Tuesday, February 11, 2014

sr01 bs01 Relativistic Black-Scholes model 石墨烯中相对论性電子 Dirac equation

Relativistic Black-Scholes model

使用超冷原子模仿的石墨烯 - 個人新聞台 - PChome

mypaper.pchome.com.tw/peregrine/post/1322859965
2012年3月28日 - 石墨烯(graphene)的一項重要電子屬性,業已首度使用超冷原子被模仿。 ... 就相對性的電子而言,這種行為可由狄拉克方程式(the Dirac equation)來 ...


[PDF]

183 石墨烯在外場下的電子特性.pdf

www.wlsh.tyc.edu.tw/ezfiles/2/1002/.../pta_2787_2817820_36159.pdf
2011年8月25日 - 不同層數及堆疊結構形成少層石墨烯,這些少層石 ... 少層石墨烯本身具有的良好特性,例如高電子遷移 .... 中的迪拉克方程式(Dirac equation)。
  • 石墨烯中的电子及其输运性质的研究_百度文库

    wendang.baidu.com/.../67b0681310a6f524ccbf8560.html 轉為繁體網頁
    2013年4月10日 - 分类号学校代码10487 学号D200977026 密级博士学位论文石墨烯中的 .... In this thesis, we use super-cell theory, Dirac equation, the transfer ...
  • 圆盘锯齿型石墨烯量子点的电子结构研究_百度文库

    wenku.baidu.com/view/540cb533cc7931b765ce1592.html 轉為繁體網頁
    2013年9月5日 - 由于石墨烯中的载流子墨烯量子点, 遵循相对论性量子力学中的Dirac .... of a perpendicular magnetic field by solving Dirac equation numerically.


  • Maciej Trzetrzelewski
    y
    Abstract

    Black-Scholes equation, after a certain coordinate transformation,

    is equivalent to the heat equation. On the other hand the relativistic

    extension of the latter, the telegraphers equation, can be derived from

    the Euclidean version of the Dirac equation. Therefore the relativistic

    extension of the Black-Scholes model follows from relativistic quantum

    mechanics quite naturally.

    We investigate this particular model for the case of European

    vanilla options. Due to the notion of locality incorporated in this way

    one nds that the volatility frown-like e
    ect appears when comparing

    to the original Black-Scholes model.
    1 Introduction
    Among many unrealistic assumptions made in the Black-Scholes model

    [1], one is particularly problematic - constant volatility . When the

    current market data are used against the Black-Scholes formula one

    nds that must in fact depend on the strike K, and time to expiry

    T, in order to make the pricing formula work. Therefore the market

    data imply that is not constant but a function

    (K; T) - called implied

    volatility. The shape of the curve
    I

    I
    (K; T) with T xed, is often

    U shaped so that it became a standard practice to call it a volatility

    smile. However that shape can also look more like a skew (a smirk)

    or a frown depending on the data/market one is considering.

    Clearly, the fact that

    (K; T) is not constant falsi es the BlackScholes

    model. However, it is also well known that this situation was
    I
    CRISIL Global Research and Analytics, Av. Libertador 1969, Olivos, Buenos Aires,

    Argentina, e-mail: trzetrzelewski@myopera.com
    y
    Opinions expressed in this document are only personal views of the author.
    1
    Out[121]=
    NORMALIZED FREQUENCY

    60

    50

    40

    30

    20

    10
    completely di
    erent before the market crash in late 80'. In the equity

    market before 1987, the implied volatility was indeed fairly constant

    - why it is not constant nowadays [2] ?

    One could explain this problem by blaming everything on yet another

    unrealistic assumption of the Black-Scholes model - that the

    underlier S

    (where W
    t

    t
    undergoes the geometric Brownian motion

    dS
    t
    =S
    t
    = dt + dW
    t
    ; 2 R; > 0 (1)

    is a Wiener process). It follows form (1) that log-returns

    (i.e. returns of ln S

    ) have Gaussian distribution. However it is very

    well known [3] that the actual log-returns are not distributed like that

    - instead they exhibit fat tails (Figure 1a). Therefore a rather nat-
    t
    aL REALHDJIL VS. GAUSSIAN DISTRIBUTIONS

    -0.04 -0.02 0.00 0.02 0.04




    0

    DAILY LOG RETURNS UNTIL 2013

    NORMALIZED FREQUENCY

    60

    50

    40

    30

    20

    10
    bL REALHDJIL VS. GAUSSIAN DISTRIBUTIONS

    -0.04 -0.02 0.00 0.02 0.04




    0

    DAILY LOG RETURNS UNTIL 1987
    Figure 1: Distribution of daily log returns for Dow Jones (dotted) and the

    corresponding Gaussian distribution (continuous). a) since 27 May 1896 to

    10 May 2013, b) since 27 May 1896 to 2 Jan 1987. The mean and the height

    of the Gaussian distribution are adjusted accordingly.
    ural way to generalize (1) is to replace W

    with the process whose

    PDF exhibits fat tails corresponding to the ones observed in the markets.

