Financial Mathematics

Lecture 13-14

Dr Wioletta Nowak

Example

• Create a three-period binominal price tree and

find the fair value of an European call and put

options and an American put option on a

nondividend-paying stock if the initial stock

price is 62 PLN, the compound risk-free

interest rate is 12% per annum, the stock

volatility is 20%, the strike price of 60 PLN is

expiring at the end of the third month (at the

end of the third week).

Period - a month

029288.11211.0

eu

971545.01 ud

01.11212.01 R

665965.0q

334035.01 q

013964.15211.0

eu

986228.01 ud

002308.15212.01 R

579736.0q

420264.01 q

Period - a week

Three months

62.00 63.82 65.68 67.61

60.24 62.00 63.82

58.52 60.24

56.86

3.88 5.00 6.28 7.61

1.76 2.59 3.82

0.16 0.24

0.00

0.11 0.00 0.00 0.00

0.34 0.00 0.00

1.04 0.00

3.14

0.16 0.00 0.00 0.00

0.49 0.00 0.00

1.48 0.00

3.14

Price tree European call option

European put option American put option

Period - three weeks

62.00 62.87 63.74 64.63

61.15 62.00 62.87

60.30 61.15

59.47

2.45 3.14 3.88 4.63

1.51 2.14 2.87

0.66 1.15

0.00

0.04 0.00 0.00 0.00

0.09 0.00 0.00

0.22 0.00

0.53

0.04 0.00 0.00 0.00

0.09 0.00 0.00

0.22 0.00

0.53

Price tree European call option

European put option American put option

The underlying asset pays continuous dividend d

u

tr Cebxu

d

tr Cebxd

du

CCx du

du

CdCueb udtr

trudtdut edu

CdCue

du

CCbexC

dd

trudtdut edu

CdCue

du

CCbexC

dd

du

CdCue

du

CCeC udtrdutr d

d

tr

u

trtr C

du

euC

du

deeC

dd

du

tr CqCqeC 1

du

deq

tr

d

Incoherent binomial option tree

(the underlying asset pays predictable income)

Example

• Find the fair value of an European call option

using the incoherent binomial option tree if the

underlying asset pays dividend of 2 PLN in

half a month. The initial stock price is 50 PLN,

the strike price of 48 PLN is expiring at the

end of the third month, the continuously

compounded risk-free interest rate is 10% per

annum, and the stock volatility is 20%

0594341

12120.eu

.

943901 .ud

00836811210 .ee .tr

du

deq

tr

5580.q

44201 .q

du

tr CqCqeC 1

48,2 KD

Example - coherent binomial option tree

(American put option)

122401112140

.eu.

89094701 .ud

00836811210 .ee .tr

du

deq

tr

5073190.q

49268101 .q

40100625052 ..r.DKS

du

tr PqPqeP 1

5.3 5T

*S

00078206210

12

153

.e.eD..

rtt

0728

3635531

554469393635

12569994955446939

77099621256999495544

06893579770996212569992249

.

..

...

....

.....

......

051435202493820175272000782 ....

1012

15010

12

15110

12

15210

12

153 ........

DeDeDeDe

S

07.28

36.355.31

55.4469.3941.37

12.56999.49598.4671.41

7.7099.6217.5802.5256.46

06.8935.7975.7201.6514.5852

American put option

9321

6414518

45531102214

06623861610

0031773866

000640162444

.

..

...

...

...

...

0818

8969

662

0

0

.

.

.

Formula K – S

49618

3110

0

98812

3529

.

.

.

.

Option strategies

Uncovered

long call, short call, long put, short put

Covered

(option position that is offset by an equal and

opposite position in underlying asset)

