Alpha

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Alpha is known as the difference between a fund’s expected returns and its actual returns. Alpha has a very close relationship with another financial term known as beta. Beta is a measure used to determine a fund’s expected returns. Along with being the term used for expected returns, beta is also associated with the level of risk.

Alpha is commonly considered the active return on an investment, working as a gauge to determine how a fund is performing against the average. In some cases, the alpha can be construed as the value a portfolio manager can bring to a fund. A smart manager will be capable of exceeding the expected returns, bringing a positive alpha. A manager who is not as successful and does not perform as expected will yield a negative alpha. However, a positive alpha can also be due to luck with the markets. There is no way to determine which is the case.

Calculating the alpha for a fund can be tricky and involves a number of factors. The formula for alpha is:

Alpha = r – Rf – beta * (Rm – Rf)

r = the security’s or portfolio’s return

Rf = the risk-free rate of return

beta = systemic risk of a portfolio

Rm = the market return

The final result is a number, either negative or positive, depending on the performance of the fund. The higher your beta, the more difficult it is for your alpha to be a positive number.

With everything else being equal, the market likely would be more efficient if all companies followed the unwritten rules – or if they were required to reward their shareholders by systematically increasing the stock price via, say, buybacks with a formulaic relation between the required buybacks and earnings (among other details). As it stands, the supply and demand is driven by what appears to be a rather random perception/interpretation of the earnings announcements (among other information) by market participants. This leads to volatility and mispricings at various time horizons. These mispricings are then arbitraged away by what can be generically termed as “mean-reversion” (or “contrarian”) strategies. For a given “mean-reversion” time horizon there might also exist opportunities to profit via what can be generically termed as “momentum” strategies on accordingly shorter (and, in some cases, longer) time horizons.

One important ingredient that is implicitly assumed in the above discussion is market impact and executions. Even if every company under the Sun followed the unwritten rules, due to a large number of market players and virtual impossibility to predict supply and demand imbalances or their precise timings, mispricings and inefficiencies are inevitable. Longer horizon strategies thereby create arbitrage opportunities on somewhat shorter horizons; strategies on such scales create arbitrage opportunities on yet shorter time scales; and so on – all the way down to HFT (high frequency trading) strategies. While on the longest horizon time scales the strategies are mostly long (mutual and pension funds, holding companies, etc.), on shorter horizons strategies can be dollar neutral, hence seemingly creating profit “out of thin air” – which does not take into account substantial monetary and human capital involved.

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No-Arbitrage & Conditional Drift from the Covariance of Fluctuations. (Didactic 4)

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From P(t,s) = exp {−∫0s−t f(t,x)dx}, we get

dlogP(t,s) = f(t,x) dt − ∫0xdy dt f(t,y) —– (1)

where x ≡ s − t. We need the expression of dP(t,s)/P(t,s) which is obtained from (1) using Ito’s calculus. In order to get Ito’s term in the drift, recall that it results from the fact that, if f is stochastic, then

dtF(f) = ∂F/df dtf + 1/2 ∫ dx ∫ dx′ ∂2F/∂f(t,x)∂f(t,x′) Cov [dtf(t,x), dtf(t,x′)] —– (2)

where Cov [dtf(t,x),dtf(t,x′)] is the covariance of the time increments of f(t,x). Using this Ito’s calculus, we obtain

dP(t,s)/P(t,s) = [dt f(t,x) − ∫0x dyEt,dtf(t,y) + 1/2 ∫0x dy ∫0x dy′ Cov dtf(t,y) dtf(t,y′)] –

0x dy [dtf(t,y) − Et,dtf(t,y)] —– (3)

We have explicitly taken into account that dtf(t,x) may have in general a non-zero drift, i.e. its expectation