    However a careful inspection shows that this cannot be the main

    reason of the volatility smile observed today. The point is that even

    before 1987 the log-return distribution revealed fat tails (see Figure

    1b; note that Mandelbrot's paper [3] was published in 1963) but at the

    same time the Black-Scholes model was working well. This is clearly

    an issue. If fat tails are the reason of all these discrepancies then why

    the constant volatility assumption was correct before 1987?
    t
    Because of practical reasons the models that consider generalizations

    of W
    t
    are not very popular and the development in this subject
    2
    1
    went in a completely di
    erent direction. Instead of changing W

    , nancial

    practitioners prefer to leave W

    unchanged and assume that

    is a function = (S; t) - called local volatility [4]. Then the smile

    is explained by assuming that increases for large j ln S
    t
    j - if this is

    the case then the tails of the Gaussian distribution will become fatter.

    There exists a way to nd the function = (S; t) directly using the

    market data [5]. However it turns out that this model also has its

    drawbacks i.e. while the smile can be accommodated, its dynamics

    (the dynamics of the smile when the strike changes) is not captured

    correctly. This brings us to further generalization by assuming that

    itself is a stochastic process [6]

    d
    t
    =
    (
    t
    ; t)dt + (
    t
    ; t)dW
    t
    (here
    and are some deterministic functions).This generalization is

    counter intuitive: the amplitude , that multiplies the random factor

    dW
    t
    , is stochastic now, but shouldn't dW

    contain all the randomness?

    Moreover, stochastic volatility models also fail in certain situations

    e.g. in the limit T ! 0 where T is the time to maturity [7]. This

    could be a motivation to generalize further and introduce jumps i.e.

    discontinuous moves of the underling S
    t

    t
    [8].
    1
    It is clear that this way of making the models more general is

    likely to have little explanation power. These models may t very

    well to the market data but in say 10 years from now they will most

    probably fail in some situations and one will have to make some other

    generalizations to t the market data again. This implies that the

    stochastic volatility models are non falsi able.

    For example, if we agree on the fact that volatility is a stochastic

    process and satis es (2) then there is a priori no reason not to go

    further and assume that is also stochastic. This would make our

    model even better calibrated to the market data. The possibilities are

    quite frankly unlimited and if it weren't for the fact that Monte Carlo

    simulations are time consuming, they would certainly be investigated.

    Because one can always augment the model in such way that it will be

    consistent with the data, it follows that the model cannot be falsi ed.

    Nevertheless most nancial practitioners prefer stochastic volatility

    models because then, one can still use Ito calculus and obtain
    t
    The reader will note that the line of reasoning presented here di
    ers from the chronological

    way these ideas were considered. Jumps were introduced in 1976, three years after

    the Black-Scholes paper, stochastic volatility in 1993, local volatility in 1994.
    3
    t
    (2)
    2
    some analytical, robust results (otherwise, when dW

    is not a Wiener

    process, little exact results/methods are known [9]). It may seem unusual,

    from the scienti c point of view, that robustness of the model

    is used as a criteria of its applicability. However quantitative nance,

    unlike Physics, is not about predicting future events but about pricing

    nancial instruments today. Therefore as long as our models are calibrated

    to the market, minimize arbitrage opportunities and are stable

    against small uctuations of the data, there is a priori no problem in

    the existence of plethora of possible models in this subject.
    t
    In Physics the situation is much di
    erent. There, we care about

    predictions and recalibration is not allowed. A theory that contains

    parameters and degrees of freedom in such amount that can explain

    any experimental data, by just appropriately tting them, cannot be

    falsi ed and hence is physically useless
    2
    . For every theory, it is absolutely

    crucial to have an example of an experiment which outcome

    may, in principle, disagree with the results of the theory. This way of

    thinking is in fact opposite to the way one proceeds in nance.

    Stochastic volatility models are clearly very successful but just like

    in the case of fat-tail distributions they will not be able to explain why

    before 1987 the Black-Scholes model was working well. In fact if one

    assumes that volatility is stochastic then clearly it must have been

    stochastic before 1987 - which seems not to be the case (one could

    still object to this point by saying that before 1987 the volatility was

    stochastic but with a tiny mean-reversion amplitude and hence the

    model could be approximated by constant volatility).

