Relative Resource Manager

Second Semester, 2011-2012

THE UNIVERSITY OF HONG KONGDEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE

STAT2807 CORPORATE FINANCE FOR ACTUARIAL SCIENCE

Tutorial 10 (Finale): Option Prices Date: April 23 – 24, 2012

What is This Final Tutorial About?

In this final tutorial, we will study the fundamentals of option pricing theory. Afterexploring the properties of option prices that the no-arbitrage assumption necessitates, wewill briefly introduce the binomial option pricing model, the method of replicating portfolioand risk-neutral valuation. The problem section includes a large number of instructiveexercises, which are collected from a wealth of sources to strengthen your understanding.

1 Key Learning Points

In the examination, candidates are expected to:

LP(1) State, prove and manipulate the put-call parity.

LP(2) Use no-arbitrage arguments to prove inequalities involving option prices.

LP(3) Detect the existence of arbitrage opportunities, and if they exist, construct a portfolio toreap risk-free profit.

LP(4) Calculate the value of an option with a possibly unfamiliar payoff structure by (i) themethod of replicating portfolio, and (ii) risk-neutral valuation.

#

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!Message from Ambrose

This is the last tutorial. Enjoy!

2 Review of Key Concepts

2.1 Identities and Inequalities of Option Prices

• The no-arbitrage assumption effectively makes option prices non-arbitrary. As functionsof the stock price, strike price, time to maturity, they have to satisfy a number of identitiesand inequalities, including:

S&AS: STAT2807 Corporate Finance for Actuarial Science Ambrose LO 2012 2

1. (IMPORTANT!) Put-call parity: CE0 +Ke−rT = S0 + PE

0 .

2. Call prices are bounded (above and below): St −Ke−r(T−t) ≤ CA/Et ≤ St.

3. Put prices are bounded: Ke−r(T−t) − St ≤ PA/Et ≤ Ke−r(T−t).

4. Call prices are decreasing functions of the strike price: Ct(K1) ≥ Ct(K2) if K1 ≤ K2.

Interpretation: This is natural as the higher the strike price K, the lower thepayoff of a call option (ST −K)+.

5. Put prices are increasing functions of the strike price: Pt(K1) ≤ Pt(K2) if K1 ≤ K2.

Interpretation: The payoff of a put option is (K − ST )+, which increases with K.

6. Option prices are increasing in the time-to-maturity: If T1 > T2,

Ct(T1) ≥ Ct(T2) and Pt(T1) ≥ Pt(T2).

7. Option prices are convex 1 in the strike price (Problem 2 (a), Section 3.3): For allλ ∈ [0, 1],

Ct[λK1 + (1− λ)K2] ≤ λCt(K1) + (1− λ)Ct(K2)

Pt[λK1 + (1− λ)K2] ≤ λPt(K1) + (1− λ)Pt(K2)

8. Option prices are Lipschitz 2 in the strike price (Assignment 3):

|C1t − C2t| ≤ e−r(T−t)|K1 −K2|,|P1t − P2t| ≤ e−r(T−t)|K1 −K2|.

• What will I be asked in the Final Exam?

In the Final Exam, you may be asked to:

– Apply put-call parity to perform some calculations, or

– Prove any of the above inequalities.

Another possibility is to present some observed option prices and ask you whether themarket is arbitrage-free. Almost surely (otherwise, what is the point of that exam ques-tion!?!?) there will be arbitrage opportunities and you can follow the steps on page 4 toconstruct an arbitrage strategy.

1Recall that a real-valued function f defined on a convex set X is said to be convex if f(λx1 + (1− λ)x2) ≤λf(x1) + (1− λ)f(x2) for all x1, x2 ∈ X and λ ∈ [0, 1].

2A real-valued function f defined on a set X is said to be Lipschitz if there exists L > 0 such that |f(x) −f(y)| ≤ L|x− y| for all x, y ∈ X.


Exercise 1. (SOA Exam MFE Sample Q1: Straightforward Put-call Parity Manipu-lations) Consider a European call option and a European put option on a nondividend-payingstock. You are given:

• The current price of the stock is 60.

• The call option currently sells for 0.15 more than the put option.

• Both the call option and put option will expire in 4 years.

• Both the call option and put option have a strike price of 70.

Calculate the continuously compounded risk-free interest rate.

Solution. Applying put-call parity, we have

C0 − P0 = S0 −Ke−rT

0.15 = 60− 70e−4r

r = 0.039 .

Exercise 2. (SOA Exam MFE Spring 2009) You are given:

• C(K,T ) denotes the current price of a K-strike T -year European call option on anondividend-paying stock.

• P (K,T ) denotes the current price of a K-strike T -year European call option on anondividend-paying stock.

• S denotes the current price of the stock.

• The continuously compounded risk-free interest rate is r.

Which of the following is (are) correct?

