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MATH3075/3975

Financial Mathematics

Week 11: Solutions

Exercise 1 We consider the CRR binomial model with the risk-free rater = 0.1 and the following values of the stock price S at times t = 0 andt = 1: S 0 = 100, S u1 = 120, S d1 = 90. We consider the American putoption with maturity date T = 3 and the strike price K = 121. The rewardprocess equals, for t = 0, 1, 2, 3,

g(S t, t) = (K − S t)+ = (121 − S t)

+.

1. We first compute the arbitrage price P at

of this option for t = 0, 1, 2, 3.

We start by noting that the unique risk-neutral probability measure P

satisfies

p = 1 + r − d

u − d =

(1 + r)S 0 − S d1S u1 − S d1

= 110 − 90

120 − 90 =

2

3.

The stock price process S t is given by

172.8

144

120

129.6

100

108

90

97.2

81

72.9

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The price process P at

is represented by the following diagram

0

0

3.9394

0

21

13

31

23.8

40

48.1

and the rational exercise decisions are given by (1 = exercise)

1

1

0

1

1

1

1

1

1

1

2

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2. The rational exercise times for the holder of the American put option

are: τ ∗0 = 0,

τ ∗1 (ω) = 2 for ω ∈ ω1, ω2, ω3, ω4,

τ ∗1 (ω) = 1 for ω ∈ ω5, ω6, ω7, ω8,

τ ∗2 = 2 and τ ∗3 = 3.

3. Suppose that the option was sold for the price P a0 = 21 and it wasnot immediately exercised by its holder. The issuer may establish thereplicating portfolio for the European claim X = (3.9394, 31) withmaturity 1 by solving the following equations

1.1 ϕ00 + 120 ϕ1

0 = 3.9394,

1.1 ϕ00 + 90 ϕ1

0 = 31.

We find that (ϕ00, ϕ0

1) = (101.9835, −0.90202) and thus the initial we-alth needed to establish this portfolio equals

V 0(ϕ) = 101.9835 − 0.90202 × 100 = 11.7815.

The difference 21 − 11.7815 is the net profit of the issuer at time 0.This argument can be extended to any date t.

Exercise 2 We consider the CRR binomial model with the risk-free rater = 0 and the following values of the stock price S at times t = 0 and t = 1:

S 0 = 100, S u1 = 120, S d1 = 90.

We examine the American call option with maturity date T = 3 and thefollowing reward process

g(S t, t) = (S t − K t)+,

where the variable strike K t satisfies

K 0 = K 1 = 100, K 2 = 105, K 3 = 110.

1. We first compute the arbitrage price X at

of this option for t = 0, 1, 2, 3.

The unique risk-neutral probability measure P satisfies

p = 1 + r − d

u − d =

(1 + r)S 0 − S d1S u1 − S d1

= 100 − 90

120 − 90 =

1

3.

3

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The stock price process S t is given by

172.8

144

120

129.6

100

108

90

97.2

81

72.9

The price process X at

of the call option is given by

62.8

39

20

19.6

8.1185

6.5333

2.1778

0

0

0

4

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The rational exercise decisions of the holder are given by

1

1

1

1

0

0

0

1

1

1

2. The rational holder should exercise the American option at time t = 1

whenever the stock price rises during the first period. Otherwise, heshould not exercise the option till time 2. Hence the rational exercisetime τ ∗0 is a stopping time τ ∗0 : Ω → 0, 1, 2 given by

τ ∗0 (ω) = 1 for ω ∈ ω1, ω2, ω3, ω4,

τ ∗0 (ω) = 2 for ω ∈ ω7, ω8,

τ ∗0 (ω) = 3 for ω ∈ ω5, ω6.

3. We now take the position of the issuer of the option:

• At t = 0, we need to solve

ϕ00 + 120 ϕ1

0 = 20,

ϕ00 + 90 ϕ1

0 = 2.1778.

