Pricing Some American Multi-Asset Options

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U.U.D.M. Project Report 2009:6

Examensarbete i matematik, 30 hp Handledare och examinator: Johan Tysk Maj 2009

Pricing Some American Multi-Asset Options

Jun Han

Department of Mathematics

Uppsala University

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List of Figures

1.1 European Call Option . . . 3

1.2 European Put Option . . . 4

2.1 At t = T , Γ1 and Γ2 degenerate to a ray: S1= S2. . . 17

2.2 At t < T , positions of Γ1and Γ2. . . 17

3.1 Direct Discretization - Explicit Method - Global Map . . . 30

3.2 Direct Discretization - Explicit Method - Part Map . . . 30

3.3 Direct Discretization - Implicit Method . . . 31

3.4 Direct Discretization - Implicit Method Compared with Brennan & Schwartz Model - Implicit Method . . . 32

3.5 Brennan & Schwartz Model - Implicit Method Compared with Brennan & Schwartz Model - Explicit Method . . . 33

3.6 Option Values for an American Better-of Option at time t = T = 1/12, r = 0.05, q1= 0.02, q2= 0.03, σ1= σ2= 0.4 . . . . 34

3.7 Option Values for an American Better-of Option at time t = 0, T = 1/12, r = 0.05, q1= 0.02, q2= 0.03, σ1= σ2= 0.4 . . . . 34

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Abstract

In the first two chapters of this thesis, we consider some basic facts about options such as the difference between American and European options. We study in more detail the American better-of and call-max options. In order to numerically solve the free boundary PDE’s corresponding to these options, we use a finite difference method developed in Chapter 3. All the programming is done in MATLAB, and the corresponding code can be found in the Appendix.

We solve these PDE’s in the original variables, without transforming them to equations on the whole space.

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Acknowledgement

I would like to express my gratitude to my supervisor professor Johan Tysk for his patience, support and guidance. He has always provided the balance of assistance, pressure and encouragement - helping me when I needed it, and leaving me on my own when I needed to figure out something by myself. More- over, in completing this thesis, he spent a lot of time helping me to correct it, even very small spelling and font mistakes. His knowledge and insight have been invaluable resources to me in my current and future career.

I am very grateful to my schoolmates, Yang Zeng, Alan, and Xuemao Zhang for the assistances about the finite difference method, MATLAB, and LATEX.

I also thank the department of mathematics of Uppsala University for giving me the opportunity to study here.

Last but not least, I thank my parents. Though they could not be with me for the last a few years, they have given me constant love, encouragement and support, and they have always been eager to help me in any way that they could.

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Chapter 1

Introduction

1.1 Some Basic Option Theory

1.1.1 The Definition of an Option

Before we value an option, we have to know what an option is. From [1] we can find the definition of a European call option:“ a European call option is a contract with the following conditions:

• At a prescribed time in the future, known as the expiry date, the owner of the option may

• Purchase a prescribed asset, known as the underlying asset or, briefly, the underlying, for a

• Prescribed amount, known as the exercise priceor strike price.”

In this definition, the author uses the word “may” to imply that the holder of the option has a right and not an obligation, but the writer does have a potential obligation, since if the holder chooses to exercise this call option, he has to sell the asset. From [2] we get another direct definiton:“ A call option gives its holder the right but not the obligation to purchase a share of stock in the underlying company at a fixed price for a fixed length of time.”

A European put option is exactly the opposite of a European call option:

the holder of a put option has the right but not the obligation to sell a prescribed asset at a prescribed time in the future.

From what has been discussed above, we may safely draw the definition of an option from [3] :“ an option is an agreement that the holder can buy from,

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or sell to, the seller of the option at a specified future time a certain amount of an underlying asset at a specified price. But the holder is under no obligation to exercise the contract.”

1.1.2 The Value of an Option

Before talking about the value of an option, we have to introduce some notation, which are used consistently throughout my thesis.

• We denote the value of an option by V , and V is a function with two variables: the current value of the underlying asset, S, and time, t : V = (S, t), such that Vt= V (St, t);

• The volatility of the underlying asset, σ;

• The exercise price, E;

• The expiry, T ;

• The interest rate, r.

From these assumptions, we know that V only depends on S and t. An option’s value at expiration date, VT, is already set, which is just the option’s payoff:

VT =

( (ST − E)⁺, (call option) (E − ST)⁺. (put option)

The value of an option, which we just want to find out, is a functionV = V (S, t),(0 ≤ S < ∞, 0 ≤ t ≤ T ), such that

V (S, T ) =

( (S − E)⁺, (call option) (E − S)⁺. (put option)

1.2 The Black-Scholes Equation

1.2.1 The Black-Scholes Equation

Before describing the Black-Scholes analysis (Black & Scholes 1973) which leads to the value of anoption, I list some assumptions, which I make for most of my thesis.

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0 2 4 6 8 10 0

1 2 3 4 5 6 7 8

ST

C

European Call Option with E=2 at time T Call Option

Figure 1.1: European Call Option

• The asset price follows the lognormal random walk.

• The risk-free interest rate r and the asset volatility σ are known functions of time over the life of the option.

• The market is ideal, which means: continuous trading, infinitely divisible assets, no transaction costs, taxes, no restrictions on short sales, and so on.

• The underlying asset pays no dividends during the life of the option.

• There are no arbitrage possibilities.

