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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 (sn) matrix of payoffs,
i 1,2, s states of nature and j 1,2, n financial assets;
x= col. (n1) 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. (s1) of the payoffs of the
portfolio in the different states: y Rx ; p= row (1n) 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.
The ith (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, yji =0). Then
its price, denoted by qi to distinguish it from the pj prices of the actual
securities, is positive qi>0: it is the price of 1 euro in the ith contingency.
The row (1s) 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 jth security [Rj the jth col. (s1) of Rij] can be
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 pq>0 p=qR.
Proof a) sufficiency: q>0p=qR AP. Assume the contrary:
px 0 and Rx 0 with strict inequality in the first or in some
components of the second. But:
1. px<0 and Rx=0 contradicts: px=qRx, since q>0.
2. px=0 and AP qRx>0 that contradicts: px=qRx, since q>0.
b) necessity: AP q>0p=qR. Let Y be the set of all payoffs y=Rx
obtainable at zero cost: Y y px 0. By hypothesis: (y=0)Y
and (y0)Y, i.e. Y contains the origin but no other point of the
positive orthant R y y 0 . Y is a convex polyhedral cone [for
any ,, yY y’Y (y+y’)Y]. Then, from a theorem on
separating hyperplanes, q>0qy=0 for yY. Thus:
qRx qR j x j 0 for all portfolios giving a payoff yY, i.e.
j
those for which px p j x j 0 . Hence: p j qR j , i.e. p=qR.
j
Example: 2 states, 2 assets with payoffs: [1,2]’ and [4,1]’; prices:
p1=p2=1. The zero-cost portfolios are those satisfying the eq. x1+x2=0.
Their payoffs can be calculated as:
y1 1 4 x1 3x1
showing that they lie on the
y2 2 1 x1 x1
straight line: y 2 / y1 1 / 3.
2
The state prices are calculated by solving p=qR, i.e.:
1 q 1 2q 2
1 4q 1 q 2
The solution is: q1 = 1/7 and q2 = 3/7. The eq. of the hyperplane is:
1 3
qy y1 y2 0
7 7
y2 R+
1
y1
-3
Y
The straight line represents both the set Y and the separating hyperplane
(in this case it is a supporting h.p.). The origin is its only intersection with
R +.
Note The price of any (existing or not existing) security can be calculated
by means of q. A 3rd asset with payoff [7,7]’ is priced:
p3 q1 R13 q 2 R23 1 / 7 7 3 / 7 7 4 .
3
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 / qi 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.
4
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.
Maxup a 0 K , 0
Note that its payoff is: Rc
Maxdp a 0 K , 0
.
Step 1: Calculate . From FT2: 1 r p R where: p pa 0 p f 0 ,
upa 0 1 r p f 0
u d and R .
dpa 0 1 r p f 0
Hence, 1 r p R can be written as the following system:
1 r pa 0 u upa 0 d dpa 0 1 r u u d d
1 r p f 0 u 1 r p f 0 d 1 r p f 0 1 u d
1 r d u 1 r
The solution is: u d
ud ud
Step 2: From FT2, the price of the call at t=0 is given by:
Maxupa 0 K , 0
1 r pc 0 u d
Maxdpa 0 K , 0
5
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 1 1 r d
pc 0 u 1 pa 0
1 r u d ud 0 1 r u d
The same result can be obtained by computing the value of a portfolio that replicates
the payoff of the call and the applying the law of one price. The portfolio 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
u1 d pa 0 u 1
The solution is: xa and xf from which:
ud 1 r p f 0 u d
u 1 d pa 0 u 1 1 1 r d
pc 0 pa 0 p f 0 u 1 pa 0 as before.
ud 1 r p f 0 u d 1 r u d
b) 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
ud ud
Only the martingale probabilities influence pc 0 . The actual probabilities * play
no role: their influence is already implicit in the 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.
6
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)
In general, with n periods the possible prices are n+1. With i increases and ni
decreases ( i 0,1,n ), the price of the stock is pa(n)= uidn-ipa(0). The martingale
probability of i increases is given by the binomial formula:
n!
i i ni i = 1,2,...n u and d already given
i! n i ! u d
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: Maxu i d ni pa 0 K, 0 . Hence
from FT2:
pc 0 1 r Rc
n
n n
1 r
0
n!
i ! n i !
i n
u d i Max ui d n ipa 0 K , 0
7
Step 4:Black-Scholes.Note that the components of Max u i d ni pa 0 K 0 with
a positive value are those for which u d p a 0 K 0 . Let m be the min number of
i ni
rises for which this is true: m Mini u i d ni pa 0 K 0 . Then we can write:
n
d i u i d n i p a 0 K
n!
p c 0 1 r
n
i ! n i !
im
i
u
n
and also:
n
n! i ni
i ni u d
n n!
[BIN.] p c 0 p a 0 u d n
K 1 r n ui dn i
i m i ! n i ! 1 r i m 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 K1 r n t . 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:
p c 0 p a 0 [Risk Factor 1] K 1 r [Risk Factor 2]
n
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 d 2 d1 n and
n
d pa t
is the standard deviation of the stock yield given by: 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).
