we do not know much about the distribution of the underlying asset’s average price A(T ) (see Eq. (2.1) and (2.5)).
A(T ) can be viewed as the sum of log-normal random variables; and the density
function of a sum of log normal random variables is currently unavailable. That is why there is no simple and exact closed-form solutions for pricing Asian options. Approximation methods suggested in the academic literature can be grouped into
r σ T S0 GE Shaw Euler PW TW MC10 MC100 SE 5.0% 50% 1 1.9 .195 .193 .194 .194 .195 .192 .196 .004 5.0% 50% 1 2.0 .248 .246 .247 .247 .250 .245 .249 .004 5.0% 50% 1 2.1 .308 .306 .307 .307 .311 .305 .309 .005 2.0% 10% 1 2.0 .058 .520 .056 .0624 .0568 .0559 .0565 .0008 18.0% 30% 1 2.0 .227 .217 .219 .219 .220 .219 .220 .003 12.5% 25% 2 2.0 .172 .172 .172 .172 .173 .173 .172 .003 5.0% 50% 2 2.0 .351 .350 .352 .352 .359 .351 .348 .007
Table 2.1: Stress Tests.
The exercise price X is 2.0. The approximation methods for comparison are from Geman and Eydeland [22] (GE), Shaw [39], Euler, Post-Widder method (PW) [1], and Turnbull-Wakeman [41] (TW). The benchmark values (MC10 and MC100) and the approximation values are from [21]. MC10 uses 10 time steps per day, whereas MC100 uses 100. Both are based on 100,000 trials. SE stands for standard error, also from [21].
three different categories: approximation analytical formulae, (quasi-)Monte Carlo simulations, and the lattice (and the related PDE) approach.
2.4.1
Approximation Analytical Formulae
This approach denotes that the value of the option is approximated by (semi-)closed form formulae. Some related academic works in this category try to approximate the probability density function of A. Turnbull and Wakeman [41] and Levy [32] try to approximate the density function of A by Edgeworth series expansion. Milevsky and Posner [37] approximate it by the reciprocal gamma distribution. Carverhill and Clewlow [12] and Benhamou [6] use Fourier transform to approximate the payoff func-tion at maturity. Geman and Yor [23] derive an analytical expression for the Laplace transform of the continuous Asian calls, and numerical inversion of this transform is considered by Geman and Eydeland [22] and Shaw [39]. Some inversion algorithms based on the Euler and Post-Widder methods are suggested in Abate and Whitt [1]. The forward starting Asian options are approximated with Taylor’s series expansion as in [8, 40]. Zhang [42] approximates the option value by combining an analytical closed form with a numerical adjustment (computed by finite difference method). The major problem of the approximation analytical formulae approach is that most suggested methods from this approach lack error control [21]. Table 2.1 illustrates that some well-known approximation methods fail in extreme cases. Besides, the American-style Asian option’s value cannot be approximated by this approach easily.
2.4.2
Monte Carlo Simulation
The Monte Carlo simulation approach denotes a pricing procedure that values a derivative by randomly sampling changes in economic variables. To value a European-style Asian option, a typical Monte Carlo simulation can divide the time interval between the option initial date and the maturity date into n time steps. Then it simulate the price path of the underlying asset by the following formula:
Si = Si−1exp[(r − 1/2σ2)∆t + σ
√
∆tω],
where Si denotes the price of the underlying asset at time i, ∆t is equal to T /n,
and ω denotes the a standard normal random variable. Note that the distribution of this n + 1-dimensional random price vector (S0, S1, S2, · · · , Sn) is the same as the
distribution of random vector (S(0), S(∆t), S(2∆t), · · · , S(n∆t)) which is governed by Eq. (2.1). The average price of each price path is computed by Eq. (2.6). Thus the payoff for each price path for the European-style Asian option can be computed by Eq. (2.7). The output option value is obtained by averaging the discounted payoffs.
