同値なマートンゲール測度と 完備市場
最近マーチンゲールの大切さを考えています。マーチンゲールの存在が無裁定の仮定を保証する事がよく書かれています。
It seems strange that the existence of a Martingale would be equivalent to a no-arbitrage space. It seems to be caused by the fact that, if there were a security v with expected price at t=1 E(v1)>v0, then an arbitrageur could buy the security at v0 and simultaneously short the futures contract (assuming no basis and storage costs). This assumes that the trader can borrow at no interest, though. Thus, perhaps the fact that a numeraire, which serves to discount the futures prices can be included as part of the Equivalent Martingale Measure is a necessary assumption. 割引ファクターを使えば裁定とマーチンゲールの関係が現れる。正式な証明を見るつもりです。
ファインマン・カッツ公式
最近よく金融工学に現れるファインマンカッツ公式を考えています。これは放物経偏微分方程式の解を確立過程と関係する。The starting point of Feynman Kac seems to be a partial differential differential equation like Black-Scholes, and the point is to relate this PDE to the conditional expectation of an integral and an expo ential, which are functions derived from a Brownian motion.
Given the PDE:
du(x,t)/dt + mu(x,t)*du(x,t)/dx + (1/2)s^2(x,t)d^2u(x,t)/dx^2-V(x,t)+f(x,t)=0
u(x,t) = EQ[INTEGRAL*1f(xr,r)dr + exp(-INTEGRAL(t to T)(V(xtau, tau))dtau)*psi(xT)|xt=x]
where EQ is the expectation under measure Q.
Where this formula is used practically is difficult to say, although it would enable easier discretization of expected future derivatives payoffs dependent on a random underlying X, some event horizon time r, and some maturity time T.
この定理な証明がわかりにくいです。 Incredibly, the proof involves beginning with applying Ito's Lemma to the interior of the expectation above, and ending with substituting u(x,t) for Y(t) (Y(0) in some texts). The initial choice of Y, as the interior of the integral above with psi replaced by u and T replaced by s, is unclear. Y(s) = INTEGRAL*2f(xr,r)dr + exp(-INTEGRAL(t to s)(V(xtau, tau))dtau)*u(Xs,s)
The proof steps involve obtaining the differential dY via Ito's lemma, integrating to obtain a stochastic process, taking the expectation, and substituting u(x,t) for Y(t) (or E(Y(t)|Xt=x)). These steps result in the desired outcome, but it is unclear what the intuition leading to the initial Y formula is. It seems that this technique of creating a formula in terms of the solution to second-order differential equations and an integral arose as a heuristic in quantum physics or thermodynamics.
今度伊藤の補題の証明を復習したいです。
ギルサノフの定理の多形式
最近ギルサノフの定理の由来と使い方を考え続けている。 One form involves integrals, while the other involves the covariance notation.
Wikipedia states that Y~ = Y - [Y,X] is a Q local martingale, given that Y is a local martingale under P. Also W~ = W - [W,X] is standard Brownian Motion under Q, where W is standard Brownian Motion under P.
シカゴ大学の統計学講座の資料によると
W~ = W - INTEGRAL(s=0 to t)(theta*ds)
http://www.stat.uchicago.edu/~lalley/Courses/391/Lecture12.pdf
以下はこの論本に基づく。http://www.qmss.jp/prob/stochasticproc/35-girsanov.pdf
The point of Girsanov's Lemma is to define, for a given market process V and measure P, under which V is not standard Brownian motion, a new measure Q exists such that, under Q, V is a standard Brownian Motion. dQ/dP is known as the Radon-Nikodym derivative and itself is equivalent to a Martingale stochastic process (確立過程).
(dQ/dP) = exp(-INTEGRAL(0 to s)(u(t))dW(t) - INTEGRAL(0 to s)(u(t)**2)dt
Q = INTEGRAL(S)(VdP), where S is a subset of the time space, it seems. The numeraire, or 基準財、is also related, as, if the numeraire for a Martingale process f/g g is the riskless asset, then the equivalent Martingale measure is the risk-neutral measure.
論文に分からなかった事は対数正規分布の過程がマーチンゲールと言う事です。
具体的にB(t)がブラウン運動であればexp(B(t)-t/2)がマーチンゲールです。
Note that the Girsanov theorem is a special case of the Cameron-Martin Theorem, which also defines a Radon-Nikodym derivative as an exponential.