Payoff diagram

Payoff diagram

Covered option strategies

covered call – short call + long share

covered put – short put + short share

short call + long share

Short

call

Long share

Profit

0 5 – 30 – 25

20 5 – 10 – 5

25 5 – 5 0

30 5 0 5

35 5 5 10

40 5 10 15

45 0 15 15

50 -5 20 15

55 -10 25 15

60 -15 30 15

40K 5C

C,SKmin T 0

30STS

30TS

SCK

short put + short share

Short

put

Short share

Profit

0 – 35 50 15

20 – 15 30 15

25 – 10 25 15

30 – 5 20 15

35 0 15 15

40 5 10 15

45 5 5 10

50 5 0 5

55 5 – 5 0

60 5 – 10 – 5

40K 5P

P,KSmin T 0

50STS

TS50

SPK

Long call + short share

Long call Short share

Profit

0 – 5 30 25

20 – 5 10 5

25 – 5 5 0

30 – 5 0 –5

35 – 5 – 5 –10

40 – 5 – 10 –15

45 0 – 15 –15

50 5 – 20 –15

55 10 – 25 –15

60 15 – 30 –15

40K 5C

C,KSmax T 0

30STS

TS30

SCK

Long put + long share

Long

put

Long share

Profit

0 35 – 50 – 15

20 15 – 30 – 15

25 10 – 25 – 15

30 5 – 20 – 15

35 0 – 15 – 15

40 – 5 – 10 – 15

45 – 5 – 5 – 10

50 – 5 0 – 5

55 – 5 5 0

60 – 5 10 5

40K 5P

P,SKmax T 0

50STS

50TS

SPK

Bullish and bearish strategies

Bull call

Bull put

Bear call

Bear put

LSLS CCKK

LSLS PPKK

LSLS CCKK

LSLS PPKK

vertical bull call

Long call Short call Profit

20 – 5 2 – 3

25 – 5 2 – 3

30 – 5 2 – 3

35 – 5 2 – 3

40 – 5 2 –3

45 0 2 2

50 5 2 7

55 10 – 3 7

60 15 – 8 7

65 20 – 13 7

40LK 5LC

STS C,SKmin 0

TS 50SK 2SC

LLT C,KSmax 0

LSLS CCKK

vertical bull put

Long put Short put Profit

20 18 – 25 – 7

25 13 – 20 – 7

30 8 – 15 – 7

35 3 – 10 – 7

40 – 2 – 5 – 7

45 – 2 0 – 2

50 – 2 5 3

55 – 2 5 3

60 – 2 5 3

65 – 2 5 3

40LK 2LP

SST P,KSmin 0

TS 50SK 5SP

LTL P,SKmax 0

LSSL PPKK

vertical bear put

Long put Short put Profit

20 25 – 18 7

25 20 – 13 7

30 15 – 8 7

35 10 – 3 7

40 5 2 7

45 0 2 2

50 – 5 2 – 3

55 – 5 2 – 3

60 – 5 2 – 3

65 – 5 2 – 3

50LK 5LP

SST P,KSmin 0

TS 40SK 2SP

LTL P,SKmax 0

LSSL PPKK

vertical bear call

Long call Short call Profit

20 – 2 5 3

25 – 2 5 3

30 – 2 5 3

35 – 2 5 3

40 – 2 5 3

45 –2 0 – 2

50 – 2 – 5 – 7

55 3 – 10 – 7

60 8 – 15 – 7

65 13 – 20 – 7

50LK 2LC

STS C,SKmin 0

TS 40SK 5SC

LLT C,KSmax 0

LSLS CCKK

Call butterfly (long + short2+long)

Long call 1 Short call x2 Long call 2 Profit

0 – 3 12=6*2 – 10 – 1

20 – 3 12 – 10 – 1

25 – 3 12 – 10 – 1

30 – 3 12 – 10 – 1

35 – 3 12 – 5 4

40 – 3 12 0 9

45 – 3 2=2*(40-45+6) 5 4

50 – 3 – 8 10 – 1

55 2 – 18 15 – 1

60 7 – 28 20 – 1

TS

350 11 LL CK 1030 22 LL CK640 SS CK

LLT C,KSmax 0 STS C,SKmin 0

Call butterfly

SLL CCC 221 )(22211 SSLLLL CKCKCK

12 LS CC

1LK2LK

Put butterfly (long + short2+long)

Long put 1 Short put x2 Long put 2 Profit

0 45 – 74 28 – 1

20 25 – 34 8 – 1

25 20 – 16 3 – 1

30 15 – 14 – 2 – 1

35 10 – 4 – 2 4

40 5 6 – 2 9

45 0 6 – 2 4

50 – 5 6 – 2 – 1

55 – 5 6 – 2 – 1

60 – 5 6 – 2 – 1

TS

550 11 LL PK 230 22 LL PK340 SS PK

LTL P,SKmax 0 SST P,KSmin 0

Put butterfly

SLL PPP 221

SLLSL PPPKK 2211

)(22211 SSLLLL KPPKPK

1LK2LK

Long straddle

Long

call

Long

put Profit

0 – 3 35 32

20 – 3 15 12

25 – 3 10 7

30 – 3 5 2

35 – 3 0 – 3

40 – 3 – 5 – 8

45 2 – 5 – 3

50 7 – 5 2

55 12 – 5 7

60 17 – 5 12

TS

540 PK

340 CK

CKST 0,max

PC

PSK T 0,max

PCKST PCKST

Short straddle - short call + short put

CSK T 0,min PKST 0,min

PC

Long strangle - long call + long put

Short strangle - short call + short put

Strip

02 PCKS 02 CPSK

Strap

02 CPSK 02 PCKS

The Black-Scholes model - assumptions

• Markets are efficient (market movements cannot be predicted)

