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					                                 Arbitrage

 There is an arbitrage opportunity when the law of one price is violated
  so that it is possible to get something for nothing (a free lunch). This
  cannot be true in conditions of equilibrium.

 Definitions. Let:   R   Rij          be the     (sn)   matrix of payoffs,
  i  1,2, s states of nature and j  1,2,n financial assets; x
  = col. (n1) vector of the quantities of assets in a portfolio. x j  0
  denotes a long position and x j  0 a short position (the holder of the
  portfolio has to pay the payoff);   y= col. (s1) of the payoffs of the
  portfolio in the different   states: y  Rx ; p= row (1n) of the
  assets’ prices.

 An arbitrage portfolio (AP) should have a non-positive cost         px  0
                     y  Rx  0 .
  and a semi-positive payoff:
  Hence: Rx  0  px  0 is the no-arbitrage condition.

       th
 The i (Arrow-Debreu) contingent security pays 1 euro in the state i and
  0 in the other states. Its payoff y is semi-positive (yi=1, yji=0). Then
  its price, denoted by qi to distinguish it from the pj prices of the actual
                                                                           th
  securities, is positive qi>0: it is the price of 1 euro in the i
  contingency. The row (1s) vector q is the state prices vector. The
  (A-D) securities represent a basis for the payoffs space (they form the
  ss unitary matrix).

 The payoff of any j security [Rj the j col. (s1) of Rij] can be
                         th                     th

  represented by a portfolio of A-D securities. Its price in no-arbitrage
  conditions is then equal to     p j   qi Rij  qR j .         In general, it
                                            i
  holds the:
                                      1
 Fundamental Theorem 1 (FT1): Linear pricing rule:
                      AP at pq>0  p=qR.
 Fundamental Theorem 2 (FT2): Implicit (or shadow or martingale)
  probabilities.
 AP         probabilities     1 ,  2  s  and a discount factor
1  r  such that: p  1  r  R  1  r  E R .
          1                          1                1



Proof: it is sufficient to define   q /  q i and 1  r    qi .
                                                             1

                                            i                      i

           0   i  1 and   i  1. Also: q  1  r  
                                                                     1
We have:                                                                   so
                                 i
that FT2 follows from FT1.


                                   Options
 Options: can be either call or put. A call is a financial instrument that
  gives its owner the right, but not the obligation, to purchase the
  underlying asset (stocks, stock indices, etc.) at a specified price (strike
  or exercise price) for a specified time. A put option gives its owner the
  right to sell the underlying at the strike price for a specified time. There
  are two kinds of options: the American option can be exercised at any
  time before or at the expiration; the European option can be exercised
  only at the expiration. We shall only deal with European options. The
  buyer of an option pays cash the option price to the seller (or writer)
  who assumes all the obligations of the contract (all the rights are of the
  buyer).
 Payoffs of the options: Let n be the date of expiration of the option,
  pa  n the price of the underlying asset at the expiration, K the strike
  price. Without considering the initial cost of the option, the payoff from
  a long position in a call is:
                      Max pa  n  K ,            0

                                      2
and the payoff from a long position in a put is:
                      MaxK  pa  n,                   0
For a short position (that of the option’s writer) in a call the payoff is:
 Max pa n  K ,                0  MinK  pa n,                    0
and for a short position in a put the payoff is:
 MaxK  pa n,                  0  Min pa n  K ,                  0

                      Option pricing: Binomial model

Assumptions: a bond priced      p f 0 yields a riskless rate r; a stock has a
price   pa 0   that either goes up   U%     or goes down     D%    [with actual
probabilities * and (1*)]. So either:
pa 1  1  U  pa 0  upa 0                            or
pa 1  1  D pa 0  dpa 0 .
Calculate the price of a 1 period call option on 1 stock with K as the strike
price. Note that its payoff is:
                      Maxupa 0  K ,                       0
                 Rc 
                      Maxdpa 0  K ,                       0
                                                                 .