The Monte Carlo simulation approach suffers from the following problems: 1. The pricing result is only probabilistic.
2. The number of simulated price paths should be large enough to obtain satisfac-tory pricing results. Thus the algorithm is not efficient enough.
3. The pricing results are significantly influenced by the random sources used to obtain the random variable ω. Biased results are produced if the random sources are unreliable.
4. The American-style option can not be handled by this approach easily.
Some related works that address the first three problems are listed below: Boyle, Broadie, and Glasserman [9], Broadie and Glasserman [10], Broadie, Glasserman, and Kou [11], Kemna and Vorst [28], and Lapeyre and Temam [31]. Briefly speaking, they try to reduce the variance of the pricing results and refine the quality of the random variable (ω) they use. Four simple methods used by them are
• Antithetic variates:
Assume that the normal random variables ˆω1, ˆω2, · · · , ˆωn are sampled. A simu-lated price path ( ˆS0, ˆS1, . . . , ˆSn) is constructed by defining ˆSi as
ˆ
Si = ˆS0e(r−σ
2/2)i∆t+σ√∆t(ˆω1+ˆω2+...ˆωi) .
A dual price path ( ˆS0, ˆS0, . . . , ˆSn) can be constructed by defining ˆS(i) as ˆ
S(i) = ˆS0e(r−σ2/2)i∆t+σ
√
∆t(−ˆω1−ˆω2−...−ˆωi)
• Moment matching:
This is done by tuning the sampled random variables so that the first few moments of the tuned sampled random variables match the moments of the distribution where the random variables are sampled from. For example, if we sample l random variables (ω1, ω2, · · · , ωl) from a normal distribution with
mean µ and standard deviation σ. The sample mean and standard error of these sampled random variables are µl and σl, respectively. Then the tuned variable
ωi is
ωi ≡ (ωi− µl)σ
σl
+ µ.
• Latin hypercube sampling:
This is a stratified sampling method that forces the cumulative probabilities of the empirical distribution (determined by the sampled random variables) to match the cumulative probabilities of the theoretical distribution (where the variables are sampled from). For example, assume that ω1, ω2, . . . , ω100 are
sampled from a normal distribution, then each observation ωi is forced to lie
between (i − 1)th and ith percentile.
• Control variates:
This method is widely applicable to reduce the variance of the output results. This is done by replacing the evaluation of an unknown expectation with the evaluation of the difference between the unknown quantity and another expec-tation whose value is known [9]. This is used by Kemna and Vorst [28] for pricing Asian options.
For pricing American-style Asian options, Longstaff and Schwartz (2001) develop a least-squares Monte Carlo approach to tackle the problem.
2.5Lattice and Related PDE Approach
A lattice is a discrete time representation of the evolution of the underlying asset’s price. It divides a certain time interval, like the interval between the option initial date (year 0) and the maturity date (year T ), into n equal time steps. The length of each time step ∆t is equal to T /n. It approximates the distribution of the underlying asset’s price at each time step. A lattice consists of nodes and branches connect them. Each node at time i can be viewed as a possible asset’s price at time i. Each branch that connects two nodes located at adjoint time steps denotes a possible evolution of the underlying asset’s price. The well-known CRR binomial lattice will be introduced next as an example to show what a lattice looks like, how it is constructed, and how an option is priced with the lattice model [15].
2.5.1
The Structure of CRR Binomial Lattice
A 2-time-step CRR binomial lattice is illustrated in Fig. 2.4(a). Recall that Si denotes
the value of the underlying asset at time i. Si+1 equals Siu with probability Pu and
Sid with probability Pd(≡ 1 − Pu), where d < u. u is equal to eσ √
∆t, where σ denotes
the annual volatility of the underlying asset’s price (see Eq. (2.1)). The identity
ud = 1 (2.11)
holds in this lattice model. The probability Pufor an up move is set to (er∆t−d)/(u−
d). Both d ≤ er∆t ≤ u and 0 < P
u < 1 must hold to avoid arbitrage. The asset’s
price at time i that results from j down moves and i − j up moves therefore equals
S0ui−jdj with probability jiPui−jPdj.
