Finance from the viewpoint of physics

What makes it more interesting is that not only the actual goods can be traded in a spot exchange, but other “financial products” as well. One of the simplest financial product is a loan. This can be money, but other assets can be lended and borrowed, too.

Who has a debt, has, in some sense, a negative property. If we owe three cows then our portfolio could be written as $-3\mathrm{cow}$. But it is not the most adequate notation, and sometimes it can lead to misunderstandings. The reason is that if we have three cows and owe three cows, the above notation would suggest writing ${\cal P}=3\mathrm{cow}-3\mathrm{cow}=0$. But it is not true that we have nothing, because we can use the benefits of the cows, for example we can drink their milk.

Thus, somewhat generalizing the concept of the loan, we will speak about general promises or liabilities. A debt can be considered as a promise that we will give (back) a certain asset if we are asked for. The loan is the opposite, somebody have promised us a payoff at some time. In fact the actual assets and the promises on actual assets are the main constituents of the more complicated financial products.

Let us denote the promise with $p$, and its argument is the asset that is promised. The loan is a positive promise, because when it is given, one will possess the given asset. This means that if we have three cows and owe three cows, then our property is

$${\cal P}=3\mathrm{cows}-3p(\mathrm{cows}).$$

(1)

Now we can not simplify this equation, this means exactly what we want to. $p$ is defined to be a linear map of the asset space

$$p:A\to A,\qquad p(\alpha a+\beta b)=\alpha p(a)+\beta p(b)$$

(2)

A promise, since it concerns future events, can have several more parameters, that is why it is worth to denote them as a function. A usual parameter is the maturity or tenor or expiration time, denoting when the promise is due. If we denote the present time as $t=0$, then

$${\cal P}=3\mathrm{cows}-3p(\mathrm{cows},T),\qquad T=1y$$

(3)

means that we should deliver 3 cows in one year from now. $T$ can be a time interval, discussed later.

In this language we can describe a lot of financial products. For example a loan with notional $X$ USD, payed back in parts, can be described as

$${\cal P}=X\mathrm{USD}-\sum_{n=1}^{N}c_{n}p(\mathrm{USD},t_{n})-X_{r}p(\mathrm{USD},T),$$

(4)

where $c_{n}$ is the interest rate to be paid at time $t_{n}$ (for example $t_{n}=n\mathrm{m}$ for monthly payoff), and $X_{r}$ is the remainder due at expiration time $T$. To determine the value of the parameters $c_{n},\,N,\,T$ and $X_{r}$ at fixed $t_{n}$, we can use different techniques discussed later. For a fixed rate loan $c_{n}$ is constant.

Another product is the futures trade when an asset ’a’ is agreed to be bought or sold at a given, strike price $K$ at maturity time $T$. If we want to buy that asset, called we are in long position, then our portfolio consists of

$${\cal P}=p(a,T)-Kp(\mathrm{USD},T).$$

(5)

If we want to sell the asset, called we are in the short position, then our portfolio is

$${\cal P}=-p(a,T)+Kp(\mathrm{USD},T).$$

(6)

Another interesting parameter of the promise can be its optionality. One of the counterparties may have the right not to fulfill or not to exercise their promise. In this case the two parties are not equivalent. We call the one who possesses the optionality to be in the long position, the other counterparty (who “sells the optionality”) is in the short position, irrespective whether the promise is about to buy or sell something.

A possible notation for the options is to multiply the possible payoffs by a number $\alpha\in\{0,1\}$. When $\alpha=1$, then the promise is fulfilled, otherwise it is denied. It is also important that who has the right to decide the value of $\alpha$, that we indicate as a $\pm$ index: if the index is $+$, then the portfolio owner has the right to set the value of $\alpha$ (i.e. she is in the long position), if the index is $-$ then someone else determines its value (so the portfolio owner is in the short position with respect to the option).