(I) 0 ≤ C(50, T )− C(55, T ) ≤ 5e−rT

(II) 50e−rT ≤ P (45, T )− C(50, T ) + S ≤ 55e−rT

(III) 45e−rT ≤ P (45, T )− C(50, T ) + S ≤ 50e−rT

Solution. • (I) is true. Call price is a decreasing function of K, so C(50, T ) ≥ C(55, T ).The second inequality follows from Assignment 3 (Lipschizity).

• (II) is incorrect, but (III) is true. By put-call parity,

P (45, T )− C(50, T ) + S = [C(45, T )− S + 45e−rT ]− C(50, T ) + S

= C(45, T )− C(50, T ) + 45e−rT .

By (I), 0 ≤ C(45, T )− C(50, T ) ≤ (50− 45)e−rT , which is equivalent to

45e−rT ≤ C(45, T )− C(50, T ) + 45e−rT ≤ 50e−rT .


• (IMPORTANT!) How to Prove These Identities/Inequalities Systematically?Suppose you want to prove

♥(<)

≤ ♠. (1)

Step 1. Assume the contrary to (1), i.e. ♥(≥)> ♠.

Step 2. Buy low and sell high.

– At time 0, enter the transactions with a cost of ♠. For example, if ♠ =Ct(St, K, T ), then you should buy a call option.

– At the same time, short the transactions which cost ♥. For example, if ♥ =St−K exp−r(T−t), then you should short sell an asset and lend K exp−r(T−t).

Step 3. Verify that arbitrage profits exist by showing that the cashflow at time 0 ispositive (non-negative), and the cashflow at the maturity date T is non-negative.For clarity, you can present your answers in a table:

Transaction 1 Transaction 2 · · · TotalST < K · · · · · · · · · (≥ 0?)ST ≥ K · · · · · · · · · (≥ 0?)

Step 4. Argue that you have constructed an arbitrage strategy, so (1) must hold.

There are no better exercises than verifying as many of the identities and inequalitiesabove on your own. Check your proof with the one in the notes. To test your under-standing, some additional inequalities are provided in Problems 1 and 2 in Section 3.3.

2.2 Binomial Option Pricing Model (OPTIONAL)

2.2.1 Basics

Using binomial trees is a general, robust, but computationally intensive method to price op-tions3. Although simple expressions of option prices may not be available, all options can bepriced theoretically.

• Construction of Binomial Trees: Under the binomial option pricing model, the stockprice in the next time period is assumed to move either up by a factor of u or down by afactor of d:

S0

Su = S0 × u

Sd = S0 × d3Black-Scholes pricing formulae will not be treated in this course.


(Note: S is not always the underlying risky stock. It only has to be an asset from whichthe value of a derivative can be derived directly. For a compound option (May 2009 ExamQuestion 10), S is actually the price of another option!)

In this course, we shall not pursue issues concerning the determination of u and d. Inother words, the values of u and d are assumed to be given.

• Objective: The time-0 price of a derivative with payoffs fu and fd in the next period.Note that this derivative need not be a standard call or put option. Any payoff structures(e.g. May 2010 Exam Question 9 and Problem 1, Section 3.4), regardless of its irregularity,can be tackled by the binomial tree method.

Case 1. One-period Binomial Tree

• Method 1: Replicating Portfolio (Good for one-period model)

This method involves solving a pair of simultaneous equations. Construct a port-folio consisting of β shares of stock and cash of α, to mimick the payoff of thederivative of interest:{

βS0u+ αerT = fu

βS0d+ αerT = fd=⇒ α = ? , β = ? .

(Note: You need not remember the expressions of β and α.) By the law of one price,the option price must be equal to the initial cost in constructing the replicatingportfolio, which is

S0β + α .

• Method 2: Risk-neutral Valuation (Good for “most” cases!)

Define the risk-neutral probability

q ,erT − du− d

.

Then the option price can be rewritten as

e−rT [qfu + (1− q)fd] = e−rTEQ[Payoff].

Interpretation The price of an option can be evaluated as a discounted expectedpayoff, where:

1. “discounted” means discounting by the risk-free rate, and

2. “expected” means the expectation when the stock moves up with a proba-bility of q (but not the true probability!).

Case 2. Multi-period Binomial Tree

It is much easier to employ risk-neutral valuation (Method 2) when we have severalperiods in the binomial tree. By working backward through the binomial tree and con-sidering each node as a single-period binomial tree model, the option can be recursivelyvalued.


• Important Special Case: For European options, early exercise is not allowed.Therefore, we can simply discount the expected payoff at the end of the binomialtree back to time 0.

1. For a two-period tree, the price of the derivative is

e−rT [q2fuu + 2q(1− q)fud + (1− q)2fdd].

2. For a three-period tree, the price becomes

e−rT [q3fuuu + 3q2(1− q)fuud + 3q(1− q)2fudd + (1− q)3fddd].

