Financial Models

Formulae

Alexander Liss

11/21/98; 02/01/99

Introduction *

Formulae *

Application of Models *

Models' Elements *

Collective Forecast *

M-process and F-process *

Approach to Modeling *

Stochastic Oscillation *

Normal Distribution Function *

Bound *

Mean Value of a Price *

Rate of Return *

Effect of the Basic Volatility *

Inflation Forecast *

Chain of Trades *

Certainty Component *

Combined Discount *

Factoring in Potential Losses *

Credit Risk *

Loss Events *

Cushion Related Price Component *

Discount *

Bonds and Loans *

Growing Interest Rate *

Benchmark Interest Rate Curve *

Specifics of Treasuries *

Short-term Debt Instruments *

Price of Bond *

Loan *

Borrowing Decisions *

Cash Flow Portfolio *

Effects of Taxes *

Invariant *

Price of Stock *

Dividends *

Stock without Dividends *

Value Based Pricing *

Forecast *

Liquidity and Collapse of Price *

Forward Contracts, Options and Futures *

Forward Contract *

Options *

Price of European Option *

Illiquid European Call and Put *

American Options *

Risk of Default and Forward *

Futures *

Directions of Models Improvement *

Trajectory Based Computation *

Time Delta *

Standard Presentation *

Cones *

Value Delta *

Jump Probabilities *

Schema *

Backward Computation *

System of Asset and Derivatives *

Stock *

Potential to be an Underlying Asset *

The Basic Volatility *

Price Correction *

Inflation and Foreign Exchange *

Exchange Coefficient *

Estimates *

Introduction

Formulae

The development of formula based language took a few centuries and now we have a good tool to describe complex concepts and methods in a compact and universally understood way. The strength of this language is such that it is said that formula is smarter than its creator is. We take advantage of this ability and continue with the description of new concepts and illustration of concepts we presented above. Our emphasis here is the presentation of models as objects of following adaptation to concrete user's conditions. Mostly, we do present final working formulae here, but approaches to make them.

Application of Models

The Application of Models requires good understanding of subject. It requires experience. Often, it requires a usage of a few different models designed from different points of view and comparing their results.

The best results with modeling can be achieved when models are used in a systematic way during a long period, when users accumulate experience and when later use of models allows the assessment of their previous use.

Application of financial models has specific difficulties.

One group of difficulties is related to the Market oscillations. This is a phenomenon, which is caused by the activity on the edge of our understanding, as is. For sure, there are no quantitative models, which can describe it well. Hence, here we need to rely on a system of doing business, which includes Cushion, and flexibility that allow survival in the time of the Market downturn. The particular size of the Cushion and degree of flexibility, needed in the current Market situation, can be estimated with models.

The other group of difficulties is related to the stream of the Market related news. Each time prices undergo drastic adjustment. It is possible to make models, which give reliable bounds that hold in spite of possible news. However these bounds are very rough and, hence, practically useless. Again, the solution lies in the system of doing business, which is flexible enough to allow fast adjustment based on news. Models can be used to estimate the needed degree of flexibility.

Models in some degree reflect both groups of factors. They reflect small Market oscillations and news, which do not lead to big changes of prices. The boundary between big and small here is not formal. It is learnt with the use of models. Mostly these factors are reflected with parts of models, which describe uncertainty of future.

Some factors are important to one group of market participants and less important to the other. For example, moderate Market oscillations are less important for long-term investors and traders than for short-term investors.

From our point of view, each application of a model should be a test of its applicability, and successful application should add to our knowledge about the area of applicability of the model. To make it possible we describe assumptions of the models explicitly, make particular steps of models simple, with intermediate results, which have good interpretation. This makes it easier to catch the situation, when the attempt is made to use the model beyond its area of applicability.

Models' Elements

Collective Forecast

We have to see the coherently working collective of market participants, which forecasts the future state of economy and society as a whole, and which reflects this forecast in prices. We call this process the Collective Forecast. We present results of such forecast not with deterministic functions but with the stochastic processes, because when the prices are set based on forecast, we have to factor in the mistakes of this forecast.

Mistakes of forecast are mostly caused by the uncertainty of future; hence, the measurements of randomness of these processes are automatically measurements of uncertainty of future.

Here we are in a position of double difficulty. First, we have to forecast, which is hard as is. Second, we have to formulate results in difficult to interpret terms of Collective Forecast. Hence, the simpler models we use, the better.

We opted for two step process of building such models. First, we choose primary parameters, which behave well from the point of view of probability of their small change. Second, based on known mathematical theory we declare that their change is M-process. We use the simplest of M-processes - F-process to forecast it. If we have some additional information about future, we use more sophisticated variant of the process. Usually we correct the trend of it, but we can correct characteristics of it randomness also.

Actual parameters, which dynamic we are interested with, are functions of our primary parameters. Their forecast is automatically a stochastic process. Mostly we are interested with trend of this process. The important part of entire construction is - the trend, we are interested with, depends not only on the trend of underlying primary parameter, but on characteristics of its randomness also. This makes result of the models so nontrivial. In the same time, the analysis of underlying primary parameters is about obvious.

M-process and F-process

If v(t) is M-process, then its main characteristics are trend

E{v(t+s) - v(t)} = m(s),

and the characteristic of randomness

Var{v(t+s) - v(t)} = D(s).

It is important to know that D(s) grows monotonously.

It is useful sometimes to analyze the derivative of D(t):

D'(s) > 0.

If we use M-process to describe a forecast, then D'(s) is a kind of "rate of uncertainty". In many cases we do not know much about future and simply assume

D'(s) - constant.

However, sometimes, we know in advance the moment in the future, when an important decision will be made, but we do not know how it will be made. We can present it as a jump in D'(s) in this moment.

In the case of F-process

m(s) = m*s

and

D(s) = D*s.

We will consistently describe M-process with the pair of functions (m(s),D(s)) and F-process with a pair of numbers (m,D).

The sum of two F-processes with parameters (m1,D1) and (m2,D2) is F-process with parameters (m1+m2,D1+D2).

Approach to Modeling

We find a primary parameter x and present parameters we need to estimate as

y = F(x,t)

where t is moment in time.

Because of Basic Volatility value of the parameter y in the moment t, y(t), is a random value. Its future value in the moment t+s is a sum of its current value y(t) and increment

y(t+s) - y(t).

This increment we forecast through the forecast of the primary parameter. In the current moment t primary parameter is concentrated in one point x0 and it is a F-process after that, x(s) (or may be it is M-process, if we have more information):

y(t+s) - y(t) = F(x(s),t+s).

Obviously,

F(x0,t) = 0

(because y(t) - y(t) = 0).

This way we achieve separation of different phenomena in the model:

Random value y(t) presents effects of the Basic Volatility. The stochastic process x(s) presents the mechanism of the working of the Collective Forecast (through the Market activity). The function F(X,T) presents deterministic relationship between the primary parameter and parameter we are interested with.

Our models differ in degree of available information. Sometimes we can estimate parameters (m,D) of the process x(s), but sometimes we know only that this is F-process. Sometimes we know function F(x,t) and sometimes we only assume that it is a "smooth" function, without big jumps. Depending on degree of available information we either build formulae with which we can estimate y(t+s) or we make a special model, which parameters we have to estimate yet.

Stochastic Oscillation

The Market as any stable system oscillates. This oscillation has many components of different period and different phase. Some of these components fade away and new components are introduced each time for different reasons.

We separate three types of oscillation.

One, of high frequency, we call the Basic Volatility, and we present it with very simple models, depending on needed degree of precision.

The other is a medium frequency natural Market oscillation; this oscillation causes many important effects in the Market activity, where the Market "hedges" against it. The Collective Forecast of this oscillation can be presented as a change of primary parameters in a form of M-process, where m(s) presents such complex oscillation. The forecast of the change of m(s) is a difficult task task, and we often forecast lower and upper bounds for it.

The other analysis of this oscillation is applied to trading and short-term investment.