For example if we agreed that trader ’A’ has the option to buy a product ’a’ at time (or time interval) $T$ for a strike price $K$ from trader ’B’ (European option), then their portfolios read

$${\cal P}_{A}=\alpha_{+}(p(a,T)-Kp(\mathrm{USD},T)),\qquad{\cal P}_{B}=\alpha_{-}(-p(a,T)+Kp(\mathrm{USD},T)).$$

(7)

The exercise date can be also optional, in American option it is any value in $[0,T]$, in Bermudan option there are some fixed dates. Similarly as in the previous case, we can denote its optionality by a subscript $\pm$. An American option can be described as

$${\cal P}_{A}=\alpha_{+}(p(a,T_{+})-Kp(\mathrm{USD},T_{+})),\qquad{\cal P}_{B}=\alpha_{-}(-p(a,T_{-})+Kp(\mathrm{USD},T_{-}))$$

(8)

where $T_{+}=T_{-}\in[0,T].$

We note that the strike price can also be a complicated construction, even depending on the price history. For example we can agree that the buyer of the option has the right to sell a given asset at the average price that was achieved in a given time interval (Asian option), or anything more exotic ones.

We also note that, although the choice of $\alpha$ is completely up to the trader in the long position, sensible traders choose $\alpha=1$ if it is beneficial to them. This makes it possible to determine the price of the option, see later.

The most simple future payoff is the zero coupon bond, which is $p(\mathrm{USD},T)$, i.e. it pays 1USD at a future time $T$ once. We also assume that it is risk free, meaning that we can count on the payoff with hundred percent certainty. For example we may think of a US government bond. Our task is to tell its value at time $t$, which is called discounting the value of the payoff.

To tell the present value, we have to compare the investment in a zero coupon bond to a bank deposit in a safe bank. If it would be more advantageous to invest into a bank deposit, then we would short the zero coupon bond now, and put the money in the bank deposit. A time $T$ the bank deposit would have a higher value, and so we could gain money with zero starting capital. If the investment into the zero coupon bond would be more advantageous, we could do the inverse: we borrow money from a bank, and put it into the bond, and realize a net profit at time $T$. To avoid these arbitrage possibilities, the present values of a risk free zero coupon bond and a risk free bank deposit must agree.

But the bank pays interest rate for all the deposits. In the most simple case it is a fixed annual interest rate $r_{1}$. Technically the paying of the interest happens periodically in each $dt$ time period, with the corresponding interest rate $r_{dt}$. $r_{dt}$ can be determined from the condition that after one year we get $r_{1}$ rate (assuming $1/dt$ is integer)

$$(1+r_{dt})^{[1/dt]}=1+r_{1}\quad\Rightarrow\quad r_{dt}=(1+r_{1})^{dt}-1$$

(10)

In case $dt\to 0$ (called continuous compounding) we denote $r_{dt}=dt\,r$. Then

$$(1+r_{dt})^{[t/dt]}=\left(1+\frac{rt\,dt}{t}\right)^{t/dt}\stackrel{{\scriptstyle dt\to 0}}{{\longrightarrow}}e^{rt}.$$

(11)

This also means that $r=\ln(1+r_{1})$.

If we deposited $X$USD in the bank at time $t$, at a later time $T$ it is worth $Xe^{r(T-t)}$ USD. This should be compared to the case, when we buy a zero coupon bond at time $t$ with maturity $T$. In an arbitrage-free fair business both should have a value of 1USD at time $T$, so we require $Xe^{r(T-t)}=1$. Thus the value of the zero coupon bond at time $t$ is

$$X=S(p(\mathrm{USD},T),t)=e^{-r(T-t)}=(1+r_{1})^{-(T-t)}.$$

(12)

This formula makes it possible to determine the value of $c$ for a fixed rate loan. The portfolio was given in (4). In a fair business the value of the portfolio is zero at all times. Let us compute it at time zero (present time), when we have

$$0=S({\cal P},0)=X-\sum_{n=1}^{N}cS(p(\mathrm{USD},t_{n}),0)-X_{r}S(p(\mathrm{USD},T),0).$$

(13)

Let us choose $t_{n}=n\,\Delta t$, $T=(N+1)\Delta t$, and denote the actual interest rate (which is the risk free interest rate plus the spread) by $r$. Then we find

$$X=c\frac{e^{-r\Delta t}-e^{-rT}}{1-e^{-r\Delta t}}+X_{r}e^{-rT},$$

(14)

and, correspondingly,

$$c=(X-X_{r}e^{-rT})\frac{1-e^{-r\Delta t}}{e^{-r\Delta t}-e^{-rT}}.$$

(15)

Therefore the condition of arbitrage freeness in the absence of risk leads to a definite price for the zero coupon bond, and a definite value of the fixed rate paying.

Let us assume that we have a promise that we are given an asset $a$ at time $T$, so our portfolio is $p(a,T)$. What is the value of the portfolio at time $t$?