Case 3. Trinomial Tree (This part is more challenging; suitable for more motivated students)

The risk-neutral probabilities for a stock to go up, stay put and go down are more difficultto obtain for a multinomial tree. In this case, the method of replicating portfolio comesto our rescue. The same idea, i.e. constructing a portfolio with available assets such thatthe payoff of the derivative can be replicated, still works, but a larger system of linearequations has to be solved. Please try Problem 3 in Section 3.4 for such a nonstandardproblem.

2.2.2 American Options

• Idea: For American options, the same kind of recursive risk-neutral valuation for Eu-ropean options can be performed. The option price at each node of the tree is givenby

max {discounted expected payoff, payoff from early exercise} .

• Important Special Case: If the underlying stock pays no dividends (as is the case inour course), then an American call is worth the same as a European call. Early exerciseneed not be accounted for.

Exercise 3. (SOA Exam MFE Sample Question 4: A Standard Exercise) For atwo-period binomial model, you are given:

• Each period is one year.

• The current price for a nondividend-paying stock is 20.

• u = 1.2840.

• d = 0.8607.

• The continuously compounded risk-free interest rate is 5%.

Calculate the price of an American call option on the stock with a strike price of 22.


Solution. The two-period binomial tree is constructed in Figure 1. The risk-neutral proba-bility for the stock price to go up is

q =e0.05 − 0.8607

1.2840− 0.8607= 0.450203.

• At node B: The value of the call option is

max

e−0.05 [q(10.9731) + (1− q)(0.1028)]︸︷︷︸=4.752922

, (25.680− 22)+︸︷︷︸=3.680

= 4.752922.

• At node C: The value of the call option becomes

max

e−0.05 [q(0.1028) + (1− q)(0)]︸︷︷︸=0.044023

, (17.214− 22)+︸︷︷︸=0

= 0.044023.

Early exercise is not optimal.

• At node A: Finally, the call price is

max

e−0.05 [q(4.752922) + (1− q)(0.044023)]︸︷︷︸=2.0584

, (20− 22)+︸︷︷︸=0

= 2.0584 .

Remark 1. As the stock pays no dividends, the American call price must be equal to theEuropean call price.

20A

25.68

B

17.214

C

32.9731 (10.9731)

22.1028 (0.1028)

14.8161 (0)

Figure 1: Binomial tree for Exercise 3.


3 Problems

Attempt ALL NINE questions in Sections 3.1 to 3.4. Marks for past paper ques-tions are shown in square brackets.

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Message from Ambrose

This problem section contains a wide variety of nonstandard questionscollected from past tests and examinations of a number of courses. In spiteof the sheer number of questions, please try as many of them as possibleduring your revision!

3.1 Miscellaneous Descriptive Questions

1. Miscellaneous Short Questions

(a) Explain how an option holder gains from the volatility of the underlying stock price. STAT2807

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Exam[4 marks]

Solution. An option holder gains from the volatility of the underlying stock price be-cause of the asymmetric payoffs of options.

• For example, if the stock price falls below the exercise price, a call option will beworthless, regardless of whether the drop in the price is only a few cents or manydollars.

• On the other hand, for every dollar stock price increase above the exercise price,the call option payoff will increase by the same amount. Hence, the option holdergains from the increased volatility on the upside, but does not lose on the downside.

(b) Explain put-call parity in words.

Solution. • The relationship between the value of a European option and the valueof an equivalent put option is called put-call parity.

• It holds only if the investor is committed to holding the options until the exercisedate. It does not hold for American options.


3.2 Put-Call Parity

1. A Simple Warm-up Question

You are given the following option prices for European puts and calls with the same time to STAT2812

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expiration:

Strike Price Put Call98 0.4394 14.3782100 0.6975 12.7575

Calculate the current price of the stock.

Solution. Applying put-call parity to the strike prices 98 and 100 respectively yields{14.3782− 0.4394 = S0 − 98e−rT

12.7575− 0.6975 = S0 − 100e−rT.

It follows that e−rT = 0.9394 and S0 = 106 .


2. (Challenging!) Further Encounters with Put-Call Parity

You are given the following four European options, all written on the same underlying non- STAT2820

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dividend paying stock and with the same maturity date which is one year later.

Call/Put Strike Price PriceCall 25 6.85Call 35 1.77Put 25 0.63Put 35 5.06

Suppose the risk-free rate is 6% per annum compounded continuously. Construct a strategyto earn risk-free profits at time 0 using the above options and/or zero-coupon bonds only.

[Total: 10 marks]

Solution. On first encounter, this question may seem intimidating.

Analysis We are given two pairs of call-put prices with different strike prices. For eachpair, we can make use of put-call parity to deduce the fair time-0 stock price.To create an arbitrage strategy, long the pair with a lower stock price and shortthe one which gives a higher stock price, i.e. buy low and sell high.