Third is high amplitude and low frequency specific oscillation, when the Markets enters into new areas in the process of its expansion, contracts to adjust, expands again and so on. These oscillations can trigger unexpected reactions, when the Market ventures beyond the area of familiar functionality. The analysis of the dynamic of these oscillations is especially difficult and especially interesting.

The Market oscillations are combined with the process of the Market expansion.

Normal Distribution Function

If x is a normally distributed random value with the zero mean value, and variance equal one, then we denote its distribution function N(x), its derivative - density of distribution, is

N'(x) = 1/sqrt(2*p )*exp(-x2/2).

The distribution function of the normally distributed random value with the mean value A, E(x)=A, and variance B, var(x)=B, is N( (x-A)/sqrt(B) ). sqrt(B) is a standard deviation of this value.

In the case of F-process, p(t+s) - p(t) has a distribution function N( (x - m*s)/sqrt(D*s) ).

Bound

If x is a normally distributed random value with the mean value A, E(x)=A, and variance B, var(x)=B, and there is a bound x0, then x > x0 with the probability R

R = N( (A - x0)/ sqrt(B) ).

When we know the probability R, we can compute the bound

x0 = A + sqrt(B)*N-1(R),

N-1(R)<0, when R<0.5.

In the case of F-process, the bound for the increment p(t+s) - p(t) with the probability R is:

x0 = m*s + sqrt(D*s)*N-1(R).

Mean Value of a Price

If x is a normally distributed random value with the mean value A, E(x)=A, and variance B, var(x)=B, then random value exp(x) has log-normal distribution and its mean value is

E( exp(x) ) = exp( A + B/2 ).

If we model the movement of the logarithm of price with F-process p(t), then in the moment t+s

p(t+s) - p(t)

is a normally distributed value with mean value m*s and variance D*s, hence

E( exp( p(t+s) - p(t) ) ) = exp( m*s + D*s/2 ),

or

E( price(t+s) / price(t) ) = exp( (m + D/2)*s ).

Rate of Return

Trade and investment produce profit; Rate of Return reflects this fact in models.

Rate of Return depends on the general Market conditions and oscillates as the Market oscillates.

The limitation on flexibility of Rate of Return comes with leverage. Highly leveraged investors need high level of profit to cover cost of borrowing. Investors, who own stock to exercise control over the company, often accept lower than average investor level of profit; their presence on the Market increases flexibility of Rate of Return.

In the absence of uncertainty, inflation and other factors the price is determined by the Rate of Return e only

p(t+s) - p(t) = e * s,

where s is a period of holding the financial instrument.

In reality, e * s is only one of the components of the price equation.

Effect of the Basic Volatility

The effect of Basic Volatility on the difference

p(t+s) - p(t)

we present as a random value z added to this difference.

The variance of this random value we denote

Db(t+s) = var(z),

where db(t+s) is our characteristic of the effect of Basic Volatility on the price.

In a short period of time, we can assume that

Db(t) = Db

is constant.

When we forecast

p(t+s) - p(t)

as a F-process with characteristic D, we estimate this characteristic based on degree of uncertainty of future. The higher the degree the bigger is characteristic D.

The variance of the price volatility caused by the Basic Volatility should follows the same pattern.

In our models, we assume that Db(t+s) and D are roughly proportional, and in the case Db is stable:

D = cba*Db,

where cb is a stable in time coefficient.

Inflation Forecast

Inflation causes change in the (logarithm of) price

p(t+s) - p(t)

of the asset, which (logarithm of) Economic Value stays the same

v(t+s) = v(t).

The collective Forecast of this change (change of the logarithm of the price) is F-process with parameters (mh,Dh).

p(t+s) - p(t) = H(s).

The change in the mean value of the price itself

exp(p(t+s))/exp(p(t)) = exp((mh+Dh/2)*s).

The mistake of forecast is present in the estimate of mean value of change.

In simplified models, we forecast inflation in the form

p(t+s) - p(t) = h*s,

h is a forecast rate of inflation.

From the above

h = mh+Dh/2.

Forecast rate of inflation can be increased by merely increasing the degree of uncertainty of its future value. This is exactly what happens when Feds' decision is anticipated but unknown.

As with the characteristic of the price forecast we should expect Dh to be proportional to characteristic of the Basic Volatility Db and in first approximation:

D = cbh*Db.

Liquidity of an asset assumes the chain of trades. On each step of this chain the traders should have profit (at least in average) to keep the holding of the asset moving. The ratio of price in the moment t+s to the price in the moment t should be big enough to cover inflation, risk of loss discount and trader's profit. Hence, the minimal value of this ratio should be

exp((g + e + h)*s)

where

g - loss rate,

e - profit rate,

h - inflation rate.

Logarithms of prices p(t) and p(t+s) are random numbers, hence we have to work with average values:

E( exp(p(t+s)) )/E( exp(p(t)) ) = exp((g + e + h)*s).

More precisely, we have to work with

E( exp(p(t+s)-p(t)) ),

but we decided to simplify the model.

When we have a chain of trades in moments

ti+1 = ti + si

t0 = t

tn + sn = t + s

we have a set of bounds

E( exp(p(ti+si)) )/E( exp(p(ti)) ) > exp((g + e + h)*si).

Note that values g, e and h are estimated in the moment of trade ti and, generally, they are different in each equation.

We multiply left sides of these equations separately and right sides separately:

E( exp(p(t+s)) )/E( exp(p(t)) ) > exp((g + e + h)*s).

Now, values g, e and h are average values for the period [t,t+s].

When we deal with the traded financial instrument of limited life (bond, option, etc.), we have to consider two interrelated phenomena. One is the additional Economic Value, which the instrument has by the virtue of its liquidity, and the other the rate of return of traders, who help moving this instrument from the place of excess to the place of shortage.

The summary effect of these phenomena is a discount to the base price, which is computed without taking in consideration the liquid nature of the instrument.

The logarithm of additional value of instrument of life period s we denote

l(s)*s.

We add this value to the logarithm of the value computed without liquidity component. This corresponds to the multiple of the Economic Value. In some cases we can assume l(s) constant, but in many cases, as we will see, l(s) decreases with s.

The rate of traders' return is et and the logarithm of the price of the traded instrument, computed without liquidity component, has to be corrected with

(et - u0(s))*s.

The rate et should be the same for all classes of instruments, but u0(s) depends on the class of instruments. Both depend on the current market conditions, in particular on the degree of uncertainty of the price dynamics. The higher is this uncertainty the lower is value of liquidity and the bigger is discount.

Certainty Component

Borrower with fixed interest rate, or buyer of a bond or derivative instrument reduce the degree of uncertainty of the decision making. This means that the instrument has an additional Economic Value; its logarithm we denote

u1(s)*s.

Usually u1(s) decreases with s.

This additional value depends on the Market conditions. The more uncertain they are the smaller is this value. Also, it depends on the class of instrument.

Combined Discount

Both rates u0(s) and u1(s) are difficult to compute. Hence, we combine them together with traders' rate of return. We denote

k0(s) = et - u0(s) - u1(s).

The higher is degree of uncertainty the bigger is this rate.

There are a few elements of financial models related to potential losses. The influence of potential losses depends on the degree of exposure to them, and it is deflected with different traditional measures.

One of such measures is maintaining the Cushion that temporary losses do not throw the market participant from the Market. The other is diversification, that particular loss event does not cause too much damage. Third is monitoring the credit risk of the company, with which one is involved (has contracts, invested, etc.).

Potential losses are reflected in pricing through the mechanism of Collective Forecast.

Credit Risk

When one is involved with a company, one has to consider the possibility of delayed payments on contracts and debt obligations or the possibility of the company bankruptcy and related losses.

In the cases of practical interest, the probabilities of such events are extremely small and data is scarce. Even if we combine companies in classes for the purpose of such study, we cannot find enough data to arrive to reasonable precision of results.

When we do not have enough data, we build more precise model. The information, which is present in the model, allows delivery of acceptable results even from small amounts of imprecise data. In this case, we build the model based on familiar M-process.

We model the amount available for payments. It is assets minus liabilities in general sense.