I assume that the market consists of two assets with dynamics given by

dB(t) = rB(t)dt, (1.2.1)

dS(t) = S(t)αdt + S(t)σdW (t), (1.2.2) where r, α, and σ are deterministic constants for convenience. Consider a simple contingent claim of the form

χ = Φ(S(T )), (1.2.3)

and assume that the claim can be traded on a ideal market and its process has the form

Π(t) = V (t, S(t)), (1.2.4)

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0 1 2 3 4 5 6 0

0.5 1 1.5 2 2.5 3 3.5 4

ST

P

European Put Option with E=2 at time T Put Option

Figure 1.2: European Put Option

for some smooth function V .

Then follow the process which is given in [4], we obtain a very important theorem.

Theorem 1.2.1 (Black-Scholes Equation). Assume that the market is specified by 1.2.1–1.2.2, and we want to price a contingent claim of the form 1.2.3. Then the only pricing function of the form 1.2.4 which is consistent with the absence of arbitrage is when V is the solution of the following boundary value problem in the domain [0, T ] × R+.

Vt(t, s) + rsVs(t, s) +1

2s²σ²(t, s)Vss(t, s) − rV (t, s) = 0, (1.2.5)

V (T, s) = Φ(s). (1.2.6)

From Theorem1.2.1, we find that if we let Φ(S(t)) be a payoff function of an option at time T , then the function V (t, s) will be price of the option.

1.2.2 Boundary and Final Conditions of European options

After deriving the Black-Scholes equation for the price of an option, we need to find out the boundary and final conditions, otherwise, the PDE does not have a unique solution.

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First, let focus on a European call option, whose value is denoted by c(S, t), with expiry date T , and exercise price E. At t = T , the value of a call is the payoff function:

c(S, T ) = max(S − E, 0), (1.2.7)

and this is our final condition. Then we need to find out the boundary condi- tions, which are applied at S = 0, and as S −→ ∞. From [1] we know that if S is ever zero, then it must remain zero, we can get the boundary conditions:

c(0, t) = 0, (1.2.8)

c(S, t) ∼ S as S −→ ∞. (1.2.9)

For a European call option, with final condition 1.2.7 and boundary conditions 1.2.8–1.2.9, we can get the Black-Scholes value of a European call option.

For a European put option, whose value is p(S, t), the final condition is also the payoff function

p(S, t) = max(E − S, 0). (1.2.10)

The boundary condition when S = 0 is

p(0, t) = Ee^{−r(T −t)}, (1.2.11) when the interest rate r is constant. As S −→ ∞, holders of put options are unlikely to exercise the contract, so

p(S, t) −→ 0 as S −→ ∞. (1.2.12)

1.2.3 The Explicit Solution of the Black-Scholes Equation

In the last section, we have established the PDE’s and their final and boundary conditions, which are satisfied by the prices of European call and put options.

In this section, we will find the explicit solution of the Black-Scholes equation.

Let us look at a European call with value c(S, t). The Black-Scholes equation and boundary conditions for it are

∂c

∂t +1

2σ²S²∂²c

∂S² + rS∂c

∂S − rc = 0, (1.2.13)

with

c(0, t) = 0, c(S, t) ∼ S − Ke^{−r(T −t)} as S −→ ∞, and

c(S, T ) = max(S − E, 0).

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First we set

S = Ee^x, t = T − τ

1

2σ², c = Eυ(x, τ ).

Then we turn Equation 1.2.13 into a new equation

∂υ

∂τ = ∂²υ

∂x² + (k1− 1)∂υ

∂x− k1υ, (1.2.14)

where k1= 1^r

2σ². The initial condition becomes υ(x, 0) = max(e^x− 1, 0).

We turn Equation 1.2.14 into a diffusion equation by a simple change of variable.

Try putting

υ = e^αx+βτu(x, τ ),

for some constants α and β to be found, then differentiation gives us βu +∂u

∂τ = α²u + 2α∂u

∂x+∂²u

∂x² + (k1− 1)(αu + ∂u

∂x) − k1u.

By choosing

β = α²+ (k1− 1)α − k1, we can get rid of all u terms, and with the choice

0 = 2α + (k1− 1)

we can eliminates the ^∂u_∂x term as well. From these equations, we get α = −1

2(k1− 1), and β = −1

4(k1+ 1)². Then we can get

υ = e⁽⁻¹²^(k¹^−1)x−¹⁴^(k¹⁺¹⁾²^{τ )}u(x, t), where

∂u

∂τ =∂²u

∂x² f or − ∞ < x < ∞, τ > 0, (1.2.15) with

u(x, 0) = u0(x) = max(e¹²^(k¹^+1)x− e¹²^(k¹^−1)x, 0). (1.2.16) The solution of the diffusion equation problem 1.2.15 is

u(x, τ ) = 1 2√

πτ Z _∞

−∞

u0(s)e^−(x−s)2^4τ ds (1.2.17)

here u0(x) is given by Equation 1.2.16.

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Now, we need to evaluate the integral in Equation 1.2.17. Making the change of variable x⁰ =^(x−s)^√_2τ , we obtain

u(x, τ ) = I1− I2, where

I1= e¹²^(k¹^+1)x+¹⁴^(k¹⁺¹⁾²^τN (d1), d1= 1

√2τ +1

2(k1+ 1)√ 2τ , and

N (d1) = 1

√2π Z _d₁

−∞

e⁻¹²^s²ds

is the cumulative distribution function for the normal distribution.