Action! • Applying put-call parity to the 25-strike pair gives

6.85− 0.63 = S0 − 25e−0.06

S25-strike0 = 29.7641. Higher!

• Applying put-call parity to the 35-strike pair yields

1.77− 5.06 = S0 − 35e−0.06

S35-strike0 = 29.6718. Lower!

Construction of Strategy

• At time 0, we long a 35-strike call, a 25-strike put, short a 35-strike put, a25-strike call and lend (35− 20)e−0.06 = 10e−0.06. The cashflow is

−(1.77 + 0.63) + (5.06 + 6.85)− 10e−0.06 = 0.0923 > 0.

• At time 1, the cashflow is as follows:

L35C L25P S35P S25C loan proceeds Total

S1 < 25 0 25− S1 −(35− S1) 0 10 0

25 ≤ S1 < 35 0 0 −(35− S1) −(S1 − 25) 10 0

S1 > 35 S1 − 35 0 0 −(S1 − 25) 10 0

The cashflow at time 1 is always zero, no matter what the stock price is. Anarbitrage strategy is thus constructed.


3. Put-call Parity for Gap Options

A gap call option is a European call option whose payoff at maturity time T is given by

payoff =

{ST −K1, if ST > K2,

0, if ST ≤ K2,

where ST is the price of the underlying non-dividend paying stock, K1 and K2 are positiveconstants, called the strike price and trigger price respectively. Similarly, the payoff of aEuropean gap put option with strike price K1 and trigger price K2 is given by

payoff =

{K1 − ST , if ST < K2,

0, if ST ≥ K2.

(a) Sketch the payoffs of the above gap call option when (i) K1 < K2; (ii) K1 ≥ K2.

(b) State the functions of the strike price K1 and the trigger price K2 in determining thepayoff of a gap option.

(c) Let Cgapt and P gap

t denote the time-t prices of a European gap call and gap put optionrespectively, both with strike price K1 and trigger price K2. For t ≤ T , prove, byno-arbitrage arguments that,

Cgapt − P gap

t = St −K1e−r(T−t).

Solution. (a) The payoff diagrams are shown below.

Payoff

STK1 K2

Payoff

STK2 K1

Figure 2: Left: K1 < K2; right: K1 ≥ K2

(b) • Strike price K1: Used to determine the size of the payoff

• Trigger price K2: Used to determine whether a payoff should be paid

(c) (Modified from page 1 of lecture notes) Construct:

Portfolio 1: Long 1 gap call option and short 1 gap put option, both with strikeprice K1 and trigger price K2. The total cashflow at maturity is:

long call short put TotalST < K2 0 −(K1 − ST ) ST −K1

ST ≥ K2 ST −K1 0 ST −K1


Portfolio 2: Hold one unit of the underlying stock and borrow $K1e−r(T−t) cash. The

total cashflow at maturity is:

long asset cash TotalST < K2 ST −K1 ST −K1

ST ≥ K2 ST −K1 ST −K1

Observe that the cashflows of the two portfolios are the same, no matter ST ≥ K2 orST < K2. By the no-arbitrage principle, the cost of constructing these two portfoliosmust be the same, i.e. Cgap

t − P gapt = St −K1e

−r(T−t).

3.3 Option Inequalities

1. Lower bound for Call Price with Dividends STAT2820

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TestLet C0(K,T ) be the time-0 price of a European call option on a stock, with strike price Kand expiry date T . Interest rate is r per annum, compounded continuously. Stock price attime 0 is S0. The stock pays discrete dividends in the time interval (0, T ). Let D be thepresent value (time-0 value) of all these dividends. Prove, by no-arbitrage arguments, that

C0(K,T ) ≥ S0 −Ke−rT −D.

[Total: 10 marks]

Proof. Assume, on the contrary, that C0(K,T ) < S0 − Ke−rT − D. At t = 0, we buy thecall option, short sell the stock and lend Ke−rT +D. The resulting cashflow is S0−Ke−rT −D − C0(K,T ) > 0.

At time T , the cashflow is as follows:

Long call Repay short sale of asset + dividends Proceeds from loan Total

ST < K 0 −ST −DerT K +DerT K − ST > 0

ST ≥ K ST −K −ST −DerT K +DerT 0

The cashflow at time T is always nonnegative, indicating that arbitrage exists. The statedinequality is then proved.


2. Suppose the current time is 0. Consider two European put options on the same underlying STAT2820

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(non-dividend paying) stock and the same maturity date T , but with different strike pricesK1 and K2, where K1 ≤ K2. The prices of the above options are denoted by P (K1) andP (K2) respectively. Use no-arbitrage arguments to show that

K2P (K1) ≤ K1P (K2).