We do not know this amount precisely even in the current moment, not to say in the future moment. Hence we estimate the distribution function F0(t,x) - the probability that amount x is available in the moment t. To make it similar to the distribution function of the random value we define

F(t,x) = 1 - F0(t,x)

which grows from zero to one when x grows from zero to infinity.

We define this function in the current moment from the analysis of the current state of the company. It can be approximated in some convenient way.

Now we need to arrive to reasonable forecast of this function. If this forecast is good enough, we can assume that the Collective Forecast of the Credit Risk related to this company gives results similar to ours.

As usual with monetary values, we forecast the change of the logarithm of this amount. We assume that it is M-process.

Sometimes we have additional information about the future financial health of the company. However in many cases, this information is not available, and we have to make simplest model - we assume that logarithm of amount available for payment is F-process. In addition, we assume that all assets and liabilities are measured in units, which do not change in time (as 1970 dollars).

The amount available for payments in initial moment is known with good precision, hence it is not important for the results, what is the form of distribution function F(t,x) in initial moment as long it has same mean value and variance. For the simplicity of the model we assume that F(t,x) correspond log-normal random value, or logarithm of our variable in initial moment is distributed normally with mean value m0 and variance D0. In the moment t+s the distribution will be

F(t+s,x) = N( (x - m0 - m*s)/sqrt(D0 + D*s) ).

We know, that in the moment t+s before we get paid, the amount b(t+s) is paid to other parties, what is left for us is

F(t+s,x) = N( (x - m0 - m*s - b(t+s))/sqrt(D0 + D*s) ).

For each amount a, which we expect to be paid to us in the moment t+s, we know the probability that there are assets available for payments

G(t+s) = N( (a - m0 - m*s - b(t+s))/sqrt(D0 + D*s) ).

The usefulness of this model lays in the ability to use different data for the estimates of different parameters of this model.

D0 reflects mistakes of our current estimate of company current ability to pay bills.

m is a trend of change of company financial health. This trend is common for many companies of the same class.

b(t+s) - is a state of current liabilities of the company.

D - is a characteristic of uncertainty of forecast. This parameter is common to many companies of the same class.

We can combine information about different companies to estimate m and D.

We can estimate parameters m0 and b(t+s) without analysis of defaults.

Loss Events

We analyze a portfolio of financial instruments and potential loss events. Each loss event causes the losses related to one instrument or to a group of instruments. It can cause either partial loss of instrument's value or complete loss.

We combine potential loss events in classes of similar events and assets in classes of similar assets. This reduces the size of the model.

Bij is a size of an asset of class j, which can be affected by events of class i. Each loss event of class i carries a loss of Aij. It is possible to have only

nij = Bij/Aij

loss events of class i for the asset of class j.

In a more general case, there is a distribution function Fij(t+s,x) - the probability that event i causes loss smaller than x in the asset j.

The probability that no loss events happened in the period [t,t+s] for the class i (the probability that Bij is intact) we present in a form

exp( gi*s )

where gi is a rate of loss events of this class.

Loss events for the portfolio are sets

K = (m1,...,mM)

where mi is a number of loss events of class i.

We assume that loss events from different classes are independent. Hence, we can compute probabilities for any loss even K. This can be done analytically or numerically.

The number of loss events K, which we have to consider, is limited. First, when mi grow the probability of K drops very fast. Second, we have a limitation on mi with maximum among nij (when it is applicable).

Generally, each portfolio loss event K can be characterized with its probability P(K,t+s) and the function of the distribution of losses F(K,t+s,x) - probability that loss is not bigger than x.

Using this information we can compute the function of distribution of losses for the portfolio F(t+s,x).

Based on this function we can find a bound: with probability R the loss will not exceed F-1(R). This gives us information for computation of the Cushion.

Also we can compute the average value of potential loss; this gives us information about required price discount, needed to compensate for the risk of losses.

Each particular investor estimates own exposure to loss events. The average market exposure can be estimated also.

Note that usually functions Fij(t+s,x) have small variance. In this case, the particular form of distribution of the corresponding random value does not affect much our results. It means that we can assume that these functions have some form convenient for computation.

Cushion Related Price Component

Size of the Cushion required to stay in business depends on the Market sector. This size can oscillate with the oscillation of the Market. One of the major factors defining the size of the Cushion as a need to survive a potential Market crash.

The need for the Cushion is reflected in prices. Lenders pay less for the future cash flow (bonds prices are discounted), investors pay less for the company shares. This discount is different for different sectors of the Market.

This discount can differ for bonds of different maturity. The nature of this discount is similar to an insurance premium. The insurance premium is established from the analysis of the entire pool of insurance policies. Hence, in the case of bonds, this discount should depend on volumes of borrowing of different maturity.

Also, there is some additional freedom in the distribution of these discounts between bonds of different maturity, and coupon paying bonds and zero coupon bonds.

Discount

When we compute market prices, we have to estimate the average Market exposure and compute the average Cushion and average risk of losses and related average price discount.

We assume that, in average, losses accumulate as interest, hence the logarithm of Economic Value left after the period s is roughly

v(t+s) - v(t) = - g*s.

where g is a rate of losses.

In the more precise model

v(t+s) - v(t) = - G(t,s),

where G(t,s) corresponds to the value of accumulated losses. As we know from the analysis of the Cushion, in the case of bonds, and debt instruments in general, we often need to use this more general model.

We can approximate the function G(t,s) according to the model's situation.

Bonds and Loans

Growing Interest Rate

There are two important components of the Economic Value of the traded Cash Flow Instruments. One is produced by an improved certainty of future, given by the fixed interest rate. The other is produced by an additional quality of the instrument - its liquidity. Both these additional components are difficult to measure. Their value depends on the degree of uncertainty of the future interest rates.

We illustrate this point on zero coupon "zero risk" bond.

If the bond has the maturity s, the logarithm of face value c and the logarithm of price p(t,s), then interest rate is defined as

i(t,s) = (c - p(t,s) )/s.

In each moment t it is a (smooth) function of s, which randomly oscillates.

Some causes of this oscillation are related to the oscillation of Market; others are results of the Basic Volatility (which is a Market's tool of setting prices).

The logarithm of Economic Value of bond has an obvious component:

c - h*s

the promised payoff, adjusted for inflation.

It has an additional Economic Value caused by its liquidity and by its reducing uncertainty property. This component depends on its price and on the Collective Forecast of the future dynamic of the interest rate curve. This component is a kind of an option.

We expect the value of this additional component to grow (as a pattern), when the period s grows. We denote it according to Model Elements

(u0(s) + u1(s))*s.

Now we can write the Economic Value of zero coupon bond as

v(t,s) = c - h*s + (u0(s) + u1(s))*s,

where we do not know yet how to compute (l(s) + c(s)).

The price of the bond should be discounted Economic Value. This discount should compensate for the expected profit, for the risk of default, etc. We ignore for now all factors except the lenders' rate of return e and the traders' rate of return et

p(t,s) = v(t,s) - (e + et)*s.

In this notation, interest rate is

i(t,s) = h + e + (et - u0(s) - u1(s)) = h + e + k0(s).

Market participants want that after period z

i(t+z,s-z) < i(t,s),

then original bond (of maturity s in the moment t) is a bargain in the moment t+z - bonds are liquid.

Market participants build into interest rate curve a safety margin (as a pattern):

i(t,s-z) < i(t,s).

This means that as a pattern

k0(s-z) < k0(s),

function k0(s) grows with s (as a pattern).

Benchmark Interest Rate Curve

Treasuries or borrowing rates between banks are ill suited as benchmark rates for the analysis and forecast of the credit system - business loans, mortgages, corporate bonds, etc. They are tied to the monetary systems of their countries and they are affected by factors relatively unimportant for the rest of credit system, or at least affecting the rest of the system indirectly and with time lags.

Hence, we had chosen, as the Benchmark Interest Rate Curve, interest rates equivalent to zero coupon interest rates of high quality corporate bonds. We extract this information from prices of coupon paying bonds.

As usual, we work with logarithm of prices and cash flows. If c is a logarithm of cash flow in moment t+s and the logarithm of its price in moment t is p(t,s), then we define our interest rate

i(t,s) = (c - p(t,s))/s,

or

p(t,s) = c - i(t,s)*s.

