Forwards and Futures

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					   Properties of
Stock Option Prices

     Chapter 9

                      1
ASSUMPTIONS:
1. The market is frictionless: No
transaction cost nor taxes exist. Trading
are executed instantly. There exists no
restrictions to short selling.
2. Market prices are synchronous
across assets. If a strategy requires the
purchase or sale of several assets in
different markets, the prices in these
markets are simultaneous. Moreover, no
bid-ask spread exist; only one trading
price.
                                        2
3.   Risk-free borrowing and lending exists
     at the unique risk-free rate.
     Risk-free borrowing is done by selling
     T-bills short and risk-free lending
     is done by purchasing T-bills.

4.   There exist no arbitrage opportunities
     in the options market




                                        3
                 NOTATIONS:
t = The current date.
St= The market price of the underlying asset.
K = The option’s exercise (strike) price.
T = The option’s expiration date.
T-t = The time remaining to expiration.
r = The annual risk-free rate.
 = The annual standard deviation of the
     returns on the underlying asset.
D = Cash dividend per share.
q = The annual dividend payout ratio.
                                          4
        Factors affecting Options Prices
              (Secs. 9.1 and 9.2)
Ct = the market premium of an American call.
ct = the market premium of an European call.
Pt = the market premium of an American put.
pt = the market premium of an European put.
  In general, we express the premiums as
  functions of the following variables:

    Ct , ct = c{St , K, T-t, , r, D },

    Pt , pt = p{St , K, T-t, , r, D }.   5
  FACTORS AFFECTING OPTIONS PRICES
  AND THE DIRECTION OF THEIR IMPACT:

Factor European European American American
          call     put     call     put
  St       +        -       +        -
  K       -        +        -        +
 T-t      ?        ?        +        +
        +         +        +        +
  r      +         -        +        -
  D       -        +        -        +

                                         6
Bounds on options market prices (Sec. 9.3)

         Call values at expiration:

 CT = cT = Max{ 0, ST – K }.

 Proof:
 At expiration the call is either
 exercised, in which case CF = ST – K,
 or
 it is left to expire worthless, in which
 case, CF = 0.
                                            7
            Minimum call value:

A call premium cannot be negative.
At any time t, prior to expiration,

                Ct , ct  0.

Proof: The current market price of a
call is
         NPV[Max{ 0, ST – K }]  0.

                                       8
            Maximum Call value:

                  Ct  St.

Proof: The call is a right to buy the stock.
Investors will not pay for this right more
than the value that the right to buy gives
them, I.e., the stock itself.



                                          9
          Put values at expiration:

  PT = pT = Max{ 0, K - ST}.

Proof:
At expiration the put is either exercised, in
which case CF = K - ST,
or it is left to expire worthless, in which
case CF = 0.


                                          10
           Minimum put value:

A put premium cannot be negative.
At any time t, prior to expiration,

                  Pt , pt  0.

Proof: The current market price of a put is

        NPV[Max{ 0, K - ST}]  0.

                                         11
       Maximum American Put value:

At any time t < T,    Pt  K.

Proof: The put is a right to sell the stock
For K, thus, the put’s price cannot exceed
the maximum value it will create: K, which
occurs if S drops to zero.




                                        12
       Maximum European Put value:

              pt  Ke-r(T-t).

Proof: The maximum gain from a European
put is K, ( in case S drops to zero). Thus, at
any time point before expiration, the
European put cannot exceed the NPV{K}.




                                            13
        Lower bound: American call value:
 At any time t, prior to expiration,

         Ct  Max{ 0, St - K}.

Proof:   Assume to the contrary that

         Ct < Max{ 0, St - K}.

 Then, buy the call and simultaneously
 exercise it for an arbitrage profit of:
 St – K – Ct > 0. a contradiction.         14
           Lower bound: European call value:

At any t, t < T,              ct  Max{ 0, St - Ke-r(T-t)}.