Et,dtf(t,x) ≡ Et [dtf(t,x)|f(t,x)] —– (4)

conditioned on f(t,x) is non-zero. The no-arbitrage condition for buying and holding bonds implies that PM is a martingale in time, for any bond price P. Technically this amounts to imposing that the drift of PM be zero:

f(t,x) = f(t,0)+ ∫0x dy Et,dtf(t,y)/dt − 1/2 ∫0x dy ∫0x dy′ c(t,y,y′) + o(1) —– (5)

assuming that dtf(t,x) is not correlated with the stochastic process driving the pricing kernel and using the definitions

c(t,y,y′)dt = Cov [dtf(t,y)dtf(t,y′)] —– (6)

and r(t) = f (t, 0). In (5), the notation o(1) designs terms of order dt taken to a positive power. Expression (5) is the fundamental constraint that a SPDE for f (t, x) must satisfy in order to obey the no-arbitrage requirement. As in other formulations, this condition relates the drift to the volatility.

It is useful to parametrize, without loss of generality,

Et,dtf(t,x)/dt = ∂f(t,x)/∂x + h(t,x) —– (7)

where h(t, x) is a priori arbitrary. The usefulness of this parametrization (7) stems from the fact that it allows us to get rid of the terms f(t,x) and f (t,0) in (5). Indeed, they cancel out with the integral over y of Et,dtf(t,y)/dt. Taking the derivative with respect dt to x of the no-arbitrage condition (5), we obtain

Black-Scholes (BS) Analysis and Arbitrage-Free Financial Economics

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The Black-Scholes (BS) analysis of derivative pricing is one of the most beautiful results in financial economics. There are several assumptions in the basis of BS analysis such as the quasi-Brownian character of the underlying price process, constant volatility and, the absence of arbitrage.

let us denote V (t, S) as the price of a derivative at time t condition to the underlying asset price equal to S. We assume that the underlying asset price follows the geometrical Brownian motion,

dS/S = μdt + σdW —– (1)

with some average return μ and the volatility σ. They can be kept constant or be arbitrary functions of S and t. The symbol dW stands for the standard Wiener process. To price the derivative one forms a portfolio which consists of the derivative and ∆ units of the underlying asset so that the price of the portfolio is equal to Π:

Π = V − ∆S —– (2)

The change in the portfolio price during a time step dt can be written as

dΠ = dV − ∆dS = (∂V/∂t + σ2S22V/2∂S2) dt + (∂V/∂S – ∆) dS —– (3)

from of Ito’s lemma. We can now chose the number of the underlying asset units ∆ to be equal to ∂V/∂S to cancel the second term on the right hand side of the last equation. Since, after cancellation, there are no risky contributions (i.e. there is no term proportional to dS) the portfolio is risk-free and hence, in the absence of the arbitrage, its price will grow with the risk-free interest rate r:

dΠ = rΠdt —– (4)

or, in other words, the price of the derivative V(t,S) shall obey the Black-Scholes equation:

(∂V/∂t + σ2S22V/2∂S2) dt + rS∂V/∂S – rV = 0 —– (5)

In what follows we use this equation in the following operator form:

LBSV = 0, LBS = ∂/∂t + σ2S22V/2∂S2 + rS∂/∂S – r —– (6)

To formulate the model we return back to Eqn(1). Let us imagine that at some moment of time τ < t a fluctuation of the return (an arbitrage opportunity) appeared in the market. It happened when the price of the underlying stock was S′ ≡ S(τ). We then denote this instantaneous arbitrage return as ν(τ, S′). Arbitragers would react to this circumstance and act in such a way that the arbitrage gradually disappears and the market returns to its equilibrium state, i.e. the absence of the arbitrage. For small enough fluctuations it is natural to assume that the arbitrage return R (in absence of other fluctuations) evolves according to the following equation:

dR/dt = −λR,   R(τ) = ν(τ,S′) —– (7)

with some parameter λ which is characteristic for the market. This parameter can be either estimated from a microscopic theory or can be found from the market using an analogue of the fluctuation-dissipation theorem. The fluctuation-dissipation theorem states that the linear response of a given system to an external perturbation is expressed in terms of fluctuation properties of the system in thermal equilibrium. This theorem may be represented by a stochastic equation describing the fluctuation, which is a generalization of the familiar Langevin equation in the classical theory of Brownian motion. In the last case the parameter λ can be estimated from the market data as