If the bond pays exp(ci) in the moments t+si, i=1,...,n, (last payment is its face value), then a sum of

exp( ci - i(t,si)*si)

is the price of the bond exp(p(t,s)), s=sn.

If exp(f) is a face value

cn = f,

and rest of ci are equal

ci = c + f, ( exp(ci) = exp(c)*exp(f) ),

then

f - p(t,s) =

- ln( [sum of exp( c - i(t,si)*si)] + exp(- i(t,s)*s) ),

s = sn.

(In the case of zero coupon bond we have

f - p(t,s) = i(t,s)*s.)

The Basic Volatility makes particular prices p(t,s) vary symmetrically around their mean value, hence this formula is well suited for the estimation of i(t,s). Now we present i(t,s) as a function with parameters and find this parameters from the available market data with least square method. Special care should be taken to reflect the distribution of the Cushion related discount between bonds of different classes, as zero coupon bonds and coupon paying bonds.

Note we work here with real prices; quoted prices should be adjusted for pro rata payment of interest.

Bonds with similar level of liquidity, but higher credit risk we can price with the help of this benchmark curve through equation

p(t,s) = c - i(t,s)*s + g*s,

where g reflects additional credit risk.

Specifics of Treasuries

The prices on Treasuries are very sensitive to some factors, which affect corporate bonds in lesser degree and with substantial time lag. Among those are Feds' trading in Treasuries, Feds' sole decisions in amounts sold on Treasury auctions, some interest rates, which Feds set at will, and other factors relevant to monetary system, which affect only short term borrowing. In addition, there is an unusual Treasury demand structure caused by the rules of exercise of derivatives with Treasuries as underlying asset.

Short-term Debt Instruments

Short-term debt instruments comprise special sector of the Market with big volume. Often, they are intended to be held to maturity; hence, their ability to be sold is not as important as with long-term instruments. Their credit quality is very important, because they are mostly tools of treasury departments, and treasury departments are less concerned with return and more concerned with quality.

In addition, the rates of short-term instruments are affected by Feds' decisions in higher degree, than long-term instruments.

Hence, short-term interest rate has to be computed with the analysis of interplay of two sectors, which collide when the original long-term instrument has a little time left to maturity.

Price of Bond

The first step is pricing of a zero coupon bond in assumption of the absence of the Cushion component of the price and its liquidity component:

c - p(t,s) = H(s) + g0*s.

where H(s) is an inflation forecast, and g0*s is a discount related to potential losses.

We apply this formula to estimate the logarithm of the price of a coupon-paying bond with face value exp(f), coupon exp(c)*exp(f) and payments in moments si (i=1,...,n):

f - p(t,s) =

- ln( [sum of exp(c - H(si) - g0*si)] + exp(- H(s) - g0*s) ),

s = sn.

Now we correct it for the Cushion discount for this class of bonds g1(s)*s and combined discount k0(s)*s:

f - p(t,s) =

- ln( [sum of exp(c - H(si) - g0*si)] + exp(- H(s) - g0*s) )

+ ( g1(s) + k0(s) )*s.

This is a random value - H(si) are random values. Random values H(si) can be presented as a sum of independent normally distributed random values z(t) with mean value mh*t and variance Dh*t:

H(si) = H(si-1) + z(si - si-1),

H(s1) = z(s1).

As an estimate of the market price of frequently traded bond, we take the mean value of the appropriate random value

E(exp(p(t)) = E( exp(f + L - ( g1(s) - k0(s) )*s ) ),

where random value L is

ln( [sum of exp(c - H(si) - g0*si)] + exp(- H(s) - g0*s) ).

Note that rate of Cushion discount g1(s) can vary according to the relative volumes of bonds of different maturity on the market.

Investor in illiquid bonds should have a diversified portfolio, hence we still should use average values and not bounds to estimate their prices. However, illiquid bond does not have price components related to trading and liquidity and discount rate k0(s) we have to replace with premium rate u1(s). The low price of illiquid bond comes usually from high credit risk and big rate g1(s). For illiquid bond we have

E(exp(p(t)) = E( exp(f + L - ( g1(s) + u1(s) )*s ) ),

with the same random value L.

When we estimate the price of Treasuries, we add additional element to the model - Treasuries' premium u2(s)*s:

f - p(t,s) =

- ln( [sum of exp(c - H(si) - g0*si)] + exp(- H(s) - g0*s) )

+ ( g1(s) + k0(s) - u2(s) )*s.

Loan

Lender lends exp(p(t)) and receives instead regular payments

exp(c + p(t)) = exp(c)*exp(p(t))

in moments si (i=1,...,n) of fixed interest rate loan.

We estimate characteristic c of the loan.

While loans usually illiquid, the lender has a portfolio of loans and the price estimate should go through average values, not bounds. It is similar to illiquid bond, only its payments are all equal.

The random value L in this case is

ln( [sum of exp(c - H(si) - g0*si)] ) =

= ln([sum of exp(c)*exp(- H(si) - g0*si)] )

and p(t) = f, hence we have an equation

0 = E( exp(L - ( g1(s) + u1(s) )*s ))

or

E([sum of exp(c)*exp(- H(si) - g0*si)])

= exp( ( g1(s) - u1(s) )*s )

From this equation, we can estimate c.

Borrowing Decisions

Computations in borrowing decisions or decisions about issue bonds rely on existing instruments on the market. We bring a few examples here, to show different discounts and their application.

If we compare loans of the same life and the same start-up conditions (some lenders charge first installment on the very first day - hidden points), then we need to compare only size of monthly payments. If we have different life of loan, different payments dates, different initial payments, then we have to discount future payments and add it up to get numbers to compare.

The trick is in the choice of discount rate.

If we borrow money to cover our needs, then discount rate is inflation rate h.

If we borrow money to invest in our business, which steady rate of return is r, then we have to roll forward until loan maturity our cash outflow and borrowed amount. Cash outflow we roll as if we invest it, say, in saving account, and the borrowed amount we roll with the business rate of return r. The difference between two gives an artificial number, which we can use to compare loans.

If we buy a house in the good area, with expanding community and improving living conditions, that the Economic Value of the house will grow, then we have a combination of two. The house is an investment (and we know its rate of return) and it is a place to live. We know how much we have to pay monthly now for renting the place to live, we project it in the future with inflation rate and subtract this from future loan payments. The rest of cash outflow we discount with the rate of return of our investment in real estate. This is a number to use, when we shop for the loan.

If the entity issues bonds in the moment s0, which mature in moment s1 to finance a profitable enterprise, which will generate a predictable cash flow starting from the moment s, s0 < s < s1, then we need to roll forward the cash outflow and cash inflow to moment s1. We roll forward both with the same rate - the rate of low risk investment.

Cash Flow Portfolio

A portfolio of bonds, loans etc. can be presented as a portfolio of Cash Flow Units. Each unit belongs to its group and has its own characteristics of time and value. In a simple case, it has a fixed moment when the payment occurs and it has a fixed value. Generally, we can take in consideration potential payment delays, possibilities of default, and dependence of the payment on some other factors as future interest rate. Delay and default characteristics of Cash Flow units of the same group are related.

When we estimate the present value of such Cash Flow portfolio, we have to take in consideration inflation. Inflation we present as a stochastic process hence discounts of different Cash Flow units are correlated.

This general presentation of different types of portfolios as a Cash Flow portfolio can be used to price a portfolio, to price swaps (exchange of such portfolios), etc.

This model does not easily lead to analytical formulae, but it can be computed with simple simulation method.

First, we simulate "external" factors - inflation trajectory, delays of payments and defaults (partial payments of a Cash Flow Unit and complete default of an entire group of units), and future interest rates. Based on this information, we compute moments and payments {si,ci} according to the particular scenario of future.

Second, we find the current price (and other characteristics) of each such Cash Flow Unit and add them up.

We compute this several times (simulation according to known probabilities) and find average value of the portfolio's price (and other characteristics).