What we certainly know is that

$$S(p(a,T),T)=S(a,T),$$

(16)

since the promise is fulfilled then, we obtain the asset, and its price is what is determined by the market at that time. We claim that it is true at other times as well, i.e.

$$S(p(a,T),t)=S(a,t),$$

(17)

it does not depend on $T$.

The reason is that if $S(p(a,T),t)>S(a,t)$, then we buy the asset now, and, at the same time we sell the promise of delivery at time $T$. Therefore we have now the asset $a$, payed its value ($-S(a,t)\,\mathrm{USD}$), we promised a delivery of $a$ at time $T$ (this is $-p(a,T)$), and we obtained the price for the promise $S(p(a,T),t)\,\mathrm{USD}$. Our portfolio therefore reads

$${\cal P}_{1}(t)=a-S(a,t)\,\mathrm{USD}-p(a,T)+S(p(a,T),t)\,\mathrm{USD}.$$

(18)

The value of the portfolio is zero at time $t$. Its value at time $T$, if the promise is fulfilled

$$S({\cal P}_{1},T)=S(a,T)-S(p(a,T),T)+(S(p(a,T),t)-S(a,t))S(\mathrm{USD},T).$$

(19)

But the first two term cancel each other by equation (16), and so what remains is

$$S({\cal P}_{1},T)=(S(p(a,T),t)-S(a,t))S(\mathrm{USD},T)>0.$$

(20)

Therefore we could gain money. If $S(p(a,T),t)<S(a,t)$, then we build a portfolio

$${\cal P}_{2}=-a+S(a,t)\,\mathrm{USD}+p(a,T)-S(p(a,T),t)\,\mathrm{USD},$$

(21)

for that $S({\cal P}_{2},0)=0$ and $S({\cal P}_{2},T)=(S(a,t)-S(p(a,T),t))S(USD,T)>0$ again. To exclude this arbitrage possibility we need to have $S(p(a,T),t)=S(a,t)$, which we wanted to demonstrate.

We remark that the two cases are somewhat different. If the price of the promise is larger than the actual price, we immediately can realize a profit without any original capital. The other case is feasible if we have the asset previously, otherwise we can not realize the $-a$ part of the portfolio. But, if the asset is liquid enough, there are enough assets in the market to forbid this arbitrage.

Using this result we can give the price of a futures trade. The portfolio of a long position is given by (5), its price is therefore

$$S(p(a,T)-Kp(\mathrm{USD},T),t)=S(a,t)-Ke^{-r(T-t)}.$$

(22)

Using the fact that the $\xi_{i}$ effects are small one-by-one, we can power expand the $\cal F$ function to first order

$$S_{n+1}=S_{n}+{\cal F}_{n}(S_{n},0)+\xi_{in}\frac{\partial{\cal F}_{n}}{\partial\xi_{in}}\biggr{|}_{(S_{n},0)}+\dots.$$

(25)

The last term is a weighted sum of the $\xi_{i}$ variables at time index $n$. Now we can argue that the distribution of the sum of mostly independent random variables (with bounded variance) is a Gaussian. This is the central limit theorem, and in fact we need to fulfil some conditions that we tacitly assume that is in fact the case here. Thus the last term can be substituted by a single term with some generic coefficient:

$$S_{n+1}=S_{n}+{\cal F}_{n}(S_{n},0)+Z_{n}(S_{n})\xi_{n},$$

(26)

where the $\xi_{n}$ variables are all Gaussian distributed random variables with zero mean and unit variance. We will assume that these random variables are independent for different times: indeed, we can argue that there are different trades throughout the world at random times, and so their interrelation is weak. But we must know that this is again an approximation, because if we do not observe all effects, the effective dynamics of the rest will contain memory effects. What we assume is that these memory effects are small.

Although all the formulae are supposed to be written for multi-component variables, it may be useful to write out the indices explicitly. In the multi-component notation the above equation can be written as

$$S_{n+1,a}=S_{na}+{\cal F}_{na}(S_{n},0)+Z_{na}(S_{n})\xi_{na}.$$

(27)

The $\xi_{na}$ random variables are not necessarily independent for different assets

$$\bm{E}\left(\xi_{na}\xi_{mb}\right)=C_{n,ab}\delta_{nm},$$

(28)

and so the covariance matrix of the complete noise term reads

$$\bm{E}Z_{na}\xi_{na}Z_{mb}\xi_{mb}=\delta_{nm}Z_{na}Z_{nb}C_{n,ab}.$$

(29)

