SPS HOLDINGS 2006Opt’95 1

Basics of Capital MarketsBasics of Capital Markets

AXA Belgium 2006AXA Belgium 2006

Options PresentationOptions Presentation

SPS HOLDINGS 2006Opt’95 2

Pricing an Option

Portfolio 1: C + XD = V1

Portfolio 2: P + S = V2

where C = Value of a Call

XD = Discounted Exercise Price

P = Value of a PutS = A share of the underlying

S*= Spot Price at Expiry

Portfolio 1 must equal Portfolio 2

V1 = V2

SPS HOLDINGS 2006Opt’95 3

Arbitrage Relationship for Put-Call Parity

S*<X S*>X

V1 = C + XD O + X (S*-X) + X

V2 = P + S (X-S*) + S* O + S*

V1 = X V1 = S*

V2 = X V2 = S*

V1 = V2 IN BOTH

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Thus:

V1 = V2

C+XD = P+S

C-(S-XD) = P

and since S-XD = FORWARD (F)

C-F = P

F = C-P

- F = P-C

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Long a Call

Long a Forward

Short a Put

F = C - P

45oS

$

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Creating the Forward

Long a Call

Long a Forward

Short a Put

F = C - P

45oS

$

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Short a Forward

Short a Call

Long a Put

S

$

-F = P-C

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Creating the Forward

Short a Forward

Short a Call

Long a Put

S

$

-F = P-C

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Valuing the Option Today - Using Intuition

Day 0 Day 1

105

95100

5 %

5 %

Share Price At Expiry

Value of Call At Expiry

$ 105 $ 5$ 95 $ 0

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CREATE AN ARBITRAGE PORTFOLIO

S C 2C S-2C

105 5 10 95 95 0 0 95

So no matter what happens - S = 105 or S = 95

S - 2C = 95

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On Day 1 S-2C = 95 But What About Day 0?

S - 2C

If (100 - 2C) day 0 > (95) day 1

Then (100 - 2C) day 0 = (95) day 1 1 + r

2C = 100 - (95) 1 + r

if r = 7.5%

C = $2.51

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Through the Arbitrage Portfolio

it is clear that as the Interest Rate Rises (r )

the Value of a Call Rises ( C ).

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Alternatively, one can see graphically

As XD < X and C = S* - XD (intrinsic value)

Then the greater r XD smaller

C = S* - XD is larger

XD X

$

S

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From this diagram, intuitively -----

As T = Time 0 the XD X therefore C if XD C the greater T

where T = time to expiry

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C = $

SXD

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Finding the Valuegiven the Volatility

C = $

SXD

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If XD, S, T are held constant, then

as the C?

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• Volatility is Why Options Exist!

• Without it We could give them Away!!!

• Can you think of examples Where someone would be so stupid???

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Finally:

0

0

0

0

0

),,,,(

Cr

CT

CX

CS

C

rTXScC

OR

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C C S X T r ( , , , , ) + - + + +

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From Put/Call ParityP = C - S + XD

,0

0

0

C

P

S

P

XD

P

C( S, X, T, r, σ )

+ - + + +

d (XD) = d ( r, T ),- -

P = P (S, X, T, r, σ )- + ? ? +

But Note

then