This price has to be corrected to reflect the Cushion needed for this particular portfolio. In the case of swaps, this is often the same correction for both portfolios and it can be ignored.

Effects of Taxes

Taxes affect interest payments and payments of principal differently. We analyze here case of bonds. Other cases can be analyzed similar.

We reduce the interest payments according to taxes and we can use our pricing formulae. While this reduction depends on the buyer's tax bracket, we can assume the average tax bracket when we estimate market price. More precise formulae should treat tax adjusted interest payments as a random value.

In the moment of payment of principal, the buyer realizes capital gain or loss. The gain is taxable and the loss can be offset by the capital gain with other investments.

For the purpose of finding the market price of the bond we assume, that capital loss is always offset with capital gain. This is equivalent to the gain equal to corresponding capital gain taxes.

The degree of influence of taxes on the price depends on the value of the price. This sounds like equation, and it is.

If F is a face value of a bond, P its price and Tax - the share of (F - P) which is paid to the government, then we are left with

F - Tax*(F - P) = F*[(1-Tax) + Tax*P/F].

This is the value of last payment, which has to be present in the formula for the price of bond, together with reduced interest payments. The logarithm of reduced interest payments we denote with the same symbol c, and we arrive to the formula

P/F = E(exp(L))* exp(( g1(s) + k0(s) )*s)

where in this case

exp(L) = [(1-Tax) + Tax*P/F]*exp(- H(s)-g0*s) +

+ sum of exp(c - H(si) - g0*si).

We have to solve this equation for P/F. In the case of zero coupon bond, it can be solved analytically, in general case it should be solved numerically.

In the case of zero coupon bond

E(-H(s)) = exp( - h*s),

now we denote

R = exp( (k0(s) - g1(s) - h)*s ),

and

P/F = (1-Tax) * R / [ 1 - Tax*R ].

Invariant

We can present the formula for the price of a bond in a form

( g1(s) + k0(s) )*s = ln( P/F ) - ln( E(exp(L) ),

where

exp(L) = [(1-Tax) + Tax*P/F]*exp(- H(s)-g0*s) +

+ sum of exp(c - H(si) - g0*si).

For our model to be useful, the left side of this equation should not vary much from bond to bond of the same credit risk class and in time. This is a test of applicability of our model.

If the function

K(s) = ( g1(s) + k0(s) )*s

is known for a given class of bonds, we can compute prices of bonds of this class by solving the equation

exp(K(s)) = (P/F) / E(exp(L))

for P/F.

We expect the expression

ln( P/F ) - ln( E(exp(L) ),

taken for different bonds of given class, to be symmetrically distributed; hence, we compute function K(s) as its mean. The standard deviation of this expression from such computed K(s) is a measure of mistake of our model.

Stock with dividends we analyze as a Cash Flow of dividends and the stock without dividends.

It is beneficial to analyze dividends separately, because we can forecast them better than business in general.

We set some period into the future, and compute price of the dividends. After, we gradually increase the period, until mistakes of forecast allow and until the change of the price of Cash Flow of dividends is substantial (this change decreases fast with the length of period).

The price of dividends is one of the components of the price of stock.

The Economic Value of business is determined by two factors - by its ability to generate profit and maintain its stock and by the value of its assets, both in the period from the current moment to some distant future.

There is no easy way to estimate either of them. This is done by specialists, who use different methods of forecast.

Mistakes of the forecast grow very fast with the forecast period; hence increasing this period beyond some value leads only to lower precision. Hence, each forecast method has a forecast horizon beyond which the qualitative forecast replaces quantitative. Therefore, we have to analyze limited period of time quantitatively and the rest qualitatively, for example, we can assume, that the Economic Value of the company grows fast enough to at least maintain the constant level of price.

Based on the forecast of the Economic Value of the company we estimate the current price of its stock. In this estimate, we have to consider following factors:

• mistakes of the forecast
• possibility of losses, in the case of company's bankruptcy
• the rate of return we should expect from our investment
• need to adjust future prices according to inflation.

Value Based Pricing

There is some future company's assets inflow and out flow, which is known in advance, as dividend payments, etc. This information can be analyzed, balanced and discounted in time separately. The rest is very much similar for all companies.

The rest we forecast with F-process. If the logarithm of the rest of Economic Value of the company is v(t), then

v(t+s) - v(t)

is an F-process, which is concentrated in the point v(t) in the moment t. Its parameters ma and Da, are parameters of forecast. ma reflects the anticipated rate of growth of the company Economic Value, and Da is a characteristic of the mistake of forecast.

The period if reasonable forecast is f. From the moment t+f we assume that the value is growing in the way that the price stays the same.

If we look at it from the point of view of frequently traded stock, then we should be concerned with the mean value of the Economic Value

E( exp(v(t+f)) ) = exp( v(t) + (ma + Da/2)*f).

If stock has low liquidity, then there is a level of risk R, which we are ready to assume with our forecast, hence we need to take an appropriate lower bound for value v(t+f):

v(t) + ma*f + sqrt(Da*f)*N-1(R).

We have to discount it with needed insurance premium g*s, and required return on investment e*s. Finally the estimate for the logarithm of price in the case of frequently traded stock is

v(t) + (ma + Da/2 - g - e)*f

and for low liquidity stock is

v(t) + (ma - g - e)*f + sqrt(Da*f)*N-1(R).

This adjusted value is a basis for the price computation.

Note that in case of frequently traded stock the uncertainty of future, which we measure with D, investors inevitably interpret with the favor for "upward potential" against "downward risk". This is not a fluke of investors' imagination - this is in the nature of the Market.

If the company currently is eating up resources instead of accumulating value, but we anticipate it will create value with high speed in the future, then we use a forecast of the change of its value in the form of M-process, and we have a a mean value of v(t+f):

v(t) + m(f) + D(f)/2

and a lower bound for value v(t+f):

v(t) + m(f) + sqrt(D(f))*N-1(R),

from which we compute the price estimate for the frequently traded stock and for low liquidity stock correspondingly.

It is important in this case, that a horizon of reliable forecast f is far beyond of the turn-around point for the company. In addition, the estimate of m(s) is difficult, and the lower bound of it should be taken.

Forecast

The future price depends on the Economic Value and inflation.

The component of the price determined by the Economic Value is

v(t+s) + (ma + Da/2 - g - e)*f

and for illiquid stock is

v(t+s) + (ma - g - e)*f + sqrt(Da*f)*N-1(R).

It has the same presentation through forecast period f as the price for the current moment t. Only coefficients ma, g, e, and Da can be different, but we do not have information about it.

The difference between components of price determined by the Economic Value is

v(t+s) - v(t).

The price should grow as the Economic Value does.

v(t+s) - v(t) is F-process with parameters (ma,Da). (Generally, we should assume that it is M-process, but for now we assume it is F-process.)

When we add inflation - F-process with parameters (mh,Dh), we get the increment of logarithm of price

p(t+s) - p(t)

F-process with parameters (ma+mh, Da+Dh).

This is our main result about price forecast. Now we can compute the mean value of the forecast of future price

E( exp(p(t+s)) ) = exp(p(t))*exp( (ma+mh)*s + (Da+Dh)/2*s ) =

= exp(p(t))* exp( ma*s + Da/2*s + h*s),

where h = mh + Dh/2 - rate of inflation

we can compute bounds for this forecast also.

If the stock is traded frequently, then

E(exp(p(t+s))/ E(exp(p(t)) > exp(( g + e + h)*s),

here coefficients g, e and h are average in the period [t,t+s].

It means

ma + Da/2 > g + e.

This is a requirement for the stock to stay liquid. Note that the Market factors uncertainty in favor of the stock.

What happens, when this condition is violated?

The first reaction of the market participants is - the stock went away from the studied area and new study is needed. It means higher level of the Basic Volatility. For awhile it lifts the value of parameter D, which reflects mistake of the Collective Forecast, and the stock stays liquid.

Gradually the situation clears; value D lowers and traders dump the stock, stock's Economic Value drops (when stock is less liquid, it has lower value), and the price drops drastically.

