introduction to finance (pcp) - keio...

2020 Fall

Lecture Notes on

Introduction to Finance (PCP)

Takuji Arai

Department of Economics, Keio University

First version: September 16, 2020

This lecture notes is written as a textbook on Introduction to Fi-nance (PCP) in Department of Economics and Graduate School ofEconomics, Keio University. Unauthorized copying and replicationof this lecture notes are prohibited.c⃝ 2020 Takuji Arai

Preface

This lecture notes is written as a textbook on Introduction to Finance (PCP)for Department of Economics and Graduate School of Economics, Keio Uni-versity. In this course, discrete time option pricing models will be discussed ina mathematical way. The course is composed of three parts. In the first part,one-period binomial models are introduced. In particular, we study how to priceoptions, and introduce the meaning of some important terminologies in optionpricing theory, e.g. arbitrage, replicating strategy, market completeness and soforth. Next, we extend one-period models to multi-period models. In this part,pricing for American options is also discussed. In the last part, general one-period models, that is, multi-asset multinomial models will be introduced. Theaim of this part is to show the fundamental theorems of asset pricing (FTAP)using some results on linear algebra. All students in this course are supposedto be familiar with calculus, linear algebra and the basic of probability theory.See exercises on the next page. This course is followed by Advanced Financeoffered in the next spring semester. In Advanced Finance, the Black-Scholesmodel, which is a representative continuous time option pricing model, will beintroduced.

i

Prerequisite

All students in this course are presumed to be able to solve the following ques-tions∗:

[1] Let A = {α, β, γ} and B = {γ, δ}. Find A ∪B and A ∩B.

[2] Let X be a random variable distributed as

P(X = −1) = 14, P(X = 0) =

1

2and P(X = 2) =

1

4.

Find P(X ≤ 0) and E[X].

[3] Show that nCk = n−1Ck + n−1Ck−1 for n ≥ 1 and n > k ≥ 1.

[4] Let A =

2 −11 0−3 4

and B = (1 −2 −53 4 0

). What is AB?

[5] Solve

(2 11 1

)(x1x2

)=

(31

).

[6] Let A =

(2 51 3

). What is A−1?

∗For solutions, see Section B.1

ii

Bibliography

The following introduces binomial asset pricing models comprehensively.[1] Shreve, S. (2004) Stochastic calculus for finance I: the binomial asset pricing

model. Springer.

To create Section 2.3, the following textbook was referred to.[2] Capinski, M., and Zastawniak, T. (2010) Mathematics for finance: An

Introduction to Financial Engineering, 2nd edition, Springer.

For Chapter 3, the following textbooks were referred to.In particular, the proof of Lemma 3.2 is based on Section 2.3 of [4].[3] Björk, T. (2009). Arbitrage theory in continuous time. 3rd edition. Oxford

university press.[4] Gale, D. (1989). The theory of linear economic models. University of

Chicago press.

In addition, the followings are textbooks for continuous time option pricing.[5] Shreve, S. E. (2008). Stochastic calculus for finance II: Continuous-time

models. Springer.[6] Roman, S. (2014) Introduction to the mathematics of finance: from risk

management to options pricing. 2nd edition, Springer.

As textbooks covering various topics in finance, the followings are useful.[7] Hull, J. C. (2018). Options futures and other derivatives. 10th edition.

Pearson Education India.[8] Luenberger, D. G. (2013). Investment science. 2nd edition. Oxford Univ

Press.

Below is a representative textbook on linear algebra.[9] Axler, S. J. (2014). Linear algebra done right. 3rd edition. New York:

Springer.

To study probability theory, the following is useful.[10] Durrett, R. (2019). Probability: theory and examples. 5th edition. Cam-

bridge university press.

iii

Contents

0 Introduction 1

1 One-period binomial models 31.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Replication . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Put-call parity . . . . . . . . . . . . . . . . . . . . . . . . 81.2.3 Martingale probability . . . . . . . . . . . . . . . . . . . . 9

1.3 FTAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.1 1st FTAP . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.2 2nd FTAP . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Linear algebraic approach to FTAP . . . . . . . . . . . . . . . . . 151.4.1 1st FTAP . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4.2 2nd FTAP . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Multi-period binomial models 212.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Cox-Ross-Rubinstein (CRR) formula . . . . . . . . . . . . . . . . 262.3 American options . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.1 American call options . . . . . . . . . . . . . . . . . . . . 292.3.2 American put options . . . . . . . . . . . . . . . . . . . . 31

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 General one-period models 353.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 1st FTAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3 2nd FTAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.1 Option pricing . . . . . . . . . . . . . . . . . . . . . . . . 44Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Appendix 46A Past exam questions . . . . . . . . . . . . . . . . . . . . . . . . . 47

A.1 AY2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

iv

A.2 AY2019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49B Solutions for exercises . . . . . . . . . . . . . . . . . . . . . . . . 51

B.1 Exercises in Prerequisite . . . . . . . . . . . . . . . . . . . 51B.2 Exercises in Chapter 1 . . . . . . . . . . . . . . . . . . . . 52B.3 Exercises in Chapter 2 . . . . . . . . . . . . . . . . . . . . 53B.4 Exercises in Chapter 3 . . . . . . . . . . . . . . . . . . . . 57

v

Chapter 0

Introduction

Financial theory is categorized into two topics: corporate finance and in-vestment theory. In particular, the latter covers modern portfolio theory,option pricing, risk management and so on. As for methods to solve prob-lems related to financial theory, there are two methods mainly: mathematicalfinance ∗ and financial engineering †. Note that we need to take different meth-ods for each aim. In this course, we focus on mathematical aspect of discretetime models in option pricing theory.

An option‡ is a contract agreed between two parties: a holder and a writer.The option holder is given the right, but not the obligation, to buy (or sell) aspecified asset, called the underlying asset, for a specified price, called thestrike price, on a specified date, called the maturity. In particular, a right tobuy (resp. sell) the underlying asset is said to be a call (resp. put) option. Forexample, we consider a call option with strike price K and maturity date T . Ifthe underlying asset price at T , denoted by ST is greater than or equal to K,the option holder exercises the option at T , and can buy the underlying assetfor K, which is less than the market price ST . Thus, selling it at the marketimmediately, she can gain the difference ST −K. In other words, the writer hasto pay ST −K to the holder in this case. Otherwise, when ST < K, the holdershould abandon the option, that is, the cashflow is 0.

Example 0.1 Consider the right to purchase one share of X corporation stockfor $200 on December 1st, 2020. At the maturity, if the stock price is greaterthan $200, then the option holder exercises the option and can purchase oneshare of the stock for $200. Otherwise, in the case where the stock price is lessthan $200, the holder should abandon the exercise of the option.

∗Mathematics used in mathematical finance includes probability theory, partial differentialequations, functional analysis and so on.

†Computational approach and empirical analysis are main ingredients in financial engi-neering.

‡The word “option” might be replaced by “derivative”, “claim” or “contingent claim” insome textbooks .

1

In addition, options are categorized into European and American ones: Eu-ropean options can only be exercised at the maturity, while American optionscan be exercised at any time before and on the maturity. Note that, throughoutthis course, we treat European options except for Section 2.3.

In summary, the holder’s cashflow at the maturity is nonnegative, whichmeans that the holder needs to pay some amount of money to the writer whenthey agree the contract. This amount is called option price or premium.The main concern in option pricing theory is how to compute option prices forgiven market models described in a probabilistic way. Firstly, tradable assetsand trading times are specified. Of course, future prices of tradable assets areunknown. Thus, the price of each tradable asset at each trading time is denotedby a random variable. The fluctuation of asset prices is illustrated as asequence of random variables with time parameter, which is called a stochasticprocess. In this course, we discuss only discrete time models, in whichtrading times are given as 0, 1, · · · , T , where the maturity T is an integer ingeneral.

2

Chapter 1

One-period binomial models

One-period binomial models are discussed as the simplest model frame-work in option pricing theory. Basically, option pricing models aredescribed in a probabilistic way. Thus, some terminologies and nota-tions in probability theory are introduced as well as terminologies infinance: arbitrage, market completeness and so on. The main goal ofthis chapter is to learn how to compute option prices. In addition,the so-called fundamental theorems of option pricing (FTAP) will beproven.

3

1.1 Model description

Consider a financial market being composed of one risky asset and one risklessasset. There are only two trading times in our market: t = 0 and T (> 0). Thatis why our market is said to be a one-period model.

Suppose that the interest rate of the riskless asset is given by r ≥ 0. Thatis, the price of the riskless asset is given by 1 at time 0 and 1 + r at time T .

t = 0 t = T

1 - 1 + r

Figure 1.1: The riskless asset price fluctuation

On the other hand, the price of the risky asset at time t is denoted by St, wheret = 0, T . Note that S0 is a positive real number, but ST is a random variable.In other words, there are multiple possible outcomes in our market. Let Ω bethe set of all possible outcomes. Remark that Ω is called the sample space.Now, we assume that Ω includes only two elements, that is, it is described as

Ω = {ωu, ωd}. (1.1)

That is why our model is called binomial. Thus, there are two possible pricesof the risky asset at time T : ST (ωu) and ST (ωd). Without loss of generality, wemay assume that ST (ωu) > ST (ωd) > 0.

t = 0 t = T

S0��

��1

PPPPPPq

ST (ωu)

ST (ωd)

Figure 1.2: The risky asset price fluctuation

t = 0 t = T Event

96��

��1

PPPPPPq

120

80

ωu

ωd

1 - 1.05

Figure 1.3: An example of one-period binomial models

4

Remark 1.1 Roughly speaking, X is a random variable if it is a real-valuedfunction defined on Ω. In addition, a sequence of random variables with timeparameter is said to be a stochastic process. Thus, {St}t=0,T is a stochasticprocess. Henceforth, we call it the risky asset price process.

Now, consider a call option with strike price K > 0. Of course, itsunderlying asset is the risky asset St, and the maturity is T . Thus, its payoffis described as

(ST −K)+(= max{ST −K, 0} = (ST −K) ∨ 0). (1.2)

That is, the payoff is also a random variable. Now, we denote it by CT =(ST −K)+. Note that this equality means

CT (ω) = (ST (ω)−K)+ for any ω ∈ Ω. (1.3)

Here, we prepare some mathematical terminologies. Let F be the family ofall subsets of Ω, that is, the power set of Ω. Since Ω = {ωu, ωd}, we have

F = {Ω, ∅, {ωu}, {ωd}}. (1.4)

Note that F is a set of sets, which is called a family. Each element of F iscalled a measurable set in Lebesgue measure theory. We regard it as a set whoseprobability is observable.