The calculation of I₂ is similar to that of I₁, just use (k₁− 1) to replace (k1+ 1) throughout.

Finally, we retrace the steps, using

υ(x, τ ) = e⁻¹²^(k¹^−1)x−¹⁴^(k¹⁺¹⁾²^τu(x, τ )

and putting x = log(_E^S), τ = ¹₂σ²(T − t), and c = Eυ(x, τ ), then we can get c(S, t),

c(S, t) = SN (d1) − Ee^{−r(T −t)}N (d2), with

d1=log(_E^S) + (r +¹₂σ²)(T − t) σp

(T − t) ,

d2=log(_E^S) + (r −¹₂σ²)(T − t) σp

(T − t) .

We can get the value function of a European put option just following the similar steps, but after having that of a European call option, we can use put-call parity formula

c − p = S − Ee^{−r(T −t)} to get the value p of a European put option.

1.3 American Options

From this section on, we focus on American options, which give their holders the right to exercise the option at any time, thus offering more opportunity to make profit than the corresponding European options. Intuitively, we conclude that the price of American option cannot be less than that of an equivalent European

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option. In mathematical theory, American option pricing corresponds to a free boundary problem. This boundary divides the domain [0, T ] × R+ into two parts: the continuation region, within which it is better to hold the option than to exercise, and the stopping region, within which it is better to exercise the option than to hold it. The dividing price between exercise and non-exercise is called the optimal exercise price Sf(t).

We will take non-dividend-paying American put option as example to establish the mathematical models.

From the discussion above, we know that for an American put option with expiration date t = T , there exist two regions: the continuation region Σ1, within which:

P (S, t) > max(E − S, 0), and the stopping region Σ2, within which:

P (S, t) = max(E − S, 0).

When S is very small, the American put option should be exercised at once, on the other hand, when S > E, the holder should continue to keep the option, because the payoff is zero. Therefore we conclude: there must exists Sf(t) ∈ (0, E), such that

Σ1= {(S, t)|Sf(t) ≤ S < ∞, 0 ≤ t ≤ ∞},

Σ2= {(S, t)|0 ≤ S ≤ Sf(t), 0 ≤ t ≤ ∞}.

And between these two regions is the optimal exercise boundary Γ : S = Sf(t).

In the continuation region Σ1, with the ∆-hedging principle and the Itˆo formula, we find that in the continuation region the option price P = P (S, t) satisfies the Black-Scholes equation:

Pt(S, t) + rsPs(S, t) +1

2s²σ²(S, t)Pss(S, t) − rP (S, t) = 0, On the optimal exercise boundary Γ,

P (Sf(t), t) = E − Sf(t),

∂P

∂S(Sf(t), t) = −1, when S −→ ∞,

P −→ 0,

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and at t = T ,

P (S, T ) = max(E − S).

This is known as the smooth fit principle. From a financial point of view it is natural, since this means that the hedging parameter ∆ is continuous across the free boundary.

In order to price an American put option, we need to find out {P (S, t), Sf(t)}

in the continuation region Σ1, such that they satisfy the Black-Scholes equation with optimal exercise boundary conditions.

At the end of this section, we will talk a little about American call option.

Since we know that the price of American option cannot be less than that of an equivalent European option, we will find that for an American call option,we always have:

C(S, t) > max(E − S, 0),

which means that the price function of American call option C(S, t) always stays in the continuation region Σ1. So we can conclude that the holder of American call options should always wait until time t = T , which means that an American call option has the same price function as an equivalent European call option.

1.4 Summary

In this chapter, we consider the definition of financial options, the Black-Scholes equation, boundary and final conditions of European options, and the free boundary conditions of American options. However, in the real world, options are not only derived from one underlying risky asset. In many cases, options are derived from two or more underlying risky assets, and they are called multi- asset options. In next chapter, we will focus on the mathematical model of American options derived from two or more underlying risky assets.

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Chapter 2

American Options on Multiple Assets

2.1 Stochastic Models of Multi-Assets Pricing

In order to price options derived from multiple assets, we need to establish the price movement model for the underlying multi-assets. From[3], let Si be the price of the i-th risky asset (i = 1, . . . , n), and Sisatisfies a stochastic differential equation with the following form:

dSi(t) = Si(t)αidt + Si(t)σidWi(t), (2.1.1)

where dWi(t)(i = 1, . . . , n) are standard Brownian motion that satisfy

E(dWi) = 0, (2.1.2)

V ar(dW_i) = dt, (2.1.3)

and

Cov(dWi, dWj) = ρijdt. (i 6= j) (2.1.4)

In this equation, Cov(., .) is the covariance, and can be expressed as

Cov(dWi, dWj) = E(dWidWj), (i 6= j)

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since

Cov(dWi, dWj) = E([dWi− E(dWi)][dWj− E(dWj)])

= E(dWidWj− dWiE(dWj) − E(dWi)dWj

+ (E(dWi)E(dWj)))

= E(dWidWj)

by the definition of covariance and Equation 2.1.2.