    In this paper we would like to approach these issues from a di
    erent

    perspective. It is well known that algorithmic trading became more

    and more popular in the 80' - increasing the changes of the prices,

    per second
    3
    . However there exists a concrete underlying limitation

    for market movements: the change of any price S(t) cannot be arbitrary

    large per unit of time i.e. there exist maximal speed c

    such

    that

    _

    S(t) < c
    M
    (market speed of light, [c
    M
    ] = s
    1

    M
    ). An obvious proof
    At this point it is worth noting that in theoretical physics there are constructions

    (such as string theory) which su
    er from making no predictions in this sense.
    3
    There is a common belief that the crash in the 80' was due to algorithmic trading. We

    do not share this point of view. Following R. Roll's argument: if the algorithmic trading

    was to blame the crash would not have started in Hong-Kong where program trading was

    not allowed yet.
    4
    of this assertion comes from the fact that the speed of information

    exchange is limited by the speed of light. It seems that this limitation

    should not be very restrictive since light travels about 30cm per

    nano-second(ns). Assuming that servers of two counter parties are,

    say, 30cm from each other, it takes at least 1ns to send an order.

    Therefore we should not see any relativistic e
    ects, unless we are considering

    situation in which there are at least billions (10

    ) orders per

    second, sent to a single server. At this point it is clear that future

    development of high frequency trading may in principle inuence the

    situation considerably.

    However there is one feature of every liquid market whose consequences

    are seen already and hence we would like to discuss it in more

    details. Any price S(t) going (say) up from S(t) to S(t + t) > S(t),

    must overcome all the o
    ers made in the interval [S(t); S(t + t)].

    This introduces a natural concept of friction/resistance in the markets

    simply because there is always somebody who thinks that the

    price is too high. This situation is similar to what happens in physical

    systems e.g. electrons in conductors. An electron can a priori move

    with arbitrary (but less than c - the speed of light) velocity. However

    due to constant collisions with atoms of the conductor the maximal

    velocity is in fact bounded even more. The drift velocity of electrons

    can be as small as e.g. 1m=h. Perhaps a better physical example is

    light traveling in a dense media where the e
    ective speed of light is

    c=n where n is the refractive index (e.g. n = 1:3; 1:5; 2:4 for water,

    glass and diamond respectively). In extreme situations, when light

    travels through the Bose-Einstein condensate, the e
    ective speed of

    light can be as small as 1m=s [10].

    To see that this resistance e
    ect is big in the markets let us consider

    the logarithm x(t) = ln S(t) and the corresponding bound on the

    derivative of x(t)

    j _x(t)j = j

    _

    Sj=S < c

    =S:

    If we assume that the order of the underlying is about 100$ and that

    c
    M
    is at least 10
    9
    s
    1

    M
    then we obtain j _xj < 10
    7
    s
    1
    . On a daily basis

    this implies that the di
    erence

    x := jx(day) x(previous day)j

    can a priori be as big as 10
    7
    3600 24 = 8:64 10
    12
    . However at the

    same time nothing alike is observed in the market. The value of x

    for any asset was, to our knowledge, never bigger than 1. We have
    5
    9
    analyzed top 100 companies (considering their market capitalization

    as of March 2012) of the SP500 index. We order them w.r.t. decreasing

    maximal absolute value of their log-returns. The list of rst 15 of

    them is presented below.

    Company log-return market move (close) date

    WMT -0.735707 0.0192 ! 0.0092 Dec 1974

    AAPL -0.730867 26.18 ! 12.60 Sep 2000

    INTC 0.698627 0.0091 ! 0.0183 Jan 1972

    C -0.494691 24.53 ! 14.96 Feb 2009

    ORCL -0.382345 0.61 ! 0.42 Mar 1990

    PG -0.360296 40.43 ! 28.20 Mar 2000

    MSFT -0.356939 0.37 ! 0.26 Oct 1987

    BAC -0.34205 7.10 ! 5.04 Jan 2009

    QCOM 0.327329 8.47 ! 11.76 Apr 1999

    JPM -0.3241 6.03 ! 4.36 Oct 1987

    MRK -0.311709 40.90 ! 29.95 Sep 2004

    AMZN -0.296181 16.03 ! 12.06 Jul 2001

    KO 0-.283731 2.37 ! 1.78 Oct 1987

    WFC 0.283415 19.37 ! 25.72 Jul 2008

    IBM -0.268241 32.35 ! 24.74 Oct 1987

    In the table we also added the corresponding movement of the stock

    price (close) and the date for reference. The biggest change of the logreturn

    is due to Walmart and Apple (note that the historical data we

    use are subject to adjustments for stock splits) whose shares dropped

    resulting in almost the same loss (in terms of log-returns). In any case

    we see that the magnitude of log-returns may be of order of 10

    , not

    10
    12
    .

