chp08a options pricing(b-s)

8/13/2019 Chp08a Options Pricing(B-S)

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Copyright K.Cuthbertson, D.Nitzsche. 1

Version 1/9/2001

FINANCIAL ENGINEERING:

DERIVATIVES AND RISK MANAGEMENT(J. Wiley, 2001)

K. Cuthbertson and D. Nitzsche

LECTURE

Options Pricing


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Valuation and Pricing (Black Scholes)

Speculation

Delta Hedging

Topics


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Valuation/Pricingof

Options


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Call Premium (price) C

depends on:

Time to maturity, T (+)

Current Spot price, relative tostrike price,

S /K (+)

Volatility spot price

(+)

Interest rate r (+)

Put Premium (price) P

depends on:

Time to maturity, T (+ or -)

Current Spot price, relative tostrike price,

S /K ( - )

Volatility spot price

(+)

Interest rate r (+ or - )

Value of Calls and Puts Prior to Expiry


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PVdNdNSC ).()(. 21

T

TrKSTdd

)2/()/ln( 2

12

PV = present value of the strike price = K e-rT

T

TrKSd

)2/()/ln( 2

1

Black-Scholes (Merton)


7/18Copyright K.Cuthbertson, D.Nitzsche.

Stock PriceK0

Value of call priorto expiry

B

A

Payoff from callat expiry

.

C0= 9.6

C1= 10

S1= 101S0= 100

C.

. D

St

Ct

CD=Intrinsic value

BC=Time value

Figure 21.5 : Black-Scholes

Call Premium



Key results are

CALLS

Call premium increases as stock price increases (butless than one-for-one)

Call premium increases if the volatility of the stockincreases

PUTS

Put premium falls as stock price increases (but less

than one-for-one)

Put premium increases if the volatility of the stockincreases

Black-Scholes


9/18Copyright K.Cuthbertson, D.Nitzsche. 9

Speculation with Options



Buy low and sell high

Expect a bull market - then buy a call and close out the positionbefore maturity

( that is, sell the call at a higher call premium)

Profit (before maturity) = C1- C0 = 5.2 - 5.0

Your net position of zero is noted by the Clearing House whichsends you the $0.2 (and cancels any delivery to you)

Sell high and buy low

Expect a bear market then sell a call at C0= 10 and close out theposition by buying back a call at C1= 9.8 (prior to maturity)

Speculation with Options (Before Maturity)



Delta Hedging with Options



Problem: You currently hold shares but you fear high

volatility of stock prices over the next next month. Youwant to protect the current value of your stock positionuntil the market returns to normal.

Can you hedge your stock position using options ?

A call option on the share is available with a delta of 0.4

which implies:

When S increases by +$1 (e.g. from 100 to 101), then Cincreases by $0.4 (e.g. from 9.6 to 10)



13/18Copyright K.Cuthbertson, D.Nitzsche.13

Note:

The contract size for one call option contract is for100 shares

But the price of the call option C is quoted as if therewas only one share underlying the option

(i.e. we need to multiply C by 100 to get the invoiceprice of the option)




Call Premium

Stock PriceK0

B

A

Payoff from callat expiry

.

C0= 9.6

C1= 10

S1= 101S0= 100

C.

. D

St

Ct

CD=Intrinsic value

BC=Time value

Delta is the slope of thiscurve

DELTA OF A CALL


15/18Copyright K.Cuthbertson, D.Nitzsche.15

Consider the following portfolio

40-sharesplus1 writ ten (sold )call option (at C0= 10)

Suppose S falls by $1 over the next month

THEN fall in C is 0.4 ( = delta of the call) So C falls to C1=

9.6

To close out you must now buy back at C1= 9.6

Loss on 40 shares = $40

Gain on initial written call = 100 (C0 -C1)= 100(0.4) = $40

DELTA HEDGING YOUR 40 SHARES WITH 1 WRITTEN CALLOPTION MEANS THAT THE VALUE OF YOUR POSITION

WILL BE UNCHANGED.




Call Premium

Stock Price100

0

D 0.4.

D 0.5

110

B

A

.

Figure : Delta Hedging: Rebalancing

As S changes then so does delta , so you have to rebalance your portfolio.

E.g. delta = 0.5, then hold 50 stocks for every written call.



END OF SLIDES


18/18Copyright K Cuthbertson D Nitzsche 18

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