If there are investors on the Market, who are ready to accept lower rate of return or carry illiquid stock, then they stop the free fall of the stock price - they buy-up the stock.

If investors are leveraged too heavily, there is no such group.

If many stocks cross this threshold for the speed of growth of the company's Economic Value, we observe market crash.

From this analysis it is obvious that heavily leveraged investors make market crash more possible.

Forward Contracts, Options and Futures

Forward Contract

With a Forward Contract, two parties agree about the future transaction of underlying asset at specified price. Forward Contract delivers better certainty in decision-making and often it can be traded. From the other hand, either party can default.

The forecast of logarithm of Economic Value of underlying asset is F-process with parameters (ma,Da). Hence the forecast of the logarithm of price is a combination of this process and F-process with parameters (mh,Dh) - forecast of inflation. The increment in price

pa(t+s) - pa(t)

we forecast as F-process with parameters (m,D):

m = ma + mh

D = Da + Dh.

The mean value of this forecast is:

E( exp(pa(t+s)) ) = exp(pa(t))*exp(m*s +D*s/2).

Usually, this value is a set future price in the contract.

To reflect the additional contract qualities - improvement of planning and tradability, and traders' rate of return, we have to discount this price to arrive to the price of the contract itself:

p(t,s) = pa(t) + m*s +D*s/2 - k0*s, (logarithm)

or

exp(p(t,s)) = exp(pa(t))*exp(m*s +D*s/2 - k0*s) (price).

If we can ignore change in value and inflation:

exp(p(t,s)) = exp(pa(t))*exp(- k0*s).

The correction of this price, which reflects the risk of parties' default, we discuss below with options.

Options

Call is a right to buy the underlying asset at the given striking price. The moment when it happens is called exercise of the option.

Producer of a call, writer of it, sells the promise to sell the asset at striking price. He exposes himself to the risk of rising price.

Standard calls in the case of the exercise of a call are randomly assigned to one of the writers, who produce the call of given class. If producer of a call bought similar call on the market he officially covered his open position and he cannot be assigned.

Put is a right to sell the underlying asset at the given striking price. Puts are exercised as calls.

Producer of a call, writer of it, sells the promise to buy the asset at striking price. He exposes himself to the risk of falling price.

Puts are randomly assigned to writers as calls, and there is a possibility to cover open position, as with calls.

All options have expiration date.

European options can be exercised only at expiration date.

American options can be exercised in any date before expiration date.

Bermudan options can be exercised only in particular dates.

Asian options are settled based on the average value of underlying asset's price in the given period of time.

Price of European Option

When traders (arbitrageurs, speculators) dominate option market, as it is with many options, we can arrive to simple formulae for the price of European option.

First, we forecast the logarithm of Economic Value of underlying asset as F-process with characteristic (m, D) in the period, when we intend to hold the option.

This model reflects the Collective Forecast and D is a measure of uncertainty, mistake of forecast. m can be zero, as in case of some investment assets, or it can be definitely non-zero, as in case of liquid stock.

The forecast of the logarithm of price of underlying asset is derived from

• the forecast of its Economic Value - F-process with characteristics (ma, Da),
• the Basic Volatility with variance Db,
• the forecast of inflation - F-process with characteristics (mh, Dh).

The price increment

p(t+s) - p(t)

we present as a sum of

• F-process with trend m=ma+mh and variance characteristic D=Da+Dh
• process presenting the Basic Volatility.

Fortunately, we do not need to describe precisely the process of the Basic Volatility - the knowledge that in the fixed moment it can be presented as a normally distributed random value with zero mean value is sufficient.

In the case of call with the logarithm of striking price ps, the payoff of the instrument in the expiration moment t+s is equal to

exp( p(t+s) ) - exp( ps ), when p(t+s) - ps > 0 or

zero otherwise.

In the case of put the payoff is

exp( ps ) - exp( p(t+s) ), when p(t+s) - ps < 0 or

zero otherwise.

From the point of view of buyer-arbitrageur engaged in statistical arbitrage, the mean value of this random payoff is a component of a reasonable option price.

The other component of a price comes from an additional value of this financial instrument as a tool of planning and liquid instrument. This value we reflect with the discount k0(s)*s similar to one we have with bonds.

Also we have to discount with h*s to arrive to current value of future payoff, h=mh + Dh/2 - rate of inflation.

This is our model of European option.

It can be computed analytically.

Random value p(t+s) - p(t) is a normally distributed value with variance D*s + Db. Now we denote

p0 = p(t)

d0 = D*s + Db

z = sqrt(d0)

d1 = [(p0 - ps) + m*s ]/z

d2 = d1 + z

price of call =

[exp(p0+m*s+d0/2)*N(d2)-exp(ps)*N(d1)]*exp(-(k0(s)+h)*s).

price of put =

[-exp(p0+m*s+d0/2)*N(-d2)+exp(ps)*N(-d1)]*exp(-(k0(s)+h)*s).

There is a parity equation:

[price of call]-[price of put] =

[exp(p0+m*s+d0/2) - exp(ps)]*exp(-(k0(s)+h)*s).

If we want to use this model separately, we use some additional information about relationship between D and Db.

We know that

D = cb*Db = (cba + cbh)*Db

and coefficients cba and cbh do not change much. We denote

cb = cba + cbh

and we can eliminate parameter D from the model:

D = cb*Db.

Parameters of this model are:

• Db - variance of the Basic Volatility,
• cb - coefficient of proportionality of the Basic Volatility and the characteristic of the mistake of Collective forecast of the price of underlying asset D
• m - trend of the change of the logarithm of price of underlying asset,
• k0(s)+h - discount rate.

Illiquid European Call and Put

If an option is not traded frequently, we cannot define its price as mean value and we do not have the discount k0(s).

If options are illiquid, but there is an extensive portfolio of different options, then it is reasonable to use average values as an estimate of the price. Hence the formulae for the price are the same as above with one correction: Instead of discount exp(-k0(s)) we have to compute premium exp(u1(s)). This premium is offset with fees of parties facilitating creation of such option.

In the case of single illiquid option (as something built-in into unique single transaction), we have to use bounds instead of average values.

We replace random value

exp( p(t+s) ) - exp( ps ), when p(t+s) - ps > 0 or

zero otherwise.

with its bound with probability R

exp( p0 + m*s + sqrt(D*s)*N-1(R) ) - exp( ps ),

when p0 + m*s + sqrt(D*s)*N-1(R) - ps > 0 or

zero otherwise.

We have to multiply this value by

exp( u1(s)*s - h*s)

which reflects the Economic Value delivered by better certainty of decision making with such option, and discount for the inflation.

This is a price of unique illiquid call.

The price of unique illiquid put is determined similar.

American Options

If there is a right to exercise an option before expiration, someone does it if the combination of prices is right. American options were invented to make it possible to move an underlying asset with the help of options in the moments preceding expiration. We can assume that there is a group of option buyers, who plan to exercise options if the conditions are right. We can assume also that these option buyers have a moment in mind, when they plan the option exercise. If conditions are not right in that moment they trade the asset in ordinary way and treat the option as an investment instrument for the rest of its life. We assume that these moments of planned exercise are distributed evenly on the period from current moment to the expiration of option.

All above is actually a model of the stream of exercises of American option. The simplest model of this type assumes that in any moment there is always the same share of options, which are ready to be exercised, if conditions are right.

Computation of American option is more difficult, than computation of European option - we have to simulate movement of the price of underlying asset and the stream of exercises.

In the case of American call, the stream of exercises often is low. It should be some extraordinary event to entice holders of American call to exercise. Sometimes such event can be an inevitable and substantial drop of the stock price in the ex-dividend date. In the absence of such events, call holders are not willing to buy the underlying asset and assume "downward risk" of the asset price.

This logic does not hold, however, when the underlying asset is consumable asset. Then American calls are often used as tools of planning. If the underlying asset is need before an expiration date, the option can be exercised instead of selling the call and buying the asset.

When stream of exercises is low, then, obviously, the price of American call can be approximated with the price of corresponding European call.

The model of American Option should include an additional model, which allows the computation of the share of options used as insurance, not as an investment instrument. These options are exercised if the time and the price are right.