P is called a probability on (Ω,F), if it is a [0, 1]-valued function definedon F such that P(Ω) = 1, and

P(A ∪B) = P(A) +P(B) (1.5)

for A, B ∈ F with A ∩B = ∅. In addition, E[X], which is the expectation ofrandom variable X, is defined as

E[X] :=∑ω∈Ω

X(ω)P({ω}). (1.6)

orE[X] :=

∑k

kP(X = k), (1.7)

where the summation is taken over all values of X. Note that P(X = k) meansP({ω ∈ Ω|X(ω) = k}).

5

1.2 Option pricing

1.2.1 Replication

We construct a portfolio being composed of a units of the riskless asset and bshares of the risky asset. Remark that a and b may take any real number, inparticular, may be negative in the case of short selling.

Remark 1.2 (Short selling (Short sales)) Borrowing an asset from some-one who owns it, you can sell it to someone else for the present price. In future,to settle your position, you pay the same amount money as the future price tothe lender.

Let Vt denote the value at time t of the portfolio (a, b). Thus, we have

V0 = a+ bS0 (1.8)

andVT (ω) = a(1 + r) + bST (ω) (1.9)

for any ω ∈ Ω.Consider a call option with strike price K. A portfolio (a, b) is said to

replicate the option, ifVT = CT , (1.10)

where CT = (ST −K)+. Remark that the above equation means

VT (ω) = CT (ω) (1.11)

holds for any ω ∈ Ω. To find a replicating portfolio, we have only to solve thefollowing simultaneous linear equations:{

a(1 + r) + bST (ωu) = CT (ωu),

a(1 + r) + bST (ωd) = CT (ωd).(1.12)

Remark that the above simultaneous equations have a unique solution (a, b) ba-sically. Actually, the price or, equivalently, the premium of the option, denotedby C0 is given by the initial cost V0 of the replicating portfolio, that is, thefollowing theorem holds:

Theorem 1.3C0 = V0 = a+ bS0. (1.13)

Example 1.4 Consider the one-period binomial model introduced in Figure1.3, and a call option with strike price K = 110. In this case, the replicatingportfolio (a, b) is given as the solution to{

1.05a+ 120b = 10,

1.05a+ 80b = 0.(1.14)

6

As a result, we obtain (a, b) = (− 201.05 ,14 ), which implies that

C0 = V0 = −20

1.05+

1

4× 96 = 104

21. (1.15)

□

Now, we show Theorem 1.3. To this end, we define arbitrage portfolios.

Definition 1.5 A portfolio with 0 initial cost is said to be an arbitrage port-folio if VT satisfies{

P(VT ≥ 0) = 1, (No risk)P(VT > 0) > 0. (Probability of capital gain is positive)

(1.16)

Suppose that there is no arbitrage opportunities∗ in the underlying market.This assumption is quite natural for the following reason: An arbitrage oppor-tunity is a good strategy for any investor, regardless of risk preference. As aresult of price mechanism, such an opportunity is considered to be vanishedimmediately. Thus, it is impossible to find any arbitrage opportunities.

Proof of Theorem 1.3. Being added the option CT , and given its priceC0, the market is enlarged to a three asset market. Thus, C0 should be givenso that the extended market still satisfy the no-arbitrage condition. Hence, inorder to see (1.13), it suffices to show that there is an arbitrage opportunity inthe extended market if C0 ̸= V0. The following table illustrates an arbitrageopportunity when C0 > V0.

t = 0 Cashflow t = T CashflowCall Selling C0 Exercise −CT

Risky asset Buying −bS0 Selling bSTRiskless asset Borrowing bS0 − C0 Payment (1 + r)(C0 − bS0)

Total 0 > 0

Table 1.1: An arbitrage opportunity when C0 > V0

The total cashflow at the maturity T is given as

−CT + bST + (1 + r)(C0 − bS0)> −a(1 + r)− bST + bST + (1 + r)(V0 − bS0)= −a(1 + r) + (1 + r)(a+ bS0 − bS0) = 0, (1.17)

since CT = VT = a(1 + r) + bST and C0 > V0 = a + bS0. Thus, this is anarbitrage. Therefore, C0 ≤ V0 holds.

∗We may replace the word “opportunity” by “strategy” or “portfolio”.

7

Similarly, the following is an arbitrage opportunity when C0 < V0:

t = 0 Cashflow t = T CashflowCall Buying −C0 Exercise CT

Risky asset Short selling bS0 Clearing −bSTRiskless asset Lending −bS0 + C0 Payment (1 + r)(bS0 − C0)

Total 0 > 0

Table 1.2: An arbitrage opportunity when C0 < V0

The total cashflow at the maturity T is given as

CT − bST + (1 + r)(bS0 − C0)> a(1 + r) + bST − bST + (1 + r)(bS0 − V0)= a(1 + r) + (1 + r)(bS0 − a− bS0) = 0, (1.18)

since C0 < V0. This is also an arbitrage. Therefore, C0 ≥ V0 holds.As a result, we have C0 = V0. This completes the proof of Theorem 1.3. □

As the main message of this subsection, we mention repeatedly that the priceof an option is given as the initial cost of the replicating portfolio.

1.2.2 Put-call parity

Consider the following one-period binomial model, and call and put options withstrike price K, where we assume that S > 0, r > d > 0 and r ≥ 0.

t = 0 t = T Event

S ��

��1

PPPPPPq

uS

dS

ωu

ωd

1 - 1 + r

Figure 1.4: Price fluctuation of one-period binomial model

We have then

C0 − P0 = S0 −K

1 + r, (1.19)

where C0 and P0 are the prices of the call and put options, respectively. Weconsider the following strategy:The total cashflow at the maturity is

−CT + PT + ST −K = −(ST −K)+ + (K − ST )+ + ST −K = 0 (1.20)

8

t = 0 Cashflow t = T CashflowCall Selling C0 Exercise −CTPut Buying −P0 Exercise PT

Risky asset Buying −S0 Selling STRiskless asset Borrowing K1+r Payment −K

Table 1.3: A strategy in the extended four assets model

for both cases ST > K and ST ≤ K. By the no-arbitrage condition, the initialcost of this strategy must be 0, that is,

C0 − P0 − S0 +K

1 + r= 0 (1.21)

holds. This relation is called the put-call parity.

1.2.3 Martingale probability

When we construct a replicating portfolio, the (underlying) probability P neverappear. Thus, we consider a convenient probability instead of P. Now, wedenote by Q such a probability, and

q := Q({ωu}). (1.22)

Note that 1 − q = Q({ωd}) automatically holds, and q must be in (0, 1). Wehope that Q satisfies

1

1 + rEQ[VT ] = V0, (1.23)

where EQ denotes the expectation under Q, that is,

EQ[VT ] = VT (ωu)Q({ωu})+VT (ωd)Q({ωd}) = qVT (ωu)+(1−q)VT (ωd). (1.24)

Note that 11+r is called the discount factor. Since

EQ[VT ] = EQ[a(1 + r) + bST ] = a(1 + r) + bEQ[ST ] (1.25)

and V0 = a+ bS0, the condition (1.23) is equivalent to

1

1 + rEQ[ST ] = S0, (1.26)

that is, the expected value of the discounted future price of the risky asset isequivalent to the present price. In general, a stochastic process whose futureexpected value is equivalent to the present value is called a martingale. Thatis, we can say that the discounted risky asset price process is a martingale underthe desirable probability Q, and call such a Q a martingale probability orrisk-neutral probability. Now, we give its precise definition and an optionpricing formula as follows:

9

Definition 1.6 A probability Q is called a martingale probability if it satisfies(1.26).

Theorem 1.7 Let Q be the martingale probability†. We have

C0 =1

1 + rEQ[CT ]. (1.27)

Remark 1.8 11+r in (1.27) is called the discount factor.

Proof. From Theorem 1.3 and (1.23), we have

C0 = V0 =1

1 + rEQ[VT ] =

1

1 + rEQ[CT ], (1.28)

since VT = CT holds. □

Example 1.9 In the model introduced in Example 1.4 (or Figure 1.3), q satis-fies

96 =120

1.05q +

80

1.05(1− q), (1.29)

that is, q = 0.52. In fact, computing C0 via Theorem 1.7, we get

C0 =1

1.05(10× 0.52 + 0× 0.48) = 5.2

1.05=

104

21. (1.30)

t = 0 t = T Event

96��

��1

PPPPPPq

120

80

ωu

ωd

1 - 1.05

Figure 1.5: The same model as Figure 1.3

Consequently, there are two ways to compute option prices under the no-arbitrage condition. One is the initial cost of the replicating portfolio. Anotheris the expectation of the discounted payoff of the option under the martingaleprobability.

†In our setting, Q exists uniquely under the no-arbitrage condition.

10

1.3 FTAP

1.3.1 1st FTAP

Consider the following one-period binomial model:

t = 0 t = T Event

S ��

��1

PPPPPPq

uS

dS

ωu

ωd

1 - 1 + r


Without loss of generality, we may assume that S > 0, u > d > 0 and r ≥ 0.The following is the 1st fundamental theorem of asset pricing (FTAP)for one-period binomial models, which asserts the equivalence between the no-arbitrage condition and the existence of martingale probabilities. Our aim is toprove it in a probabilistic way. Note that a linear algebraic proof will be givenin the next section.

Theorem 1.10 (1st FTAP) The underlying market is arbitrage-free if andonly if there exists a martingale probability.

Remark 1.11 We can extend Theorem 1.10 to more general models.

Proof. If u ≤ 1 + r, the market includes arbitrage opportunities as follows:At t = 0, an investor sells one share of the risky asset short, and lend $S‡ tosomeone. At the maturity, the investor gets

(1 + r)S − ST =

{(1 + r)S − uS ≥ 0, if ωu occurs,(1 + r)S − dS > 0, if ωd occurs.