• There are no transaction costs in buying the option

• The risk-free rate and volatility of the underlying asset are known and constant

• The returns on the underlying are normally distributed

• The option is European and can only be exercised at expiration

• No dividends are paid out during the life of the option

Pricing options in the Black-Scholes framework

• T – an expiration time, r – a risk-free interest rate, – a stock’s volatility (a standard deviation), K – a strike price, S(t) – price of the stock at t, N(x) – the cumulative standard normal distribution (see a standard normal distribution table).

21 dNeKdNtSt,SC tTr

21 dNeKdNtSt,SP tTr

tT

tTrKtSlnd

22

1tTdd 12

Example

• You are given:

• The Back-Scholes framework holds

• The stock is currently selling for 50 PLN

• The option will expire in 5 months with a strike price of 48 PLN

• The stock’s volatility is 20%

• The continuously compounded risk-free interest rate is 10%

• Calculate the price of the European call and put options on a nondividend-paying stock.

Example

12510204850 T.r.KS

tT

tTrKtSlnd

22

1

70350414012520

12

5

2

20104850

2

1 ..

..ln

d

5744047012 .tTdd

Example

75912328301 .dN 71714621602 .dN

21 dNeKdNtSt,SC tTr

937968471710487591050 12

510

..e.t,SC.

24066549501 .dN 282749802 .dN

984835021 .dNeKdNtSt,SP tTr

Example

European call

option

European put

option

The Black-Scholes

model 4.937968 0.984835

Binominal option tree

(continuously

compounded interest)

4.932961 0.988305

The Merton model

• T – an expiration time, r – a risk-free interest rate, – a stock’s volatility (a standard deviation), K – a strike price, S(t) – price of the stock at t, N(x) – the cumulative standard normal distribution d – the dividend yield of the stock.

21 dNeKdNetSt,SC tTrtT d

21 dNeKdNetSt,SP tTrtT d

tT

tTrKtSlnd

d 22

1tTdd 12

Example (the Merton model)

• The stock is currently selling for 90 PLN,

• The stock’s volatility is 20%

• The strike price is 100 PLN, risk-free interest rate is10%

• The option expired at 3 months.

• The stock pays dividend continuously at a rate proportional to

its price. The dividend yield is 3%.

030123102010090 .T.r.KS d

003451.t,SC 177489.t,SP

Appendix (u, d and p)

knkk

n duSS

knk ppk

n

1

Binomial coefficient

!k!kn

!n

k

n

!n - factorial of n

dlnknulnkSlnSln k

n

10 S dlnknulnkSln k

n

knkk

n duSS

1S10 S

dlnpulnpSlnE 11

2

11 SlnESlnpSlnvar i

i

i

2

2

1

11

1

dlnpulnpSlnp

dlnpulnpSlnpSlnvar

d

u

2

2

1

11

1

dlnpulnpdlnp

dlnpulnpulnpSlnvar

2

2

1

11

1

dlnpulnpdlnp

dlnpulnpulnpSlnvar

2222

1 11 dlnulnppdlnulnppSlnvar

222

1 121 dlnulnpppppSlnvar

2

1 1 dlnulnppSlnvar

0S

SlnE T

0

2

S

Slnvar T

T

T eSS

0

0S

SlnT T

1SlnEt

1

2 lnvar S

tdlnpulnpSlnE 11

tdlnulnppSlnvar 22

1 1

UD

tDUpp

tDpUp

221

1

dlnD

ulnU

dlnuln

DU

du 1

tUpp

tUpUp

2214

1

UD

tDUpp

tDpUp

221

1

222

12

ttU

tUp

222

2

2

2

12

ttU

p

tU

22

2

2

12tt

p

t

1

12

12

2

2

t

t

p

1

12

12

2

t

t

p

111

2

12

2

t

tp

11

2

2

t

t

tp 1

2

1

tUp 12

12

p

tuln

1

12

12

2

t

t

p

12

2

t

tuln

11

2

2

t

t

tuln

teu ted