The problem of optimal behavior of an investor in high-developed option markets is studied. On a basis of a specific investor's risk-preferences function, the continuous VaR-criterion (CC-VaR) as a continuous generalization of well-known common VaR-criterion is introduced. This criterion suggests the investor to construct a portfolio that ensures maximizing the average income and satisfying continuous set of restrictions on income from bottom. In fact, the investor specifies in advance the distribution function and finds the portfolio, which generates random income that satisfies all these restriction. NewmanPearson procedure as the essence of methods that construct optimal on CC-VaR portfolios is given and their properties in relation to CC-VaR problems are studied. The theoretical δ-market and the theoretical option market as its equivalent are being investigated. As usual, the investor has own forecast on probability properties of future behavior of option underlier. The optimal portfolio in theoretical one-period option market with the given prices picture is constructed. The method is illustrated by an example with two-sided exponential probability distributions.
Any instrument (portfolio) that an investor acquires in the option market generates
some random income. The type of randomness forecasts the investor himself. The
result of investment is estimated thru the average yield (or income) and risk – the first
has to be maximized, and the second is measured usually by the variation or so-called
VaR-criterion. In markets with restricted choice of instruments this can suffice.
However in markets with broad toolkits, one can construct portfolios from common calls
and puts that generate random income as the function of arbitrary future underlier’s
price. In this case usual measures of risk can be non-adequate [
So, the application of VaR-criterion in the problem of maximizing the average
income produces an income that is no more than critical income level of criterion
(usually less than investment amount) with the probability near the one (at the dense
lattice of strikes). This result hardly suits the rational investor. Also, using the variation,
as in the theory of Markowitz [
So we arrive at an idea of using continuous VaR-criterion (CC-VaR), when the priority for the investor is to solve market problems with fulfilling continuous set of restrictions, which provides the distribution function of incomes given by the investor in advance. Formally, we introduce the CC-VaR as follows.
The risk interests of an investor are described by monotone increasing and
nonnegative risk-preferences function (r.p.f.) ϕ(ε), ε[
Questions about forming the forecast of the investor remain beyond our consideration. The forecast probability distribution of the future underlier’s price as well as investor’s risk preferences are subjects of the responsibility of the investor and are set in the model from outside. And optimization algorithm has to work with arbitrary forecast and risk-preferences functions.
It is easy to conceive that a portfolio of, for instance, butterflies, which generates the income rather well approximating arbitrary (non-negative) function of future underlier’s price, can be constructed in this market. It is sure that in this case butterflies with mutually rather nearby arranged strikes can be needed.
These considerations lead us to the construction of theoretical market where so called δ-instruments that are marginal instruments for the set of all butterflies form a basis, from which we can receive arbitrary portfolios. The offered construction can be conceived also as marginal for generalized roulette whose positions correspond to future prices of underlier, and probabilities of stopping form arbitrary distributions. 2
For simplicity and elucidating peculiarities of the offered criterion, ideal theoretical one-period option market generated by some underlier X, e.g. a stock, is considered. We call the market theoretical because strikes of options can be arbitrary real numbers, and ideal because all bid-offer spreads and all commissions are equal to zero, and then all instruments can be traded in any quantities (even though fractional). At the beginning of the period, the underlier's price is known and equal, say, to μ0, but this value is for our purposes inessential. At the end of period, this price is the variate Χ, that takes on values x from a continuous set X+ (or even ), is the set of all real numbers, + is that of all positive numbers. The value x is the income from X at the end of period, too. Investor elaborates own forecast for the future price of underlier in the form of forecast probability density p(x), xX.
In the market, the arbitrary instrument generating income g(x), xX, which is a non-negative measurable function, can be traded. We call the function g(x) the payoff of instrument and write g(x) = π(x; G); in particular, π(x; X) = x. Prices of instruments are formed by the market towards the beginning of period. We denote |G| the price of G, and ||G|| its average income.