(1.31)

which means that this is an arbitrage opportunity. Now, suppose that thereexists a martingale probability Q. We have then

1

1 + r(quS + (1− q)dS) = S, (1.32)

where q := Q({ωu}) and q ∈ (0, 1). Since d < u ≤ 1 + r, the left hand side of(1.32) is less than qS + (1 − q)S = S. This is a contradiction. Hence, there isno martingale probability.

‡This means that buying S units of the riskless asset.

11

Secondly, when d ≥ 1+ r, we can find an arbitrage opportunity by the samesort argument as the case of u ≤ 1 + r, and can see that there is no martingaleprobability.

Lastly, we consider the case where u > 1 + r > d. Assume that there is anarbitrage portfolio (a, b), that is, the following hold:

V0 = a+ bS = 0,

a(1 + r) + buS ≥ 0,a(1 + r) + bdS ≥ 0,Either of the above two inequalities holds strictly.

(1.33)

We have then a = −bS from the first line of (1.33). Substituting it for thesecond line, we get

−(1 + r)bS + buS = bS(u− 1− r) ≥ 0. (1.34)

Since u− 1− r > 0, we have b ≥ 0. On the other hand, the third line, togetherwith b− 1− r < 0, gives b ≤ 0. As a result, b = 0 holds. This is a contradictionto the last line of (1.33). Hence, the market satisfies the no-arbitrage condition.In this case, denoting

q :=1 + r − du− d

, (1.35)

q belongs to (0, 1), that is, q gives a martingale probability. This completes theproof of Theorem 1.10. □

Remark 1.12 By the above proof, we can say that a one-period binomial modelis arbitrage-free if and only if u > 1 + r > d holds.

1.3.2 2nd FTAP

We begin with the definition of “market completeness”.

Definition 1.13 A market is complete if all tradable options are replicable.

Actually, arbitrage-free one-period binomial models are complete. To see thisfact, we consider the following model and option:Note that Cu and Cd represent the payoff of the option when the event ωu andωd occur, respectively. Let (a, b) be a replicating portfolio, which is a solutionto the following: {

a(1 + r) + buS = Cu,

a(1 + r) + bdS = Cd.(1.36)

Hence, we get

a =1

1 + r

uCd − dCuu− d

, b =Cu − Cd(u− d)S

. (1.37)

12

t = 0 t = T Payoff Event

S ��

��1

PPPPPPq

uS

dS

Cu

Cd

ωu

ωd

1 - 1 + r

Figure 1.7: One-period binomial model and option payoff

Note that u > d is assumed. Hence, any option is replicable.From the 1st FTAP, the existence of martingale probabilities is guaranteed.

Letting Q be a martingale probability, and denoting q := Q({ωu}), we can seethat q satisfies

1

1 + r(quS + (1− q)dS) = S. (1.38)

This is a linear equation, that is, q is determined uniquely. As a result, amartingale probability exists uniquely. The following theorem, which is calledthe 2nd fundamental theorem of asset pricing (FTAP), holds.

Theorem 1.14 (2nd FTAP) When the underlying market is arbitrage-free,it is complete if and only if there exists a martingale probability uniquely.

Remark 1.15 1. As well as Theorem 1.10, we can show Theorem 1.14 formore general cases.

2. From the view of Theorem 1.14, for any option, its price is determineduniquely if the market is complete and arbitrage-free.

Next, we extend binomial models to the trinomial case as follows:

t = 0 t = T Payoff Event

S ��

��1

-PPPPPPq

uS

mS

dS

Cu

Cm

Cd

ωu

ωm

ωd

1 - 1 + r

Figure 1.8: Price fluctuation of trinomial models and option payoff

13

Without loss of generality, we may assume that u > m > d, S > 0 and r ≥ 0.The sample space Ω is denoted as

Ω = {ωu, ωm, ωd}. (1.39)

As long as u > 1 + r > d, the above model is arbitrage-free. Consider anoption with payoff Cu, Cm, Cd. Let (a, b) be a replicating portfolio. Thus, it isa solution to the following simultaneous equations:

a(1 + r) + buS = Cu,

a(1 + r) + bmS = Cm,

a(1 + r) + bdS = Cd.

(1.40)

There are 3 equations, but only 2 unknown variables. In general, (1.40) does nothave a solution. In other words, one-period trinomial models are not complete.From the view of the FTAPs, there are multiple (infinitely many) martingaleprobabilities in trinomial models. Indeed, letting Q be a martingale probability,and denoting

qu := Q({ωu}), qm := Q({ωm}), qd := Q({ωd}), (1.41)

where qu + qm + qd = 1 and 0 < qu, qm, qd < 1, we obtain

1

1 + r(quuS + qmmS + qddS) = S, (1.42)

equivalently,

quu− d

1 + r − d+ qm

m− d1 + r − d

= 1, and qd = 1− qu − qm. (1.43)

Consequently, we cannot determine the option price uniquely in trinomial mod-els.

Remark 1.16 In general, when the underlying market is arbitrage-free andincomplete, no-arbitrage option prices form the open interval(

infQ∈Q

1

1 + rEQ[CT ], sup

Q∈Q

1

1 + rEQ[CT ]

), (1.44)

where CT is the option payoff, and Q is the set of all martingale probabilities.That is, if the option price C0 is less than infQ∈Q

11+rEQ[CT ], then any investor

can construct an arbitrage strategy by purchasing the option. Similarly, whenC0 > supQ∈Q

11+rEQ[CT ], any investor can find an arbitrage opportunity by

selling the option.

14

1.4 Linear algebraic approach to FTAP

Consider the same one-period binomial model as Figure 1.6.

t = 0 t = T Event

S ��

��1

PPPPPPq

uS

dS

ωu

ωd

1 - 1 + r


Suppose that u > d > 0, S > 0 and r ≥ 0 as well as Figure 1.6.. In this section,we shall give proofs of the 1st and 2nd FTAPs using linear algebra. To this end,we describe the model in linear algebraic way, that is, the price fluctuation isdenoted by a 2-by-2 matrix D as follows:

D =

(1 + r uS1 + r dS

).← ωu← ωd

(1.45)

↑ ↑riskless asset risky asset

A portfolio is described by a 2-dimensional column vector

x =

(ab

), (1.46)

where a, b ∈ R, and the value of the portfolio x at the maturity T is denoted by(VT (ωu)VT (ωd)

)= Dx =

(a(1 + r) + buSa(1 + r) + bdS

). (1.47)

1.4.1 1st FTAP

To see the 1st FTAP, we show the following proposition firstly:

Proposition 1.17 Our market is arbitrage-free if and only if u > 1 + r > d.

Proof. Let x =

(ab

)be a portfolio with 0 initial cost, that is, a+ bS = 0

or, equivalently, a = −bS holds. We have then

Dx = bS

(u− 1− rd− 1− r

). (1.48)

15

Note that u > d > 0 is assumed. First of all, x is not an arbitrage portfolioas long as b = 0. Next, x is an arbitrage portfolio with b > 0 if and only ifu − 1 − r > 0 and d − 1 − r ≥ 0. In addition, x is an arbitrage portfolio withb < 0 if and only if u − 1 − r ≤ 0 and d − 1 − r < 0. As a result, there is noarbitrage portfolio in our market if and only if u− 1− r > 0 and d− 1− r < 0.□

Next, we discuss martingale probabilities. Note that a probability P is

identified with a vector

(pupd

)with pu + pd = 1, 0 < pu, pd < 1. Set

P :=

{(pupd

) ∣∣∣∣∣pu + pd = 1, 0 < pu, pd < 1}, (1.49)

which is corresponding to the set of all probabilities. A probability

(quqd

)∈ P

is a martingale probability if

1

1 + r

{quuS + qddS

}= S, (1.50)

which implies

quu+ qdd = 1 + r ⇐⇒ qu(u− 1− r) + qd(d− 1− r) = 0

⇐⇒(quqd

)⊥

(u− 1− rd− 1− r

). (1.51)

To complete the proof of the 1st FTAP, we show one more proposition.

Proposition 1.18 There exists a martingale probability if and only if u >1 + r > d.

Proof. From the view of (1.51), there exists a martingale probability if

and only if the vector

(u− 1− rd− 1− r

)is orthogonal to some

(quqd

)∈ P. Since

any vector in P lies in the interior of the 1st orthant, the above condition is

equivalent to that the vector

(u− 1− rd− 1− r

)lies in the interior of the 4th orthant,

that is, u− 1− r > 0 and d− 1− r < 0 hold simultaneously. □

1.4.2 2nd FTAP

Consider an option

c =

(cucd

), (1.52)

16

where cu, cd ∈ R. A portfolio x is called a replicating portfolio of c if it is asolution to

Dx = c. (1.53)

That is, if D is invertible, we have

x = D−1c, (1.54)

where

D−1 =1

(1 + r)S(d− u)

(dS −uS−1− r 1 + r

). (1.55)

Remark that D is invertible ⇐⇒ detD ̸= 0⇐⇒

(1 + r)dS − (1 + r)uS ̸= 0⇐⇒ u ̸= d. (1.56)

Thus, D is always invertible. In other words, any one-period binomial modelis complete. In addition, the uniqueness of martingale probabilities in binomialmodels holds clearly, since the linear equation on qu(

1

1 + rEQ[ST ] =

)1

1 + r

(qu 1− qu

)(uSdS

)= S (1.57)

has a unique solution. Remark that the price c0 of the option is given as

c0 =(1 S

)(ab

)=

(1 S

)D−1c. (1.58)

17

Exercises§

[1] Consider a one-period binomial model being composed of one risky assetand one riskless asset with 0 interest rate. Suppose that the fluctuationof the risky asset is described as follows:

t = 0 t = T

60��

��1

PPPPPPq

80

50

(a) Show the martingale probability.

(b) Show the replicating portfolio of a call option with strike price 55.

(c) Find its price.

(d) What is the price of a put option with strike price 60?

[2] Consider a one-period binomial model being composed of one risky assetand one riskless asset with 0 interest rate. Suppose that the fluctuationof the risky asset is described as follows:

t = 0 t = T

1000��

��1

PPPPPPq

SU

900

(a) Prove that our market is arbitrage-free whenever SU > 1000.(Hint: The first fundamental theorem of asset pricing is useful.)

(b) Construct an arbitrage portfolio when SU = 1000.

Let SU = 1050.

(c) Show the martingale probability.

(d) Show the replicating portfolio of a call option with strike price 980.

(e) Find its price.

