options by stariya

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```									                               Options
A Primer on Option
An option gives its holder the right to buy or sell a specified asset at a
specified price on (or before) a specified date

＊call vs. put
＊underlying asset
＊exercise (strike) price
＊expiration date
European Options vs. American Options
Options on Physical vs. Options on Futures

 Value at Expiration of a European Option
CT = max｛ST – K , 0｝
PT = max｛K – ST , 0｝

 Put-Call Parity
C – P = S – Ke-rT
C－P = f
P－C = -f

(Fig.10-4)
Option Valuation
 Simplified Approach to Option Valuation
(Binomial OPM)
One day option
t0                       t1
100                 105 (CU = 5)
95 (CD = 0)
Q: value of C today w/ K=100?
Form arbitrage portfolio
Let 105－(N×5) = 95－(N×0)
N=2
SU  S D
or N =    CU  C D
(define hedge ratio Δ= 1/N )
Value of arbitrage portfolio S-2C at t1 is
105－2×5 = 95－2×0 = 95
discounted at rf = 7.5%  VH today
100－2C = 95/1.0002 ∴C = \$2.51
Two day option
Day 0    Day 1                    Day 2
110.25      C = 10.25
105
100                          99.75      C=0
95
90.25      C=0

Form hedge portfolio on day 1

Form hedge portfolio on day 0

Day 0                Day 1                    Day 2
110.25    C = 10.25
C = 5.14
105
Δ= 0.976
C = 2.58
100                                      99.75    C = 0.0
Δ= 0.514
C = 0.0
95
Δ= 0.0
90.25   C = 0.0
C = C (S, K, T, r, σ)

From put-call parity
P = C (S, K, T, r, σ)－S＋Ke-rT

P = P (S, K, T, r, σ)
 Note on American Options
American Calls
(i) div = 0
∵ C = S－ Ke-rT＋ time value ＞ S － K
( early exercise )

(ii) div≠0

only     SX－K＋ div＞ C (SX , K, T)
or      div ＞ C (SX , K, T) － (SX－ K)

American puts
Early exercise only for deep-in-the-M put
ex. K = \$100, T = 1 yr., r = 20%
(i) Suppose S = \$10

(ii) Suppose S = \$20
 Black-Scholes Model
Assumptions
1. div = 0
2. TC = 0, Tax = 0
3. r constant
4. short sale no penalties
5. S follows continuous Ito process
6. S～log normal
Derivation
Value of hedge portfolio
VH = QSS＋QCC           (1)
dVH = QSdS＋QCdC        (2)
dS
S = udt＋σdZ
Z ～ N ( 0, dt )
dS = uSdt＋σSdZ
and E ( dS ) = uSdt, Var( dS ) =σ2S2dt
since C = C ( S, t )
from Ito’s Lemma
C       C 1  2C 2 2 
 t  2 S 2 S   dt
dS                   
dC = S                                  (3)
                 

(Fig.11-2)

(Fig.11-3)
Sub eq (3) into (2):
C       C 1  2C 2 2 
dHV = QSdS + QC                 t  2 S 2 S   dt
dS                         (4)
S                       

Note:
QS           C
Chose        QC = S
 dVH deterministic riskless
dVH
 V   = rdt
H
dVH = rVHdt
C
setting QC = -1,        QS = S

From eq (4) :
 C 1  2C 2 2 
dVH = －  t  2 S 2 S   dt
                                     (5)
                 

From eq (1) :
C
VH = S S - C
C
r
dVH = S Sdt - rCdt                       (6)

Setting (5) = (6),     differential eq :
C        C      1  2C 2 2
 rC -    rS -        (S  )           (7)
t        S      2 S 2

Boundary Condition:
CT = max｛ST – K , 0｝
To solve eq (7), let preference “risk neutral”
C = e-rT E (CT)
Ass ST～lognormal

 ( S - K ) L' ( S )dS
- rT
C= e          k                         (8)

Result in Black-Scholes solution:
C = SN(d1) - e-rT KN(d2)
S         σ2
ln( )  ( r  ) T
K          2
d1 =       σ T

d2 = d1 -σ T

Again,
C = C (S , K , T , r ,σ)

Note:
Form eq (8)
C = e-rT E (ST\ST＞K) Prob(ST＞K) - e-rTKProb(ST＞K)

For extremely out-of-the-M
ln ( S/K ) ＜＜0 → N (ln ( S/K )) → 0
∴ C → 0
For extremely in-the-M
ln ( S/K )＞0  N (ln ( S/K )) → 1
∴ C → S – e-rT K
Analytical Models
 Generalization of B-S Model

(Table 11-2)

Merton (1973) div≠0

Ingersoll (1976)

Merton (1973) : variable r

Merton (1976) : Jump-diffusion model
Ass. size of jumps distributed lognormally,

 Extensions to B-S Model
Options on Futures
Black (1976)

Options on Currencies
Garman-Kohlhagen (1983)
Grabbe
Compound Options
Geske (1979)

Path-Dependent Options

American-Style Options
Roll (1977), Geske (1979), Whaley (1981)

Numerical Models
 Binomial Models—Lattice Approach
Sharpe (1978), Cox-Ross-Rubinstein (1979)

 Finite Difference Methodology
Swartz (1977), Courtandon (1982) convert differential eq. into a set of
difference, and solved iteratively

Brennan-Schwartz:
trinomial lattice approach

Hull-White (1990):
modification that ensures the trinomial lattice methodology converges
to solution of underlying differential eq.
 Monte Carlo Simulation
Value of Option  Expected value of ST

Boyle (1977)
Simulate future movements in S  distribution of ST
Application

 Hedging Interest Rate Risk

 Caps – call options on forward r

 Floors – put options on forward r

 Collars – buy cap + sell floor

 Case: Falcon Cable

 Case: L.L Bean

 Case: First Union Bank

 Case: Muzak

 Using Options to Reduce Funding Costs
 Sell floor
[ex.] Firm’s borrowing rate = LIBOR+ 50bp

(i) Sell 7% floor, earn premium 35bp/yr

LIBOR
Floor Cost            LIBOR
Cap Cost

10       0                      10         3
9        0                      9          2
8        0                      8          1
7        0                      7          0
6        1                      6          1
5        2                      5          2
4        3                      4          3

 Issue hybrid debts
[ex.] Bond w/warrant

[ex.] Indexed bonds

 Sell “embedded option” bonds

 Hedging FX Risk
 Case: US food products firm contracted to sell DM 19M goods
to German supermarket chain over next year.

Delivery＆payment in DM on 15th of mar. Jun. Sep. ＆ Dec
(\$2.8M (DM4.75M) worth of receivable each quarter)

Current rate = \$0.59/DM

Hedging strategies:

[III] P(K=0.58) – C(K=0.60)

[IV]    PL  PH

[V] Scale hedge

 Case: Burns & McBride
 Case: Chase Manhattan Bank

 Hedge Commodity Price Risk

 Case: Qualex – use commodity collar

 Case: Govn’t bodies use options to hedge energy
prices

 Using Options to Manage Equity Price Risk
Share Repurchase Program

 Case: M&A activities
 Monetizing an Equity Position

To help manage employee stock option programs
Buy Call on its own stock

Investors protect stock portfolio
(Figure12-4)
(Figure12-5)
   Enchance yield of portfolio
(Figure12-6)

 Using Options to Increase Debt Capacity

 Using Options as A Competitive Tool

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