S✈0 1 ✟✟✟✟ ✟✟✟✟✯ ❍❍ ❍❍ ❍❍❍❍❥ S0✈u Pu ✟✟✟✟ ✟✟✟✟✯ ❍❍ ❍❍ ❍❍❍❍❥ S0✈d Pd ✟✟✟✟ ✟✟✟✟✯ ❍❍ ❍❍ ❍❍❍❍❥ S0✈u2 Pu2 S✈0 2PuPd S0✈d2 Pd2 (a) N (0, 0) ✈ ✟✟✟✟ ✟✟✟✟✯ ❍❍ ❍❍ ❍❍❍❍❥ N (1, 0) ✈ ✟✟✟✟ ✟✟✟✟✯ ❍❍ ❍❍ ❍❍❍❍❥ N (1, 1) ✈ ✟✟✟✟ ✟✟✟✟✯ ❍❍ ❍❍ ❍❍❍❍❥ N (2, 0) ✈ N (2, 1) ✈ N (2, 2) ✈ 1 P Pd p2 2PuPd Pd2 (b) Figure 2.4: The CRR Binomial Lattice.
A 2-time-step CRR binomial lattice model are illustrated in (a) and (b). The price of the underlying asset for each node is illustrated in (a) and the alias (N (∗, ∗)) for each node is illustrated in (b). N (i, j) stands for the node at time i with j cumulative down moves. The probability of reaching each node is listed under the node.
We now map the asset’s prices to nodes on the CRR binomial lattice used for pricing. Node N(i, j) stands for the node at time i with j cumulative down moves. Its associated asset’s price is hence S0ui−jdj. The asset’s price can move from N(i, j)
to N(i + 1, j) with probability Pu and to N(i + 1, j + 1) with probability Pd. As a
consequence, node N(i, j) can be reached from the root with probability jiPui−jPdj.
2.5.2
How to Construct a Lattice
We use the well-known Cox-Ross-Rubinstein (CRR) binomial lattice to illustrate how a lattice is constructed in principle. The logarithmic asset’s price mean (µ) and variance (Var) one time step from now are derived from Eq. (2.1) as
µ ≡ (r − 0.5σ2)∆t, (2.12)
Var ≡ σ2∆t. (2.13)
To make sure that the lattice converges to the continuous-time asset’s price process, the mean and the variance of the logarithmic price process should be calibrated by matching those of the lattice and those of the continuous-time model:
Puln u + Pdln d = µ, (2.14)
Pu(ln u − µ)2+ Pd(ln d − µ)2 = Var. (2.15)
Note that
Pu+ Pd= 1. (2.16)
The 4 parameters (Pu, Pd, u, and d) are uniquely obtained by solving Eqs. (2.11),
(2.14)–(2.16). The branching probabilities Pu and Pd should be between 0 and 1 to
meet the no-arbitrage requirements. In the CRR lattice, this demand can always be met by suitably increasing n [35].
Construction of a Multinomial Lattice
If each node in a lattice can branch to # nodes at the next time step, we call it an
#-nomial lattice. The above idea can be applied to construct an #-nomial lattice. Note
that 2# degrees of freedoms are provided by an #-nomial lattice. They include # price multiplicative factors (like u and d in the CRR binomial lattice) and # branching probabilities (like Pu and Pd in the CRR binomial lattice). These branching
proba-bilities must be between 0 and 1 to meet the no-arbitrage requirements. We need 2# independent equations to determine these 2# variables uniquely. The calibration of mean and variance gives 2 equations. The branching probabilities sum to 1, giving another one. Additional 2# − 3 equations must be added. For example, Eq. (2.11) is the additional equation used in the CRR binomial model.
Generally speaking, most of these extra equations are enforced to meet specific requirements. The number of these extra equations determines the lattice model that will be constructed. In Chapter 3, a trinomial lattice (# = 3) will be constructed to price an Asian option.