(f) What is the price of a put option with strike price 930?

§For solutions, see Section B.2

18

[3] Consider a one-period binomial financial market model being composed ofone risky asset and one riskless asset whose interest rate is given by r > 0.The risky asset price process is assumed to be given by: Suppose that S,

t = 0 t = T

S��

��1

PPPPPPq

uS

dS

u and d are positive real numbers satisfying d < 1 + r < u.


(b) Show the replicating portfolio of a call option with strike price K,where dS < K < uS.

(c) Find the price of the call option.

[4] Consider the following one-period binomial model with interest rate 0.03.

t = 0 t = T

100��

��1

PPPPPPq

110

100


(b) Show the replicating portfolio of a call option whose strike price is109.

(c) Find its price.

(d) What is the price of a put option with strike price 101?

19

[5] Consider the following one-period binomial model with interest rate r > 0.

t = 0 t = T

90 ��

��1

PPPPPPq

108

94.5

(a) What is an equivalent condition on r > 0 to the arbitrage-free con-dition?

Let r = 0.1.

(b) Show the martingale probability.

(c) Show the replicating portfolio of a call option whose strike price is100.

(d) Find its price.

(e) What is the price of a put option with strike price 102?

[6] Consider the following one-period trinomial model with 0 interest rate.Suppose that the fluctuation of the risky asset is described as follows:

t = 0 t = T

100 ��

��

PPPPPP

120

90

80

(a) Show that the market is arbitrage-free.

(b) Show that the market is not complete.

(c) Describe all martingale probabilities.

(d) Consider a call option with strike price 85. Find an arbitrage portfoliofor the seller who sells the option for 352 .

20

Chapter 2

Multi-period binomialmodels

One period binomial models will be extended to multi-period modelsin this chapter. Option prices for multi-period models are obtainedby solving one-period models repeatedly. A formula on option pric-ing for multi-period binomial models, which is called the Cox-Ross-Rubinstein (CRR) formula, will be introduced. In the last section,we study pricing and hedging problems for American option, whichholders can exercise before the maturity. In particular, possibility ofearly exercise for American put options will be discussed.

21


Consider a multi-period binomial model being composed of one risky assetand one riskless asset with trading times t = 0, 1, . . . , T , where the maturity T isan integer. The price of the riskless asset at time t, denoted by S0t , is describedas

S0t = (1 + r)t (2.1)

for t = 0, 1, · · · , T , where r ≥ 0 is the interest rate (short rate) of our market.For t = 1, · · · , T , let Zt be a random variable taking only the two values uand d with u > d > 0. The distribution of Zt is independent of t, that is,P(Zt = u) is not depending on t. Moreover, Z1, Z2, · · · , ZT are assumed tobe independent. In this case, {Zt}t=1,··· ,T is called an i.i.d. (independent andidentically distributed) sequence.

Remark 2.1 (Independence) Two subsets (events) A and B ⊂ Ω are said tobe independent if

P(A ∩B) = P(A)P(B). (2.2)

On the other hand, two random variables X and Y are said to be independentif

P({X = x} ∩ {Y = y}) = P(X = x)P(Y = y). (2.3)

□

Example 2.2 Let Ω = {ω1, · · · , ω6}, and P({ωi}) = 16 for any i = 1, · · · , 6.Then, A = {ω1, ω3, ω5} and B = {ω3, ω6} are independent, since P(A ∩ B) =P({ω3}) = 16 , and P(A)P(B) =

12 ·

13 =

16 . □

Suppose that the risky asset price at time t, denoted by St, satisfy

St = St−1Zt (2.4)

for t = 1, · · · , T , and S0 > 0 is a constant. Note that there are 2T paths, thatis, #Ω = 2T . For example, when T = 3, Ω is denoted as

Ω = {ωuuu, ωuud, ωudu, ωduu, ωudd, ωdud, ωddu, ωddd}, (2.5)

where “uud” means that the 1st and 2nd movements are up, but the 3rd is down,that is, Z1(ω

uud) = u,Z2(ωuud) = u,Z3(ω

uud) = d and S3(ωuud) = u2dS0 hold.

Note that the distribution of Z1 is given as

P(Z1 = u) = P({ω ∈ Ω|Z1(ω) = u}) = P({ωuuu, ωuud, ωudu, ωudd}). (2.6)

22

t = 0 t = 1 t = 2 t = 3

S ��

�1PPPPq

uS ��

�1PPPPq

dS ��

�1PPPPq

u2S ��

�1PPPPq

udS ��

�1PPPPq

d2S ��

�1PPPPq

u3S

u2dS

ud2S

d3S

1 - 1 + r - (1 + r)2 - (1 + r)3

Figure 2.1: Price fluctuation of 3-period binomial model

Consider an investor rebalancing her position at each time t = 0, 1, · · · , T−1,using the risky asset prices S0, S1, · · · , St. Note that S0(= S) is a constant. Fort = 1, · · · , T , at represents the number of units of the riskless asset whichthe investor holds from time t − 1 until t. Similarly, bt denotes the numberof the shares of the risky asset held by the investor from time t − 1 until t.Thus, a portfolio is given by a pair of two stochastic processes {at}t=1,··· ,T and{bt}t=1,··· ,T . Note that the values of at and bt are functions on S1, · · · , St−1.For example, when T = 2, a2 and b2 are depending on the value of S1. In thiscase, a2(ω

uu) = a2(ωud) and a2(ω

du) = a2(ωdd) hold, but a2(ω

uu) might bedifferent from a2(ω

du). Moreover, the value of the portfolio (at, bt) at time t isdescribed as

Vt = at(1 + r)t + btSt (2.7)

for t = 1, · · · , T . Note that V0 = a1 + b1S holds.

t− 1 t

at and bt are constructed at+1 and bt+1 are constructed

riskless asset (1 + r)t−1 - (1 + r)t

risky asset St−1 - St

Figure 2.2: Portfolio dynamics from t− 1 to t

A portfolio (at, bt) is said to be self-financing if it satisfies that

at(1 + r)t + btSt = at+1(1 + r)

t + bt+1St (2.8)

for any t = 1, · · · , T . This means that the position is recombined at each timewithout injection of external funds or withdrawal of money. An arbitrageportfolio is defined as a self-financing one whose value process satisfies

V0 = 0,

P(VT ≥ 0) = 1,P(VT > 0) > 0.

(2.9)

23

Let Q be a probability on Ω. For simplicity, suppose that Q({ω}) > 0 forany ω ∈ Ω. Q is called a martingale probability if it satisfies

s =1

1 + rEQ[St+1|St = s] (2.10)

for t = 0, 1, · · · , T − 1, and any s ∈ R with P(St = s) > 0. Remark that we canrewrite (2.10) as

St =1

1 + rEQ[St+1|St]. (2.11)

Remark 2.3 (Conditional probability and conditional expectation) LetA, B be subsets of Ω (in other words, events) with P(B) > 0. The conditionalprobability of A given B (or the probability of A under the condition B),denoted by P(A|B), is defined as

P(A|B) := P(A ∩B)P(B)

. (2.12)

Let X be a random variable. The conditional expectation of X given B isdefined as

E(X|B) :=∑k

P(X = k|B) =∑k

P({X = k} ∩B)P(B)

. (2.13)

□

Example 2.4 Considering the same model as Example 2.2, we have

P(A|B) = P(A ∩B)P(B)

=1/6

1/2=

1

3. (2.14)

□

Since St+1 takes the value of either us or ds when St = s, we have

EQ[St+1|St = s] = usQ(St+1 = us|St = s) + dsQ(St+1 = ds|St = s)= usQ(Zt+1 = u) + dsQ(Zt+1 = d) = sEQ[Zt+1], (2.15)

that is,EQ[St+1|St] = StEQ[Zt+1]. (2.16)

Remark that

Q(St+1 = us|St = s) =Q(St+1 = us, St = s)

Q(St = s)=

Q(Zt+1 = u, St = s)

Q(St = s)

=Q(Zt+1 = u)Q(St = s)

Q(St = s)= Q(Zt+1 = u). (2.17)

24

Denoting qu = Q(Zt = u) and qd = Q(Zt = d)∗, we have the following equiv-

alence: Q is a martingale probability if and only if EQ[Zt] = 1 + r, that is,uqu + dqd = 1 + r. As a result, we obtain

qu =1 + r − du− d

. (2.18)

By the same sort argument as one-period binomial models, we can see thefollowing theorem:

Theorem 2.5 A multi-period binomial model is arbitrage-free if and only ifu > 1 + r > d.

∗qu and qd are independent of t.

25

2.2 Cox-Ross-Rubinstein (CRR) formula

We start with the following example:

Example 2.6 We consider a multi-period binomial model with T = 3, S0 = 80,u = 1.5, d = 0.5 and r = 0. The risky asset price dynamics is illustrated asfollows:

t = 0 t = 1 t = 2 t = 3

80 ��

�1PPPPq

120��

�1PPPPq

40 ��

�1PPPPq

180 ��

�1PPPPq

60 ��

�1PPPPq

20 ��

�1PPPPq

270

90

30

10

Figure 2.3: Price dynamics of risky asset

Consider a call option with strike price 80. That is, its payoff is described asC3 = (S3 − 80)+. To compute the price C0, we firstly focus on the followingone-period binomial model:

t = 2 t = 3 C3

180��

��1

PPPPPPq

270

90

190

10

Figure 2.4: Upper-right corner of Figure 2.3 and option payoff

Since

qu =1− 0.51.5− 0.5

=1

2, (2.19)

C2, which is the price of the option at time t = 2, under the condition “S2 =180”, is given as

C2 = EQ[C3|S2 = 180] =1

2190 +

1

210 = 100. (2.20)

By similar calculations, the values of C2 for other cases are given. As a result,we obtain

C2 =

100 if S2 = 180,5 if S2 = 60,0 if S2 = 20.

(2.21)

26

Next, to compute C1 when S1 = 120, we focus on the following one-periodbinomial model:

t = 1 t = 2 C2

120��

��1

PPPPPPq

180

60

100

5

Figure 2.5: A part of Figure 2.3 and option price at time 2

By the same calculation for C2, C1 is given as

C1 =

{52.5 if S1 = 120,2.5 if S1 = 40.

