ACT 4000, MIDTERM #1ADVANCED ACTUARIAL TOPICS

FEBRUARY 8, 2007HAL W. PEDERSEN

You have 70 minutes to complete this exam. When the invigilator instructs you tostop writing you must do so immediately. If you do not abide by this instruction youwill be penalised. All invigilators have full authority to disqualify your paper if, intheir judgement, you are found to have violated the code of academic honesty.

Each question is worth 10 points. Provide sufficient reasoning to back up your answerbut do not write more than necessary.

This exam consists of 8 questions. Answer each question on a separate page of theexam book. Write your name and student number on each exam book that you useto answer the questions. Good luck!

Question 1. Consider a single-period binomial market. There are two assets availablefor trade. The first asset pays 2 in the upstate and 1 in the downstate. The secondasset pays 2 in the upstate and 3 in the downstate. The price of asset 1 is PI = 1.39.The price of asset 1 is P2 = 2.29. The market is arbitrage-free.

(1) [6 points] Compute the price of a risky asset that pays 10 in the upstate and 0in the downstate.

(2) [4 points] What is the implicit effective interest rate in this model?

Question 2. An equity securities market model follows a multi-period binomialmodel. At each node of the binomial tree the current stock price S will branchto uS in the upstate and dS in the downstate. You are given that the initial stockprice is 10, u = 1.25, d = 0.85 and the interest-rate is 5% effective per period.

(i) (5 points) Compute the price of a European put option on the stock which expiresin 4 periods and has a strike price of 8.5.

(ii) (5 points) Compute the price of an American put option on the stock whichexpires in 4 periods and has a strike price of 8.5 and describe the optimal exercisepolicy for this American put option.

2 ACT 4000 - MIDTERM # 1

Question 3. In North America, it is common. to issue investment products withreturn guarantees. For instance, a product may offer the returns of the S&P 500index except the investor is guaranteed a 0% return over the investment horizon andthe maximum total return the investor will be credited over the investment horizonis 30%.

You are the pricing actuary for a large insurance company. Your company has decidedto sell an investment product for which the investments will be credited a return equalto that on an index that will experience either a 20% gain or a 10% loss at the endof one period (i.e. the single-period binomial model). The continuously compoundedinterest-rate (force of interest) for the period is 0.08. The customer is guaranteed a5% return and you are to set the maximum total return the customer will receiveover the period (denoted a) so that your insurance company will break even on theproduct.

Determine a.

Suppose that the exchange rate is 0.79 €I$. Let r($) = 3%, r(€) = 5%, u = 1.1850, d = 0.8581,T = 15 months, n = 3, and K = 0.8 €.

(a) What is the price of a 15-month European call?(b) What is the price of a 15-month European put?

The price of a 6-month dollar-denominated call option on the euro with a $0.90strike is $0.0404. The price of an otherwise equivalent put option is $0.0141. Theannual continuously compounded dollar interest rate is 5%.

a. What is the 6-month dollar-euro forward price?

b. If the euro-denominated annual continuously compounded interest rate

is 3.5%, what is the spot exchange rate?

A one-period arbitrage-free model has two assets with the following price dynamics.

Asset 1

10.4

12

8

Asset 2

10

2.5

A call option on asset 1 with a strike price of 9 has a price of 1.8.

(i) Compute the state prices implicit in this model.

(ii) Compute the risk neutral probabilities for this model.

(iii) Compute the implied short rate for this model. [Compute the implied short rate as a

force of interest.]

(iv) The price of a call option on asset 2 is 2.4. What is the strike price of this call

option?

7

Suppose call and put prices are given by

Strike

Call premiumPut premium

50 5518 147 10.75

609.50

14.45

Find the convexity violations. What spread would you use to effect arbitrage?Demonstrate that the spread position is an arbitrage.

810 . If )

Suppose the S&P 500 futures price is 1000, a = 30%, r = 5%, 8 = 5%, T = I,and n = 3.

a. What are the prices of European calls and puts for K = $1 OOO?Why doyou find the prices to be equal?

b. What are the prices of American calls and puts for K = $1000?

c. What are the time-O replicating portfolios for the European call and put?

1.)~ ­Z..L1 :;

2 ir 1JZ 1", + 5'ld

1, ~ . LfS ) 1..= .'1-7

/oivt:;' lor·if?) =- '1·70

~~,~<~ ~ ~ ~

(1.)I·~ ("'

'lV\ i i :;-.f L

Q~'..-<r" 'g". 7".9

~ J = , 0070 .

QL.~

I. c5 - . ~..s

1.25-.~S

IS

$-foe-r. f r I Co <-.5 :

ll.f. 'II Lf<J&

'--

I~. (,olSb

,~.5I'. L 89 D b

-. ,. " ~ .,.-.... ~~--/0 ------ ~

---- ----7.fo1'Sb- .----

71, 'LS ..