Proof: If, to the contrary, ct < Max{ 0, St - Ke-r(T-t)}

then, ct < St - Ke-r(T-t)    0 < St - Ke-r(T-t) - ct
   P/L PROFILE OF                     AT        EXPIRATION
     STRATEGY:                ICF     ST < K        ST > K
   SELL THE STOCK             St       - ST          - ST
       SHORT
       BUY CALL            -ct           0          ST - K
    LEND NPV OF K        -Ke-r(t-t)      K            K
        TOTAL               ?          K - ST         0 15
      American vs European Calls

The market value of an American call is
at least as high as the market value of a
European call.

  Ct  ct  Max{ 0, St - Ke-r(T-t)}.

Proof: An American call may be
exercised at any time, t, prior to
expiration, t<T, while the European call
holder may exercise it only at expiration.
                                        16
      Lower bound: American put value:

At any time t, prior to expiration,
         Pt  Max{ 0, K - St}.

Proof:    Assume to the contrary that
          Pt < Max{ 0, K - St}.
Then, buy the put and simultaneously
exercise it for an arbitrage profit of:
K - St – Pt > 0. A contradiction of the
no arbitrage profits assumption.
                                          17
    American vs European Puts (Sec. 9.6)

         Pt  pt  Max{0, Ke-r(T-t) - St}.

Proof: First, An American put may be
exercised at any time, t; t < T. A European put
may be Exercise only at T. If the price of the
underlying asset fall below some price, it
becomes optimal to exercise the American
put. At that very same moment the European
put holder wants to exercise the put but
cannot because it is European.              18
  Second, the other side of the inequality:
At any t, t < T,       pt  Max{ 0, Ke-r(T-t) - St }.

Proof: If, to the contrary, pt < Max{ 0, Ke-r(T-t) - St }

then, pt < Ke-r(T-t) – St  0 < Ke-r(T-t) - St - pt

   P/L PROFILE OF                     AT       EXPIRATION
     STRATEGY:              ICF       ST < K      ST > K
     Buy the stock         - St         ST          ST
      Buy the put          - pt       K - ST         0
   Borrow NPV OF K        Ke-r(t-t)     -K          -K
        TOTAL               ?            0              K
                                                  ST - 19
American put is always priced higher than its
     European counterpart. Pt  pt

     P/L
             For S< S** the European put
             premium is less than the put’s
             intrinsic value. For S< S* the
      K
             American put premium
Ke-r(T-t)    coincides with the put’s
                       intrinsic value.
                              P
                         p
              S*    S** K                     S

                                         20
          The put-call parity (Sec. 9.4)
European options:
The premiums of European calls and puts
written on the same non dividend paying
stock for the same expiration and the same
strike price must satisfy:

            ct - pt = St - Ke-r(T-t).

The parity may be rewritten as:

            ct + Ke-r(T-t) = St + pt.   21
   P/L                                     AT       EXPIRATION
PROFILE of           ICF
STRATEGY
                                           ST < K      ST > K
BUY STOCK           - St                     ST          ST
 BUY PUT            - pt                   K - ST        0
  TOTAL         - [St + pt]                  K           ST
   P/L                                     AT       EXPIRATION
PROFILE of
STRATEGY            ICF                    ST < K      ST > K
BUY CALL            - ct                        0      ST - K
                     – r(T – t)
  LEND          - Ke                            K        K
 NPV(K)
                          – r(T – t)
  TOTAL      - [ct + Ke                ]        K        ST
                                                          22
      Synthetic European options:

The put-call parity
          ct + Ke-r(T-t = St + pt
can be rewritten as a synthetic call:

          ct = pt + St - Ke-r(T-t),

or as a synthetic put:

           pt = ct - St + Ke-r(T-t).
                                        23
           Synthetic Risk-free rate