Equation 2.1.1 can be modeled in another form:

dSi= Siµidt + Si

Xm j=1

σijdWj, (i = 1, . . . , n) (2.1.5)

where dWi(i = 1, . . . , n) are one-dimensional Brownian motions, such that:

E(dWi) = 0, (2.1.6)

V ar(dWi) = dt, (2.1.7)

Cov(dWi, dWj) = 0, (i 6= j) (2.1.8) In many cases, using Equation 2.1.5 will be easier for us.

In order to price options on multiple assets, we need to find out the Black- Scholes equation for multiple assets. Let S1, . . . , Sn be n risky assets, which satisfy Equation 2.1.5, and let V (S1, . . . , Sn, t) be an option derived from these risky assets.

With 4-hedging principle, we choose 4i shares of asset Si(i = 1, . . . , n) to construct a portfolio Π:

Π = V (S1, . . . , Sn, t) − Xn

i=1

∆iSi,

such that this portfolio is risk-free in (t, t + dt).

With the Itˆo formula for the multivariate stochastic process, we will have

dΠ = dV − Xn i=1

∆idSi− Xn i=1

∆iSiqidt

= Ã

∂V

∂t +1 2

Xn i,j=1

αijSiSj ∂²V

∂Si∂Sj

! dt

+ Xn

i=1

∂V

∂SidSi− Xn i=1

∆idSi− Xn i=1

∆iSiqidt,

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where qi is the dividend rate of asset Si, and αij =

Xm k=1

σikσjk. (i, j = 1, . . . , n)

Since we need Π bo be risk-free in time (t, t + dt), we have dΠ = rΠdt = r(V −

Xn i=1

∆iSi)dt, (2.1.9)

We use ∆i=_∂S^∂V

i to replace ∆iin Equation 2.1.9, and eliminate dt, then we will get the equation:

∂V

∂t +1 2

Xn i,j=1

αijSiSj ∂²V

∂Si∂Sj

+ Xn i=1

(r − qi)Si∂V

∂Si

− rV = 0. (2.1.10)

Equation 2.1.10 is the Black-Scholes equation for multi-asset options.

2.2 The Mathematical Model of American Multi- Asset Option

From [3], we know that the mathematical model of American multi-asset option can be seen as a variational inequality.

The domain of it is:

Σ =©

(S1, . . . , Sn, t)|(S1, . . . , Sn) ∈ Rⁿ₊, 0 ≤ t ≤ Tª , where

Rⁿ₊=©

(S1, . . . , Sn)|0 ≤ Si< ∞, i = 1, . . . , nª .

Let V = V (S1, . . . , Sn, t) be the American option price, then V is the solution of the following problem in Σ:



 minn

−∂V

∂t − LV, V − Φ(S1, . . . , Sn)o

= 0, (Σ) (2.2.1) V (S1, . . . , Sn) = Φ(S1, . . . , Sn), (S1, . . . , Sn∈ R₊ⁿ). (2.2.2) where L is the multivariate Black-Scholes differential operator, which is given as:

LV = 1 2

Xn i,j=1

αijSiSj ∂²V

∂Si∂Sj

+ Xn i=1

(r − qi)Si∂V

∂Si

− rV, and Φ(S1, . . . , Sn) is the payoff function.

From Equation 2.2.1, we conclude that: in the continuation region Σ1:

∂V

∂t + LV = 0,

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V > Φ(S1, . . . , Sn);

and in the stopping region Σ2:

∂V

∂t + LV ≤ 0, V = Φ(S1, . . . , Sn).

Between Σ1 and Σ2is the optimal exercise boundary Γ.

From [3], we have the following theorem:

Theorem 2.2.1. Suppose the payoff function Φ satisfies:

(1) Φ is a Lipschitz continuous function;

(2) There exists Φε∈ C^∞(Rⁿ₊), such that:

ε→0limΦε= Φ,

and is uniformly convergent in any finite bounded domain within Rⁿ₊; (3)

∂Φε

∂Si > 0 (or∂Φε

∂Si < 0), (2.2.3)

and in any finite bounded domain D within (Rⁿ₊), there exists a constant CD which depends on D only and is independent of ε, such that

LΦε≥ CD, (2.2.4)

then the solution of the American multi-asset option pricing problem 2.2.1–2.2.2, V (S1, . . . , Sn, t), has the following properties:

(1) V (S₁, . . . , S_n, t) is a nondecreasing (monotone increasing) function of S_i(i = 1, . . . , n),

(2) V (S1, . . . , Sn, t) is a nonincreasing function of t.

2.3 American Options with Two Assets

2.3.1 American Better-of Option on Two Assets

From[3],the holder of better-of option receives exercise payoff associated with the better performer of the underlying assets.

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There are two kinds of better-of option: one is based on the price, the payoff function of which is:

payof f = max(α1S1(T ), . . . , αnSn(T )),

where Si(T ) is the price of the i-th risky asset at time t = T , and αi is a coefficient so that all risky asset prices are at the same level, and the other one is based on the growth rate, the payoff function of which is:

payof f (rate) = max( ˆS1(T ), . . . , ˆSn(T )), where ˆSi is the growth rate of the i-th risky asset at time t = T .

Under the transformation:

Sˆi(T ) = Si(T ) − Si(0)

Si(0) = Si(T ) Si(0) − 1,

we will find these two kinds of better-of options have an relationship, since their payoff functions have an relationship, which is:

max( ˆS1(T ), . . . , ˆSn(T )) = max(α1S1(T ), . . . , αnSn(T )) − 1, where

αi= 1

Si(0), (i = 1, . . . , n).