2.5.3
Pricing an Ordinary Option with the Lattice Approach
Options can be priced by the backward induction method on a lattice. Let us focus on pricing a European-style call on the CRR binomial lattice first. Recall that option
value can be obtained by taking expectation of the future discounted payoffs as in Eq. (2.9), which can be decomposed into a combination of numerous formulae as follows: C = e−r∆t(PuCu+ PdCd), (2.17)
where C denotes the option value of an arbitrary node N, Cu denotes the option value
of the node that can be reached from N by a upward movement, and Cd denotes the
option value of the node that can be reached from N by a downward movement. The option value for each node at the maturity date (at time n) is defined by Eq. (2.2). The option value for each node at time i (0 ≤ i ≤ n − 1) is evaluated by substituting the option values of the nodes at time i + 1 into the right hand side of Eq. (2.17). The pricing result is the option value at the root node.
100 112.5 200 200 50 25 400 350 100 50 25 0
Figure 2.5: Pricing an Ordinary Option on a CRR Lattice.
The upper cell of each node (colored of gray) denotes the price of the underlying asset, and the lower cell of each node denotes the option value at that node.
Take a simple 2-time-step CRR binomial lattice model illustrated in Fig. 2.5 as an example. The upward factor u and the downward factor d are 2 and 0.5, respectively. The upward branching probability Pu and the downward branching probability Pd
are both 0.5. The risk-free rate is set as 0, and the strike price is 50. The upper cell of each node (colored of gray) denotes the price of the underlying asset, and the lower cell of each node denotes the option value at that node. The option value of a node at time 2 (the maturity date) is computed by Eq. (2.2). For example, the option value for the uppermost node at time 2 is
max(400− 50, 0) = 350.
For each node at time 0 or time 1 can be evaluated by applying Eq. (2.17). For example, the option value for the upper node at time 1 is
The final pricing result is 112.5.
An American-style option can be exercised before the maturity date. An option will be exercised early if the option holder finds that it is more beneficial to exercise the option than to hold the option. The lattice approach can handle this by modifying Eq. (2.17) as follows:
C = max(Exercise value, e−r∆t(PuCu + PdCd)), (2.18)
where the “Exercise value” denotes the payoff to exercise the option immediately (see Eq. (2.3)).
In the above method, the option value of each node is evaluated once by applying Eq. (2.2) (for the nodes at time n), Eq. (2.17) (for pricing European-style options), or Eq. (2.18) (for pricing American-style option). Since the above mentioned equations can be evaluated in constant time and there are O(n2) nodes in the lattice, the computational time complexity is O(n2).
2.5.4
The PDE Approach
The value of the option may be represented as the solution of a PDE. When the PDE cannot be solved analytically, the solution can be approximated by the finite difference method numerically. A finite difference method places a grid of points on the space over which the desired function takes value and then approximates the function value at each of these points. The approximation method is similar to the backward induction used in the lattice approach.
2.6
Pricing the Asian option with Lattice Approach
and Its Problems
2.6.0.1 The Lattice Approach and and the Inefficiency Problem
The Asian option can be priced by the lattice approach. However, the pricing algo-rithm is much more complex as the value of the Asian option is influenced by the historical average price of the underlying asset (see Eq. (2.7)). For most nodes, there is more than one possible option value at a node since there is more than one price paths reaching this node and most of these price paths carry distinct historical aver-age prices. For convenience, some terms are introduced as follows: A path prefix is defined as a partial price path (S0, S1, . . . , Sj) that starts at time 0 and ends at time
j. The sum of this path prefix is defined by ji=0Si. We call it a prefix sum. For
an arbitrary node N illustrated in Fig. 2.6, we can find a path prefix that has the maximum prefix sum among the path prefixs that end at N. This maximum path prefix is denoted by the upper path (in thick lines) that ends at N. Similarly, the path prefix that has the minimum prefix sum is denoted by the lower path (in thick
N
Maximum prefix sum
Minimum prefix sum
Figure 2.6: Prefix Sums and the Prefix-Sum Range.
lines) that ends at N. The prefix sum range for N is defined as the range between the maximum and the minimum prefix sums for N.