(2.22)

Finally, we get

C0 =1

2(52.5 + 2.5) = 27.5. (2.23)

In summary, the dynamics of Ct is given as follows:

t = 0 t = 1 t = 2 t = 3

27.5��

�1PPPPq

52.5��

�1PPPPq

2.5 ��

�1PPPPq

100 ��

�1PPPPq

5 ��

�1PPPPq

0 ��

�1PPPPq

190

10

0

0

Figure 2.6: Dynamics of option prices

By the same way as the above, we can calculate the replicating portfolio.For example, when S2 = 180, the values of a3 and b3 are given as a solution tothe following equation: {

a3 + 270b3 = 190,

a3 + 90b3 = 10.(2.24)

Thus, we obtaina3 = −80, b3 = 1. (2.25)

□

Considering a general multi-period binomial model, we shall generalize theresult of Example 2.6. By the same argument as the above example, we can

27

provide the price dynamics and the replicating portfolio for any option whosepayoff is given as a function of ST , that is, CT = f(ST ), where f is an R-valuedfunction on R. Note that a self-financing portfolio ({at}t=1,··· ,T , {bt}t=1,··· ,T )is said to replicate the option f(ST ) if its value process {Vt}t=0,··· ,T satisfiesVT = f(ST ). Since multi-period binomial models are complete, we have then

Ct = (1 + r)−(T−t)EQ[f(ST )|St](= Vt). (2.26)

Now, we give a formula for Ct, which is called the Cox-Ross-Rubinstein(CRR) formula. Consider the following model and an option f(ST ):

t = 0 t = 1 · · · · · · · · · t = T Payoff

S ��

�1PPPPq

uS ��

�1PPPPq

dS ��

�1PPPPq · · · · · · · · ·

· · · · · · · · ·

· · · · · · · · ·

��1

PPPPq

��1

PPPPq

uTS

uT−1dS

uT−2d2S.........

PPPPq dTS

f(uTS)

f(uT−1dS)

f(uT−2d2S)...

...

...

f(dTS)

Figure 2.7: Dynamics of general multi-period binomial models and option payoff

LetQ be the martingale probability, qu := Q(St+1 = us|St = s) and qd := 1−qu.We have the following:

Theorem 2.7 (CRR formula) For t = 0, 1, · · · , T − 1, the price Ct whenSt = s is given as follow:

Ct = (1 + r)−(T−t)

T−t∑k=0

(T − tk

)qT−t−ku q

kdf(u

T−t−kdks), (2.27)

where (T − tk

)= T−tCk =

(T − t)!(T − t− k)!k!

. (2.28)

Note that

(T − tk

)qT−t−ku q

kd is equivalent to Q(ST = u

T−t−kdks|St = s).

Remark 2.8 (Combination or binomial coefficient) nCk gives the num-ber of ways to choose k elements from a set of n elements, or the coefficient ofxk term in polynomial expansion of (1 + x)n.

28

2.3 American options

A European option gives its holder the right to exercise it at the maturity. Onthe other hand, a holder of an American option has the right to exercise theoption at any time up to the maturity.

2.3.1 American call options

Consider an American call option with strike price K in a multi-period binomialmodel with maturity T and short rate r. If the holder exercises the option attime t, its payoff is given as (St−K)+. In particular, the payoff at the maturityT is (ST −K)+, which is the same as the European call option with the samestrike price.

Let CAt and CEt be the prices of the American and European call options

with strike price K at time t. We have then

CAT = CET . (2.29)

Contrary to intuition, we can generalize this result to all t = 0, 1, · · · , T asfollows:

Theorem 2.9 CAt = CEt holds for any t = 0, 1, · · · , T .

Proof. We have already seen CAT = CET . Supposing that

CAs < CEs (2.30)

holds for some s = 0, 1, · · · , T − 1, we can construct an arbitrage portfolio asfollows: At time s, an investor sells one unit of the European call option, andbuys one unit of American call option and CEs −CAs amount of the riskless asset.Note that her cashflow at time t is 0. Keeping the American option until thematurity T , she gets (ST − K)+ at the maturity by exercising the Americanoption, and has to pay (ST −K)+ simultaneously, since the European option isexercised. As a result, the balance of her bank account is

(1 + r)T−s(CEs − CAs ), (2.31)

which is positive, that is, this is an arbitrage portfolio. Consequently, CAt ≥ CEtholds for any t = 0, 1, · · · , T − 1.

t = s Cashflow t = T Cashflow

European Call Selling CEs Exercise −(ST −K)+American Call Buying −CAs Exercise (ST −K)+Riskless asset Lending −CEs + CAs Return (1 + r)T−s(CEs − CAs )

Total 0 > 0

29

Next, suppose thatCAs > C

Es (2.32)

holds for some s = 0, 1, · · · , T − 1. We can find an arbitrage portfolio again bya similar way. At time s, an investor sells one unit of the American option, andbuys one unit of European option and CAs − CEs amount of the riskless asset.If the American option is exercised at time u = s, s+ 1, · · · , T , then she has topay (Su −K)+ at time u. To do it, she sells short one unit of the risky assetat time u. Without loss of generality, we may assume Su ≥ K. Otherwise, theshort-selling is not needed. As a result, the balance of her bank account at timeu is

(1 + r)u−s(CAs − CEs ) +K (2.33)

At the maturity, settling her short-selling position and exercising the Europeanoption, she obtains

−ST + (ST −K)+. (2.34)

Consequently, the balance of her bank account at the maturity is

(1 + r)T−t(CAt − CEt ) + (1 + r)T−uK − ST + (ST −K)+

≥ (1 + r)T−t(CAt − CEt ) +K − ST + (ST −K)+

≥ (1 + r)T−t(CAt − CEt ) > 0. (2.35)

This is an arbitrage portfolio. Therefore, Theorem 2.9 follows.

t = s Cashflow t = u Cashflow t = T Cashflow

European Call Buying −CEs Exercise (ST −K)+American Call Selling CAs Exercise −(Su −K)+Risky asset Shortsale Su Clearing −ST

Riskless asset Lending CEs − CAs Lending −K Return X, see belowTotal 0 0 > 0

Note: X = (1 + r)T−s(−CEs + CAs ) + (1 + r)T−uK.

□

Remark that

CAt = CEt =

1

(1 + r)T−tEQ[(ST −K)+|St] (2.36)

for any t = 0, 1, · · · , T , where Q is the martingale probability. For example, ifthe holder exercises the option at time T −1, its payoff is (ST−1−K)+, but thevalue of the option CAT−1 is

1

1 + rEQ[(ST −K)+|ST−1], (2.37)

which means

CAT−1 = max

{(ST−1 −K)+,

1

1 + rEQ[(ST −K)+|ST−1]

}

30

=1

1 + rEQ[(ST −K)+|ST−1], (2.38)

that is,

(St −K)+ ≤1

(1 + r)T−tEQ[(ST −K)+|St] (2.39)

for any t = 0, 1, · · · , T . Actually, we can generalize this result as follows: Con-sider an American option described by a function f : R→ R, that is, the payoffat time t is f(St), e.g. the call option with strike price K is corresponding tothe case of f(x) = (x−K)+. Assume that f is a convex function with f(0) = 0.

Remark 2.10 (Convex functions) A function f is convex if

f(αx+ (1− α)y) ≤ αf(x) + (1− α)f(y) (2.40)

holds for any x, y ∈ R and any α ∈ [0, 1].

Now, we introduce the following inequality:

Lemma 2.11 (Jensen’s inequality) When f : R → R is a convex function,we have

E[f(X)] ≥ f(E[X]). (2.41)

Lemma 2.11, together with the convexity of f , implies that

1

1 + rEQ[f(St+1)|St]

= EQ

[1

1 + rf(St+1) +

(1− 1

1 + r

)f(0)|St

]≥ EQ

[f

(1

1 + rSt+1 + 0

)|St

]= EQ

[f

(1

1 + rSt+1

)|St

]≥ f

(EQ

[1

1 + rSt+1|St

])= f(St), (2.42)

which means that the holder should not exercise the option at time t, andpossess it until time t+1. Thus, any American option whose payoff is expressedby a convex function f with f(0) = 0 has the same price as the correspondingEuropean option.

2.3.2 American put options

Next, consider American put options. Note that the payoff function of a putoption is given as a convex function, but f(0) ̸= 0. Indeed, as seen in thefollowing example, American put options might be exercised earlier than thematurity.

31

t = 0 t = 1 t = 2 t = 3 Payoff

80 ��

�1PPPPq

120��

�1PPPPq

40 ��

�1PPPPq

180 ��

�1PPPPq

60 ��

�1PPPPq

20 ��

�1PPPPq

270

90

30

10

0

0

50

70

Figure 2.8: Price dynamics of Figure 2.3 and payoff of the put option

Example 2.12 Consider the following 3 period binomial model and an Amer-ican put option with strike price 80:

Let r > 0 be the short rate. We need to assume r < 12 additionally for theno-arbitrage condition.

When S2 = 20, PA2 , the price of the American put option at time 2, is given

as

PA2 = max

{(K − S2)+,

1

1 + rEQ[(K − S3)+|S2 = 20]

}= max

{60,

50qu + 70(1− qu)1 + r

}= max{60, 60− 20r

1 + r} = 60, (2.43)

where Q is the martingale probability, and qu := Q(S1 = 120) =12 + r. In this

case, the holder should exercise the option at time 2.On the other hand, when S2 = 60, we have

PA2 = max

{(K − S2)+,

1

1 + rEQ[(K − S3)+|S2 = 60]

}= max

{20,

0× qu + 50(1− qu)1 + r

}= max

{20,

25− 50r1 + r

}. (2.44)

Thus, early exercise occurs if r > 570 .

32

Exercises†

[1] We consider the following two-period binomial model with short rate r =0.1.

t = 0 t = 1 t = 2

20 ��

��1

PPPPPPq

30

10

��

��1

PPPPPPq

45

15

��

��1

PPPPPPq 5

(a) Find the distributions of S1 and S2 under the martingale probability.

(b) Find the price process and the replicating portfolio of the (European)call option with strike price 40.

(c) What is the price process of the American put option with strike price14? In addition, when should a holder exercise the option before thematurity?

[2] Consider the following three-period binomial model with 0 short rate.

t = 0 t = 1 t = 2 t = 3

270��

�1PPPPq

360��

�1PPPPq

180��

�1PPPPq

480��

�1PPPPq

240��

�1PPPPq

120��

�1PPPPq

640

320

160

80

(a) Let Q be the martingale probability. What is Q(S3 = 320)?