From now on, we will talk about American better-of option on two assets, the payoff function of which is max(S1, S2), in this section.

The mathematical model is: (from[3])



 min

n

−∂V

∂t − LV, V − max(S1, S2) o

= 0, (Σ) (2.3.1) V |t=T = max(S1, S2), (S1, S2∈ R²₊). (2.3.2) where

LV = 1

2 h

σ²₁S₁²∂²V

∂S²₁ + 2ρσ1σ2S1S2 ∂²V

∂S1∂S2 + σ₂²S₂²∂²V

∂S₂² i

+(r − q₁)S₁∂V

∂S1

+ (r − q₂)S₂∂V

∂S2

− rV, (2.3.3)

and this is the variational inequality model for American better-of option on two assets.

We introduce the transformation:

ξ = S1

S₂, u(ξ, t) = V (S1, S2, t)

S₂ , (2.3.4)

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then we have:

∂V

∂S1

= S₂∂u

∂ξ

∂S1

=∂u

∂ξ,

∂V

∂S2 = u + S2∂u

∂ξ

∂S2 = u − ξ∂u

∂ξ,

∂²V

∂S₁² = 1 S2

∂²u

∂ξ²,

∂²V

∂S1∂S2 = ∂²u

∂ξ²

∂ξ

∂S2 = − ξ S2

∂²u

∂ξ²,

∂²V

∂S₂² = ∂u

∂ξ

∂S2 −∂u

∂ξ

∂S2 − ξ∂²u

∂ξ²

∂ξ

∂S2 = ξ²∂²u S2∂ξ².

Substituting these into Equation 2.2.1–2.2.2,we will have a problem in one dimensional:









 min

n

−∂u

∂t −1

2σˆ²ξ²∂²u

∂ξ² − (q2− q1)ξ∂u

∂ξ +q2u, u − max(ξ, 1)o

= 0, (2.3.5)

u|t=T = max(ξ, 1). (2.3.6)

where the domain is ˆΣ = {(ξ, t)|0 ≤ ξ ≤ ∞, 0 ≤ t ≤ T } and ˆσ²= σ₁²+ 2ρσ1σ2+ σ₂².

Since q1, q2are dividend rate, in the real world, the dividend rates are always nonnegative, so we can assume that the dividend rate are positive (q1, q2 > 0) for simple. Then we can write the domain ˆΣ to be ˆΣ1∪ ˆΓ ∪ ˆΣ2, where ˆΣ1is the continuation region of the option:

u > max(ξ, 1),

∂u

∂t +1

2ˆσ²ξ²∂²u

∂ξ² + (q2− q1)ξ∂u

∂ξ − q2u = 0;

Σˆ2is the stopping region of the option:

u = max(ξ, 1),

∂u

∂t +1

2ˆσ²ξ²∂²u

∂ξ² + (q2− q1)ξ∂u

∂ξ − q2u ≤ 0;

Γ is the interface of ˆˆ Σ1 and ˆΣ2.

Theorem 2.3.1. If q1, q2 > 0, then for the problem 2.3.5–2.3.6, the continua- tion region is:

Σˆ1=©

(ξ, t)|ξ1(t) ≤ ξ ≤ ξ2(t), 0 ≤ t ≤ Tª ,

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where

ξ1(T ) = ξ2(T ) = 1,

ξ1(t) nondecreasing,

ξ2(t) nonincreasing.

In ˆΣ1 , u satisfies the free boundary problem:











∂u

∂t +1

2ˆσ²ξ²∂²u

∂ξ² + (q2− q1)ξ∂u

∂ξ − q2u = 0, ( ˆΣ1) (2.3.7)

u(ξ1(t), t) = 1, (0 ≤ t ≤ T ) (2.3.8)

∂u

∂ξ(ξ1(t), t) = 0, (0 ≤ t ≤ T ) (2.3.9)

u(ξ2(t), t) = ξ, (0 ≤ t ≤ T ) (2.3.10)

∂u

∂ξ(ξ2(t), t) = 1, (0 ≤ t ≤ T ) (2.3.11) We omit the proof of the theorem. The complete proof is given in [3].

Now, we need to go back to the original function V (S1, S2, t), and we will get the price of the American better-of option:

V (S1, S2, t) = S2u(S1

S2) =









S2, 0 ≤ ^S_S¹

2 ≤ ξ1(t), S2u(^S_S¹

2, t), ξ1(t) ≤ ^S_S¹

2 ≤ ξ2(t), S1, ξ2(t) ≤ ^S_S¹

2 < ∞,

(2.3.12)

and the optimal boundary consists of surfaces Γ1and Γ2:

Γ1: S1= ξ1(t)S2, (2.3.13)

and

Γ2: S2= ξ2(t)S2, (2.3.14) where ξ1(t) = ξ2(t) = 1, ξ1(t) ↑, ξ2(t) ↓.

Then we can get the figures at t = T and t < T .

From the Figure 2.1 and 2.2, we conclude that at time t = T , the optimal boundary lines Γ1 and Γ2 coincide to the line S1 = S2; at time 0 < t < T , the optimal boundary lines separate, and the optimal consists of two lines:

S1 = ξ1(t)S2 and S1 = ξ2(t)S2, between these two lines lies the continuation region, and beyond them lies the stopping region; and at time t = 0, the optimal boundary lines are S1= ξ1(0)S2and S1= ξ2(0)S2.