λ = -1/(t -t’) log [〈LBSV/(V – S∂V/∂S) (t) LBSV/(V – S∂V/∂S) (t’)〉market / 〈(LBSV/(V – S∂V/∂S)2 (t)〉market] —– (8)

and may well be a function of time and the price of the underlying asset. We consider λ as a constant to get simple analytical formulas for derivative prices. The generalization to the case of time-dependent parameters is straightforward.

The solution of Equation 7 gives us R(t,S) = ν(τ,S)e−λ(t−τ) which, after summing over all possible fluctuations with the corresponding frequencies, leads us to the following expression for the arbitrage return at time t:

R (t, S) = ∫0t dτ ∫0 dS’ P(t, S|τ, S’) e−λ(t−τ) ν (τ, S’), t < T —– (9)

where T is the expiration date for the derivative contract started at time t = 0 and the function P (t, S|τ, S′) is the conditional probability for the underlying price. To specify the stochastic process ν(t,S) we assume that the fluctuations at different times and underlying prices are independent and form the white noise with a variance Σ2 · f (t):

⟨ν(t, S)⟩ = 0 , ⟨ν(t, S) ν (t′, S′)⟩ = Σ2 · θ(T − t) f(t) δ(t − t′) δ(S − S′) —– (10)

The function f(t) is introduced here to smooth out the transition to the zero virtual arbitrage at the expiration date. The quantity Σ2 · f (t) can be estimated from the market data as:

∑2/2λ· f (t) = 〈(LBSV/(V – S∂V/∂S)) 2 (t)⟩ market —– (11)

and has to vanish as time tends to the expiration date. Since we introduced the stochastic arbitrage return R(t, S), equation 4 has to be substituted with the following equation:

dΠ = [r + R(t, S)]Πdt, which can be rewritten as

LBSV = R (t, S) V – (S∂V/∂S) —– (12)

using the operator LBS. 

It is worth noting that the model reduces to the pure BS analysis in the case of infinitely fast market reaction, i.e. λ → ∞. It also returns to the BS model when there are no arbitrage opportunities at all, i.e. when Σ = 0. In the presence of the random arbitrage fluctuations R(t, S), the only objects which can be calculated are the average value and other higher moments of the derivative price.

Phenomenological Model for Stock Portfolios. Note Quote.

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The data analysis and modeling of financial markets have been hot research subjects for physicists as well as economists and mathematicians in recent years. The non-Gaussian property of the probability distributions of price changes, in stock markets and foreign exchange markets, has been one of main problems in this field. From the analysis of the high-frequency time series of market indices, a universal property was found in the probability distributions. The central part of the distribution agrees well with Levy stable distribution, while the tail deviate from it and shows another power law asymptotic behavior. In probability theory, a distribution or a random variable is said to be stable if a linear combination of two independent copies of a random sample has the same distributionup to location and scale parameters. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it. The scaling property on the sampling time interval of data is also well described by the crossover of the two distributions. Several stochastic models of the fluctuation dynamics of stock prices are proposed, which reproduce power law behavior of the probability density. The auto-correlation of financial time series is also an important problem for markets. There is no time correlation of price changes in daily scale, while from more detailed data analysis an exponential decay with a characteristic time τ = 4 minutes was found. The fact that there is no auto-correlation in daily scale is not equal to the independence of the time series in the scale. In fact there is auto-correlation of volatility (absolute value of price change) with a power law tail.