(b) Find the price process of the American call option with strike price120.

(c) Find the price process of the American put option with strike price130.

(d) For the above call and put options, confirm that there is no possibilityof early exercise.

[3] Express the CRR formula for call options for T -period binomial models.

†For solutions, see Section B.3

33

[4] (Put-call parity for American options) Consider a T -period binomialmodel with short rate r ≥ 0, and American call and put options withthe same strike price K. Show the following inequality:

S0 −K

(1 + r)T≥ C0 − P0 ≥ S0 −K, (2.45)

where S0, C0 and P0 are the risky asset price at t = 0, the call optionpremium and the put option premium, respectively.

[5] Consider a multi-period binomial model with short rate r = 0. Show thatearly exercise for American put options never occur.

34

Chapter 3

General one-period models

In this chapter, we consider general one period models in which mul-tiple risky assets are traded, and multiple possible outcomes exist.The asset price fluctuation for such models is described by a matrix.Our goal of this chapter is to show the 1st and 2nd FTAPs for suchmodels using linear algebra.

35


Consider a one-period model being composed of N(≥ 2) tradable assets, thatis, one riskless asset and N − 1 risky assets. Let Ω be the sample space, whichis given as

Ω = {ω1, · · · , ωM}. (3.1)Without loss of generality, we may suppose that P({ωj}) > 0 for any j =1, · · · ,M . For any i = 1, · · · , N and j = 1, · · · ,M , SiT (ωj) denotes the price ofthe i-th asset at the maturity T when the event ωj occurs. Suppose that the1st asset is riskless, that is, its price at the maturity is given as

S1T (ωj) = 1 + r (3.2)

for any j = 1, · · · ,M , where r(≥ 0) is the interest rate of our market. Fori = 2, · · · , N , the price of the i-th asset at time 0, denoted by Si0, is a constant.In addition, set S10 = 1, and denote

S0 = (1, S20 , · · · , SN0 )⊤, (3.3)

where ⊤ denotes the transposed vector, that is, S0 is an N -dimensional columnvector. The asset price at the maturity is denoted by an N -by-M matrix Ddefined as

D :=

S1T (ω1) S

1T (ω2) · · · S1T (ωM )

S2T (ω1) S2T (ω2) · · · S2T (ωM )

......

. . ....

SNT (ω1) SNT (ω2) · · · SNT (ωM )

=

1 + r 1 + r · · · 1 + rS2T (ω1) S

2T (ω2) · · · S2T (ωM )

......

. . ....

SNT (ω1) SNT (ω2) · · · SNT (ωM )

.(3.4)

On the other hand, a portfolio is described as an N -dimensional row vector

x = (x1, · · · , xN ). (3.5)

Note that xi represents the number of units of the i-th asset held by the investorfrom time 0 to the maturity T . The corresponding value process to the portfoliox, denoted by V x, is given as

V x0 = x · S0 =N∑i=1

xiSi0, (3.6)

V xT (ωj) = x · dωj =N∑i=1

xiSiT (ωj) (3.7)

for any j = 1, · · · ,M , where

dωj := (S1T (ωj), · · · , SNT (ωj))⊤. (3.8)

In addition, we define a vector VxT as

VxT = (VxT (ω1), · · · , V xT (ωN )). (3.9)

36

Recall that x ·S0 is the inner product between the two vectors x and S0, andthe matrix D is also expressed as

D =(dω1 · · · dωM

). (3.10)

We need more three preparations. A portfolio x with 0 initial cost, that is,x · S0 = 0, is said to be an arbitrage portfolio if{

P(V xT ≥ 0) = 1,P(V xT > 0) > 0.

(3.11)

Remark that the above 1st and 2nd conditions are equivalent to

V xT (ωj) ≥ 0 for any j ∈ {1, · · · ,M}, (3.12)

andV xT (ωj) > 0 for some j ∈ {1, · · · ,M}, (3.13)

respectively. For later use, we define the discounted price matrix D̂ as

D̂ :=1

1 + rD =

1 1 · · · 1

S2T (ω1)1+r

S2T (ω2)1+r · · ·

S2T (ωM )1+r

......

. . ....

SNT (ω1)1+r

SNT (ω2)1+r · · ·

SNT (ωM )1+r

. (3.14)Lastly, a probability Q on Ω is said to be a martingale probability if itsatisfies

1. Q({ωj}) > 0 for any j = 1, · · · ,M ,

2. EQ

[SiT1 + r

]= Si0 for any i = 1, · · · , N .

37

3.2 1st FTAP

In this section, we give a proof of the following 1st FTAP for general one-periodmodels.

Theorem 3.1 (1st FTAP) The underlying market model is arbitrage-free ifand only if a martingale probability exists.

To see the above theorem, we show the following lemma firstly:

Lemma 3.2 (Farkas) Let A be an n-by-m matrix, b an n-dimensional columnvector. Then exactly one of the following two problems possesses a solution.

Problem 1 Find a nonnegative m-dimensional column vector∗ u such that

b = Au (3.15)

Problem 2 Find an n–dimensional row vector v such that{v · b < 0vA ≥ 0, that is, vA is a nonnegative m-dimensional row vector,

(3.16)where 0, different from 0, is the vector whose all elements are zero.

Proof of Lemma 3.2. For j = 1, · · · ,m, we denote the j-th column of Aby aj , that is,

A =

↑ ↑ ↑a1 a2 · · · am↓ ↓ ↓

. (3.17)Note that aj is an n-dimensional column vector. Letting aij be the (i, j)-entryof A, we can describe the vector aj as

aj = (a1j , a2j , · · · , anj)⊤. (3.18)

Denoting u = (u1, · · · , um)⊤, we have

Au =

a11 a12 · · · a1ma21 a22 · · · a2m...

.... . .

...an1 an2 · · · anm

u1u2...

um

=

a11u1 + a12u2 + · · ·+ a1muma21u1 + a22u2 + · · ·+ a2mum

...an1u1 + an2u2 + · · ·+ anmum

=

∑m

j=1 a1juj∑mj=1 a2juj

...∑mj=1 anjuj

=m∑j=1

uj

a1ja2j...

anj

=m∑j=1

ujaj . (3.19)

∗A vector is said to be nonnegative if all elements are nonnegative.

38

Thus, (3.15) in Problem 1 is rewritten as

b =

m∑j=1

ujaj . (3.20)

Next, we have

vA = v

↑ ↑ ↑a1 a2 · · · am↓ ↓ ↓

= (v · a1,v · a2, · · · ,v · am) . (3.21)Then, (3.16) in Problem 2 is rewritten as

v · b < 0, v · aj ≥ 0 for any j = 1, · · · ,m. (3.22)

Now, define

K :=

m∑j=1

cjaj

∣∣∣∣ c1 ≥ 0, c2 ≥ 0 · · · , cm ≥ 0 . (3.23)

Note that each element of K is an n-dimensional column vector, and K isa convex cone, that is, the set of all linear combinations with nonnegativecoefficients. In particular, a1, · · · ,am and 0 are included in K.

If b ∈ K, then there exist coefficients c1 ≥ 0, c2 ≥ 0 · · · , cm ≥ 0 satisfying

b =

m∑j=1

cjaj , (3.24)

that is, c := (c1, c2, · · · , cm)⊤ is a solution to Problem 1 from the view of (3.20).On the other hand, taking arbitrarily an n-dimensional row vector v satisfyingvA ≥ 0, that is, v · aj ≥ 0 for any j = 1, · · · ,m, we have

v · b = v ·m∑j=1

cjaj =

m∑j=1

cj(v · aj

)≥ 0. (3.25)

Thus, such a vector v is not a solution to Problem 2. Next, taking v arbitrarilysuch that v · b < 0, we can find a j ∈ {1, · · · ,m} such that v · aj < 0, whichmeans that v is not a solution to Problem 2. As a result, there is no solutionto Problem 2 if b ∈ K.

If b /∈ K, then Problem 1 does not have a solution. On the other hand, bythe separation theorem for convex sets, there exists a hyperplane H whichseparates b from K, namely, there exists a vector v such that v · b < 0 andv · k ≥ 0 for any k ∈ K†. Thus, v · aj ≥ 0 holds for any j ∈ {1, · · · ,m}. Thus,v is a solution to Problem 2. This completes the proof of Lemma 3.2. □

†v ·k ≥ 0 (resp. < 0) is equivalent to the angle formed by v and k is acute (resp. obtuse),and a normal vector of H might be a candidate of such vectors v.

39

Remark 3.3 (Separation theorem for convex sets) A set H ⊂ Rn is saidto be a hyperplane, if it is described as

H = {x ∈ Rn|v · x = α} (3.26)

for some α ∈ R and v ∈ Rn. The separation theorem asserts that, for any(closed) convex set K and any vector b ∈ Rn\K, we can find a hyperplane Hseparating K from b. In other words, there exist α ∈ R and v ∈ Rn such that

K ⊂ {x ∈ Rn|v · x > α}, and b ∈ {x ∈ Rn|v · x < α}. (3.27)

In particular, when K is a convex cone, the convex set K\{0} is separated fromb ∈ Rn\K by a hyperplane H with α = 0. □

Let us go back to the proof of Theorem 3.1. First of all, note that theunderlying market is arbitrage-free if and only if there is no N -dimensional rowvector x such that

x · S0 = 0, (0 initial cost)

xD̂ ≥ 0⇐⇒ (xD̂)j ≥ 0 for j = 1, · · · ,M, (no risk)

(xD̂)j > 0 for some j ∈ {1, · · · ,M}. (possibility of capital gain)

(3.28)

Denoting 1 = (1, · · · , 1)⊤ (M -dimensional vector), we can rewrite (3.28) asfollows:

x · S0 = 0,

xD̂ ≥ 0

xD̂ · 1 > 0.