[Total: 7 marks]

Proof. Suppose, on the contrary, that K2P (K1) > K1P (K2).

• At time 0, we buy K1 put options with a strike price of K2 and sell K2 put optionswith a strike price of K1, realizing a positive cashflow of K2P (K1)−K1P (K2) > 0.

• At the maturity date T , the cashflow is as follows:

Long K1 K2-strike puts Short K2 K1-strike puts TotalST ≤ K1 K1(K2 − ST ) −K2(K1 − ST ) (K2 −K1)ST ≥ 0

K1 < ST ≤ K2 K1(K2 − ST ) 0 K1(K2 − ST ) ≥ 0K2 < ST 0 0 0

As the payoff at time T is always non-negative, arbitrage profits exist. Thus we have

K2P (K1) ≤ K1P (K2).


3. Convexity of Option Prices

(a) Suppose that c1, c2 and c3 are the prices of European call options written on the same STAT2807

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(Ex-

tended)

underlying stock. Their strike prices are K1, K2 and K3 respectively, where K2 > K3 >K1, and they satisfy

(n1 + n2)K3 = n1K1 + n2K2, n1 and n2 are some positive integers.

All options have the same maturity.

(i) Show that

c3 <n1

n1 + n2

c1 +n2

n1 + n2

c2.

[8 marks]

(ii) New! Explain in words the meaning of the result in (i).

(iii) New! Hence or otherwise, prove the corresponding inequality for put options.

(b) Near market closing time on a given day, you lose access to stock prices, but some STAT2812

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Exam

SOA

MFE

Sample

Q2

European call and put prices for a stock are available as follows:

Strike Price Call Price Put Price$40 $11 $3$50 $6 $8$55 $3 $11

All six options have the same expiration date.

Construct two strategies based on different properties of option prices to exploitarbitrage profits.

Solution. (a) (i) The stated inequality is equivalent to

(n1 + n2)c3 < n1c1 + n2c2.

Assume the contrary, i.e. (n1+n2)c3 ≥ n1c1+n2c2. Consider a portfolio in which n1

K1-strike calls and n2 K2-strike calls are purchased and (n1+n2) K3-strike calls aresold. At time 0, we receive a non-negative cashflow of (n1+n2)c3−n1c1−n2c2 ≥ 0.At the maturity time T , the cashflow is as follows:

Long K1-strike Long K2-strike Short K3-strike Total

call call call

ST < K1 0 0 0 0

K1 ≤ ST < K3 n1(ST −K1) 0 0 n1(ST −K1)

K3 ≤ ST < K2 n1(ST −K1) 0 (n1 + n2)(K3 − ST ) n2(K2 − ST ) > 0

K2 ≤ ST n1(ST −K1) n2(ST −K2) (n1 + n2)(K3 − ST ) 0

The payoff is non-negative and even positive when ST ∈ [K3, K2). Hence arbitrageexists, and we have

c3 <n1

n1 + n2

c1 +n2

n1 + n2

c2.

(ii) The inequality in (a) implies that call price, as a function of the strike price K, isconvex.


(iii) By put-call parity, the inequality in (i) can be expressed in terms of put prices:

p3 + S0 −K3e−rT <

n1

n1 + n2

(p1 + S0 −K1e−rT ) +

n2

n1 + n2

(p2 + S0 −K2e−rT ).

Since by assumption, (n1 + n2)K3 = n1K1 + n2K2, all terms involving asset pricesand strike prices cancel and we finally get

p3 <n1

n1 + n2

p1 +n2

n1 + n2

p2 .

(b) (i) First Property: ConvexityWith n1 = 1, n2 = 2, K1 = 40, K3 = 50, K2 = 55, the inequalities in (a)(i) and(a)(iii) take the form

c3 <1

3(11) +

2

3(3) =

17

3= 5.6667 and p3 <

1

3(3) +

2

3(11) =

25

3= 8.3333.

Since c3 = 6, the first inequality is violated4. To exploit arbitrage profits, weconstruct the following portfolio:

• At time 0, we sell three 50-strike calls and buy one 40-strike call and two55-strike calls, resulting in a positive cashflow of 6(3)− 11− 2(3) = 1.

• At the expiration date, the cashflow is always non-negative:

S50C L40C L55C TotalST < 40 0 0 0 040 ≤ ST < 50 0 ST − 40 0 ST − 40 ≥ 050 ≤ ST < 55 −3(ST − 50) ST − 40 0 −2ST + 110 > 0K2 ≤ 55 −3(ST − 50) ST − 40 2(ST − 55) 0

4The put prices, however, satisfy convexity.