We compute an American option through simulation.

The results of computation of an American option are:

• its price
• probability it can be profitably exercised in a particular moment
• projected stream of exercises.

Risk of Default and Forward

There is a probability PB that asset buyer defaults on his promise, then seller has to sell the asset at market price and potentially loose an amount LB, which is similar to one we estimated with European put. There is a probability PS that asset seller defaults and asset buyer has to buy at market price and potentially loose the amount LS, which is similar to one we estimated with European call. Note that these loss amounts should be discounted with inflation rate, not with rate k0.

The difference

PA*LA - PB*LB

shows which party of the contract is at higher risk and it gives the forward contract price correction.

Futures

Futures freed parties of forward contract from the need to trust each other. The difference between market price and forward price is moved from one account to the other through clearing house. Accounts (margin accounts) are held by brokers, who have their own agreements with the parties when cash has to be added to the account, when it can be withdrawn, and how much interest is paid if excess of cash sits in the account. These procedures are strictly regulated, which make the entire system very reliable.

There are standardized futures, this helps futures be liquid, and there is an army of traders-speculators, who make them liquid. The designers of standardized futures made life of the party, which has to deliver an asset easier - there is a choice of delivery variants and actual delivery date in the delivery month. This makes computation of the price more difficult.

While formally futures look similar to forward contracts, they have different working and different application. They are useful for someone who is in a daily business of buying or selling of underlying asset, and who can benefit from lowering the uncertainty of daily volatility of assets price. Futures transfer the risk related to daily price volatility to the other party (which is paid for it) and allow better planning (and benefiting from better planning).

Speculating in futures requires daily attention to the Market and frequent trading in related derivatives to hedge the risk.

The computation of futures' prices involves simulation of the movement of price of underlying asset, movement of the cash between margin accounts, accrued interest, and simulation of delivery of asset.

Because futures have liquidity and they are used as tools of (sophisticated) planning, we should use in their price computation discount exp(-k0(s)*s) similar to one we use with options.

The result of computation of future contract is:

• its price
• state of margin account in a particular moment.

Directions of Models Improvement

These models can be corrected (improved) in different directions.

For example, if we use a model of the price of stock, which pays dividends according to a schedule, and we know the drop in price of stock at each ex-dividend date, we can substitute trend

ma*s

in the model with the function

ma(t+s)

and compute needed model value

m*s = ma(t+s) + mh*s.

Similar, we can anticipate growth of uncertainty from some moment in the future t+s1. This leads to change in the value Db and value D. We need to present Db as a function of time Db(t+s) and instead of value

D*s

we need to use monotonously growing function of time

D(s),

and we need to analyze more carefully the relation between Db(t+s) and D(s).

This approach can be helpful in the case of anticipated future events, which time is known, but outcome is not known, as Feds decisions.

The Collective Forecast works through the Market trading. Hence, when markets are closed, our models are, actually, not applicable. When these periods are small relatively to the period in the model, we simply ignore this fact, as we model discrete process of changes in prices with continuous process. However sometimes, especially when the period to expiration is small, we have to use models that are more precise. For example, we can assume that function D(s) does not grow when markets are closed.

Trajectory Based Computation

With many derivative instruments, we can arrive to decent estimates only if we can follow each trajectory (sample path) of stochastic process, which we use as forecast of price. There is infinite number of trajectories and we need to cut them to manageable finite number of examples, which we have to study. Generally, this is a numerical procedure and it has to be treated as such - we have to set parameters of precision and from them derive particular computational method.

We start with definition. Our primary parameter y(s) is a stochastic M-process. There is a function defined on its trajectories G({y(t)}) which we have to evaluate. Usually, it is defined as follows: For each trajectory, we can evaluate a number. This is a definition of some random number, because trajectories are random. We evaluate the mean value of this random number or the bound for it. This is our function G({y(t)}).

Time Delta

We have to emphasize one particular property of the function G({y(t)}), which holds for all functions we work with in financial models.

In each case there is some (small) "time delta" that movements of y(t) inside the time interval of this length do not affect the value of G({y(t)}) as long the beginning and the end of this movement is the same. Usually this is related to the time interval, which is needed to execute market decisions, as exercise of option.

This property goes a long way with our forecast using M-process. M-process theoretically can make huge movements in any time interval as small as one wants. If G({y(t)}) were sensitive to this movements we would not be able to use this combination of forecast and function.

This "time delta" gives us first step toward our goal of selecting small set of representatives of y(t). We divide the forecast interval [t,t+s] with points si withstanding not further than this "time delta" and consider instead of process y(t) defined on [t,t+s] its approximation y(i) = y(si). We do not loose precision here, because this is how G({y(t)}) is defined.

Standard Presentation

As a technical trick we replace y(t) with the "standard" F-process x(t) with parameters (0,1) (this is a model of simple Brownian process).

Actually, we work only with processes of a form

y(t) = m0(t) +y1(t),

where m0(t) is a known function, and y1(t) is F-process. Hence, we describe the conversion only for F-process y1(t).

y1(t+s) with parameters (m, D) can be presented as

y1(t+s) = y1(t) + m*s + sqrt(D)*x(s)

Hence, we can present

G({y(t+s)}) = G1({x(s)}) =

= G({y(t) + m0(s) + m*s + sqrt(D)*x(s)})

and we can work with initial random value y(t), stochastic process x(s) and the function G1({x(s)}).

This presentation has its inconveniences, for example if G({y(t+s)}) is defines through some bound y1 which is the same for all moments t+s, we need to define a similar bound for G1({x(s)}), which now is a function of s

(y1 - m(s) - m*s)/sqrt(D),

but this is a small price to pay for the simplicity of stochastic process modeling.

Cones

We define a subset of trajectories of x(si), which present can occur with the given (big) probability P. Our goal is to ignore the rest of trajectories, however before we make this step, we have to discuss when we can actually ignore them.

This is an important subject, the subject of applicability of this numeric method to computation of particular model. The problem follows: while we ignore some trajectories, which likelihood is extremely small, we could open ourselves to the rare but extremely damaging events. For example, if on one of these ignored trajectories we enter a situation where to stay on the market we have to come up with a sum, which exceeds our abilities, no estimated future profit can help us. Poker players know this effect.

Hence, before we cut off these trajectories with low probability, we have to make sure, that this does not move us beyond the area of model applicability. Sometimes the best way to do that is not through the model but through special trading strategy, which provides exit rout (cut losses) in the case we cannot hold the position. This strategy will be reflected in the function G({y(t)}) - it will allow this selection of trajectories as numerical procedure.

Now we go to our selection. We define Pi probabilities for each jump

x(si+1) - x(si)

that their product is equal P.

Usually si+1 - si are equal and Pi are equal.

Because x(si+1) - x(si) is a normally distributed random value with parameters (0,1) it is only natural to select symmetric interval [-a,a], where this random value is with the probability Pi. The trajectories x(si), which make jumps from one such interval to the other are trajectories which we take as our approximation of x(s).

Value Delta

We need the estimate up to some level of precision. If in some moment si the trajectory goes through point x+dx instead of point x, we have a mistake in our estimate, but if dx is small the mistake is small also. We find the biggest dx, that such variation in any moment si or in all of them, causes an acceptable mistake in our estimate of G({y(t)}).

This allows us to set a few points xj in the interval

[-a,a],

which we had defined above. These points stand apart not more than dx.

Now we take only trajectories, which go through these points. This is a finite number of trajectories.

Jump Probabilities

After all these efforts we have a discrete process, and we need to present the probabilities of its jumps from the point xi,j0 in the moment si to the point xi+1,j1 in the moment si+1, and we know, that these points belong to the cone:

-a < xi+1,j1 - xi,j0 < a.

We assign these probabilities according to distance between xi+1,j1 and xi+1,j1+1 - there are only a few such points in the cone. The optimal assignment is - probability for xi+1,j1 according to distance between

(xi+1,j1 + xi+1,j1+1)/2 and (xi+1,j1 + xi+1,j1-1)/2.