To simplify the treatment, we diagonalize the correlation matrix (which is a symmetric regular real matrix) as

$$C_{n,ab}=\sum_{k=1}^{N}\lambda_{nk}v_{a}^{(nk)}v_{b}^{(nk)},$$

(30)

where the $\bm{v}^{(nk)}$ vectors are eigenvectors of the covariance matrix $\bm{C}_{n}$, and they are orthonormal: $\bm{v}^{(nk)}\bm{v}^{(n\ell)}=\delta_{k\ell}$. Then we can write (27) as

$$S_{n+1,a}=S_{na}+{\cal F}_{na}(S_{n},0)+\sum_{k=1}^{N}Z_{n,ak}(S_{n})\xi_{nk}$$

(31)

with the volatility matrix

$$Z_{n,ak}=Z_{na}\sqrt{\lambda_{nk}}v_{a}^{(nk)},$$

(32)

and uncorrelated noise terms

$$\bm{E}\left(\xi_{nk}\xi_{m\ell}\right)=\delta_{k\ell}\delta_{nm}.$$

(33)

Indeed, the correlation of the noise term reads now as

$$\bm{E}\left(\sum_{k=1}^{N}Z_{n,ak}\xi_{nk}\right)\left(\sum_{\ell=1}^{N}Z_{m,b\ell}\xi_{m\ell}\right)=\delta_{nm}Z_{na}Z_{nb}\sum_{k=1}^{N}\lambda_{nk}v_{a}^{(k)}v_{b}^{(k)}$$

(34)

which is exactly the complete covariance matrix (29).

All the above means that it is enough to have as many random Gaussian variables, as the number of the assets on the market (originally we had much more). These variables can be thought to be independent, and appear in the evolution equations multiplied by the volatility matrix $Z_{n,ak}$. Thus the cumulative distribution of the random variables is

$${\cal P}(\{\xi\})=\prod_{nk}{\cal P}_{G}(\xi_{nk}),\qquad{\cal P}_{G}(\xi)=\frac{1}{\sqrt{2\pi}}e^{-\frac{\xi^{2}}{2}}.$$

(35)

From now on we suppress the multidimensional indices, treat $Z$ as a matrix $Z_{n,ak}$, and $\xi$ as a vector $\xi_{nk}$.

In the above discussion the value of $d\tau$ could be chosen arbitrarily. Our first guess was $1\mu$sec, but just as well could it be $2\mu$sec or even $0.5\mu$sec. What effect does it have on the form of the dynamic equation?

Let us first assume that we want to work with $dt=2d\tau$. This can be thought that we want to tell $S_{n+2}$ from $S_{n}$. When we recursively substitute the equation of $S_{n+1}$ into the equation of $S_{n+2}$ we have a lengthy expression. But the price changes are so very little in this time interval that in the argument of $\mu$ and $\sigma$ functions we can use the previous value. This simplifies the discussion to

$$S_{n+2}=S_{n}+2{\cal F}_{n}(S_{n},0)+\sqrt{2}Z_{n}(S_{n})\frac{\xi_{n}+\xi_{n+1}}{\sqrt{2}},$$

(36)

where in the last expression we divided and multiplied by $\sqrt{2}$. The distribution of the sum of independent Gaussian random variables is a Gaussian random variable. The correlation matrix coming from the the last expression is thus

$$\frac{1}{2}\bm{E}(\xi_{na}+\xi_{n+1,a})(\xi_{n,b}+\xi_{n+1,b})=\frac{1}{2}\bm{E}(\xi_{na}\xi_{nb})+\frac{1}{2}\bm{E}(\xi_{n+1,a}\xi_{n+1,b})=\delta_{ab}$$

(37)

is the same as for $\xi_{n}$. Thus we may write

$$S_{n+2}=S_{n}+2{\cal F}_{n}(S_{n},0)+\sqrt{2}Z_{n}(S_{n})\xi_{n}.$$

(38)

This can be generalized to arbitrary $dt$ (as far the change of the prices in this time interval is negligible): the first term is multiplied by $dt/d\tau$, the second term, on the other hand, by $\sqrt{dt/d\tau}$.

$$S_{n+dt/d\tau}=S_{n}+\frac{dt}{d\tau}{\cal F}_{n}(S_{n},0)+\sqrt{\frac{dt}{d\tau}}Z_{n}(S_{n})\xi_{n}.$$

(39)

We may introduce the notations

$$\mu_{n}(S_{n})=\frac{1}{d\tau}{\cal F}_{n}(S_{n},0),\qquad\sigma_{n}(S_{n})=\frac{1}{\sqrt{d\tau}}Z_{n}(S_{n}),\qquad dS_{n}=S_{n+dt/d\tau}-S_{n},$$

(40)

and then we can write for the $dt$ discretization time

$$dS_{n}=\mu_{n}(S_{n})\,dt+\sigma_{n}(S_{n})\sqrt{dt}\,\xi_{n}.$$

(41)

We remark that in the multi-dimensional case $\sigma$ is a matrix in the asset price space.