(ii) Second Property: Put-Call Parity

Starting Point: Unlike Problem 2 in Section 3.2, here we are not even giventhe continuously compounded interest rate r and the maturity time T . However,put-call parity implies that we have

8 = S0 − 40e−rT

−2 = S0 − 50e−rT

−8 = S0 − 55e−rT,

which is an inconsistent linear system in the variables (S0, e−rT ). In fact, the

second and third equations imply that S0 = 58 and e−rT = 1.2, which, uponsubstitution into the first equation, give

S0 − 40e−rT = 58− 40(1.2) = 10 > 8.

For the ease of presentation, we define the following three options to be the posi-tions obtained by longing a call and shorting a put, both with the same strike priceof 40, 50 and 55 respectively:

Strike Price Option Cost of Option$40 1 11− 3 = 8$50 2 6− 8 = −2$55 3 3− 11 = −8

Put-call parity asserts that the cost of Option 1 is also equal to S0− 40e−rT . Notethat

S0 − 40e−rT = 3(S0 − 50e−rT ) + (−2)(S0 − 55e−rT ).

[How to Get This? You can set up a(S0− 50e−rT ) + b(S0− 55e−rT ) = S0− 40e−rT ,which gives {

a+ b = 1

−50a− 55b = −40,

and solve for a and b.]The cost of buying 3 units of Option 2 and selling 2 units of Option 3 is −2(3) +(−8)(−2) = 10︸︷︷︸

high!

> 8︸︷︷︸low!

. We are now ready to engage in the buy-low-sell-high

strategy!

• At time 0, we buy 1 unit of Option 1, buy 2(= −b) units of Option 3, sell 3 (orbuy −a = −3) units of Option 2. The cashflow is

−8− 2(−8) + 3(−2) = 2 > 0.

• At the maturity time T , the cashflow of Option i is ST −Ki, where i = 1, 2, 3and K1 = 40, K2 = 50, K3 = 55. Hence the overall cashflow is given by

(ST − 40) + 2(ST − 55)− 3(ST − 50) = 0.


We have thus constructed an arbitrage strategy.

Remark 2. In Problem 2, Section 3.2, we compared the implied values of S0, whilein this question, we compare the implied values of S0 − 40e−rT .

Remark 3. If you have conquered Problem 2 in Section 3.2 as well as this question,...congratulations! Put-call parity is now nothing to you!

3.4 Two More Challenging Questions

1. Replicating an Unfamiliar Derivative STAT2820

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TestThe payoff of Derivative X maturing at the time T is given by

Payoff =

5, if 0 ≤ ST < 10,

3ST − 25, if 10 ≤ ST < 20,

ST + 15, if ST ≥ 20,

where ST is the price of the underlying stock at time T .

(a) Sketch the payoff of X against ST . [2 marks]

(b) Decompose the above payoff into the payoffs of put option(s), the underlying stock,and/or zero-coupon bond only. [5 marks]

(c) You are given:

• The time to maturity of X is 1 year.

• The risk-free interest rate is 5%.

• At time 0, the stock price is 15.

• The following put option prices are observed:

Strike Price Put Option Price10 0.120 4.5

• The time-0 price of X is 19.

Determine whether any arbitrage opportunity exists. If there is, construct a strategyto earn risk-free profits; otherwise, explain your answer. [5 marks]

[Total: 12 marks]

Solution. (a) The payoff of X against ST is sketched below:

Payoff

ST

5

10

35

20


(b) Note that the turning points of the payoff function are at ST = 10 and ST = 20, whichmust be the strike prices of the put options involved. Since put options are worthlessif ST ≥ 20, the term ST + 15 must come from the stock and bond.

Let payoff = A(10− ST )+ +B(20− ST )+ + ST + 15, where A and B are coefficients tobe determined. Then{

A(10− ST ) +B(20− ST ) + ST + 15 = 5 if 0 ≤ ST ≤ 10,

B(20− ST ) + ST + 15 = 3ST − 25 if 10 < ST ≤ 20.

Solving yields A = 3 and B = −2. In conclusion, the payoff can be decomposed as 3(long) put options with strike 10, plus 2 short put options with strike 20, plus 1 stockand bond with face value 15.

(c) The value of the replicating portfolio constructed in (b) is 3(0.1)−2(4.5)+15+15e−0.05 =20.5684, which is higher than the price of X. Thus an arbitrage opportunity exists.

• At time 0, we short the replicating portfolio, i.e. short 3 10-strike put options,long 2 20-strike put options, short 1 stock and borrow 15e−0.05, and long X. Thecashflow is 20.5684− 19 = 1.5684 > 0.

• At time T , the cashflows are as follows:

long X short 3 10-strike put long 2 20-strike put short stock repay loan Total

0 ≤ ST < 10 5 −3(10− ST ) 2(20− ST ) −ST −15 0

10 ≤ ST < 20 3ST − 25 0 2(20− ST ) −ST −15 0

ST ≥ 20 ST + 15 0 0 −ST −15 0

The cashflow at time T is always zero, irrespective of the stock price at the maturitydate. Hence this is an arbitrage.