 Step 1: Calculate        .   From FT2:        1  r  p  R          where:

   p  pa 0         p f 0 ,              u             d        and

                       upa 0             1  r  p 0
                   R
                                                     f

                       dpa 0             1  r  p 0
                                                          .
                                                     f


                                       3
Hence,   1  r  p  R can be written as the following system:
1  r  p 0   up 0   dp 0
             a            u       a             d   a


1  r  p 0   1  r  p 0   1  r  p 0
             f            u               f             d          f
                                         or:
1 r   u u   d d
1  u  d

                        1 r  d                             u  1  r 
The solution is:   u                                  d 
                         ud                                   ud
Step 2: From FT2, the price of the call at t=0 is given by:
                                               Maxupa 0  K ,   0
          1  r  pc 0   u       d 
                                               Maxdpa 0  K ,   0
Footnote and example:

a) To simplify, assume that       K  pa 0 so that the call is at the money.
We have:


          1 1 r  d                     u  1  r  u  1 pa 0
pc 0                                                             
         1 r u  d                        ud              0
     1 1 r  d
               u  1 pa 0
    1 r u  d



                                          4
Alternative solutions by replicating portfolios

The same result can be obtained by computing the value of a portfolio that
replicates the payoff of the call and then applying the law of one price.
 1st replicating portfolio: it is given by the vector [xa , xf]’ calculated as
  the solution of the system:
upa 0 x a  1  r  p f 0 x f  u  1 pa 0
dpa 0 x a  1  r  p f 0 x f  0
The solution is:
     u 1                         d pa 0 u  1
xa                      xf                                    from which:
     ud                        1  r p f 0 u  d

                u 1            d pa 0 u  1
pc 0  pa 0       p f 0                   
                ud            1 r pf 0 u  d
   1 1 r  d
             u  1 pa 0
  1 r u  d
     as before.
 Note that at the martingale probabilities, the expected returns of both the
  stock and the bond are equal:
E pa 1   u upa 0   d dpa 0 
 1  r  d    u  1  r                             .
          u             d  pa 0  1  r  pa 0
  ud           ud         
Only the martingale probabilities             influence   pc 0 .   The actual
probabilities   *   play no role: their influence is already implicit in the



                                       5
value of   pa 0 . If investors are risk-neutral, it holds: =* i.e. the ’s
define the market equilibrium under the hypothesis of risk-neutrality.
 2nd replicating portfolio. Consider:
1. A long position in  shares of a stock
2. A short position in 1 call option on the stock.
The value of    is calculated as to make the porfolio riskless (this is an ex.
of  hedging)
 At the end of the period the value of the portfolio is:
upa 0  Maxupa 0  K ,                     0   if the stock went up

dpa  0  Maxdpa  0  K ,                   0   if the stock went down
The two values are equal when:
     Maxup a  0  K , 0  Maxdpa  0  K ,                                  0

                           upa  0  dpa  0
                                  u1
For K  pa  0 as before:  
                                  u d
Note that  is the ratio between the change in the value of the call and the
change in the value of the stock when the stock changes from      dpa  0 to
upa  0 . As we shall see it the DELTA of the Black-Scholes formula.
 The cost of the portfolio is: pa  0  pc  0
 The present value of the portfolio is:
1  r  1 upa  0  Maxupa  0  K ,                   0 
 1  r 
             1
                   up  0   u  1 p  0
                      a                      a




                                       6
The cost and present value must be equal:
        u1                               u u  1  u  1 
pa  0       pc  0  1  r  pa  0           
                                 1

        u d                               u d           
from which it is easy to obtain:
                           1 r  d
pc  0  1  r                    u  1 pa  0
                      1
                                                          which is the same
                            u d
result obtained before.
 Note that the cost of the portfolio at t=0 is:
        u1            u  1 1 r  d           u1 d
pa  0       pa  0                 pa  0
        u d           u  d 1 r               u  d 1 r
while the value of the portfolio, which is certain, at t=1 is:

upa  0
         u1
                u  1 pa  0  pa  0
                                            u  1d
         u d                                u d
Hence, it can be seen that the rate of return on the investment in the
portfolio is r as it should be since the investment is riskless.


 Step 3: Calculate the price of a n period call. The price of the stock at

  t=0,1,2 is:

 t=0                                t=1                          t=2
                                                                 u2pa (0)
                                     upa (0)
                                                                 udpa (0)
pa (0)
                                                                 dupa (0)
                                     dpa (0)
                                                                 d2pa (0)
                                      7
 In general, with n periods the possible prices are n+1. With i increases
  and ni decreases ( i  0,1,n ), the price of the stock is
pa  n  ui d n i pa  0 . The martingale probability of i increases is
given by the binomial formula:
              n!
 i                 ui  dn i       i = 1,2,...n        u   and    d
         i !n  i !
already given.
Since we are considering n periods, we have to apply the n period
discount factor to the eq. of FT2. Also, we have to consider that the payoff
of the call is now a col. vector with the n+1 components given by:
Maxu i d n i pa 0  K ,              0 . Hence from FT2:
pc  0  1  r  Rc 
                  n




                          uidni Maxui d nipa 0  K ,                    0
                  n!
 1  r  0
                  n
             n
                  
             i !n  i !
 Step 4:Black-Scholes.Note that the components of
 Maxu i d n i pa 0  K         0 with a positive value are those
           i
for which u d
               n i
                    pa 0  K  0 . Let m be the min number of rises
for which this is true: m  Mini u d            pa 0  K  0 .
                                         i  n i


Then we can write:


                                      ui  dn i u i d n i pa 0  K 
                            n!
pc 0  1  r  i m
                           n
                      n
                     i ! n  i  !