This form shows that the continuous time limit is not trivial: not all the variables scale like $dt$, and so in the $dt\to 0$ limit the above equation does not go to a differential equation. Indeed, the continuous limit is known as a stochastic differential equation.

In the practical point of view the treatment of (41) looks like the following. First we find the solution $S_{n+1}$ depending on the time series $\xi=\{\xi_{0},\xi_{1},\dots,\xi_{n}\}$ and on the initial condition $S_{0}$. Let us denote it

$$S_{n+1}(S_{0},\xi).$$

(42)

Here we have used the fact that $S_{n+1}$ can depend only on the past events. Then we should calculate the expected value of any function of $S_{n+1}$ by averaging over the possible $\xi$ series over independent Gaussian distributions. In formula this reads

$$\bm{E}f(S_{n+1})=\int\limits_{-\infty}^{\infty}\frac{d^{N}\xi_{0}}{(2\pi)^{N/2}}e^{-\frac{1}{2}\xi^{2}_{0}}\dots\frac{d^{N}\xi_{n}}{(2\pi)^{N/2}}e^{-\frac{1}{2}\xi^{2}_{n}}f(S_{n+1}(S_{0},\xi)).$$

(43)

The two equations (41) and (43) provide a well defined numerical framework to solve any stochastic problem numerically.

Often we use a momentum generation function that is defined as

$$\bm{E}e^{JS}=\int\limits_{-\infty}^{\infty}\frac{d^{N}\xi_{0}}{(2\pi)^{N/2}}e^{-\frac{1}{2}\xi^{2}_{0}+J_{0}S_{0}}\dots\frac{d^{N}\xi_{n}}{(2\pi)^{N/2}}e^{-\frac{1}{2}\xi^{2}_{n}+J_{n}S_{n}},$$

(44)

where $JS=\sum_{a,n}J_{na}S_{na}$ and the $S$ series satisfy (41).

A very interesting consequence of the different scaling properties of the various terms in (41) is that, in case of a variable change, a nontrivial factor appears.

Let us assume that we have a new variable $X=f(t,S)$, where $f$ is a smooth function. In discretized case it reads $X_{n}=f_{n}(S_{n})$. Let us consider the change in $X$ up to ${\cal O}(dt^{3/2})$:

$$dX_{n}=f_{n+1}(S_{n+1})-f_{n}(S_{n})=\partial_{t}f_{n}(S_{n})dt+f_{n}(S_{n}+dS_{n})-f_{n}(S_{n}).$$

(45)

If all terms were scale as $dt$ in $dS$, then we could power expand $f$ to first order. But the different terms scale in different ways, so we must go until the second order term:

$$dX_{n}=\partial_{t}f_{n}(S_{n})dt+\partial_{S}f_{n}(S_{n})dS_{n}+\frac{1}{2}\partial_{S}^{2}f_{n}(S_{n})dS_{n}^{2}+{\cal O}(dS_{n}^{3}).$$

(46)

Here we can use (41) for the value of $dS_{n}$. We remark that in $dS_{n}^{2}$ there is a single term that is proportional to $dt$, all other terms are of higher order. Thus we shall write

$$dX=\partial_{t}fdt+\partial_{S}f\left(\mu dt+\sigma\sqrt{dt}\,\xi\right)+\frac{1}{2}\sigma^{2}\partial_{S}^{2}fdt\xi^{2}+{\cal O}(dt^{3/2}),$$

(47)

where we omitted the arguments for brevity (note that $f(t,S)$ is a differentiable funciton, so it is sensible to speak about $\partial_{t}f$ even if the time steps are discrete). We rewrite this formula as

$$dX=\partial_{t}fdt+\partial_{S}f\mu dt+\frac{1}{2}\sigma^{2}\partial_{S}^{2}fdt+\partial_{S}f\sigma\sqrt{dt}\,\bar{\xi},$$