Hence (ϕ00, ϕ1

0) = (−51.2888, 0.5941) for all ωs.

• If the stock price has risen during the first period, the option isexercised by its holder. Hence we do not need to compute thestrategy at time 1 for ω ∈ ω1, ω2.

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• If the stock price has fallen during the first period, we need to

solve

ϕ01 + 108 ϕ1

1 = 6.5333,

ϕ01 + 81 ϕ1

1 = 0.

Hence (ϕ01, ϕ1

1) = (−19.599, 0.2420).

• If the stock price has fallen twice then the option is exercised andhas value 0.

• If the stock price has fallen during the first period and has risenduring the second period then we need to solve

ϕ02 + 129.6 ϕ1

2 = 19.6,

ϕ02 + 97.2 ϕ1

2 = 0.

Hence (ϕ01, ϕ1

1) = (−58.8, 0.6050).

We conclude that the replicating strategy ϕ = (ϕ0, ϕ1) is representedby the following diagram

V u

1 (ϕ) = 20

(ϕ00, ϕ1

0) = (−51.29, 0.59)

(ϕ02, ϕ1

2) = (−58.8, 0.61)

(ϕ01, ϕ1

1) = (−19.60, 0.24)

V dd2 (ϕ) = 0

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Exercise 3 (MATH3975) We consider the European call option with strike

price K = 10 and maturity date T = 5 years. We assume that the initialstock price S 0 = 9, the risk-free interest rate is r = 0.01 and the stock pricevolatility equals σ = 0.1 per annum.

We use the CRR parametrization for u and d with ∆t = 1. We obtain

u = eσ√ ∆t = 1.105171, d =

1

u = 0.904837.

and thus

˜ p = 1 + r − d

u − d

= 0.524938, ˆ p = ˜ pu

1 + r

= 0.574402.

Using the backward induction method, we get the following results.

1. The price at 0 of the European call option equals C 0 = 0.5522. Thedetailed computations are summarised on the next page.

2. The price at 0 of the European put option equals P 0 = 1.0669. Fordetails, see the next page.

3. The put-call parity at time t = 0 reads

C 0 − P 0 = 0.5522 − 1.0669 = −0.5147 = 9 − 10(1.01)5

= S 0 − K (1 + r)T

.

4. The price at 0 of the American put option equals P a0 = 1.2112. Theoption should not be exercised at time 0, but it should be exercisedby its holder at time 1 if the stock price falls during the first period.For the full description of rational decisions of the holder, see the nextpage.

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Week 11

Exercise 3 Interest

Volatility

Stock at 0 Strike price Maturity

0.01 0.1 9

10 5

up down

tilde p 1 tilde p

1.105171 0.904837 0 .524938 0 .475062

Year

0

1

2 3 4

5

Stock

9 9.946538

10.99262 12.14873

l3 42642

14.83849

price

8.143537 9 9.946538 10 .99262 12.14873

7.368577 8.143537

9

9.946538

6.667364 7.368577 8.143537

6.

03288

6.667364

5.458776

European 0.552247

0.920649 1.498351 2.357597 3.525432 4.838491

call price 0.156793 0 .301676 0.580436 1.116781 2.148729

0 0 0 0

0 0 0

0

0

0

European

1.066904

0.583914 0.211627

0.011828

0 0

put price 1.62306 1.007577 0.436858

0.025146

0

2.337325 1.659424

0 .90099

0.053462

3 l35597 2 5324l3 1.856463

3.

86811

3.332636

4.541224

Put call parity

0 51466

American 1.211179 0.650276 0 .233532

0 011828

0 0

p

ut price

1.856463

1.124461

0.483428 0.025146 0

2.631423

l 856463

1

0.053462

3.332636 2.631423

l 856463

3.96712 3.332636

4.541224

Exercise o

o

o o

Yes Yes

decision Yes

o

o

o

Yes

Yes Yes

Yes Yes

Yes Yes Yes

Yes

Yes

Yes