We consider so-called δ-instruments D(s), sX, as options of special kind, and s is its strike. The income of the instrument D(s) is singular and defined as δ-function δ(x − s), s, xX, which is equal to zero if x ≠ s and infinity if x = s, with integral within X in x equal to the one. The price and the average income of D(s) are equal, respectively, to c(s) and p(s), sX.
In terms of δ-instruments, we can describe practically any continuous-in-s portfolios:
G = X g ( s) D( s) ds (= g ( X )) . (1)
Therefore we can consider a collection of δ-instruments {D(s), sX} as a basis of the market, and we call it δ-market. These instruments can be considered as marginal for such instruments of real markets as simplest common butterflies properly normalized. The price and the average income of δ-instruments D(s) are, respectively, D ( s) = c ( s ) , D ( s) = p ( s) , sX.
Non-negative function c(x), xX, has to be produced by the market. Also, G = X g ( s) D( s) ds = X g (s)c (s) ds , G = X g (s) D(s) ds = X g (s) p (s)ds .
In theoretical markets, such instruments as indicators of sets play significant role. The payoff of the indicator H{М} of the set МX is its characteristic function: π(x; H{М}) ≡ χM(x) ≡ {1, xM; 0, xM}. There is among indicators also the riskless asset of unit size U = H{X}. We have the representations
H M = M D(s)ds , U = H X = X D(s)ds , U = X c ( x) dx = 1 r , U = X p ( x) dx = 1 .
The parameter r denotes the riskless return relative for period, and r − 1 is then the riskless yield. Without loss of generality we can set r ≡ 1 and consider the price function c(x), xX, as probability density. We call it the cost density. Usual calls Cs and puts Ps of strikes sX are given, respectively, by relations
Cs = xs( x − s) D( x) dx , Ps = xs( s − x) D( x) dx , x, s X.
By analogy, representations of other known option instruments such as spreads, butterflies, condors et cetera can be given.
The methods of constructing optimal portfolios expounded further have to consider
divergences between densities c(∙) and p(∙) (or measures C{∙} and P{∙}). The special
significance is therefore attached to the function of return relative ρ(x) = p(x)/c(x),
xX. Functions of such type are considered in the theory of mathematical statistics;
they are called likelihood functions and are at the heart of the so-called
NewmanPearson procedure [
In the market under consideration, we can state investment problems differently. Some most important settings are as follows:
Problem CO (original). Given investment amount and r.p.f. of the investor ϕ(ε), the
portfolio that achieves max Eq (E is the symbol of the expectation) under condition
P{q ϕ(ε)} 1 − ε for all ε[
Problem CB (basic). Investment amount is not given, and ϕ(ε) is investor's r.p.f..
The regular portfolio that achieves min A under condition P{q ϕ(ε)} 1 − ε for all
ε[
Problem CH (homogenious). Given investment amount S (> 0) and r.p.f. in the
form bϕ(ε), where b is a scale parameter, the regular portfolio and the value of
parameter b that achieve max Eq under condition P{q ϕ(ε)} 1 − ε for all ε[
Emphasize that for original problem CO, singular solutions are inherent. The problem CB, on the contrary, straight requires constructing the regular portfolio, but with minimum investment amount, and hence is a base for solving many other problems. The problem CH is an important example of applying problem CB to solving problem CO in case of r.p.f. that is homogeneous in scale of investment. 3
As we have said above, the problem CB is of special interest. And all subsequent constructions just aim at solving this problem. If we already know its solution then the solution, particularly, of the problem CO is derived if we add to the investment amount received for the problem CB all the rest from given original amount investment, which investor directs to acquiring D(s*) where s* is an element on X that achieves the maximum of ρ(s). And we receive a singular solution. But this is just the reason why we have refused using VaR-criterion in favor of CC-VaR.
So we suggest that reasonable approach is in adequate setting the problem. For example, the problem CH is may be adequate just in fully homogeneous case. And problem CO can be transformed into adequate one if investor choices the r.p.f. in a form ϕ(b; ε), where b is a scale parameter, and find its value to nullify the singular component of portfolio.
