3 An Example of Single-Period Asset Pricing: Risk-Neutral Probabilities and Arbitrage
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5.4 Arbitrage Detection and the Formation of Arbitrage Portfolios
319
Say the call option C has strike price of 100 dollars and expires exactly at time t + 1. This
option then has pay-off of zero and 50 dollars, respectively. If we denote the price of the call
option today as C, then equations (5.12) to (5.14) give
˜1+ ˜2 = 1
0 5 ˜ 1 + 1 5 ˜ 2 = 1 07
50 ˜ 2 = 1 07C
(5.15)
(5.16)
(5.17)
By satisfying the first two equations we actually obtain the arbitrage-free price for C by
substituing the resulting risk-neutral values ˜ 1 = 0 43, ˜ 2 = 0 57 into the third equation.
The correct (no-arbitrage) price is therefore C = 26 6355 dollars. If, however, we are given
a market price for C = 25 dollars and we wish to answer the question of whether there
is arbitrage or not in this case, then we solve equation (5.17), giving ˜ 2 = 0 535, and
then equation (5.16) gives ˜ 1 = 0 535. These values, however, do not satisfy probability
conservation equation (5.15), therefore, one concludes that there is indeed arbitrage at that
market price.
5.4 Arbitrage Detection and the Formation of Arbitrage Portfolios
in the N-Dimensional Case
The preceding example involves an overdetermined system of linear equations. Now, however,
we shall consider a uniquely specified system where the number of unknowns is equal to the
number of equations. Hence, we consider the case of N states and N assets, i.e., M = N .
We shall assume that one of the assets always corresponds to a bond with fixed rate of
return R. The payoff matrix has the form given in equation (5.9), where the first row has
all equal elements of value 1 + R . The corresponding system of N equations is given in
equation (5.18). The problem is then the following. Generate an arbitrary price vector in
one of two fashions: Set p1 t = 1 and then generate independent components pi t (i ≥ 2)
distributed either (i) uniformly as integers lying within some given minimum and maximum
integer values or (ii) continuously using some standard normal distribtuion, say, pi t N 0 1
(i ≥ 2). Similarly, generate N N − 1 arbitrary payoff matrix elements Dij (i ≥ 2) in the
discrete or continuous cases, respectively. The numerical library called MFRangen is useful
for random-number and random-matrix generation. For a given generated pair of price vector
p t and payoff matrix D one obtains the vector of risk-neutral probabilities ˜ = 1 + R
by solving the linear system
⎛ ⎞ ⎛
⎞⎛ ⎞
1+R ··· 1+R
1
1
⎜ p2 ⎟ ⎜
⎟⎜ ⎟
D
·
·
·
D
21
2N
⎜ ⎟ ⎜
⎟⎜ ⎟
⎟⎜ ⎟
⎜ ⎟=⎜
·
·
(5.18)
⎟⎜ ⎟
⎜ ⎟ ⎜
⎠⎝ ⎠
⎝ ⎠ ⎝
·
·
pN
DN 1 · · · DNN
N
In practice, one can solve this system numerically by inverting the payoff matrix using a
routine based on the singular value decomposition. Note that the first equation in the system is
that of probability conservation [this is equation (5.10) with M = N ]. Arbitrage then exists if
the solution gives at least one nonpositive component, that is, if for any given i, i ≤ 0 (since
we are enforcing probability conservation). For every such i we then have a corresponding ith
state, which we can use to form an arbitrage portfolio that we denote by i with components
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CHAPTER 5
i
1
. Project: Arbitrage theory
i
, N . According to the discussion on equations (5.5) and (5.6), then, the payoff vector
corresponding to the ith state alone can be obtained by setting only the ith component to a
i
nonzero positive value, j = 1 for j = i (i.e., picking a number greater than zero) and setting
all other j components to zero. Note that this corresponds to the pay-off of an Arrow–Debreu
security, yet with nonpositive initial value. The transpose of this N -dimensional pay-off
0 1 0
0 , where
column vector, denoted by i , has a row representation of 0
unity occurs in the ith position only. The arbitrage portfolio is then obtained by solving the
i
N , as in equation (5.5) or,
linear system of N equations in the N unknowns j , j = 1
in matrix form:
D·
i
=
i
(5.19)
To obtain more arbitrage portfolios, one can repeat the preceding steps for the other state
components that led to arbitrage, i.e., for the other nonpositive i components. To see
why i is an arbitrage portfolio note that pt = D . So the portfolio has present value
i
i
Vt = i pt = D i
= i = i . Since i ≤ 0, then Vt ≤ 0 and by construction the
i
i
terminal value or pay-off of this portfolio is given by Vt+1 i = i = 1 (hence greater than
i
0) when j = i yet Vt+1 j = 0 (hence ≥ 0) for all other states. From the definition of
single-period arbitrage we conclude that the portfolio i is indeed an arbitrage.