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0 1 2 3 4 5 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

S₁ S2

Γ₁

Γ₂ V=S₂

V=S₁ S₁=S₂

Figure 2.1: At t = T , Γ1 and Γ2 degenerate to a ray: S1= S2.

0 1 2 3 4 5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

S₁ S2

V=S₂

V=S₁ Γ₁

Γ₂ V=S₂u(S₁/S₂,t)

Figure 2.2: At t < T , positions of Γ1and Γ2.

2.3.2 American Call-max Option on Two Risky Assets

In this section, we will talk about American call-max option on two risky assets, the exercise payoff function is:

(max(S1, S2) − E)⁺, (2.3.15) where the notationx⁺ is short for max(x, 0). This option can be regarded as a generalization of the American better-of option, since the American better-of option is the E = 0 case of American call-max option on two risky assets.

We should set up the mathematical model for American call-max options on two risky assets. From [3], we get the mathematical model: In domain

Σ : {(S1, S2) ∈ R²₊, 0 ≤ t ≤ T }

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solve the variational inequality



 min

n

−∂V

∂t − LV, V − (max(S1, S2) − E)⁺ o

= 0, (Σ) (2.3.16) V |t=T = (max(S1, S2) − E)⁺, (S1, S2∈ R₊²). (2.3.17) where

LV = 1

2 h

σ²₁S₁²∂²V

∂S²₁ + 2ρσ1σ2S1S2 ∂²V

∂S1∂S2 + σ₂²S₂²∂²V

∂S₂² i

+(r − q1)S1∂V

∂S1 + (r − q2)S2∂V

∂S2− rV. (2.3.18) From the Theorem 2.2.1, we infer some properties of the price function of an American call-max option, V (S1(t), S2(t), t).

(1) If ˆSi≥ Si, for i = 1, 2, then

V ( ˆS1(t), ˆS2(t), t) ≥ V (S1(t), S2(t), t); (2.3.19)

(2) If ˆE ≥ E, then

0 ≤ V (S1(t), S2(t), t, E) − V (S1(t), S2(t), t, ˆE) ≤ ˆE − E; (2.3.20)

(3) If ˆt ≥ t, then

V (S1(t), S2(t), ˆt) − V (S1(t), S2(t), t) ≤ 0. (2.3.21)

Standard American Options

Before we proceed further, we review some basic results for a standard American call option derived from a single underlying risky asset. Let Γ be the optimal exercise boundary, that is Γ = inf{S(t) : (S(t), t) ∈ Σ2}.

From [9], we know that Van Moerbeke(1976) and Jacka(1991) show that Γ is continuous; Kim(1990) and Jacka(1991) show that Γ is decreasing in t;

Kim(1990) shows that Γ_T⁻ ≡ limt→T Γt= max((^r_qE, E)); Merton(1973) shows that Γ is bounded above and derives a formula for Γ−∞≡ limt→−∞Γt; Jacka(1991) shows that the option value C_t(S_t)(the theoretical value of an American call option) is continuous and the stopping region Σ2is closed.

Proposition 2.3.1. For a standard American call option, (S(t), t) ∈ Σ2implies that (λS(t), t) ∈ Σ2 for all λ ≥ 1.

Proof. (S(t), t) ∈ Σ2 implies that C(S(t), t) = S(t) − E > 0. Since λ ≥ 1, we have λS(t) − E > 0. Then we have C(λS(t), t) = λS(t) − E > 0, which means that (λS(t), t) ∈ Σ2.

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Stopping Region of American Call-max Option on Two Risky Assets From Proposition 2.3.1, we will have similar conjectures for American call-max option on two risky assets:

Conjecture 2.3.1. (S1(t), S2(t), t) ∈ Σ2 implies (λ1S1(t), λ2S2(t), t) ∈ Σ2) for all λ₁≥ 1 and λ₂≥ 1.

Conjecture 2.3.2. If q₁> 0 and q₂> 0 then there exist constants M₁ and M₂ such that (S1(t), S2(t), t) ∈ Σ2 for all S − 1(t) ≥ M1 and S2(t) ≥ M2.

Conjecture 2.3.3. (S1(t), S2(t), t) ∈ Σ2 and ( ˜S1(t), ˜S2(t), t) ∈ Σ2 implies λ(S1(t), S2(t), t) + (1 − λ)( ˜S1(t), ˜S2(t), t) ∈ Σ2 for all 0 ≤ λ ≤ 1.

From [9], we know that all these three conjectures are false. However, also from [9], we know that properties similar to those on one risky asset do hold on certain subregions of Σ₂ (stopping region of American call-max option on two risky assets).

Define the subregion Σⁱ₂ of the stopping region Σ2 by Σⁱ₂ = Σ2∩ Gi where Gi≡ {(S1(t), S2(t), t) : Si(t) = max(S1(t), S2(t)} for i = 1, 2

Proposition 2.3.2. If S1(t) = S2(t) > 0 and t < T then (S1(t), S2(t), t) /∈ Σ₂. That is, prior to maturity exercise is not optimal when the prices of the underlying assets are equal.