Portfolio is a set of stock issues. The Hamiltonian of the system is introduced and is expressed by spin-spin interactions as in spin glass models of disordered magnetic systems. The interaction coefficients between two stocks are phenomenologically determined by empirical data. They are derived from the covariance of sequences of up and down spins using fluctuation-response theorem. We start with the Hamiltonian expression of our system that contain N stock issues. It is a function of the configuration S consisting of N coded price changes Si (i = 1, 2, …, N ) at equal trading time. The interaction coefficients are also dynamical variables, because the interactions between stocks are thought to change from time to time. We divide a coefficient into two parts, the constant part Jij, which will be phenomenologically determined later, and the dynamical part δJij. The Hamiltonian including the interaction with external fields hi (i = 1,2,…,N) is defined as

H [S, δ, J, h] = ∑<i,j>[δJij2/2Δij – (Jij + δJij)SiSj] – ∑ihiSi —– (1)

The summation is taken over all pairs of stock issues. This form of Hamiltonian is that of annealed spin glass. The fluctuations δJij are assumed to distribute according to Gaussian function. The main part of statistical physics is the evaluation of partition function that is given by the following functional in this case

Z[h] = ∑{si} ∫∏<i,j> dδJij/√(2πΔij) e-H [S, δ, J, h] —– (2)

The integration over the variables δJij is easily performed and gives

Z[h] = A {si} e-Heff[S, h] —– (3)

Here the effective Hamiltonian Heff[S,h] is defined as

Heff[S, h] = – <i,j>JijSiSj – ∑ihiSi —– (4)

and A = e(1/2 ∆ij) is just a normalization factor which is irrelevant to the following step. This form of Hamiltonian with constant Jij is that of quenched spin glass.

The constant interaction coefficients Jij are still undetermined. We use fluctuation-response theorem which relates the susceptibility χij with the covariance Cij between dynamical variables in order to determine those constants, which is given by the equation,

χij = ∂mi/∂hj |h=0 = Cij —– (5)

Thouless-Anderson-Palmer (TAP) equation for quenched spin glass is

mi =tanh(∑jJijmj + hi – ∑jJij2(1 – mj2)mi —– (6)

Equation (5) and the linear approximation of the equation (6) yield the equation

kik − Jik)Ckj = δij —– (7)

Interpreting Cij as the time average of empirical data over a observation time rather than ensemble average, the constant interaction coefficients Jij is phenomenologically determined by the equation (7).

The energy spectra of the system, simply the portfolio energy, is defined as the eigenvalues of the Hamiltonian Heff[S,0]. The probability density of the portfolio energy can be obtained in two ways. We can calculate the probability density from data by the equation

p(E) ΔE = p(E – ΔE/2 ≤ E ≤ E + ΔE/2) —– (8)

This is a fully consistent phenomenological model for stock portfolios, which is expressed by the effective Hamiltonian (4). This model will be also applicable to other financial markets that show collective time evolutions, e.g., foreign exchange market, options markets, inter-market interactions.

Forward Pricing in Commodity Markets. Note Quote.

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We use the Hilbert space

Hα := {f ∈ AC(R+,C) : ∫0 |f′(x)|2 eαx dx < ∞}

where AC(R+,C) denotes the space of complex-valued absolutely continuous functions on R+. We endow Hα with the scalar product ⟨f,g⟩α := f(0) g(0) + ∫0 f′(x) g(x) eαx dx, and denote the associated norm by ∥ · ∥αFilipović shows that (Hα, ∥ · ∥α) is a separable Hilbert space. This space has been used in Filipović for term structure modelling of bonds and many mathematical properties have been derived therein. We will frequently refer to Hα as the Filipović space.