8
Put-Call Parity
A put with strike price K has, for the holder, the payoff: Max0, 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.
The payoffs af A and B are equal:
payoff of A = pa n Max0, K pa n Maxpa n , K
payoff of B = 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 pa n 1 r E Max0, K pa n 1 r E Max pa n, K
n n
and, from the 2nd and the 4th, we have:
1 r n K pc 0 1 r n E K 1 r n E Max pa n K , 0
1 r E Max pa n, K
n
Since the second members are equal, so are the firsts.
9
Use of capand floor to cover mismatchings
A cap is a series of call options on an interest rate, typically the Libor. Conversely,
a floor is a series of put interest rate options.
If the interest rate increases, the buyer of a cap receives from the seller the
difference between the Libor and the strike price, at each payment of the option.
The strike price is the cap that locks in a maximum interest rate. The payoff of the
t
cap is: F Max Libor K , 0
360
A. Protecting interest income
Consider a liability sensitive bank with a Gap 10 mln euro. At t=0, let
Libor= 5%. In order to protect its interest income, the bank can buy caps for a
notional value 10 mln (F = Gap) at a strike price K = 5%. With yearly
payments (t = 360), if the Libor goes to 6% the bank will exercise the option
and receive 10 mln 0.01 = 100,000 euro i.e. the amount of the fall of its
interest income.
Note that the bank is covered against interest rate increases but still maintains its
interest gains if the Libor goes down. It is for these privileges that the bank pays
the price of the cap.
An asset sensitive bank can lock in a minimum interest rate by buying floors for
an amount F Gap . The payoff of a floor is:
t
F Max K Libor , 0 . In the case of Libor rises, the bank will
360
cash from the option the amount of lost interests on its assets.
10
B. Protecting net worth
The net worth of a bank that operates in options is given by its equity E (assets
liabilities) and by the value O of the options it holds (these have a value since the
beginning). Hence: NW E O and hence: NW 0 E / O 1.
For a change i , we already know that E MDG Ai . As for O , let
No be the notional value of the capital involved in the option contract of a bank.
Hence: O N o po where po is the price of the option. It follows that:
O N o po and, since the underlying instrument of the option is the interest
po
rate, that: O N oi where (remember that the of an option is
i
the derivative of the option price with respect to the underlying instrument price).
MDG A
We then have that: NW 0 N o . A bank with MDG 0
should buy (NO > 0) caps (remember that for a call > 0) or sell (NO < 0)
floors (since for a put < 0). The contrary conclusions hold for banks with
MDG< 0.
11
Arbitrage Pricing Theory (APT) Model
The APT model assumes that different risk factors influence the assets’
payoffs (returns). A factor is a variable (ex. the rate of inflation) that
assumes certain values in the different states of nature.
Let k 1,2, K be the index of factors and j 1,2, J the index
of financial assets. The APT model assumes that the assets’ payoffs X j
are linear functions of the factors:
[1] X j j kj Gk
k
where j and kj are constant and Gk is the value of the factor k.
With any arbitrary probability distribution for the values of the factors, it
is possible to calculate the expected value of [1]:
[2] EX j j kj EGk
k
and subtracting [2] from [1]:
[3] X j EX j kj H k
k
where: Hk Gk EGk so that: EHk 0
From the FT2 of the arbitrage, we know that:
p j 1 RF E X j
1
[4] and taking account of [3]:
p j 1 RF EX j kj 1 RF E Hk
1 1
[5]
k
and by letting: Lk 1 RF
1
E Hk we obtain:
[6] p j 1 RF 1EX j kj L k
k
Note that Lk can be written as Lk 1 RF EHk E Hk
1
because
EHk 0. Hence Lk is the discounted value of the difference between the
expected values of the factor k at the arbitrary and at the martingale
probabilities. So Lk can be interpreted as the risk of the factor k.
12
Xj
Define: R j 1 and substitute into [6] to get:
pj
[7]
1 RF E 1 R j kj / p j 1 RF Lk
k
and, by letting 1 RF Lk Fk and kj / p j kj we obtain:
[8] ER j RF kj Fk
k
i.e. the APT Securities Market Line. Fk is the value of the factor k
(expressed as the deviation from the mean). In equilibrium the assets’
returns have to lie on the APT line [8].
The APT is consistent with a variety of equilibrium models. In fact the
CAPM can be seen as a particular case of [8] when only one factor is
considered.
The APT factors are exogenous but unspecified. Much empirical work
had been done to determine a suitable set of factors. Very often the
statistical factor analysis had been utilized, increasing the number of
factors up to the point of finding that the unsystematic risk of each asset
is incorrelated with the unsystematic risk of any other asset. Among the
variables highly correlated with the factors so determined, some
empirical works have detected: industrial production, unexpected
inflation, chamges in actual inflation, default premia (difference in
returns of bonds of different rating), interest rates premia, etc.
Are we confident that one factor identified as important in one set of
data will still be important in another set? This criticism, frequently
raised against the APT model, comes from the fact that the factors are
theoretically unspecified.
13