CHAPTER
.6
Project: The Black–Scholes
(Lognormal) Model
The purpose of this project is to develop pricing routines for plotting and analyzing the
Black–Scholes price for European calls, puts, and butterfly spreads as well as for the corresponding sensitivities — delta, gamma, rho, vega, and theta — as a function of the five basic
parameters that make up the plain-vanilla Black–Scholes pricing formula.
Worksheet: bs
Required Libraries: MFioxl, MFFuncs, MFStat
6.1 Black–Scholes Pricing Formula
The celebrated Black–Scholes pricing formula is quite straightforward since it makes use of
the standard normal distribution. Building the necessary Visual Basic code for this spreadsheet
will, however, quickly familiarize the user with the use of ActiveX numerical library methods
for input/output to Excel. One of the features of the spreadsheet is to allow the user the
flexibility of inputting any values for the fixed parameters while also allowing a choice for
the range of plotting.
Although symmetries of the Black–Scholes formula can be used to reduce the number of
dependent functional parameters, the price of a call option can be most explicitly written (as
seen in Chapter 1) as a function of five variables (or parameters): the interest rate r (assumed
constant), the stock price S, the time to maturity ≡ T − t t = current calendar time and T =
maturity calendar time , the volatility
(assumed constant), and the strike price K. The
Black–Scholes formula for the value of a plain-vanilla European call option is
CS K r
= SN d+ − Ke−r N d−
(6.1)
where
d± = log S/K + r ± 21
2
/
√
(6.2)
321
322
CHAPTER 6
. Project: The Black–Scholes (lognormal) model
√
where d− = d+ −
. The function N x is the cumulative standard normal distribution
at x.
As an example of the functionality built into the bs spreadsheet, a plot of the value of
a call option C as a function of S in the range Smin (the minimum spot price) to Smax (the
maximum spot price) is generated via equation (6.1) while holding r, K, , and
fixed.
A plot of the option price as a function of varying the interest rate while holding the other
four variables constant is generated in a similar manner. The same plotting functionality is
also generated for varying volatility, time-to-maturity, and strike price while simultaneously
making use of the Black–Scholes formula at appropriate interval points. The interface for the
bs spreadsheet also allows for the choice of plotting a variable input number of points for
each graph.
Put-call parity
P = C − S + Ke−r
(6.3)
can also be used to study the corresponding prices and sensitivities of puts. The dimensionality
of the variables is worth emphasizing and is as follows. Volatility refers to a per annum (i.e.,
√
is dimensionless.
yearly) time scale and has units of year −1/2 . Maturity is in years, so
The interest rate is per annum and has units of year −1 , making r dimensionless. Both strike
and spot are in units of currency (e.g., dollars). One noteworthy property of the Black–Scholes
formula is its so-called numeraire invariance. This essentially implies that prices can be made
dimensionless so that the formula is invariant with respect to the underlying currency. This
is easily seen by dividing equation (6.1) throughout by the strike, giving
C/K = S/K N d+ − e−r N d+ −
√
(6.4)
where d+ is also a function of the dimensionless quantity S/K.
From the vanilla call or put options one can construct many other options with various
payoff structures, as was discussed with the theory of static hedging in Chapter 1. One important pay-off that was discussed explicitly is the butterfly spread, as given by equation (1.228).
Here we reconsider this option, with pay-off defined in a similar manner except for a trivial
normalization constant. Namely, the pay-off is peaked at strike K and has a nonzero width
of 2 K. This pay-off is statically replicated by taking a long position in a vanilla call struck
at K + K, another long position in a vanilla call struck at K − K, and two short positions
in a vanilla call struck at K:
S = S− K+ K
=
++
S− K− K
K + K −S
S− K− K
+
S≤K
+
S>K
+ −2
S−K
+
(6.5)
Note that from put-call parity one can also construct such a pay-off with a combination of
puts. The exact analytical expression for the Black–Scholes price of such a butterfly contract
maturing in time is hence
B
K
S K r
= C S K+ K r
−2C S K r
+C S K − K r
(6.6)