(48)

where we introduced a new random variable having zero mean as

$$\bar{\xi}=\xi+\frac{\sigma\partial_{S}^{2}f}{2\partial_{S}f}\sqrt{dt}(\xi^{2}-1).$$

(49)

As we see, the change of $X$ is not Gaussian distributed, so $X$ is not a Brownian motion any more. But the difference from a Brownian motion vanishes like $\sim\sqrt{dt}$ as $dt\to 0$. So in the limit we can omit the difference of $\bar{\xi}$ and $\xi$. Then we find

$$dX=\left(\partial_{t}f+\mu\partial_{S}f+\frac{1}{2}\sigma^{2}\partial_{S}^{2}f\right)dt+\sigma\partial_{S}f\sqrt{dt}\,\xi.$$

(50)

This is the It$\hat{\mathrm{o}}$-formula. In case of any number of correlated assets it reads

$$dX_{c}=\left(\partial_{t}f_{c}+\mu_{a}\frac{\partial f_{c}}{\partial S_{a}}+\frac{1}{2}(\sigma^{T}\sigma)_{ab}\frac{\partial^{2}f_{c}}{\partial S_{a}\partial S_{b}}\right)dt+\frac{\partial f_{a}}{\partial S_{a}}\sigma_{ab}\xi_{b}\sqrt{dt}.$$

(51)

Let us assume that we have a statistical information about the price at present, we know its distribution function ${\cal P}_{0}(S)$. Then what will be the distribution function at later times?

To give a formal definition for the distribution function we realize that for any quantity $g(\xi)$ depending on a real valued random variable the expected value can be written with the help of the Dirac-delta

$$\bm{E}_{\xi}g(\xi)=\int\limits_{-\infty}^{\infty}dx\,\bm{E}\delta(\xi-x)g(x).$$

(52)

The $g(x)$ does not depend on $\xi$, so we can take it out from the scope of the expected value and obtain

$$\bm{E}_{\xi}g(\xi)=\int\limits_{-\infty}^{\infty}dx\,{\cal P}(x)g(x),$$

(53)

where ${\cal P}$ is the distribution function

$${\cal P}(x)=\bm{E}_{\xi}\delta(\xi-x).$$

(54)

The question we want to answer is that what is the distribution function of the prices at time $t=ndt$ if we know the distribution ${\cal P}_{m}(S)$ at time $t=mdt$. What we have to do is to solve the price motion using the equation (41), starting from some $S=S_{m}$ initial condition at $t=m$, and assuming given $\xi=\{\xi_{m},\dots,\xi_{n-1}\}$. Having obtained a solution $S_{n}(S_{m},\xi)$, finally we have to average over all $\xi$ and $S_{m}$.

Then we can write for the expected value of any $f(S_{n})$ function

$$\bm{E}f(S_{n})=\bm{E}_{S_{m}}\bm{E}_{\xi}f(S_{n}(S_{m},\xi))=\!\int\limits_{-\infty}^{\infty}\!dSdS^{\prime}\bm{E}_{S_{m}}\delta(S^{\prime}-S_{m})\bm{E}_{\xi}\delta(S-S_{n}(S^{\prime},\xi))f(S).$$

(55)

With the distribution functions we can write this expression as

$$\bm{E}f(S_{n})=\int\limits_{-\infty}^{\infty}\!dSdS^{\prime}{\cal P}_{m}(S^{\prime}){\cal P}_{mn}(S^{\prime},S)f(S),$$

(56)

where

$${\cal P}_{mn}(S^{\prime},S)=\bm{E}_{\xi}\delta(S-S_{n}(S^{\prime},\xi)).$$

(57)

There are different methods to derive this quantity, here we will use the It$\hat{\mathrm{o}}$ formula, applied to the expectation value of the $f(S_{n})$ function above. First let us fix the initial distribution to

$${\cal P}_{m}(S^{\prime})\to\delta(S^{\prime}-S_{m}),$$

(58)

then the $S^{\prime}$ integral disappears. Now we change $n$, and write up the change in the expected value in two ways. At the one hand we have

$$d(\bm{E}f(S_{n}))=\int\limits_{-\infty}^{\infty}dS\,d{\cal P}_{mn}(S_{m},S)f(S).$$

(59)