At the heart of solving problem CB is the Newman-Pearson procedure, which is
broadly used in mathematical statistics [
Newman-Pearson procedure. The set family Z = {Z(τ), τ0} is constructed by the
rule: xZ(τ) if and only if ρ(x) τ, where ρ(x) is defined above and τ ≥ 0. The function
fP(τ) = P{Z(τ)}, τ[τ′, τ″], τ′ = minxρ(x), τ″ = maxxρ(x), is introduced. The family Z
non-decreases in τ, hence fP(τ) is a non-decreasing function. Moreover, 0 fP(τ) 1
and fP() = 1. For each ε[
The next assertions can also be proved: • The functions fP(τ) and fC(τ) are distribution functions of ρ(Χ) for measures P{∙} and C{∙}, respectively, where Χ is the random future price of the underlier.
• The dissonant γ(ε), ε[
• The ordering function w(x) defined by equivalence rule
w( x) = (= fP ( ( x))) x = lim ( X − X ) , ε[
• The distribution function of the random income q = g(Χ) = ϕ(w(Χ)), is equal to
the function ϕ←(z), z[ϕ(0), ϕ(1)), and the identity P{q ≥ ϕ(ε)} ≡ 1 − ε holds for all
ε[
We submit the method connected with the discretization of the problem and the passage to the limit continuous scheme. In discrete case, the discretization parameter n and levels εi, γi, ϕi, iI = {1, 2, …, n}, of continuous parameters ε, γ, ϕ are given. Applying the Newman-Pearson procedure is readily seen to give the next representations of the optimal portfolio at b = 1 and its characteristic:
G(n) = in=1(i −i−1 ) H X − Xi , A(n) = in=1(i −i−1 )(1 − i ) = in=1i ( i − i−1 ), 0 = 0, 0 = 0 , R(n) = in=1(i −i−1 )(1− i ) = in=1i ( i − i−1 ), 0 = 0, 0 = 0 , Pq i 1− i for all iI.
By replacing differences in these formulae by differentials in ε, γ and ϕ, and sums by integrals, we obtain in continuous case
A = 01(1− ( )) d ( ) = 01 ( ) d ( ) , R = 01(1− ) d ( ) = 01 ( ) d . The next assertion can be proved:
Theorem 1. The family of sets Z = {Xε, ε[
Theorem 2. The average income R and the investment amount A for the optimal theoretical portfolio at b = 1 are defined by formulae
R = 1 ( ) d , 0
A = 1 ( ) d ( ) = 1 ( ) ( ) d .
0 0 4
The second Laplace distribution (also called two-sided exponential distribution), which we denote Exp(μ, α), has the density
21 exp(− x − ), x , with expectation EΧ = μ, variation DΧ = 2α2 and kurtosis equal to 6 (excess = 3). Let's set p(x) ~ Exp(μ, α), c(x) ~ Exp(μ, β) и β ≠ α. We may set μ = 0. So
Let's set κ = α/β. If κ < 1 the function ρ(x) increases at x < 0, decreases at x > 0, and
at x = 0 is maximized; the investor «sells volatility». If κ > 1 the ρ(x) decreases at
x < 0, increases at x > 0, and at x = 0 is minimized; the investor «buys volatility». We
set ϕ(ε) = ελ, ε[
Volatility selling. At κ < 1 the investor supposes the market to be less volatile than option prices speak about it. Algorithm generates the next symmetric contractions Z ( ) = x x ( ) = − ln ( ) ,
X = x x x , −x = x = x = ( ) 0 , = ( x ) = ( x ) = ( x ) , P X = 2x p ( x ) dx = −1 x e−x dx = exp ( − x ) = , x = − ln is (1 − ε/2)-quantile of distribution Exp(0, α), = x , x = −x , x is the set of all (here, two) limit points of Xε.