One difficulty with Girsanov's lemma is that it is defined only for transitioning between equivalent probability measures. In addition, it seems to provide no exact formula for translating from a P expectation to a Q expectation. In fact, though taking an expectation wrt P is integrating wrt to P of f, as P is the cumulative distribution function and dP is the probability density, or . Thus E(wrt Q)f = INTEGRAL(fdP(dQ/dP)), so the Radon-Nikodym derivative enables measure change.
ラドンニコディム微分を見つければ測度変換が出来る。
Application domains are swaptions (risk neutral with respect to the annuit numeraire) and the LIBOR Market Model (rolling forward risk neutral world). Note that, in the case of a swaption, the change of measure makes the swap rate, not the swaption price, be a Martingale. It seems that the numeraire determines the measure, so it must be related to the Radon-Nikodym derivative. Change of measure seems to be due to the change in the timing of the uncertain future cash flows. In addition, the measure changes when converting from risk-neutral to historical probabilities.
斉次な微分方程式
最近微分方程式の解き方を学んでいます。一つな問題は斉次な微分方程式です。It seems strange that, although an equation is homogeneous, it turns out that the variables cannot be separated to allow simple integration to occur. 斉次と言うのは方程式をM(x,y)ｄｘ＋N(x,y)ｄｙ ＝０ の形にする。
ｘとｙの代わりにｃｘとｃｙを入れ替えればM(cx,cy)＝ｃ＾ｎM(x,y)とN（ｃｘ、ｃｙ）＝ｃ＾ｎN（ｘ、ｙ）。
However, a homogeneous diff eq which cannot be solved satisfactorily (it seems) is
x(dy/dx)-2y = -x
Solving this equation by assuming it is homogeneous (using y=vx substitution) yielded a different result than solving it by assuming a nonhomogeneous equation (substituting y=uv and initially setting the v coefficient to 0 in u(dv/dx)+v(du/dx+Pu)=Q.
斉次じゃない常微分方程式を解けば積分因数を使用する。
以上な方程式は斉次にしても不斉次にしても正解の算出が出来るらしい。Solving x(dy/dx)-2y=-x using the y=vx change of variable, integrating the separated variables (using (dx/x)=(dv/(f(v)-v))) and taking the exponent of both sides (as the integral outputs a logarithm), the result is:
y = c(x*x)+x
This agrees with the WikiHow example, which does not assume homogeneity.
ギルサノフの定理とハーディ空間
最近よく金融工学に使用されているギルサノフの定理を学んでいます。最初の習うべき用語は確率空間です。The probability space is an (W, F, P) triple consisting of a set of outcomes W, a sigma algebra of filtration events F, and the probability function P. もう一つ
な大事な用語は関数期間です。 Function space basically represents functions as vectors and defines a metric for distance, usually "Linf", which , for a function f, is MAX(t)|f(t)|, where |f(t)| is the norm of a vector, usually the L2. The H2 space is similar to the L2 space (it is a subset). The point of L2 in the case of functions, though, seems to refer to square-integrable functions, and the use in the Novikov and Girsanov theorems seems to be that a process Z is defined in terms of another process theta, and this process is in H2 (why H2 is chosen instead of L2 is unclear). ギルサノフの定理は確率分配とブラウン運動を関係する法則です。However, this brings up another strange definition, namely that of a standard Brownian motion under a measure. Brownian motion is governed by a normal distribution, which itself would be a measure, it seems.
Numerous problems remain. Among them are: why proving Z(t) = INTEGRAL(0 to t)(Z(s)*theta(s)dWs) means that Z(t) is an L2 Martingale, how Girsanov's Theorem enables pricing of assets under different probability distribution assumptions, and how to incorporate the Radon Nikodym derivative into pricing computation.
もう一つな問題は積分の書き方です。 An expectation of a function f can be written as INTEGRAL(0 to t)(f(t)dP), where P is a probability density. This is due to the fact that the integral is a Lesbegue integral, which partitions the range of the probability density and sums the width of the f value range corresponding to each probability over the range. The Riemann integral form would be INTEGRAL(0 to t)(f(t)P(t)dt).
次に金融工学によく使われている言葉です。 A martingale is a zero-drift stochastic process. In other words, E(Z(t+1)|Z(0)...Z(t)) = Z(t).