The put-call parity ct + Ke-r(T-t) = St + pt

      1          K        
  r      ln              
     T  t  St  p t  ct 
 For another synthetic risk-free rate we next
      analyze the Box Spread strategy: 24
          Stock XYZ, TH SEP 21, 2007. All prices $/share.
S=61.48              calls                          puts
  K       OCT    NOV     JAN      APR    OCT     NOV    JAN     APR
  40       22                23                          .56
  50       12                             .63
  55      8.13           11.5            1.25           3.63
  60      4.75           8.75            2.88    4.00   5.75
  65      2.50   3.98    5.75     8.63   6.00    6.63   8.38    10.00
  70       .94           3.88            9.25           11.25
  75       .31                    5.13   13.38                  16.79
  80                     1.63
  90                     .81
  95                     .44




                                                                  25
  1.       Stock and options markets

       P/L of strategy        ICF         AT           EXPIRATION
                                           ST < 75          ST > 75
       Buy the STOCK        -61.48             ST             ST
       Buy APR 75 PUT       -16.79          75 - ST            0
    Sell APR 75 CALL         5.13              0            75 - ST
           TOTAL            -73.14             75             75
            P/L                              1.86            1.86


This strategy guarantees its holder a sure profit of $1.86/share for an
investment of $73.14/share in a 7 months period.
For this strategy to work all the initial prices – the stock the put and the
call must be available for the investor at the same instant.
If the strategy is possible, it creates a RISK-FREE rate:
                                                                      26
            7
         r
73.14e     12
                 75
    1      75 
r     ln       4.305%
   7/12  73.14



                            27
 Suppose that the yield on T-bills that mature on the option’s
 expiration is r = 5.17%. Then, to make arbitrage profit:


  P/L of strategy            ICF           AT          EXPIRATION
                                           ST < 75        ST > 75
sell the STOCK short     61.48                  -ST         -ST
 short APR 75 PUT        16.79            -[75 – ST]         0
 long APR 75 CALL         -5.13                 0        -[75 – ST]
    Buy T-bills          -73.14             75.38          75.38
      TOTAL                  0               0.38          0.38
                                             0.38          0.38

When the T-bills mature, the GOV pays you:
                          .0517[7/12]
                    73.14e              = 75.38.
The above strategy guarantees you an ARBITRAGE PROFIT of
38 cents per share.                                                   28
       Another Synthetic Risk-free rate
     A Box spread (p. 235):     K1 < K 2 .
   P/L
PROFILE of
                        AT      EXPIRATION
             ICF
STRATEGY            ST < K1 K1<ST < K2    ST > K2
Buy p(K2)    - p2   K2 - ST   K2 - ST        0
Sell p(K1)   p1 ST - K1          0            0
Sell c(K2)   c2    0             0       - [ST - K2]
Buy c(K1)    -c1   0          ST – K1      ST - K1

 TOTAL        ?     K2 - K1   K2 - K1     K2 - K1
                                                29
   The initial cost of the box spread is:

               c1 - c2 + p2 - p1

The certain income from the box spread at
        the options’ expiration, T, is:

                   K2 - K1

                    Thus:


      c1 - c2 + p2 - p1 = (K2-K1)e-r(T-t)
                                            30
                  Reiterating:
For the Box spread strategy:
An initial investment of c1 - c2 + p2 - p1
yields a sure income of K2-K1regardless of the
underlying asset’s market price. Thus, solving

          c1 - c2 + p2 - p1 = (K2-K1)e-r(T-t)

for r, yields a risk-free rate:

      1         K 2  K1      
  r      ln                  
     T  t  c1  c2  p2  p1                 31
  2.     Options markets only. A BOX SPREAD. (p. 235)

   P/L of strategy          ICF            AT          EXPIRATION
                                      ST < 55    55 < ST < 60    ST > 60
Buy the JAN 55 CALL        - 11.50       0          ST - 55      ST - 55
 Buy the JAN 60 PUT        - 5.75      60 - ST      60 - ST         0
Sell the JAN 60 CALL        8.75         0             0         60 - ST
 Sell the JAN 55 PUT        3.63       ST - 55         0            0
        TOTAL              - 4.87       5.00         5.00         5.00


This strategy guarantees its holder a sure profit of $.13/share for an
investment of $4.87/share in a 4 months period.
For this strategy to work all the initial prices – the CALLS and the PUTS
must be available for the investor at the same instant.
If the strategy is possible, it creates a RISK-FREE rate:
                                                                   32
                  4
               r
  4.87e          12
                       5
      1      5 
  r     ln       7.903%
     4/12  4.87 
Of course, an annual risk-free rate of 7.903% is large and it
indicates that one could NOT have been able to create this
strategies with the prices given in the table.