The ordering function with two symmetric branches, the weight function and the dissonant with its derivative have views u1,2 ( ) = x , x = x = ln , w( x) = u1 ( x), x 0; u2 ( x), x 0 = exp (− x ) , g ( x) = ( w( x )) = e− x , g( x ) = e− x , g ( x ) = 2e− x , ( ) = C X = −1 − ln e−x dx = ,
( ) = −1 .
The main characteristics of investment A = 01 ( ) ( ) d == ( + )−1 , R = 01 ( ) d = (1 + )−1 , y = ((11+− )) 0 .
The optimal portfolios both in put-spreads and call-spreads and both in puts and calls themselves are given by mixed representations, respectively,
0 e xdP ( x) + e− xdC ( x)) , G = (U − − 0 G = U − ( P (0) + C (0)) + 2 −0 e x P ( x) dx + 2 0 e− xC ( x) dx .
The payoffs of these portfolios are defined uniquely by parameter θ. Hence the less α and the greater λ the more acute is the peak of the function g(x) at x = 0 and the greater is the yield y. We can consider this portfolio as some continuous expansion of the long butterfly, which realizes in real markets the same volatility selling.
Volatility buying. At κ > 1 the investor supposes the market to be more volatile than option prices speak about it. The analogical computations in this case give sequentially
Z ( ) = x x ( ) = − ln ( ) ,
X = x x x , −x = x = x = ( ) , = ( x ) = ( x ) = ( x ) , = x , x = −x , x is the set of all limit points of Xε.
The ordering function with two symmetric branches, the weight function and the dissonant with its derivative have views u1,2 ( ) = x , x = x = ln (1− ) , w( x) = u1 ( x), x 0; u2 ( x), x 0 =1− exp(− x ) , g ( x) = ( w( x)) = (1 − exp (− x )) , g( x) = e− x (1− e− x ) −1 , g( x) = 22 e− x (1− e− x ) −2 (e− x − 1 ) ,
g(0) = 0, −1, at 1, =1, 1, ( ) = CX = −1 − ln(1− ) e−x dx = 1 − (1 − ) , ( ) = (1− ) −1 .
0 The main characteristics of investment A = 1 ( ) ( ) d = (1+(1+ )+(1+) ) , 0
R = 1 ( ) d = (1 + )−1 ,
0 y = R A −1 = (1+(1+ )+(2+) ) −1 0 , ( ) = 0 x −1e−xdx .
The optimal portfolio both in put-spreads and call-spreads is given by mixed representation
G = −0 g( x) dP ( x) − 0 g( x) dC ( x) .
The character of the optimal portfolio in puts and calls themselves depends on the parameter λ, and corresponds to the behavior of the function g(x) near zero. We have at λ > 1, λ = 1, λ < 1, respectively,
G = −0 g( x) P ( x) dx +0 g( x)C ( x) dx , G = 1 ( P (0) + C (0)) − 12 −0 e− x P ( x) dx − 12 0 e− x C ( x) dx , G = −0 g( x)( P ( x) − P (0)) dx +0 g( x)(C ( x) − C (0)) dx .
These portfolios are some continuous expansions of the short butterflies or condors
in dependency on the value of λ, which realize in real markets the same volatility
buying. At λ < 1 the function g(x) has at x = 0 the singularity of the type "spike", and
the less λ, the more the instrument looks like the riskless asset. At λ = 1 the function
g(x) has at x = 0 a fracture and reminds the payoff of the short butterfly. At λ > 1 the
function g(x) at x = 0 is smooth with g′(0) = 0, and reminds the payoff of the condor;
here qualitative analogy with the real markets can be readily traced, too.
The consideration of illustrative examples demonstrates the rationality of the
optimization problem setting with the continuous VaR-criterion and efficiency of theoretical
algorithm. However, very few problems can be solved by such analytical way. Hence
the method of the solution has to be expanded on discrete models, which could be
considered as approximation to the continuous ones. All the more so that it is better to
interpret real markets as discrete. Indeed, they are such in many aspects. So, to more
adequate formalize the optimization problems, we ought to introduce the so called
scenario markets (see, e.g., [