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

 Reading the Options Quotes
 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)
 If S follow Ito process,
   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

    (ii) Buy 7% cap, pay 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:

  [I] Buy at-the-M puts

  [II] Buy out-of-the-M puts

  [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: P&L Railroad

  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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