                                                                33
                   Summary
We have seen that there are strategies that
yield synthetic Risk-free rates.
1. The put-call parity yields a risk-free rate
   that requires inputs from the options
   market and the stock market.
2. The Box spread yields a risk-free rate
   that requires inputs from the options
   market ONLY.
Of course, T-bill rates are risk-free.
IN AN EFFICIENT ECONOMIC MARKETS ALL
THESE RATES MUST BE EQUAL.
                                          34
                 Summary
If the above rates are not equal arbitrage
profit exists. You may use a strategy to
create a positive, risk-free cash flow; i.e.,
borrow at the lower risk-free rate, and
invest the proceeds in the strategy that
yields the higher risk-free rate.

The above is exactly what professional
arbitrageurs do, mainly, using the Box
Spread strategy.
                                           35
                      Example
Suppose that calculating the risk-free rate
from a Box spread on slide 26 yields r = 3%.
The options are for .5yrs and the 6-month
T-bill yield a risk-free rate of 5%.
Arbitrage:
Borrow money employing the reverse Box
Spread ( effectively borrowing at 3%) and
invest it in the 6-month T-bill. At expiration
receive 5% from the GOV and repay your 3%
Debt for An ARBITRAGE PROFIT of 2%.
                                          36
                DIVIDEND FACTS:
Firms announce their intent to pay dividends
   on a specific future day – the X-dividend
   day. Any investor who holds shares before
   the stock goes – X-dividend will receive
   the dividend. The checks go out about one
   week after the X-dividend day.



          S  CDIV        S
                         XDIV

 tAnnouncement           t
                         XDIV      t
                                   PAYMENT   Time line


4.           S   XDIV   =SCDIV   - D.           37
                DIVIDEND FACTS:
1. The share price drops by $D/share when
   the stock goes x-dividend.
2. The call value decreases when the price
   per share falls.
3. The exchanges do not compensate call
   holders for the loss of value that ensues
   the price drop on the x-dividend date.


          S  CDIV        S
                         XDIV

 tAnnouncement           t
                         XDIV      t
                                   PAYMENT   Time line


4.           S   XDIV   =SCDIV   - D.           38
             The dividend effect
Early exercise of Unprotected American calls
   on a cash dividend paying stock:
Consider an American call on a cash
dividend paying stock. It may be optimal to
exercise this American call an instant before
the stock goes x-dividend. Two condition
must hold for the early exercise to be optimal:
First, the call must be in-the-money. Second,
the $[dividend/share], D, must exceed the
time value of the call at the X-dividend
instant. To see this result consider:      39
            The dividend effect
The call holder goal is to maximize the Cash
  flow from the call. Thus, at any moment in
  time, exercising the call is inferior to
  selling the call. This conclusion may
  change, however, an instant before the
  stock goes x-dividend:

              Exercise     Do not exercise
Cash flow:    SCD – K      c{SXD, K, T - tXD}
Substitute:   SCD = SXD + D.
Cash flow:    SXD –K + D SXD – K + TV.
                                           40
                 The dividend effect
   Early exercise of American calls may be
                    Optimal if:
1. The call is in the money

2.               D > TV.
     In this case, the call should be (optimally)
         exercised an instant before the stock
     goes x-dividend and the cash flow will be:
                  SCD – K = SXD –K + D.