2. Chooser Option STAT2820

11-12

ExamSuppose the current time is 0. Consider a chooser option on a (non-dividend paying) stock.At time t1 (where t1 ≥ 0), the owner of the option can choose whether the option becomesa European call option or a European put option, both with the same strike price K andmaturity date T (where T ≥ t1). The risk-free rate per annum compounded continuously isconstant at r.

(a) For the special case where t1 = T , what option strategy is a chooser option equivalentto? Discuss briefly one advantage and one disadvantage of the strategy. [6 marks]

(b) Determine the time-0 price of the chooser option in terms of the time-0 price(s) ofappropriate call and/or put option(s). Specify the strike price(s) and maturity date(s)of the option(s) involved. [5 marks]

(c) Suppose r = 0, K = $30, t1 = 2 years and T = 6 years. Now you are given the followingtime-0 prices: the stock is selling at $32; the price of the chooser option is $10.48; and acall option with strike price $30 and maturity date 2 years from now is selling at $4.89.Determine the time-0 price of a put option with strike price $30 and maturity date 6years from now. [3 marks]

[Total: 14 marks]

Solution. (a) The payoff at time t1 = T of the chooser option is

max{(ST −K)+, (K − ST )+} =

{K − ST , if ST < K

ST −K, if ST ≥ K= |ST −K|,

which is the payoff of longing a European call option as well as a European put option,both with the same strike price K and maturity date T , or a straddle.

Advantage: Payoff is independent of the direction of the stock price movement / Theoption holder can profit from price movements in both directions.

Disadvantage: It requires heavier investment (both call and put need to be bought).

(b) Hint: Try to rewrite the “unfamiliar” payoff function of the chooser option in terms of“familiar” payoff functions, e.g. those of call and put.

Let C(St, t, T ) and P (St, t, T ) denote the time-t prices of a European call and Europeanput option on the stock, both with maturity date T and strike price K. The payoff ofthe chooser option at time t1 is

max {C(St1 , t1, T ), P (St1 , t1, T )} = max {C(St1 , t1, T )− P (St1 , t1, T ), 0}+ P (St1 , t1, T )

= max{St1 −Ke−r(T−t1)

}+ P (St1 , t1, T ), (2)

where the last equality follows from the put-call parity. Note that max{St1 −Ke−r(T−t1)

}is the payoff of a European call option written on the stock expiring at time t1 with astrike price of Ke−r(T−t1). By discounting (2) to time 0, the time-0 price of the chooseroption is a sum of:

• The time-0 price of the above European call option, i.e. written on the stock

expiring at time t1 with a strike price of Ke−r(T−t1) .

• The time-0 price of a European put option written on the stock expiring at timeT with a strike price of K .


Remark 4. Why did I write

max {C(St1 , t1, T ), P (St1 , t1, T )} = max {C(St1 , t1, T )− P (St1 , t1, T ), 0}+ P (St1 , t1, T )

instead of

max {C(St1 , t1, T ), P (St1 , t1, T )} = C(St1 , t1, T )+max {P (St1 , t1, T )− C(St1 , t1, T ), 0}?

Because...I peeked at part (c) and saw that the price of a put expiring at time T willbe calculated at the end! So, the first writing will be more useful later!

(c) Because r = 0, Ke−r(T−t1) = K = 30, so that the call option and put option comprisingthe price of the chooser option (result in (b)) have the same strike price. Using thegiven option prices,

10.48 = 4.89 + P

P = 5.59

Remark 5. The fact that S0 = 32 is not needed here.

Remark 6. A similar question is Question 4 (actually an Exam MFE sample question) inSTAT2820 December 2009 Examination.


3.5 The Binomial Tree Model (OPTIONAL)

1. Straddle

For a one-year straddle on a nondividend-paying stock, you are given: SOA

MFE

Spring

2007 Q14

(Adapted)

• The straddle is constructed by a long position in a call option and a long position ina put option, both of which are written on the same underlying stock, have the samematurity of 1 year and the same strike price of $50.

• The straddle can only be exercised at the end of one year.

• The stock currently sells for $60.


• In one year, the stock will either sell for $70 or $45.

Calculate the current price of the straddle by

(a) replicating the straddle by a portfolio of stock and cash;

(b) using risk-neutral valuation.

Solution. The payoff of the straddle is given by the absolute value of the difference betweenthe strike price and the stock price at expiration date:

(S(1)− 50)+ + (50− S(1))+ = |S(1)− 50|.

The binomial tree is constructed in Figure 3 with the payoffs of the derivative shown inparenthesis.