    This implies that there is a huge resistance in the market for the

    price to move up or down (notice that some of the log-returns in our

    table are positive, e.g. for Wells Fargo). Therefore one may conclude

    that the e
    ective maximal velocity of S(t) is much smaller than c

    .

    For completeness we also performed the same analysis for other markets.

    Below we present the results for Forex majors, some precious

    metals and major indices.
    6
    0

    M
    Forex major log-return market move (close) date

    AUDUSD -0.192451 1.2304 ! 1.015 Nov 1976

    UDSCHF 0.103529 1.9949 ! 2.2125 Dec 1982

    EURUSD 0.0619917 0.6192 ! 0.6588 Feb 1973

    GBPUSD 0.0459699 1.1819 ! 1.2375 Mar 1985

    USDCAD -0.0388492 1.2688 ! 1.2218 Oct 2008

    USDJPY -0.0950101 293.26 ! 266.68 Feb 1973

    Commodity

    XAGUSD -0.222755 12.87 ! 10.20 Feb 1983

    XPTUSD -0.221841 594.6 ! 476.3 Mar 1980

    XAUUSD -0.203157 809.9 ! 661 Jan 1980

    Index

    DJI -0.256325 2246.7 ! 1738.7 Oct 1987

    SPX -0.228997 282.7 ! 224.84 Oct 1987

    NKX 0.200503 135.89 ! 166.06 Aug 1951

    NDX 0.17203 2128.78 ! 2528.38 Jan 2001

    DAX -0.137061 1589.28 ! 1385.72 Oct 1989

    FTM -0.119613 2528.55 ! 2243.49 Oct 1987

    Again, all the log-returns are small. This con rms our claim that

    the maximal value of j _xj is smaller than 1 per day. We will use the

    notation c

    for the upper bound of j _xj.

    In the next section we present a basic idea investigated in this
    m
    paper - the existence of the bound on log-returns implies that the corresponding

    PDF, p(x; t), cannot be positive everywhere but must be

    0 for jxj > x
    max
    := c

    t. This generically introduces a skew/smirk of

    the volatility when comparing to the Gaussian distribution. Based on

    the market data analysed above we claim that this e
    ect can in fact

    be noticeable. The main question is then, in what way we can generalize

    the Black-Scholes model so that the niteness of c
    m
    is taken

    into account. Towards this direction it seems natural we study the

    relativistic generalization of the di
    usion equation. One could object

    that such relativistic extension is a bit arti cial. After all, using the

    analogy of an electron in the conductor, the electron is only slowed

    down to drift velocity and no relativistic e
    ects occur at this speed.

    This argument is of course true in generic cases. However there are

    examples of conductors (graphene surfaces, for a review see e.g. [11])

    for which description of electrons is e
    ectively given by the massless

    Dirac equation i.e the description is relativistic even though the elec-
    7
    m
    tron's speed is still not even close to the speed of light c (it is about

    1% of c). This is due to a particular honey-comb lattice structure of

    the graphene. It is therefore a physical example of the non-relativistic

    processes whose e
    ective description nevertheless requires relativistic

    equations due to the speci c structure of the environment. We see no

    reason why a phenomena of this kind could not take place in nancial

    markets.

    In Section 3 we review the correspondence between the relativistic

    di
    usion equation, the telegraphers equation and the Dirac equation

    found a few decades ago [12, 13, 14, 15, 16, 17]. The di
    usion equation

    can be obtained from the telegrapher equation in the limit v ! 1,

    where v is the velocity of a particle. Since the Black-Scholes equation

    is equivalent to the di
    usion equation in the x = ln S variables,

    we make a proposal that its proper relativistic extension is given by

    the telegrapher equation with v replaced by c

    (Section 4). As a

    result, in Section 5, we arrive at a pricing formula for options and

    present numerical analysis for option prices, put-call parity and implied

    volatility. In particular we nd that the proposed formula allows

    for arbitrage opportunities. In the region of parameters where put-call

    parity is not violated signi cantly we calculated the implied volatility

    and nd a volatility-frown like e
    ect. Lastly we perform the 1=c

    expansions and nd exact formula for 1=c
    2

    m

    m
    corrections (1=c

    terms

    give no contribution). This result can then be used to evaluate the

    implied volatility exactly when c
    2 Basic idea
    m
    is large

    No comments:

    Post a Comment