This proposition states that: prior to maturity, stopping is suboptimal when the prices of the underlying assets are equal, no matter how large the prices are and no matter how large the dividend rates are. We can prove this proposition by intuition: we can exercise this option at some fixed time t₁where t < t₁< T , then at least we will have

P V (t1− t) = S1(t)e^−q¹^(t¹^−t)− Ee^−r(t¹^−t)

plus a European option to exchange asset 2 for asset 1 with a maturity date t1, and the value of this European option is

E_t^∗[e^−r(t¹^−t)(S2(t1) − S1(t1))⁺].

As t1 converges to t, P V (t1− t) converges to S1(t) − E at a finite rate, but at the same time, the value of that European exchange option converges to 0 at an increasing rate that approaches infinity in the limit. So, there must be some time t1> t such that waiting until t1 gives a strictly positive return relative to stopping at time t. you can find the strict proof in [9].

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Proposition 2.3.3 (Subregion Convexity). Let S = (S1, S2) and ˜S = ( ˜S1, ˜S2).

Suppose (S, t) ∈ Σⁱ₂ and ( ˜S, t) ∈ Σⁱ₂ for a fixed i = 1 or 2. Given λ, with 0 ≤ λ ≤ 1, define S(λ) = λS + (1 − λ) ˜S. Then (S(λ), t) ∈ Σⁱ₂. That means, if stopping is optimal at S and ˜S and if (S, t) ∈ Gi and ( ˜S, t) ∈ Gi then stopping is optimal at S(λ).

Since the payoff function is convex with respect to (S1, S2) and the mul- tiplicative structure of the uncertainty in Equation 2.1.1 when i = 1, 2, the stopping region is convex.

Proposition 2.3.4. Let Σ2 represent the stopping region for a max-option.

Then Σ2 satisfies the following properties.

(1) (S1(t), S2(t), t) ∈ Σ2 implies (S1(t), S2(t), s) ∈ Σ2 for all t ≤ s ≤ T . (2) (S1(t), S2(t), t, E1) ∈ Σ2 implies (S1(t), S2(t), t, E2) ∈ Σ2 for all E2> E1. (3) (S₁(t), S₂(t), t) ∈ Σ¹₂ implies (λS₁(t), S₂(t), t) ∈ Σ¹₂ for all λ ≥ 1.

(4) (S1(t), S2(t), t) ∈ Σ¹₂ implies (S1(t), λS2(t), t) ∈ Σ¹₂ for all 0 ≤ λ ≤ 1.

(5) (S1(t), 0, t) ∈ Σ¹₂ implies S1(t) ≥ Γ1(t).

where Γ1(t) is the optimal exercise boundary for a standard American option derived from a single underlying risky asset. In (3), (4), (5), analogous results hold for the subregion Σ²₂.

From property (1), we can conclude that the stopping region expands as time moves forward, since a short maturity option cannot be more valuable than the longer maturity option,which gives holder more opportunity to make profit.

Property (2) implies that the stopping region shrinks as the exercise price E increases. Property (3) implies that the stopping subregion is connected in the direction of increasing S1, and property (4) implies that the stopping subregion is connceted in the direction of decreasing S2.

Now, I will give the proof of the property (1), you can find these two proof in [3]. Since the stopping region expands as time moves forward, we conclude that if t2> t1, then

Σ2(t2, E) ⊃ Σ2(t1, E) (2.3.22) Proof. Relation 2.3.22 is a corollary of Equation 2.3.21. Suppose otherwise:

there exist (S⁰₁(t), S₂⁰(t)) and t = t⁰₁, t⁰₂ with t⁰₂> t⁰₁, such that:

(S₁⁰(t), S₂⁰(t), t⁰₁) ∈ Σ2(t⁰₁, E),

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but

(S₁⁰(t), S₂⁰(t), t⁰₂) ∈ Σ1(t⁰₂, E).

Then we will have

V (S₁⁰(t), S₂⁰(t), t⁰₂) > Φ(S₁⁰(t), S₂⁰(t)) = V (S₁⁰(t), S₂⁰(t), t⁰₁), which contradicts to Equation 2.3.21.

Next, I will prove the property (2). Since the stopping region shrinks as the exercise price E increases, we have: if E2> E1, then

Σ2(t, E2) ⊂ Σ2(t, E1). (2.3.23) Proof. Assume otherwise, then we will have (S₁⁰(t), S₂⁰(t)) and t = t0, such that when E2> E1,

(S₁⁰(t), S₂⁰(t), t0) ∈ Σ2(t, E2), but

(S₁⁰(t), S₂⁰(t), t0) ∈ Σ1(t, E1).

Due to Equation 2.3.20, we have

Φ(S₁⁰(t), S₂⁰(t); E2) = V (S₁⁰(t), S₂⁰(t), t0; E2)

≥ V (S₁⁰(t), S₂⁰(t), t0; E1) − E2+ E1

> Φ(S₁⁰(t), S⁰₂(t); E1) − E2+ E1, i.e.

(max(S₁⁰(t), S⁰₂(t)) − E2)⁺+ E2> (max(S₁⁰(t), S⁰₂(t)) − E1)⁺+ E1. (2.3.24) Thus there must be

max(S₁⁰(t), S₂⁰(t)) < E₂, (2.3.25) otherwise, if

max(S₁⁰(t), S₂⁰(t)) ≥ E2> E1, then we will have

(max(S₁⁰(t), S₂⁰(t))−E2)⁺+E2= max(S₁⁰(t), S₂⁰(t)) = (max(S₁⁰(t), S₂⁰(t))−E1)⁺+E1. This contradicts to Equation 2.3.24.