(3.29)

Since x · S0 = 0 is equivalent to “x · S0 ≥ 0 and x · (−S0) ≥ 0”, denotingd := −D̂1, (N -dimensional column vector)D := (D̂ S0 −S0) , (N -by-(M + 2) matrix) (3.30)we see that the market is arbitrage-free if and only if{

x · d < 0,xD ≥ 0

(3.31)

has no solution x.By Lemma 3.2, the market is arbitrage-free if and only if there is a nonneg-

ative (M + 2)-dimensional column vector u such that

d = Du. (3.32)

Defineβ := (u1, · · · , uM )⊤, and α := uM+2 − uM+1, (3.33)

40

where u = (u1, · · · , uM , uM+1, uM+2)⊤. We have then

d = D̂β − αS0, (3.34)

that is,−D̂1 = D̂β − αS0. (3.35)

As a result, we obtainαS0 = D̂(β + 1). (3.36)

Note that β is a nonnegative vector. Recalling that the first entry of S0 is 1,and the first row of D̂ is 1, we have

α =

M∑j=1

(uj + 1) > 0. (3.37)

Thus, defining

qj :=1

α(uj + 1) (3.38)

for j = 1, · · · ,M , we have qj > 0 for any j = 1, · · · ,M and∑M

j=1 qj = 1.

Moreover, q := (q1, · · · , qM )⊤ satisfies

S0 = D̂q. (3.39)

Hence, q gives a martingale probability Q via

Q({ωj}) = qj (3.40)

for j = 1, · · · ,M . This completes the proof of Theorem 3.1. □

41

3.3 2nd FTAP

In this section, the 2nd FTAP and representations of option prices for generalone-period models will be discussed.

For any option traded in our market, its payoff is given by a random variableC defined on the sample space Ω. In addition, we can identify C with an M -dimensional row vector C through

C := (C(ω1), · · · , C(ωM )). (3.41)

Remark that the set of all options coincides with the set of all M -dimensionalrow vectors RM . An option C is said to be a replicable if there exists aportfolio x, which is an N -dimensional row vector, such that

VxT = C, that is, VxT (ω) = C(ω) for any ω ∈ Ω. (3.42)

Recall that

V xT (ωj) =

N∑i=1

xiSiT (ωj). (3.43)

In addition, the underlying market is complete if any option is replicable.Now, we show the following 2nd FTAP:

Theorem 3.4 (2nd FTAP) When the underlying market is arbitrage-free,the market is complete if and only if the martingale probability exists uniquely.

Proof. By the definition of the market completeness, the market is completeif and only if the equation

xD = C (3.44)

has a solution for any M -dimensional row vector C, which is equivalent to

Im f = RM , (3.45)

wheref(x) := xD, (linear mapping from RN to RM ) (3.46)

andIm f := {xD|x ∈ RN}. (3.47)

Next, we consider a linear mapping g : RM → RN defined as

g(y) := Dy, (3.48)

where y is an M -dimensional column vector‡. Moreover, we define the kernelof g as

Ker g := {y ∈ RM |g(y) = 0}. (3.49)

‡Dy is an N dimensional column vector.

42

We can see thatz · y = 0 (3.50)

for any y ∈ Ker g and any z ∈ Im f . Indeed, taking z ∈ Im f arbitrarily, we canfind x ∈ RN satisfying z = xD, which implies that

z · y = xDy = x · (Dy) = 0, (3.51)

since y ∈ Ker g.Now, we show the following lemma:

Lemma 3.5 Im f = RM is equivalent to Ker g = {0}

Proof of Lemma 3.5. Recall that Im f and Ker g both are linear spacesin RM . Assume that Im f = RM , and there is nonzero y ∈ Ker g. We have thenthat y ∈ RM = Im f and y · y = 0. This is a contradiction. Thus, Ker g = {0}holds when Im f = RM .

Next, suppose Ker g = {0} and dim Im(f) < M , that is, there is y /∈Im f\{0} such that y ⊥ Im f . We have then xDy = 0 for any x ∈ RN ,which means that Dy = 0, that is, y ∈ Ker g. This is a contradiction. Hence,Im f = RM follows when Ker g = {0}. □

As a result, the market is complete if and only if Ker g = {0}.On the other hand, the market is assumed to be arbitrage-free. Thus, there

exists a martingale probability Q. Let qj := Q({ωj}) for j = 1, · · · ,M , andq := (q1, · · · , qM )⊤. We have

D̂q = S0, equivalently, Dq = (1 + r)S0. (3.52)

Let q1 and q2 be two martingale probabilities. We have

Dq1 = (1 + r)S0, and Dq2 = (1 + r)S0, (3.53)

which imply D(q1−q2) = 0. Thus, q1−q2 ∈ Ker g, which implies that Ker g ={0} if and only if a martingale probability exists uniquely. This completes theproof of Theorem 3.4. □

Remark 3.6 (Image, kernel and dimension) Let T be a linear mappingfrom Rn to Rm. The image of T , denoted by ImT , is defined as a subset ofRm such that

ImT := T (Rn) = {y ∈ Rm|y = T (x) for some x ∈ Rn}. (3.54)

On the other hand, the kernel of T is defined as

KerT := {x ∈ Rn|T (x) = 0}, (3.55)

which is a subset of Rn. Note that both ImT and KerT form linear spaces.

43

Let V be a linear space. Its dimension, denoted by dimV , is defined asthe number of linearly independent vectors. Of course, dimRn = n holds. Fora linear mapping T : Rn → Rm, the following relation between ImT and KerTholds:

dim ImT + dimKerT = n. (3.56)

□

3.3.1 Option pricing

Consider an arbitrage-free complete general one-period model, and an optionC. The price of C, denoted by C0, is given by

C0 =1

1 + rEQ[C], (3.57)

where Q is the martingale probability. Remark that EQ[C] is defined as theinner product between C and q. More precisely,

EQ[C] =∑ω∈Ω

C(ω)Q({ω}) =M∑j=1

C(ωj)Q({ωj}) =M∑j=1

Cjqj = C · q, (3.58)

whereC = (C1, · · · , CM ) = (C(ω1), · · · , C(ωM )) (3.59)

andq = (q1, · · · , qM ) = (Q({ω1}), · · · ,Q({ωM})). (3.60)

Now, we show (3.57).Since the market is complete, any option is replicable, that is, we can find a

portfolio x satisfying

C = VxT =

N∑i=1

xiSiT = xD, (3.61)

where SiT = (SiT (ω1), S

iT (ω2), · · · , SiT (ωM )). Thus,

C0 = Vx0 =

N∑i=1

xiSi0 (3.62)

holds. On the other hand, there exists a martingale probability Q uniquely, andwe have

S0 = D̂q. (3.63)

Then,

C0 =

N∑i=1

xi(D̂q)i = x · D̂q =1

1 + rxDq

=1

1 + rC · q = 1

1 + rEQ[C], (3.64)

from which (3.57) follows.

44

Exercises§

In what follows, an (N,M)-model represents a one-period (t = 0, T ) model withN assets (including one riskless asset) and M scenarios(#Ω = M).

[1] We consider a (3, 3)-model with the initial price vector S0 and the pricematrix D given as

S0 =

13040

, D = 1 1 110 10 70100 10 10

. (3.65)(a) Show that the market is arbitrage-free.

(b) Find the martingale probability.

(c) Show the market completeness.

(d) Find the replicating portfolio of the option C = (10, 0, 0), and itsprice.

[2] We consider a (3, 3)-model with

S0 =

15040

, D =1.1 1.1 1.144 55 77

55 44 22

. (3.66)(a) Show that the market is arbitrage-free.

(b) Show that the market is not complete.

[3] We consider a (3, 4)-model with

S0 =

11012

, D = 1 1 1 18 9 9 u10 10 14 14

. (3.67)(a) Let u = 12. Show that the market is arbitrage-free, but not complete.

(b) Let u = 11. Show that the market is not arbitrage-free, and find anarbitrage opportunity.

(c) Show that the market is not complete for any u > 0.

§For solutions, see Section B.4

45

Appendix

46

A Past exam questions

A.1 AY2018

1 (10∗4 = 40 points) Consider the following one-period binomial model withinterest rate 0.1.

t = 0 t = T

S0��

��

PPPPPP

121

88

1. What is an equivalent condition on S0 > 0 to the arbitrage-free condition?

Let S0 = 100.

2. Find the replicating portfolio of the call option with strike price 110.

3. Find its price.

4. What is the price of the put option with strike price 99?

2 (10∗2 = 20 points) Consider the following two-period binomial model withshort rate 0.1.

t = 0 t = 1 t = 2

640��

��

PPPPPP

960

320

��

��

PPPPPP

1440

480

160

��

��

PPPPPP

1. Find the distribution of S1 (the risky asset price at time 1) under themartingale probability.

2. What is the price at time 0 of the American put option with strike price460?

47

3 (10 ∗ 3 = 30 points) Consider a one-period model with the following initialprice vector S0 and price matrix D:

S0 =

12020

, D = 1 1 116 16 24

28 4 16

.1. Show that the market is arbitrage-free and complete.

2. Find the martingale probability.

3. For the option C = (0, 0, 4), find its replicating portfolio and price.

Solutions

1 1. 80 < S0 < 110, 2. a = −80

3, b =

1

3, 3. C0 =

20

3, 4. P0 =

103 .

2 1. Q(S1 = 960) =3

5, 2. P0 =

560

11.

3 1. Omitted, 2.

(5

12,1

12,1

2

), 3. x =

(−8, 1

2, 0

), C0 = 2.

48

A.2 AY2019

1 (10∗4 = 40 points) Consider the following one-period binomial model withinterest rate 0.

t = 0 t = T

200��

��

PPPPPP

220

195

1. Show the martingale probability.

2. Find the replicating portfolio of the call option with strike price 205.

3. Find its price.

4. Find the price of the put option with strike price 200.

2 (10∗2 = 20 points) Consider the following two-period binomial model withshort rate 0.05.

t = 0 t = 1 t = 2

100��

��

PPPPPP

110

90

��

��

PPPPPP

121

99

81

��

��

PPPPPP

1. Find the price at time 0 of the American call option with strike price 120.

2. Find the price at time 0 of the American put option with strike price 98.

49

3 (10 ∗ 3 = 30 points) Consider a one-period model with the following initialprice vector S0 and price matrix D:

S0 =

11510

, D = 1 1 110 20 x

5 10 15

.1. Show that the market is not arbitrage-free when x = 30.

Let x = 15.

2. Find the martingale probability.

3. For the option C = (1, 0, 0), find its replicating portfolio.

Solutions

1 1. Q(ST = 220) =1

5, 2. a = −117, b = 3

5, 3. C0 = 3, 4. P0 = 4.