                                              41
    Early exercise of Unprotected American
     calls on a cash dividend paying stock:
The previous result means that an investor is
 indifferent to exercising the call an instant
   before the stock goes x-dividend if the
    x- dividend stock price S*XD satisfies:

      S*XD –K + D = c{S*XD , K, T - tXD}.

 It can be shown that this implies that the
            Price, S*XD ,exists if:

             D > K[1 – e-r(T – t)].         42
  Early exercise of Unprotected American
   calls on a cash dividend paying stock:

            D > K[1 – e-r(T – t)].
Example:
r = .05
T – t = .5yr.
K = 30.
           30[1 – e-.05(.5)] = $.74.
 Thus, if the dividend is greater than 74
 cent per share, the possibility of early
 exercise exists.                         43
                    Explanation

  C XD                                             SXD  Ke    r(T t)
                          *
                      S   XD




                                                          SXD
      SXD  K  D    Ke    r(Tt)
                                      K
                                      -r(T-t)
 K D              -K+D> Ke
      r(T t)                       -r(T-t)
- Ke                D> K[1-e                   ]                 44
  Early exercise: Non dividend paying stock

It is not optimal to exercise an American call
  prior to its expiration if the underlying stock
  does not pay any dividend during the life of
  the option.
  Proof: If an American call holder wishes to
rid of the option at any time prior to its
expiration, the market premium is greater
than the intrinsic value because the time
value is always positive.
                                            45
Early exercise: Non dividend paying stock

The American feature is worthless if the
 underlying stock does not pay out any
 dividend during the life of the call.
 Mathematically: Ct = ct.

 Proof: Follows from the previous result.




                                            46
American put on a non dividend paying stock

It may be optimal to exercise a put on a non
  dividend paying stock prematurely.

Proof: There is still time to expiration and the
 stock price fell to 0. An American put holder
 will definitely exercise the put. It follows
 that early exercise of an American put may
 be optimal if the put is enough in-the
 money.

                                            47
   The put-call parity of European options
               with dividends:

      Consider European puts and calls are
        written on a dividend paying stock.
The stock will pay dividend in the amounts Dj
    on dates tj; j = 1,…,n, and      tn < T.
  rj = the risk-free during tj – t; j=1,…,n.
Then:
                                     n
                                     D je
                      rT (T t)              rj (t j  t)
c t  p t  St  Ke
                                     j1
                                                   48
    The put-call parity with dividends
   P/L                         AT     EXPIRATION
PROFILE of        ICF
STRATEGY                      tj    ST < K ST < K
SELL STOCK        St          -Dj    - ST     - ST
  SHORT
 SELL PUT         pt                ST - K     0
BUY CALL         - ct                 0      ST - K
  LEND       - Ke-rT(T-t)             K        K
 NPV(K)
  LEND       - Dje-rj(tj-t)   Dj
 NPV(DJ)
  TOTAL            0          0       0        0
                                                   49
  The put-call “parity” for American options
      on a non dividend paying stock:
                 (p. 220)

    At any time point, t, the premiums of
              American options
on a non dividend paying stock, must satisfy
         the following inequalities:

                                  -r(T-t)
   St - K < Ct - Pt < St - Ke
                                            50
        Proof: Rewrite the inequality:
         St - K < Ct - Pt < St - Ke-r(T-t).
The RHS of the inequality follows from the
parity for European options: Ct = ct because
the stock does not pay dividend. Moreover,
For the puts Pt > pt.
Next, on the LHS, suppose that:
St - K > Ct - Pt   i.e.,     St - K - Ct + Pt > 0.
It can be easily shown that this is an
arbitrage profit making strategy, and hence
Cannot hold.
                                             51
 The put-call “parity” for American options
          on dividend paying stock:
Let NPV(D) denote the present value of the
   dividend payments during the life of the
   options.
                          n
       NPV(D)   D e
                                  rj (t j  t)
                               j
                         j1
Then:

                                         -r(T-t)
   St – NPV(D) – K < Ct - Pt < St - Ke

                                               52

				
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