Corporate finance chapter 015 option and contingent claims

•To show how the law of one price maybe used to derive prices of options •To show how to infer implied volatility from option prices... Chapter 15 Contents15.1 How Options Work 15.2 Inve

•To show how the law of one price may

be used to derive prices of options

•To show how to infer implied volatility from option

prices

Chapter 15 Contents

15.1 How Options Work

15.2 Investing with Options

15.3 The Put-Call Parity

Relationship

15.4 Volatility & Option Prices

15.5 Two-State Option Pricing

15.6 Dynamic Replication & the

Table 15.1 List of IBM Option Prices

(Source: Wall Street Journal Interactive Edition, May 29, 1998)

IBM (IBM) Underlying stock price 120 1/16

Table 15.2 List of Index Option Prices

(Source: Wall Street Journal Interactive Edition, June 6, 1998)

Change 31-Dec Change

S&P500 1113.88 1084.28 1113.86 19.03 143.43 14.8

Strike Volume Last Change Interest

Jun 1110 call 2,081 17 1/4 8 1/2 15,754 Jun 1110 put 1,077 10 -11 17,104 Jul 1110 call 1,278 33 1/2 9 1/2 3,712 Jul 1110 put 152 23 3/8 -12 1/8 1,040 Jun 1120 call 80 12 7 16,585 Jun 1120 put 211 17 -11 9,947 Jul 1120 call 67 27 1/4 8 1/4 5,546 Jul 1120 put 10 27 1/2 -11 4,033

Terninal or Boundary Conditions for Call and Put Options

Stock, Call, Put, Bond

Call_BondShare

Put-Call Parity Equation

( rf ) Put Strike Maturity Share

Strike Maturity

(

Synthetic Securities

• The put-call parity relationship may be solved for any of the four

security variables to create synthetic securities:

C=S+P-B

S=C-P+B

P=C-S+B

B=S+P-C

Options and Forwards

• We saw in the last chapter that the discounted value of the forward

was equal to the current spot

• The relationship becomes

( )Maturity ( )Maturity

rf

Forward Maturity

Strike

Put rf

Strike Maturity

+

1

) ,

( 1

) ,

(

Implications for European

Options

• If (F > E) then (C > P)

• If (F = E) then (C = P)

• If (F < E) then (C < P)

• E is the common strike price

• F is the forward price of underlying share

• C is the call price

• P is the put price

Strike = Forward

Call = Put

Put and Call as Function of Share Price

PV Strike

Strik e

Volatility and Option Prices, P0 = $100, Strike = $100

Stock Price Call Payoff Put Payoff Low Volatility Case

Binary Model: Call

• Implementation:

– the synthetic call, C, is created by

• buying a fraction x of shares, of the stock, S,

and simultaneously selling short risk free bonds with a market value y

• the fraction x is called the hedge ratio

y xS

Binary Model: Call

• Specification:

– We have an equation, and given the value of

the terminal share price, we know the

terminal option value for two cases:

– By inspection, the solution is x=1/2, y = 40 x y

120 20

Binary Model: Call

• Solution:

– We now substitute the value of the

parameters x=1/2, y = 40 into the equation

– C to obtain: = xS − y

10

$ 40

100 2

Binary Model: Put

• Implementation:

– the synthetic put, P, is created by

• sell short a fraction x of shares, of the stock,

S, and simultaneously buy risk free bonds with a market value y

• the fraction x is called the hedge ratio

y xS

Binary Model: Put

• Specification:

– We have an equation, and given the value of

the terminal share price, we know the terminal option value for two cases:

– By inspection, the solution is x=1/2, y = 60 x y

120 20

Binary Model: Put

• Solution:

– We now substitute the value of the

parameters x=1/2, y = 60 into the equation

– P to obtain: = − xS + y

10

$ 60

100 2

1

= +

−

=

P

Decision Tree for Dynamic

Replication of a Call Option

The Black-Scholes Model:

• N(.) = cum norm dist’n

• The following are annual,

compounded continuously:

• r = domestic risk free rate of interest

• d = foreign risk free rate

or constant dividend yield

• σ = volatility

2 2

2 1

2

1 ln

2

1 ln

d N

Ee d

N Se

P

d N Ee

d N Se

C

T

d T

T d

r E

S d

T

T d

r E

S d

rT dT

rT dT

− +

σ σ

σ

The Black-Scholes Model: Equations (Forward Form)