ルベーグ積分と同じように抽象的で難しい物は測度です。Measure theory is fairly broad, but, in the context of probability, the measure seems to be nothing more than the cumulative distribution function (CDF), and the Radon-Nikodym derivative, dQ/dP is the ratio of density functions. In addition, the measure Q can be written as the expectation of the R-N derivative with respect to P over the filtration, i.e. it seems that you would sum (q(x)/p(x))*p(x) over all x containing the specified set of events F (not verified).
Girsanov's Theorem supposedly indicates that, under a change of measure, the drift will change while the variance remains the same. It is used for options pricing, quantos, and the LIBOR Market Model. ギルサノフ;の定理はWがPに関する標準ブラウン運動である場合にQと言う測度が存在するしQがW~にかんする標準ブラウン運動です。W~はW-INTEGRAL(0,ｔ）（theta*ds)です。Here, theta is defined as the adapted process such that Z is defined as an integral in terms of theta, and Z is exp(INTEGRAL(0,t)(theta*dW)-INTEGRAL(0,t)*1. dQ/dP = Z. Thus, a process that is a standard Brownian Motion on P is not a standard Brownian Motion on Q, due to a change of drift, but the variances will be the same.
*1:theta*theta/2)ds
フーリエ変換アダマール変換の似ているところ
最近ウオルシュ関数とアダマール行列を学んでいます。最初に離散フーリエ変換（DFT）を考慮します。DFTの計算式は
Fk(x) = SIGMA(n=0:N-1)(f(xn)exp(-2*pi*k*n/N)
ウオルシュの変換は同じようにWkの関数族を定義する。
Wk(x) = -1^(SIGMA(kj*x(j-1)))
The Fourier transform converts the time domain variable x to the frequency domain k. The Walsh function provides a similar transform, albeit in the narrower range [-1,1]. However, whether the Walsh transform has any usage in filtering, or how the DFT can be utilized in lattice option pricing, remains to be verified.
利率模型の違い
最近ライボー・マーケット・モデルとヒース・ジャロー・モートン・フレームワーク（HJM)の差を考えています。ライボー・マーケット・モデルは市場に現れる先物の利率を算出する。HJMは瞬間的なフォワードレートの模型です。
分かりにくかった点はライボーとヒース・ジャロー・モートンの瞬間的な相等性です。
(Based on Hull's Solutions Manual)
In the limiting case, the LMM formula converges to the HJM dF forward rate process, but the proof is non-intuitive.
The LIBOR formula is:
dFk = FkLkSIGMA(i=m(t) to k)*1dt + Lk*Fk*dz
Here, di is the time period (t(i+1)-ti). Firstly, the point is to demonstrate that the above is equivalent to the abstracted HJM formula dF(t) = s(t)*INTEGRAL(m(t) to k)(s(tau)dtau)*dt + s(t)dz
Here, s(t) is the instantaneous absolute volatility of the forward rate, which (as L must be the percentage volatility of the forward rate) means lim(dk->0)FkLk = s(t)
Thus, the dz terms can be equated. The logic to equate the dt terms is somewhat more strained. 時間とともに動く因数のを導き出すのが簡単ではありません。The denominator within the sum of the LMM formula must be assumed to go to 1 as di goes to 0. then, the SIGMA factor becomes a summation of (di*Fi*Li) terms as di->0, which is equivalent to an integral over tau of s(tau). QED.
The next step is to determine how the above
dF(t) = s(t)*INTEGRAL(m(t) to k)(s(tau)dtau)*dt + s(t)dz
is equivalent to
dF(t) = (dv(t,T)/dT)*v(t,T)*dt + (dv(t,T)/dT)dz
Here, v is the volatility of a zero-coupon bond price.
This requires expressing the forward rate as follows.
f(t,T1,T2) = (lnP(t,T1)-lnP(t,T2))/(T2-T1)
Here, the rate is seen at time t and holds between T1 and T2. P is the price of the zero-coupon bond with maturities T1 or T2.
One can use Ito's Lemma to solve for dlnP(t,T1) and dln(t,T2), and the value for dlnf when df is known is a well-known case.
df(t,T1,T2) = *2/(T2-T1))dz
Taking the limit as T2-T1 -> 0, or for a rate holding from T to T+delta, we obtain the derivatives of v^2 and v.
df(t,T) = (1/2)(dv^2(t,T)/dT)dt-(dv(t,T)/dT)dz
Simplifying dv^2 yields
df(t,T) = v(t,T)(dv(t,T)/dT)dt -(dv(t,T)/dT)dz QED