A 3-time-step CRR lattice is illustrated in Fig. 2.7 to show how an Asian option is priced on a lattice. We focus on the European-style call option first. The settings of the multiplication factors and the branching probabilities are the same as in Fig. 2.5 except that there may be more than one option value at each node. The uppermost cell for each node (colored of gray) denotes the underlying asset’s price for that node. The prefix sum for each path prefix is marked directly on the branch. The option value for a path prefix is denoted in the cell connected by the branch that is marked by the prefix sum of the path prefix. For example, the option value for a path prefix (100, 200, 100) is 81.25.
The option value for each path prefix that reaches the maturity date (time 3) can be computed by Eq. (2.7). For example, the payoff for the path prefix (100, 50, 25, 12.5) is max(100+50+25+12.54 , 0) = 0. Define the option value of a path prefix (S0, S1, · · · , Si) as V (i, Si, M), where M =
i
j=0Sj. Assume that the underlying asset’s price can
either move up to Si+1+ or move down to Si+1− . Then the value of V (i, Si, M) can be
expressed as
V (i, Si, M) = e−r∆t
PuV (i + 1, Si+1+ , M + Si+1+ ) + PdV (i + 1, Si+1− , M + Si+1− )
.
(2.19) For example, the option value of the path prefix (100, 200, 100) (≡ V (2, 100, 400)) is
V (2, 100, 400) = e0(0.5 × V (3, 200, 600) + 0.5 × V (3, 50, 450)) = 100 + 62.5 = 81.25. When pricing American-style option, the backward induction formula is
V (i, Si, M ) = max(E, e−r∆t
PuV (i + 1, Si+1+ , M + Si+1+ ) + PdV (i + 1, Si+1− , M + Si+1− )
), (2.20) where “E” denotes the value to exercise the option immediately (see Eq. (2.8)).
100 94.5313 200 165.625 50 23.4375 400 250 800 325 50 62.5 300 150 700 1500 250 100 81.25 200 175 600 175 12.5 0 25 3.125 225 187.5 400 900 300 43.75 100 25 62.5 6.25 450 450 Asset's price
Option's value at each state
Figure 2.7: Pricing an Asian Call Option on a CRR Lattice.
The uppermost cell for each node (colored of gray) denotes the underlying asset’s price at that node. The other cell(s) denote(s) possible option value(s) at that node.
The option value for each path prefix can be evaluated in constant time. Unfor-tunately, the number of path prefix in a n-time step lattice is
20+ 21+ 22+· · · + 2n= 2n+1− 1.
The time complexity is exponential in n. Although some of the path prefixes may have the same option value, no one has reduced the time complexity significantly without approximation.
2.6.1
Approximation Algorithms and Its Problems
A successful approximation paradigm is suggested by Hull and White [26], and this paradigm is widely incorporated by other approximation pricing algorithms. We call this paradigm the Hull-White paradigm. This paradigm limits the number of possible prefix sums at each node to a manageable magnitude k. The option values for the missing prefix sums are estimated by interpolation. Take Fig. 2.8 as an example. There are three nodes in the figure. The leftmost node is at time i, and the two other nodes are at time i + 1. Max and Min denote the maximum and the minimum prefix sums for each node, respectively. Each line segment at node N denotes the one of the possible option value at N. The corresponding prefix sum for each possible option value is marked in the right (or left) of the line segment. The prices of the underlying asset of the upper-right and the lower-right nodes are 4 and 2, respectively.