Probabilities for leftmost and rightmost points have to be adjusted (increased) to make the sum of probabilities of jumps equal one.

This procedure of replacing the continues process with discrete one and adjustment of probabilities is equivalent to approximation of the function G({y(t)}). All careful analysis above we had done to be sure, that we could make such approximation.

Schema

We choose points globally: for each i (si) we set points xij that they satisfy above conditions, and we take only trajectories, which go through these points and fit into cones, which we described above.

Obviously, it is desirable to have dx as big as possible and P as small as possible - the number of trajectories, which we need to consider grows fast, when degree of needed precision grows.

Some of these schemas are well known and widely used, they are called trees. Especially popular are binomial trees, where interval [-a,a] includes only two points.

The other choice of the computational schema is the time step (which is usually much longer than time delta).

Backward Computation

In real situation, the number of trajectories, which we have to consider is huge, but, luckily, we do not to keep all information in memory at once.

Consider computations for an American option.

In the last moment, we know the price of the option when price of the asset is xij.

Now we move to the previous moment.

We know that the option either exercised during time delta or it is exercised just before expiration.

If option is exercised, the payoff is obvious. If we deal with an investment asset as underlying asset, then it is not exercised if payoff of exercise is smaller than average price of option in the next step. This average price we can compute immediately, because we know all potential changes in the price, their probabilities and related option prices.

In the case of underlying asset, which is not an investment asset, we have to come up with some model describing the dynamic of exercises.

If the option is carried to the next step, then its current price can be computed from the price in the next step by discounting with k0(s)*s, where s is our time delta. We already had explained why this is an appropriate discount for the traded option - it reflects special properties of the option and traders rate of return.

In any case, we know now the payoff of the option in the moment preceding the moment of expiration for all potential values of the price of asset xi-1,j.

We can continue to roll back in time until we reach current moment and find current price of the option.

System of Asset and Derivatives

The Market sets the prices of an asset and its derivatives together. There are a few important consequences of this phenomenon, which we discuss here.

Stock

If there are traded derivatives with given stock as underlying asset, then the stock is priced in a special way.

In the moment of the split of the stock, the ability of the company to generate profit and its assets and liabilities do not change. However, the number of shares available on the Market increases. The potential to write the derivatives increases accordingly.

The value of a derivative does not depend on the value of underlying asset; it depends on the change of this value only. Hence, with the split of stock the Economic Value of the stock, as financial instrument, increases, therefore, the Economic Value of the company increases.

We observe this phenomenon on the stock market. It is possible even to measure corresponding increase in the price of stock.

When the company issues new stock, each share has a claim on lesser value of the company, and this should drive the price of a share down. However, this tendency is partially compensated by the fact that there is more shares on the Market to serve as underlying asset for derivatives.

Potential to be an Underlying Asset

When exchanges introduce new types of contracts (new striking prices) the potential of the asset to serve as an underlying asset for derivatives increases, its Economic Value increases and the price follows. When some derivatives go out of existence (contracts expire), this potential decreases, the Economic Value decreases and the price follows.

Short selling we can interpret as a kind of derivative. When the asset becomes a subject of unusual interest of short-sellers, its Economic Value increases and the price follows.

If the asset is stock, then all this has little to do with activity of the company, which stock is traded.

The Basic Volatility

The Market can "test hypotheses" related to the asset (stock) by varying the prices of derivatives with this asset as underlying asset and vice versa. The Basic Volatility of the price of asset and prices of derivatives is independent, but results of these "testing" are shared between market participants. Hence, if there are derivatives with given underlying asset, then assets Basic Volatility should be smaller, than otherwise.

Price Correction

In the above models, we had assumed that the logarithm of price of an asset should be taken as a primary parameter and should be forecast with F-process (or with M-process when we have some additional information). In the presence of derivatives this assumption could lead to big mistakes, and we have to go for more general assumption, that there is some unknown to us yet primary parameter x, which we can forecast as F-process x(s) with parameters (m,D), and a function F(X,T) that the forecast of logarithm of price is

p(t+s) - p(t) = F(x(s),t+s).

As a first step, we approximate F(X,T) with linear function and get a formula for the price forecast, which works for small s. This formula can be used, for example, in trajectory based computation.

When s=0, F(x(0),t) = p(t) - p(t) = 0, and we take a Taylor series of F(X,T) in the point X0=x0=x(0) and T0=t.

F(X,T) = F'T*(T-T0) + F'X*(X-X0) + 0.5*F''XX*(X-X0)2 + ...,

F(x(s),t+s) = F'T*s + F'X*(x(s) - x0) + 0.5*F''XX*D*s + ....

The rest of series (deterministic and random part) reduces with the reduction of s with the speed sa, with a>1, and we ignore them in our approximation. Finally, we have an approximation for small s:

p(t+s) - p(t) = (F'T + F''XX * D/2)*s + F'X * (x(s) - x0).

Inflation and Foreign Exchange

Both concepts - Inflation and Foreign Exchange depend on the concept of buying power of the currency. There are methods of measuring the buying power through prices of a basket of goods, which are used in standard government statistics. However, in our models, we have to define buying power broader: we have to include financial instruments, debt instruments in particular.

We cannot wait for the government statistics for the data related to inflation - the computation is done too infrequently, its precision low and its definition differs from what we need. We rather extract the inflation data from available market data based on the models, which show how prices (especially prices of debt instruments) and currency exchange coefficient depend on the characteristics of inflation.

Exchange Coefficient

We will work with the logarithm of the currency exchange coefficient c(t). If the logarithm of the price of an easily tradable product in the other currency is p1(t) while in the base currency it is p0(t), then the logarithm of currency exchange coefficient

c(t) = p1(t) - p0(t).

If H1(s) and H0(s) are forecasts of inflation processes for corresponding currencies, with parameters (mh1,Dh1), (mh0,Dh0), then in the forecast of the exchange coefficient in moment t+s is:

c(t+s) = p1(t) - p0(t) + H1(s) - H0(s) =

= c(t) + [H1(s) - H0(s)]

F-process

H(s) = H1(s) - H0(s)

has parameters

(mh1 - mh0, Dh1 + Dh0).

This is our model of the currency exchange coefficient:

c(t+s) - c(t) = H(s).

Estimates

We will make estimates of parameters (mh,Dh) for different currencies from the observation of currency exchange rates and interest rates indifferent currencies.

From observation of exchange coefficients we form equations for mh1 - mh0 and Dh1 + Dh0 for different pairs of currencies.

From observation of prices of bonds of similar credit rating and liquidity traded in different currencies we establish other set of equations for parameters of inflation.

If we have a bond of face value F and price P, then from the formula for the price of bond we have:

P/F = exp(- g1(s) - k0(s))*E( exp(L) ),

where L is a random value, which depends on inflation, coupon and payment schedule.

If two bonds have similar maturity, and Cushion level and distribution, then they have similar functions g1(s) and k0(s). If we divide the ratio P1/F1 for one of them by similar ratio for the other P0/F0, then we have an equation, which does not depend on these functions:

(P1/F1) / (P0/F0) = E( exp(L1) ) / E( exp(L0) ).

We can take different pairs of bonds in the same currency or in different currencies. Each such pair produces an equation.

L = ln([sum of exp(c - H(si) - g0*si)] + exp(- H(s) - g0*s)).

Bonds of similar credit ratings have similar coefficients g0.

In the case of zero coupon bond

L = - H(s) - g0*s.

If zero coupon bonds have similar credit ratings, then

(P1/F1) / (P0/F0) = E( exp(- H1(s)) ) / E( exp(- H0(s)) ) =

= exp((-mh1 + Dh1/2)*s) / exp((-mh0 + Dh0/2)*s).

In terms of logarithms of prices and face values

[(p1 - f1) - (p0 - f0)]/s = - (mh1 - mh0) + (Dh1 - Dh0)/2.

Note that these equations allow estimates of differences (mh1 - mh0); we need some other means to estimate at least one value mh separately. In the same time, an exchange rate gives us an equation for a sum (Dh1 + Dh0), while bonds give as an equation for a difference (Dh1 - Dh0), this allows us the computation of all values Dh.