At the other hand from (41) we have

$$\begin{split}d(\bm{E}f(S_{n}))&=\bm{E}df(S_{n})=\bm{E}\left(\mu\partial_{S}f+\frac{1}{2}\sigma^{2}\partial_{S}^{2}f\right)dt=\\ &=\int\limits_{-\infty}^{\infty}dS\,\left(\mu\partial_{S}f+\frac{1}{2}\sigma^{2}\partial_{S}^{2}f\right){\cal P}_{mn}(S_{m},S)dt=\\ &=\int\limits_{-\infty}^{\infty}dS\,f(S)\left(-\partial_{S}(\mu{\cal P})+\frac{1}{2}\partial_{S}^{2}(\sigma^{2}{\cal P})\right)dt,\end{split}$$

(60)

where in the last line we performed partial integration, and omitted the arguments of ${\cal P}$ for brevity. Since the two expressions are equal for any $f$ function, we can conclude

$$d{\cal P}=\left(-\partial_{S}(\mu{\cal P})+\frac{1}{2}\partial_{S}^{2}(\sigma^{2}{\cal P})\right)dt.$$

(61)

In continuous time this leads to a partial differential equation known as the Fokker-Planck-equation or Kolmogorov-PDE:

$$\partial_{t}{\cal P}=-\partial_{S}(\mu{\cal P})+\frac{1}{2}\partial_{S}^{2}(\sigma^{2}{\cal P}).$$

(62)

If we wanted to write out the indices explicitly we would write

$$\frac{\partial}{\partial t}{\cal P}_{a}=-\frac{\partial}{\partial S_{b}}(\mu_{b}{\cal P}_{a})+\frac{1}{2}\frac{\partial^{2}}{\partial S_{b}\partial S_{c}}((\sigma^{T}\sigma)_{bc}{\cal P}_{a}).$$

(63)

In (43) we have seen how to compute an expectation value numerically. Here we continue this line of thought, rewriting that formula.

To treat (43) we have to know the additional information of how to determine $S$, i.e. we need the equation (41). We may work out a formula which is self-contained, i.e. it contains both the time evolution as well as the averaging. The key is that we can represent a recursion through an integral over a Dirac-delta

$$f(S_{m+1})=\int\delta\left(S_{m}+g_{m}(S_{m},\xi_{m})-S_{m+1}\right)f(S_{m+1})dS_{m+1},$$

(72)

where in the present case (41) corresponds to $g_{m}(S,\xi)=\mu_{m}(S)dt+\sigma_{m}(S)\sqrt{dt}\xi_{m}$. This form can be applied for all $m=1,2\dots n$, and obtain

$$\bm{E}f(S_{n})=\int\prod_{m=1}^{n}\left[e^{-\frac{1}{2}\xi^{2}_{m}}\delta(S_{m}-S_{m-1}-g_{m-1}(S_{m-1},\xi_{m-1}))\right]\frac{f(S_{n})\,{\cal D}\xi{\cal D}S}{(2\pi)^{Nn/2}},$$

(73)

where with initial condition $S_{0}=$given, and we also introduced the notation

$${\cal D}h=dh_{1}dh_{2}\dots dh_{n}$$

(74)

for $h=\xi$ and $S$.

In order to simplify the formulae, and get rid of the disturbing constant factors, we may introduce

$${\left\langle{f(S_{n})}\right\rangle}=\int\prod_{m=1}^{n}\left[e^{-\frac{1}{2}\xi^{2}_{m}}\delta(S_{m}-S_{m-1}-g_{m-1}(S_{m-1},\xi_{m-1}))\right]f(S_{n})\,{\cal D}\xi{\cal D}S,$$

(75)

and then

$$\bm{E}f(s_{n})=\frac{1}{{\left\langle{1}\right\rangle}}{\left\langle{f(S_{n})}\right\rangle}.$$

(76)

We can also introduce the generator functional

$$Z[S_{0};J]=\left.{\left\langle{e^{JS}}\right\rangle}\right|_{S_{0}}$$

(77)

where we also indicated the initial condition. We usually denote $Z(S_{0})=Z[S_{0};0]={\left\langle{1}\right\rangle}$, it is sometimes called partition function in physics. Then

$$\bm{E}f(s_{n})=\frac{1}{Z(S_{0})}\left.{\left\langle{f(S_{n})}\right\rangle}\right|_{S_{0}}.$$

(78)

In the sequel we will omit all constant factors in all expected values, the division with the corresponding $Z$ will take care of the correct normalization.