But Equation 2.3.25 and (S₁⁰(t), S₂⁰(t), t0) ∈ Σ2(t, E2) are contradictory, since Equation 2.3.23, therefore (S₁⁰(t), S₂⁰(t), t0) must belong to Σ2(t, E1).

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Proposition 2.3.5 (Divergence of the stopping region). For a fixed time t, satisfying t < T . There exists λ1 and λ2 with λ2< 1 < λ1 such that

Σ2(t) ∩ R(λ1, λ2) = ∅.

where

R(λ1, λ2) ≡©

(S1, S2) ∈ R²₊: λ2S1< S2< λ1S1

ª

and for λ2 < λ1, R(λ1, λ2) denotes the open cone defined by the price rations λ1 and λ2.

With this result, we can draw the shape of a typical stopping region Σ2. Proposition 2.3.6. Let V (S1(t), S2(t), t) be the value of the American call-max option, we have:

(1) V (S1(t), S2(t), t) is continuous on R⁺× R⁺× [0, T ];

(2) V (·, S2(t), t) and V (S1(t), ·, t) are nondecreasing on R⁺for all S1(t), S2(t) in R⁺ and all t in [0, T ];

(3) V (S1(t), S2(t), ·) is nonincreasing on [0, T ] for all S1(t) and S2(t) in R;

(4) V (·, ·, t) is convex on R⁺× R⁺ for all t in [0, T ].

The continuity of the function can be proved by the continuity of the payoff function and the continuity of Equation 2.1.1. The monotonicity of the function follows the monotonicity of the payoff function. The property (3) is right since a shorter maturity option cannot be more valuable. The convexity is implied by the convexity of the payoff function.

Remark 1 (American call-max option with more than two underlying assets).

We will talk about some properties of American call-max option with more than two underlying assets. Since they are in more general situation, I think they are correct in case of two underlying assets.

First, we introduce some notations:

• The stopping region of the American call-max option derived from n assets, Σⁿ₂.

• The corresponding price, V (S1(t), . . . , S2(t), t).

• The vector of underlying asset prices, S ≡ (S1(t), . . . , Sn(t)).

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• G_iⁿ≡©

(S, t) : Si(t) = max(S1(t), . . . , Sn(t))ª

for i = 1, . . . , n.

Proposition 2.3.7. If max(S₁(t), . . . , S_n(t)) = S_i(t) = S_j(t) for i 6= j, i ∈ {1, . . . , n}, j ∈ {1, . . . , n} and if t < T then (S, t) /∈ Σⁿ₂. Which means prior to maturity stopping is suboptimal if the maximum is achieved by two or more asset prices.

This property parallels Proposition 2.3.2, and implies that stopping is subop- timal on all regions where the maximum asset price is achieved by two or more asset prices.

I will give some properties that parallel the properties in Section 2.3.2, and I omit the proofs of these properties, which can be found in [9].

Proposition 2.3.8 (Subregion Convexity). Consider two vectors S ∈ Rⁿ₊ and S ∈ R˜ ⁿ₊. Suppose that (S, t) ∈ Σ^n,i₂ and ( ˜S, t) ∈ Σ^n,i₂ for the same i ∈ {1, . . . , n}, where Σ^n,i₂ ≡ Σ2∩ Gⁿ_i. Given λ with 0 ≤ λ ≤ 1 denote S(λ) = λS + (1 − λ) ˜S.

Then (S(λ), t) ∈ Σ^n,i₂ . That is, if stopping is optimal at S and ˜S and if (S, t) ∈ G_iⁿ and ( ˜S, t) ∈ G_iⁿ then stopping is optimal at S(λ).

Proposition 2.3.9. Σⁿ₂ satisfies the following properties.

(1) (S, t) ∈ Σⁿ₂ implies (S, s) ∈ Σⁿ₂ for all t ≤ s ≤ T ;

(2) (S, t) ∈ Σ^n,i₂ implies (S1(t), . . . , λSi(t), . . . , Sn(t), t) ∈ Σ^n,i₂ for all λ ≥ 1;

(3) (S, t) ∈ Σ^n,i₂ implies (λ1S1(t), λ2S2(t), . . . , Si(t), λi+1Si+1(t), . . . , λnSn(t)) ∈ Σ^n,i₂ for all 0 ≤ λj≤ 1, j = 1, . . . , i − 1, i + 1, . . . , n;

(4) Si(t) = 0 and (S, t) ∈ Σ^n,i₂ implies (S1(t), . . . , Si−1(t), Si+1(t), . . . , Sn(t), t) ∈ Σ^n−1,i₂ .

2.4 Summary

In this chapter, we describe mathematical models of American multi-asset options, especially American better-of options on two assets and American call- max option on two assets. In Remark 1, we discuss some properties of American call-max options with n > 2 underlying assets. Since American option pricing is a free boundary problem to the Black-Scholes equation, there is no explicit solution of the option price in general, except for the perpetual American option.

Therefore we have to use numerical methods to find the solution of American option price, and we will discuss the numerical method in the next chapter.

Pricing Some American Multi-Asset Options

U.U.D.M. Project Report 2009:6

Examensarbete i matematik, 30 hp Handledare och examinator: Johan Tysk Maj 2009