S0 = 60

Su = 70 (20)

q = 0.7999

Sd = 45 (5)

1− q = 0.2001

Figure 3: Binomial tree for Question 1

(a) Consider a replicating portfolio consisting of α shares of stock and β amount of cash.Matching the payoffs in the up and down movements,{

70α + e0.08β = 20,

45α + e0.08β = 5,

which gives α = 0.6 and β = −20.3086. The current price of the derivative is the sameas the portfolio value at time 0, which is

60(0.6) + (−20.3086) = 15.6914 .


(b) The risk-neutral probability of an “up” movement is

q =60e0.08 − 45

70− 45= 0.7999.

Using risk-neutral valuation, the current price of the straddle is

e−0.08 [20q + 5(1− q)] = 15.6914 .

2. Modification to the Binomial Tree

The following one-period binomial stock price model was used to calculate the price of a SOA

MFE

Spring

2009 Q7

(Extended)

one-year 10-strike call option on the stock.

S0 = 10

Su = 12

Sd = 8

You are given:

• The period is one year.

• The true probability of an up-move is 0.75.

• The stock pays no dividends.

• The price of the one-year 10-strike call is $1.13.

Upon review, the analyst realizes that there was an error in the model construction andthat Sd, the value of the stock on a down-move, should have been 6 rather than 8. The trueprobability of an up-move does not change in the new model, and all other assumptions werecorrect.

(a) Recalculate the price of the call option.

(b) Comment on the new price in (a).

Solution. (a) We can perform risk-neutral valuation to deduce the value of e−r, whichremains the same after the model review:

e−r[q(2) + (1− q)(0)] = 1.13

e−r[2

(10er − 8

12− 8

)]= 1.13

e−r = 0.9675.

After the model review, the risk-neutral probability becomes

q′ =10er − 6

12− 6= 0.7226.

The new price of the call option is then e−r(2q′) = 1.3983 .


(b) The new price is higher. Note that q, as a function of the down movement, d, isdecreasing, because

q =er − du− d

=er − uu− d

+ 1,

which decreases with d as er < u. This means a decrease in d will result in an increasein q. With a higher probability (in the risk-neutral world) of being in the money, thecall will have a higher price.

Remark 7. The true probability of an up-move is designed by the SOA to distract you!

3. Three-State World SOA

Exam

MFE

Sample

Q27

You are given the following information about a securities market:

• There are two nondividend-paying stocks, X and Y .

• The current prices for X and Y are both $100.


• There are three possible outcomes for the prices of X and Y one year from now:

Outcome X Y1 $200 $02 $50 $03 $0 $300

Let CX be the price of a European call option on X, and PY be the price of a European putoption on Y . Both options expire in one year and have a strike price of $95.

Using the method of replicating portfolio, calculate PY − CX .

Solution. Although there are three possible outcomes, the philosophy of the method of repli-cating portfolio is exactly the same as that in the binomial model.

The payoff associated with buying a put option on Y and selling a call option on X, bothexpiring in one year with strike price of $95, is given by:

Outcome Payoff of Put Payoff of Call Total Payoff1 (95− 0)+ = 95 (200− 95)+ = 105 −102 (95− 0)+ = 95 (50− 95)+ = 0 953 (95− 300)+ = 0 (0− 95)+ = 0 0

Consider a portfolio consisting of α amount of cash, β shares of X and γ shares of Y . Toreplicate the above payoff structure, we solve

e0.1α + 200β = −10,

e0.1α + 50β = 95,

e0.1α + 300γ = 0,

which gives α = 117.6289, β = −0.7 and γ = −13/30. Hence

PY − CX = α + 100β + 100γ = 4.2955 .

Remark 8. Stocks X and Y can be said to be mutually exclusive, in that whenever one ofthem has a positive price, then the price of other must be zero.


Epilogue

Option pricing is indisputably a topic of paramount importance in risk management and actu-arial science. For this reason, many courses in our department cover option pricing, though todifferent extent. If you are interested, you may try the following additional past paper questionsfrom other STAT courses.

Course Title Year QuestionSTAT2309 The Statistics of Investment Risk December 2008 4STAT2807 Corporate Finance for Actuarial Science May 2007 7

May 2006 9STAT2808 Derivatives Markets December 2008 3STAT2812 Financial Economics I December 2009 1STAT3303 Derivatives and Risk Management December 2010 1, 2STAT3308 Financial Engineering December 2009 2

December 2008 3May 2006 2 (a), 3 (b)

Table 1: Additional Relevant Past Examination Questions in STAT courses. Numerical answersare available from the Statistics and Actuarial Science Society.


4 Farewell

Figure 4: So much for the hardship in my tutorials. For Year 2 students, see you in the nextsemester if possible; for final-year students, congratulations (in advance) on graduation!

Ambrose LOTutor of STAT2807 Corporate Finance for Actuarial Science (2011-2012)

********** END OF TUTORIAL 10 & THE WHOLE COURSE **********

♦ ♣ ♥ ♠ , Good Luck for Final Examination , ♠ ♥ ♣ ♦

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