Option Valuation

1
Option Valuation

• Arbitrage should force two assets with identical
  cash flows to have equal prices
• Example 1
• A Buy call (EX100, P100, T1)
• B Buy stock (P100) Borrow PV(100)(r.08)

25
Call
0
125-100
Stock borrowing
100-100
2

• In part B, we start by borrowing 100/1.08
  92.59
• Since both investments lead to cash flows of
  either 25 or 0, their initial investments have
  to equal, otherwise arbitrage opportunity exists
• Call P PV(100) 100 92.59 7.41

3

• Example 2
• A Buy call (EX40, P32, T3 mos ¼)
• B Buy stock (P32) Borrow PV(16)
• (r.02/quarter) ? borrow 16/1.02 15.69

4

• 2 calls will lead to same payoff
• Value of 2 x Call 32 15.69 16.31
• Call 8.16
• In Example 1, rate of return on stock
• E(r) Pu(.25)Pd(0)
• Assuming that investors ignore risk, E(r)
  Pu(.25)(1-Pu)(0) .08
• Pu must be .32
• Pd .68

25
0
5

• For the call option,
• E(CF) .32(25) .68(0) 8
• E(CF) Emax(P-EX,0)
• PVE(CF) PVE(max(P-EX,0) 8/1.08 7.41
• In Example 2
• E(r) Pu (100) Pd(-50) .02 (under risk
  neutrality)
• Pu .3467 Pd .6533
• Emax(P-EX,0) .3467(24) .6533(0) 8.32
• PVE(CF) 8.321.02 8.16

24
Call
0
6

• Back to Example 1
• Note 1 Call is equivalent to 1 share borrowing

dP 25
dC 25
7

• In Example 1
• Note 1 Call is equivalent to ½ (share
  borrowing)
• 8.16 ½ (32-15.69)

dP 48
dC 24
8
Delta

• ? dC/dP
• Rule One long call is equivalent to long ?
  shares borrow (?P Call)
• Example 2
• ? 0.5, P 32, Call 8.16
• Long ? shares 0.5(32) 16
• Borrow (?P Call) borrow (16-8.16) 7.84
• Pay back (7.84)(1.02) 8.00

9

• Example 1
• ? 1, P 100, Call 7.41
• Long 1 share 100
• Borrow (?P-Call) 100 7.41 92.59
• Pay back (92.59)(1.08) 100

10

• One long call is equivalent to long ? shares
  borrow (?P Call)
• Call ?P loan (? - Call)
• Loan (? - Call) ?P Call
• Both sides give risk free return
• delta hedging
• Recall riskless hedge through
• 1 share1 put- 1 call
• Delta hedge through
• Long ?P 1 call
• Or long ?P 1 put
• dynamic hedging because ? is not constant

11

• So,
• Call ?P Loan
• Loan ?P Call
• Put -?P Loan
• Loan ?P Put ? - Call

12
Black Scholes Merton Model

• Call P N(d1) EX e-rT N(d2)
• Put EX e-rT N(-d2) - P N(-d1)

13

• Example
• P 100, EX 100, r .06, T 1, ? .35
• From Normal tables, N(d1) .6368 N(d2) .5
• in Excel NORMDIST(0.35,0,1,TRUE)
• Call 100(.6368)-100(.94176)(.5) 16.59

14

• For riskless hedge, we would need to buy .6368
  shares per each call written
• Buy 6368 shares _at_ 100 636,800
• Write 10,000 calls _at_ 16.59 (165,900)
• 470,900
• If stock goes up to 101, dP 1
• What should be the effect on Call value?
• dC/dP ?C/?P x dP ? x dP .6368 x 1 .6368
• So, Call 16.59 .6368 17.2268
• Portfolio
• Long 6368 shares x 101 643,168
• Short 10,000 calls x 17.2268 (172,268)
• 470,900

15

• Notes on delta hedging
• In our example, we had a small change in P,
  therefore Call left portfolio value unchanged
• Since P T change continuously, delta varies
• If delta (hedge ratio) is maintained throughout
  T, portfolio will grow by riskfree rate

16
Static hedge

• 1 share 1 put 1 call
• A put w/ EX 100 in previous example
• Put EX e-rT N(-d2) - P N(-d1)
• 100(.94176)(.5) 100(.3632) 10.77
• Riskless hedge
• Buy 5000 shares x 100 500,000
• Buy 5000 puts x 10.77 53,850
• Write 5000 calls x 16.59 (82,950)
• 470,900

17

• One year holding period
• 5000 x 5.82 29,100
• In one year, our 470,000 investment grows by
  29,100

18

• Portfolio values
• Portfolio value will be 500,000 regardless of
  the stock price
• Portfolio return will depend on put call prices