                                     8
and also:
                             n        n!                    u i d n i 
            pc 0  pa 0 i m               ui  dn i               
                               i ! n  i !              1  r  n 
[BIN.]
                        n         n!                   
             K 1  r  i m
                             n
                                             ui  dn i 
                             i !n  i !              
To understand this formula, we can recall a property of a call: its value
before expiration (t<n) is never lower than the price of the stock less the

                                                  
present value of the strike price: pc t  pa t  K 1  r               
                                                                     n  t 
                                                                               .
It is equal for options in-the-money if the stock price is certain to remain
unchanged until expiration. Otherwise arbitrage opportunities would arise.
The factors in the square brakets (that depend on the martingale
probabilities) can be interpreted as risk factors that can push the price of
the call above the difference between the price of the stock and the present
value of the strike price. We can intuitively write [BIN.] as:


pc 0  pa 0 [Risk Factor 1]  K 1  r 
                                                       n
                                                            [Risk Factor 2]

Its structure is that of the Black-Scholes formula:

[B-S]       pc 0  pa 0 N d1   Ke rn N d 2                   where:


N  is the cumulative normal distribution function,
      ln pa t  / K   r  0,5 2 n
d1 
                       n
d 2  d1   n


                                       9
   is the standard deviation of the stock yield given by:
         d             pa t 
                       
ra t   ln pa t  
         dt            pa t 

[B-S] can be derived as the lim [BIN.] as     n , u  1, d  1.
N d 1  and    N d 2  are the risk factors.In particular, N d 2  is the
probability that the stock price at maturity be greater than the strike price.
Hence the second term of [B-S] is the present value of the payment for the
exercise of the call.   N d 1  is instead the present value of the stock price
at maturity conditional on its being greater than K. Therefore, the [B-S]
price of the call is measured by the present value of its payoff at the
martingale probabilities, as it should be in order to avoid arbitrage (FT2).

                                 Put-Call Parity

 A put with strike price K has, for the holder, the payoff:
Max0,     K  pa n. Its present price is then:
  p p 0  1  r  E  Max0,           K  pa n
                    n



 Consider the portfolios A and B:

A contains 1 stock and 1 put option on the stock expiring at t=n, with
strike price K
B contains 1 bond that pays K at t=n and 1 call on the stock, n periods,
strike price K.




                                        10
 The payoffs af A and B are equal:
The payoff of A is equal to:
pa n  Max0,                K  pa n  Max pa n,             K
The payoff of B is equal to:
K  Max pa n  K ,                 0  Max pa n,           K
 Hence, the two portfolios are worth the same:


                   pa 0  p p 0  1  r  K  pc 0
                                                  n
[P-C P]

Exercise. It is possible to derive the put-call parity relation by direct
application of the FT2:

p p 0  1  r  E  Max0,     K  pa n
                  n



pc 0  1  r  E  Max pa n  K , 0
                  n


pa 0  1  r  E  pa n
                        n



p f 0  1  r  E  K  1  r  K
                        n                  n




Now, from the 1st and the 3rd, we have:

pa 0  p p 0 
 1  r 
             n
                   E p n  E
                       a            Max0,           K  pa n 
 1  r  E  Max pa n,                K
             n




                                      11
and, from the 2nd and the 4th, we have:

1  r  K  p 0 
          n
                       c


 1  r  E K  1  r  E Max p n  K ,                          0 
               n                     n
                                                         a


 1  r  E Max p n,     K
               n
                                 a


Since the second members are equal, so are the firsts.

                    Black-Scholes: pricing of a put option

 From the continuous-time version of the [P-C P] we have:
                    pp  0  pc  0  pa  0  Ke  rn
Substituting the [B-S] formula for the call, we get:
pp  0  pa  0 N  d1   Ke  rn N  d2   pa  0  Ke  rn
and, collecting the like terms, we have:

                                                 
pp  0  pa  0 N  d1   1  Ke  rn 1  N  d 2               
Since the Normal distribution is symmetrical, N  x   N   x   1
Therefore:
             pp  0  Ke  rn N   d 2   pa  0 N   d1 
so that the price of a put has a structure similar to that of a call.




                                           12