We next introduce our dynamics for the term structure of forward prices in a commodity market. Denote by f (t, x) the price at time t of a forward contract where time to delivery of the underlying commodity is x ≥ 0. We treat f as a stochastic process in time with values in the Filipović space Hα. More specifically, we assume that the process {f(t)}t≥0 follows the HJM-Musiela model which we formalize next. The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of interest rate curve – instantaneous forward rate curve in particular (as opposed to simple forward rates). When the volatility and drift of the instantaneous forward rate are assumed to be deterministic, this is known as the Gaussian Heath–Jarrow–Morton (HJM) model of forward rates. For direct modeling of simple forward rates the Brace–Gatarek–Musiela model represents an example.

On a complete filtered probability space (Ω,{Ft}t≥0,F,P), where the filtration is assumed to be complete and right continuous, we work with an Hα-valued Lévy process {L(t)}t≥0 for the construction of Hα-valued Lévy processes). In mathematical finance, Lévy processes are becoming extremely fashionable because they can describe the observed reality of financial markets in a more accurate way than models based on Brownian motion. In the ‘real’ world, we observe that asset price processes have jumps or spikes, and risk managers have to take them into consideration. Moreover, the empirical distribution of asset returns exhibits fat tails and skewness, behavior that deviates from normality. Hence, models that accurately fit return distributions are essential for the estimation of profit and loss (P&L) distributions. Similarly, in the ‘risk-neutral’ world, we observe that implied volatilities are constant neither across strike nor across maturities as stipulated by the Black and Scholes. Therefore, traders need models that can capture the behavior of the implied volatility smiles more accurately, in order to handle the risk of trades. Lévy processes provide us with the appropriate tools to adequately and consistently describe all these observations, both in the ‘real’ and in the ‘risk-neutral’ world. We assume that L has finite variance and mean equal to zero, and denote its covariance operator by Q. Let f0 ∈ Hα and f be the solution of the stochastic partial differential equation (SPDE)

df(t) = ∂xf(t)dt + β(t)dt + Ψ(t)dL(t), t≥0,f(0)=f

where β ∈ L ((Ω × R+, P, P ⊗ λ), Hα), P being the predictable σ-field, and

Ψ ∈ L2L(Hα) := ∪T>0 L2L,T (Hα)

where the latter space is defined as in Peszat and Zabczyk. For t ≥ 0, denote by Ut the shift semigroup on Hα defined by Utf = f(t + ·) for f ∈ Hα. It is shown in Filipović that {Ut}t≥0 is a C0-semigroup on Hα, with generator ∂x. Recall, that any C0-semigroup admits the bound ∥Utop ≤ Mewt for some w, M > 0 and any t ≥ 0. Here, ∥ · ∥op denotes the operator norm. Thus s → Ut−s β(s) is Bochner-integrable (The Bochner integral, named for Salomon Bochner, extends the definition of Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions). and s → Ut−s Ψ(s) is integrable with respect to L. The unique mild solution of SPDE is

f(t) = Utf0 + ∫t0 Ut−s β(s)ds+ ∫t0 Ut−s Ψ(s)dL(s)

If we model the forward price dynamics f in a risk-neutral setting, the drift coefficient β(t) will naturally be zero in order to ensure the (local) martingale property (In probability theory, a martingale is a model of a fair game where knowledge of past events never helps predict the mean of the future winnings and only the current event matters. In particular, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values.) of the process t → f(t, τ − t), where τ ≥ t is the time of delivery of the forward. In this case, the probability P is to be interpreted as the equivalent martingale measure (also called the pricing measure). However, with a non-zero drift, the forward model is stated under the market probability and β can be related to the risk premium in the market. In energy markets like power and gas, the forward contracts deliver over a period, and forward prices can be expressed by integral operators on the Filipović space applied on f. The dynamics of f can also be considered as a model for the forward rate in fixed-income theory. This is indeed the traditional application area and point of analysis of the SPDE. Note, however, that the original no-arbitrage condition in the HJM approach for interest rate markets is different from the no-arbitrage condition. If f is understood as the forward rate modelled in the risk-neutral setting, there is a no-arbitrage relationship between the drift β, the volatility σ and the covariance of the driving noise L.