2 1.25

49, 2.

40

21.

3 1. Omitted, 2.

(1

3,1

3,1

3

), 3.

(2,− 1

15,− 1

15

).

50

B Solutions for exercises

B.1 Exercises in Prerequisite

[1] A ∪B = {α, β, γ, δ}, A ∩B = {γ}.

[2] P(X ≤ 0) = 34 , E[X] =14 .

[3] Omitted.

[4] AB =

−1 −8 −101 −2 −59 22 15

.[5]

(x1x2

)=

(2−1

).

[6] A−1 =

(3 −5−1 2

).

51

B.2 Exercises in Chapter 1

[1] (a) Q(ST = 80) =13 , Q(ST = 50) =

23 . (b) a = −

1253 , b =

56 . (c)

253 .

(d) 203 .

[2] (a) From the 1st FTAP, it suffices to show the existence of martingaleprobabilities. In other words, for any SU > 1000, we have only to findq ∈ (0, 1) satisfying qSU + (1 − q) × 900 = 1000. Actually, this equationholds when q = 100SU−900 , which is positive and less than

1001000−900 = 1.

(b) Selling one share of the risky asset short at t = 0, and lending 1000 tosomeone, the investor’s cashflow at the maturity T is either 0 or 100.(c) Q(ST = 1050) =

23 , Q(ST = 900) =

13 . 4. a = −420, b =

715 . 5.

1403 .

6. 10.

[3] (a) Q(ST = uS) =1+r−du−d , Q(ST = dS) =

u−1−ru−d .

(b) a = − d1+ruS−Ku−d , b =

uS−K(u−d)S . (c)

(1+r−d)(uS−K)(1+r)(u−d) .

[4] (a) Q(ST = 110) =310 , Q(ST = 100) =

710 . (b) a = −

1000103 , b =

110 .

(c) 30103 . (d)70103 .

[5] (a) 0.05 < r < 0.2. (b) Q(ST = 108) =13 , Q(ST = 94.5) =

23 .

(c) a = − 56011 , b =1627 . (d)

8033 . (e)

5011 .

[6] (a) (b) Omitted.(c) Let Q be a martingale probability, and denote qu := Q(ST = 120),qm := Q(ST = 90) and qd := Q(ST = 80). Note that Q can be identifiedwith a vector (qu, qm, qd). Then, the set of all martingale probabilities isdescribed as{

(qu, qm, qd)∣∣∣ 13< qu <

1

2, 4qu + qm = 2, qu + qm + qd = 1

}. (3.1)

(d) a = −70, b = 78 .

52


[1] (a) Q(S1 = 30) =35 , Q(S1 = 10) =

25 ;

Q(S2 = 45) =925 , Q(S2 = 15) =

1225 , Q(S2 = 5) =

425 ,

(b) The left and right figures give the price process and the replicationportfolio, respectively. In the right figure, (a, b) represents the portfoliocomposed of a shares of the riskless asset and b shares of the risky assets.

t = 0 t = 1 t = 2

180121

��

��

PPPPPP

3011

0

��

��

PPPPPP

��

��

PPPPPP

5

0

0

t = 0 t = 1

(− 150121 ,322 )

��

��

PPPPPP

(− 2511 ,16 )

(0, 0)

(c) The price process is given in the following figure, and early exercise isimplemented at the node S1 = 10.

t = 0 t = 1 t = 2

1611

��

��

PPPPPP

0

4

��

��

PPPPPP

0

0

��

��

PPPPPP 9

53

[2] (a) Q(S3 = 320) =38 ,

(b)

155��

�PPPP

240��

�PPPP

70 ��

�PPPP

360��

�PPPP

120��

�PPPP

20 ��

�PPPP

520

200

40

0

(c)

6.25��

�PPPP

0 ��

�PPPP

12.5��

�PPPP

0 ��

�PPPP

0 ��

�PPPP

25��

�PPPP

0

0

0

50

(d) Early exercise never occur.

[3] We consider a T -period binomial model with short rate r ≥ 0. Let(St)t=0,...,T be the risky asset price process such that St+1/St takes thevalue of either u or d for any t = 0, . . . , T − 1, where u > 1 + r > d > 0.Moreover, q denotes Q(S1 = uS0). Then, Ct the price of the call optionwith strike price K at time t is given by

Ct = (1 + r)−(T−t)

T−t∑k=m

(T − tk

)qk(1− q)T−t−k(ukdT−t−kSt −K), (3.2)

where m is the least integer satisfying ukdT−t−kSt ≥ K. Note that Ct isgiven as a function of St.

54

[4] To see the first inequality, we suppose that

C0 − P0 − S0 +K(1 + r)−T > 0 (3.3)

holds. It suffices to find an arbitrage portfolio under (3.3). Now, weconstruct the following portfolio:

t = 0 t = s t = TCall Selling C0 Exercise∗ −(Ss −K)Put Buying −P0 Exercise (K − ST )+

Risky asset Buying −S0 Selling SsRiskless asset Borrowing† −C0 + P0 + S0 Lending −K See below

Total 0 0 > 0

At the maturity, the investor receives (C0−P0−S0)(1+r)T +K(1+r)T−sshares of the riskless asset. (3.3) implies that this amount is greater than−K+K(1+ r)T−s, which is nonnegative. As a result, the above portfoliois an arbitrage. This contradicts to the no-arbitrage condition. We canconclude that (3.3) does not hold.

Next, we prove the second inequality by the same sort argument as theabove. Supposing that

−C0 + P0 + S0 −K > 0 (3.4)

holds, we construct the following portfolio:

t = 0 t = s t = TCall Buying −C0 Exercise (ST −K)+Put Selling P0 Exercise‡ −(K − Ss)

Risky asset Selling S0 Clearing −SsRiskless asset Lending C0 − P0 − S0 Borrowing K See below

Total 0 0 > 0

At the maturity, the investor receives (−C0+P0+S0)(1+r)T−K(1+r)T−sshares of the riskless asset. (3.4) implies that this amount is greater thanK(1 + r)T −K(1 + r)T−s ≥ 0. Thus, the above portfolio is an arbitrage.This contradicts to the no-arbitrage condition. We can conclude that (3.4)does not hold.

∗We suppose that the holder of the call option exercises the option at time s. Withoutloss of generality, we may assume Ss −K ≥ 0. If the call option is not exercised at all, thenthe investor sells the risky asset she holds at the maturity. In this case, her cashflow at thematurity is given as (K − ST )+ + ST + (C0 − P0 − S0)(1 + r)T > (K − ST )+ + ST −K ≥ 0.

†If −C0 + P0 + S0 is negative, the investor lends C0 − P0 − S0 at t = 0.‡We suppose that the holder of the put option exercises the option at time s. Without loss

of generality, we may assume K − Ss ≥ 0. If the put option is not exercised at all, then theinvestor settles her position of the risky asset at the maturity. In this case, her cashflow at thematurity is given as (ST−K)+−ST+(−C0+P0+S0)(1+r)T > (ST−K)+−ST+K(1+r)T ≥ 0.

55

[5] Let K be the strike price. Since the function f(x) = (K − x)+ is convex,Jensen’s inequality implies

EQ[(K − St+1)+|St] ≥ (K −EQ[St+1|St])+ = (K − St)+ (3.5)

for any t = 0, . . . , T −1, where EQ means the expectation under Q. Thus,early exercise never occur.

56


[1] (a)-(c) From the view of the 1st and 2nd FTAPs, it suffices to see thatthe equation Dq = S0 has a unique solution which is positive, where q =(q1, q2, q3)

⊤. By simple calculation, we have q = (1/3, 1/3, 1/3)⊤ is theunique solution to the above equation. Thus, the market is arbitrage-freeand complete. The martingale probability Q is given by Q({ωj}) = 1/3for any j = 1, 2, 3.(d) Noting that the replicating portfolio x is given as the solution to theequation xD = C, we have x = (−10/9, 0, 1/9). On the other hand, theprice of C, denoted by C0, is given as C0 = x ·S0 or C0 = EQ[C] = q ·C,from which we obtain C0 = 10/3.

[2] (a)-(b) Note that D̂ = 11.1D. Let q = (q1, q2, q3)⊤. Solving D̂q = S0, we

can see that q =(23 (1− a), a,

13 (1− a)

)⊤is a solution for any a ∈ (0, 1) .

Thus, the equation D̂q = S0 has infinitely many positive solutions q. Inother words, the market is arbitrage-free, but not complete.

[3] Letting q = (q1, q2, q3, q4)⊤ be a solution to the equation Dq = S0, we

have q1 + q2 = 1/2, q3 + q4 = 1/2 and q2 + (9 − u)q3 = 6 − u2 , regardlessof the value of u > 0.(a) When u = 12, q is a solution to Dq = S0 if and only if q1 + q2 = 1/2,q3 + q4 = 1/2 and q2− 3q3 = 0 hold. Thus, there are infinitely many posi-tive solutions. As a result, the market is arbitrage-free, but not complete.(b) When u = 11, q2− 2q3 = 1/2, that is, −2q3 = q1 holds. Thus, there isno positive solution. Then the no-arbitrage condition does not hold.Let x = (x1, x2, x3) be an arbitrage portfolio. Thus, x · S0 = 0 holds, andxD is a nonnegative vector having at least one positive entry. Now, wehave xD = (−2x2 − 2x3,−x2 − 2x3,−x2 + 2x3, x2 + 2x3)⊤. Since xD isnonnegative, x2 + 2x3 = 0 holds. Thus, we have xD = (2x3, 0, 4x3, 0)

⊤,which implies x3 > 0 follows. For example, x = (8,−2, 1) is an exampleof arbitrage portfolios. (c) Define g(y) := Dy, where y = (y1, y2, y3, y4)

⊤.By the proof of the 2nd FTAP, the market is complete if and only ifKer(g) = {0}, where Ker(g) := {y|g(y) = 0}. Now, we have y ∈Ker(g)⇐⇒ Dy = 0⇐⇒ “y1+y2 = 0, y3+y4 = 0 and y2+(u−9)y4 = 0”.Thus, we obtain Ker(g) = {((u − 9)a,−(u − 9)a,−a, a)⊤|a ∈ R}, fromwhich the market is not complete for any u > 0.

57

introduction to finance (pcp) - keio...

Documents