To solve the option value whose corresponding prefix sum is 20 at the leftmost node, two different option values from the next time step— one with corresponding prefix sum 24 (= 20 + 4) from the upper-right node (V (i + 1, 4, 24)) and the other one with corresponding prefix sum 22 (= 20 + 2) from the lower-right node (V (i + 1, 2, 22))— are required. However, these two required prefix sums are missing since the Hull-White paradigm limits the number of possible prefix sums. These two option values can be estimated by using interpolation. For example, V (i+1, 4, 24) can be estimated by V (i + 1, 4, 25) and V (i + 1, 4, 23). V (i + 1, 2, 22) can be estimated by V (i + 1, 2, 23) and V (i + 1, 4, 21). The time complexity of this approximation algorithm is O(n2k),
which is much more efficient than the original exact pricing algorithm.
❅ ❅ ❅ ❅ ❅ ❅ ❅ max .. . 20 .. . min i ✛ max .. . 25 23 .. . min max .. . 23 21 .. . min i + 1 4 2
Figure 2.8: The Approximation in Hull-White Paradigm.
A problem of the Hull-White paradigm is that it lacks error control. Since the interpolation errors are introduced and accumulated at each time step, we are no longer sure whether the pricing results converge to the desired option value computed by the exact pricing algorithm as n → ∞. The relationship is listed in Fig. 1.4, where
e denotes the maximum error between the approximation algorithm and the exact
pricing algorithm. Since more interpolation errors are introduced as n increases, the error e does not necessarily converge to 0. Thus the approximation pricing algorithm does not necessarily converge to the true option value (see Fig. 1.5.)
The Hull-White paradigm has been analyzed and extended [4, 20, 29]. Barraquand and Pudet apply similar idea to the PDE approach [4]. Klassen assumes that the pric-ing errors of the approximation algorithm converge in O(1/n) and apply extrapolation to eliminate the error [29]. Forsyth et al. argue that an improper interpolation scheme
results in divergence or incorrect convergence [20]. They also prove that the modified Hull-White approximation algorithm converges under some ad hoc assumptions when
k and n are related in a certain manner. The pricing error control problem is not
The Integral Trinomial Lattice
The difficulty for pricing the Asian options with the lattice approach lies in its com-putational intractability: The time complexity of any existing exact pricing algorithm is exponential in the number (n) of time steps of the lattice. This chapter suggests a new lattice model to tackle this problem. The resulting exact pricing algorithm is proved to break the exponential-time barrier [18].
3.1
A Simple Intuition
Existing exact pricing algorithms for Asian options are computationally intractable since the total number of prefix sums for any existing lattice grows exponentially in n. To solve this problem without approximations, a new lattice with fewer prefix sums is proposed. Imagine a new lattice composed of integral asset prices. A hypothetical example is illustrated in Fig. 1.3. In such a integral lattice, all possible prefix sums are restricted to be integers. A simple calculation shows us that there might be fewer prefix sums in an integral lattice.
A rough analysis is done by comparing the number of prefix sums at the last time step (time n) between a integral lattice and a usual binomial lattice. There are 2n
possible prefix sums at time n in a usual binomial lattice. Estimating the number of prefix sums at time n in an integral lattice is heuristic. Assume that the maximum prefix sum at time n of an integral lattice is α. Thus any possible prefix sum must be some integer between 1 and α. There must be O(nα) possible prefix sums at time
n as there are O(n) nodes at time n. α can be roughly estimated by the maximum
prefix sum of the CRR binomial lattice as follows:
α ≈ S0+ S0u + S0u2+ . . . + S0un = S0[1 + u + u2+ . . . + un] = S0u n+1− 1 u − 1 ≈ S0 e σ√T n− 1 eσ√T /n− 1 30
![Table 3.3 tabulates American-style Asian option values generated by various algo- algo-rithms: the Hull-White algorithm [26, 27], the upper- and lower-bound algorithms of Chalasani et al](https://thumb-ap.123doks.com/thumbv2/9libinfo/8766526.209931/39.892.144.356.207.656/table-tabulates-american-generated-various-algorithm-algorithms-chalasani.webp)