---5>~lZ-bch

~ (.5) if (~ ..5 - 5. 1.1-f/t)6)li.OJ) -1

+ if _, (.5)'1(~.5-7.b1b5cJ(t.05)1f

o

o

VtiJ", ,

~~~~3.s

l~rLo

.7/7...c.?

Q.~~t t. 1..1s] 1/1.5 !lf1

~ 2.y, ~7-'>-D

Q'"Or, ~ •• f"t fn"~ c; • >& 'Y

cr

-,) ~ e:.~" f; c~:J~

~C/L arret-.3 <:e:o "'=:J C-c c::.",,- ~-v L

NS:JL (~c;}J/

I. 1t:J

I ~ If"--'.0 _S

i. L..:. - (/-t~) =- . 1- .••. - ~

. r 0 - I. c .5 ;:. -. IS

.o~

e - .10 -- , ~/o9i:;

IL

to, i"l~

..Sa L f r " L ·t ~(...5011.< it ~j>.J MCI~""iI {

[ c-k, {.. ~ 7 J 4 ,.. (I

5f) Iw -t . c~ /:. 0 ·tk~5

" ~ .f ~!.. Sf) IIA +I 0;\5 -t (;>

f0j,4.r-~u<.tl ~~-tb'dk Vro ~/!f>'-J

',5

lOb" to{.

9,(,.

frc h/~~~ <: /;r

P1 tjld~A I .

J?~to.j::: /z..tj", + 8idI.r::: ~j~

1d =- .'1

,f:;-.0+· t

- . fa

-('"%-;1&(II " )

e.- ---

<)

r ;:;. 0c ~,t+-· b -==- I

C\v) }o :;:..,0 X+'L.5 (. If)L-J

~

·t~ .~

==-7

X -:= IS

oOf)

i 0

J 7 " •...• eJ . -

15

L...$

Tl,~

0" 50 )" L . 0 ( 15 - k.) =- l. Y

==3> K~ II·

lk, S I> -f l- 'f. -t ~0" Jt-

S £ '- -t l ~_ 50/" +-. .,.1.j

fJ '-< t"-1eor t..ll Te): + h~0 k

Q "< ~f.OIA 9. 10 .

t:c- bvt!V1

U. 8.- -rl~i...$f.(1--t boe It-

{;( j,( <! h...,.../0./6 .

~..5l~tJ.~5~JMt~O"'J -t-e:>h~....,

rvV'-t" ~u~d

T{j\ + ~"'''Jt-fr .. Ide~.5(Vld;"~/ .

1"

t tI

7. (a) We can calculate the price of the call currency option in a very similar way to our previouscalculations. Please pay attention to the fact that we have a strike price and exchange rate inEuros, therefore the foreign interest rate is the $ interest rate!

For the European call option, we have:

Time (yrs)0.79

0.095828

0.416667

0.936117 ­

0.17665

0.677895

0.03079

0.833333

1.10926

0.31197

0.803277

0.06837

0.581698

o

1.25

1.314427

0.51443

0.95185

0.15185

0.689288

o

0.499152

o

The value of the option is 0.096€.

(b) for the European put option, we have:

Time (yrs)0.79

0.086435

0.416667

0.936117

0.031

0.677895

0.13698

0.833333

1.10926

o

0.803277

0.05858

0.581698

0.20903

1.25

1.314427

o '

0.95185

o

0.689288

0.11071

0.499152

0.30085

The value of the option is 0.086€.

Question 9.6.

a) We can use put-call-parity to determine the forward price:

+ C (K, T) - P (K, T) = PV (forward price) - PV (strike) = e-rT FO.T _ Ke-rT

'* FO,T = erT [+C (K, T) -. P (K, T) + Ke-rT]

= eO.05*O.5[$0.0404 - $0.0141 + $0.ge-O.05*O.5]

'* Fo. T = $0.92697.

b) Given the forward price from above and the pricing formula for the forward price, we canfind the current spot rate:

FO.T = xoe(r-rf)T

-¢} xo = Fo. Te-(r-rf)T = $0.92697e-(O.05-0.035)O.5 = $0.92.

'.¢'quations(9.17) and (9.18) of the textbook are violated. To see this, let us calculate the values.,b;>'t

!~ve:

}1 violates equation (9.18) .

.a!culate lambda in order to know how many options to buy and sell when we construct therfly spread that exploits this form of mispricing.Because the strike prices are symmetric around

~~mbdais equal to 0.5.I..,

Pdore, we use a call and put butterfly spread to profit from these arbitrage opportunities.

'ch violates equation (9. I7) and

ST > 60ST - 50110 - 2 x ST

ST - 60o

14 - 9.50 = 0.9,60 - 55

14.45 - 10.75 = 0.74,60 - 55

55 ::: ST ::: 60ST - 50110 - 2 X ST

o60 - ST 2: 0

C (Kz) - C (K3)

K3 - Kz

P (K3) - P (Kz)

K3 - K2

50 ::: ST ::: 55ST - 50ooST - 50 2: 0

and

and

ST < 50oooo

t=O-18+28-9.50+0.50

10.75 - 7 = 0.7555 - 50

. (K_1_) _-_C_( K_z_) = _18_-_1_4= 0.8Kz - K) 55 - 50

Transaction

"Buy 1 50 strike call.Sell 2 55 strike calls

,Buy 1 60 strike callI'~'," TOTALr~

Transaction t=OST < 5050 :S ST :S 5555 :S ST :S 60ST >60Buy 1 50 strike put

-750 - ST000-

Sell 2 55 strike puts

21.502 x ST - 1102 x ST - 11000

Buy] 60 strike put

-14.4560 - ST60 - ST60 - ST0TOTAL

+0.050 ST - 50 ~ 060 - ST ~ 00

Please note that we initially receive money and have non-negative future payoffs. Therefore w, e

have found an arbitrage possibility, independent of the prevailing interest rate.

Question 10.18.

a) We have to use the formulas of the textbook to calculate the stock tree and the prices of theoptions. Remember that while it is possible to calculate a delta, the option price is just the value

of B, because it does not cost anything to enter into a futures contract. In particula~, this yields thefollowing prices: For the European call and put, we have: premium = 122.9537. The prices mustbe equal due to put-call-parity.

b) We can calculate for the American call option: premium = 124.3347 and for the Americanput option: premium = 124.3347.

c) .' We have the following time e replicating portfolios:

For the European call option:

Buy 0.5371 futures contracts.Borrow 122.9537

For the European put option:

Sell 0.4141 futures contracts.Borrow 122.9537