E d

N Se

d N e

C

T

T E

Se d

T

T E

Se d

T d r rT

T d r rT

T d r

T d r

2 1

2 1

2 2

2 1

2

1 ln

2

1 ln

σ

S P

C

d N d

N S P

C d

P d

N d

N Se

C

T d

T d

Se E

dT

T d r

σ

σ π

σ σ

39886

0 2

0 If

2

1

; 2

1

If

2 1

2 1

2 1

Determinants of Option Prices

Increases in: Call Put

Stock Price, S Increase Decrease

Exercise Price, E Decrease Increase

Volatility, sigma Increase Increase

Time to Expiration, T Ambiguous Ambiguous

Interest Rate, r Increase Decrease

Cash Dividends, d Decrease Increase

Value of a Call and Put Options with Strike =

Current Stock Price

01234567891011

0.00.1

0.20.3

0.40.5

0.60.7

0.80.9

Call and Put Prices as a Function of Volatility

Computing Implied Volatility

n_d_1 =NORMSDIST(d_1)

n_d_2 =NORMSDIST(d_2)

call_part_1 =n_d_1*share*EXP(-rate_for*maturity)

call_part_2 =- n_d_2*strike*EXP(-rate_dom*maturity)

Payoffs for Bond and Stock Issues

Debtco Security Payoff Table

Debtco’s Replicating Portfolio

• Let

– x be the fraction of the firm in replicator

– Y be the borrowings at the risk-free rate in

the replicator

– In $’000,000 the following equations must

be satisfied

308 ,

692 ,

57

$

; 7 6

04 1 70

0

; 04 1 140

Y x

Y x

Debtco’s Replicating Portfolio

($’000)

Position Immediate Case a Case b 6/7 assets -85,714 120,000 60,000 Bond (rf) 57,692 -60,000 -60,000

Debtco’s Replicating Portfolio

• We know value of the firm is $1,000,000, and the value of the total

equity is $28,021,978, so the market value of the debt with a face of 80,000,000 is $71,978,022

• The yield on this debt is (80…/71…) - 1 = 11.14%

Another View of Debtco’s

Replicating Portfolio (‘$000)

Security Total

market Value

Equivalent Amount

of Firm

Equivalent Amount

of Rf Debt Bonds 71,978 14,286 57,692

Stock 28,022 85,714 -57,692

Bonds +

Stock

100,000 100,000 0

Valuing Bonds

– We can replicate the firm’s equity using x =

6/7 of the firm, and about Y = $58 million riskless borrowing (earlier analysis)

– The implied value of the bonds is then

$90,641,026 - $20,000,000 = $70,641,026 &

the yield is (80.00-70.64)/70.64 = 13.25%

026 ,

641 ,

90

$ 7

6

308 ,

692 ,

57 000

, 000 ,

Y xV

E

Determining the Weight of

Firm Invested in Bond, x, and the Value of the R.F.-Bond, Y

308 ,

692 ,

57

$

; 7 1

04

1 140

80

04

1 70

=

+

=

Y x

Y x

Y x

Valuing Stock

– We can replicate the bond by purchasing 1/7

of the company, and $57,692,308 of free 1-year bonds

default-– The market value of the bonds is $909.0909 * 80,000 = $72,727,273

– The value of the stock is therefore E=V -D =

$105,244,753 - $72,727,273= $32,517,480

753,244,105

$7

1

308,692,57273,727,72

xV D

Outline Decision Tree

$110MM Node-E

$90MM Node-G

Node-A

$100MM

State-Contingent Security #1

1

$

495 505

494

0 04

1

2 000

, 000 ,

70

000 ,

000 ,

100 000

, 000

,

1

04 1

2

; 000 ,

000 ,

70

1 0

04 1 000

, 000

,

140

1 04

1 000

, 000

,

70

#2 S.

C.

S.

967 032

467

0 04

1

1 000

, 000 ,

70

000 ,

000 ,

100 000

, 000

,

1

04 1

1

; 000 ,

000 ,

70

1 1

04 1 000

, 000

,

140

0 04

1 000

, 000

,

70

#1 S.

−

= +

= +

=

−

= +

= +

Y x P

Y

x Y

x

Y x

Y x P

Y

x Y

x

Y x

SCS Conformation of Guarantee’s Price

• Guarantee’s price is 125 * $0.494505 = $61.81