10.2.1 On the Use and Meaningfulness of Models

Throughout this book we presented and worked with a series of economic models. When working with them, you should always keep in mind the following caveats.

Models are never an exact description of what actually happens, but rather logical frameworks that help structure your thinking when interpreting real data. Giving models a mathematical formulation often allows us to derive implications of a certain setting/scenario that are not necessarily visible at first sight. Often we also are able to exclude certain interpretations that are not consistent with the chosen framework.

These implications/exclusions are, however, valid only if the assumptions of the model framework are correct. Correct means that they adequately reflect the driving forces of actors’ motivations and interactional logic.

The ultimate test for every model is when it is confronted with real data or tested. Still, not being able to reject a model base on real data does not necessarily mean that it is true. It is simply not (yet) falsified. With Karl Popper we say that there is no true model, but only models or theories that have not yet been falsified.

It is important to recall this fundamental scientific wisdom, as—particularly in practical work—models all too often are taken for granted, providing “the truth.” When working with models, you should always be wary of them, taking the rich help they can give you but never really trusting them.

10.2.2 Random Variables

A random variable or “statistic” is a concept that allows you to capture the fact that for many aspects in life we have to deal with uncertainty. It represents a conceptual shell for a measure of some aspect of real life that may take on any of a predefined set of possible, mutually exclusive values, its domain. The assignment of a concrete value to a previously random variable, the realisation, is dependent on a specific event, usually determined by the definition of the random variable. Each of the possible values of a domain may be assigned a probability according to which might become the realisation of the random variable. The distribution of probabilities over possible values is called the probability distribution of the random variable. Hence, a random variable may be fully characterised by its domain and the corresponding probability distribution.

Let’s make this a bit more lively: Think of the random variable “weather tomorrow” that may take on the values “same as today,” “better,” or “worse.” The event that will assign the value to that random variable is the emergence of “tomorrow,” because then we will clearly know how the weather is in relation to “today” (then “yesterday”).

This example also nicely shows the importance of the domain definition for the meaningfulness of a random variable, as you equally could have assigned the possible values “rain,” “sunshine,” and “snow.” Although the event in both cases will bring along the information you need to assign a value to the random variable, the transformation from observed weather tomorrow into one of the measures defined by the two domains fundamentally determines the meaning and information content represented by the variable.

The example also illustrates the importance of the mutually exclusive property of the values in the domain. This important specification means that nature at any one time can only take on one state of the random variables domain. Think of a domain for the random variable “weather tomorrow” that consists of the states “rain,” “fog,” and “sunshine.” Even though “sunshine” is well enough a state on its own, the state “fog” may imply additionally that it rains! Hence, the information content of the domain is flawed.

The most common example for a random variable is the value of a die. Here, the domain is clearly one of the numbers {1,2,3,4,5,6} and the event determining which of the numbers will be the value is the throwing of a die. The example usually is used in statistics text books to illustrate random variables but its experimental nature does, by construction, not exhibit the typical problems encountered with random variables in financial practice.

We may say that a lot (if not every) of financial aspects of a financial instrument may (need to) be expressed as a random variable, as even promised cash flows may need be paid out due to defaulting to the counterparty or other unforeseen events.

In finance, typical random variables are the:

• Future price of an asset
• Next cash flow on an asset
• Actual discount factor

Which distribution is the right one for a given random variable is the subject of an academic field of its own: probability or stochastic theory. In practice, it is often not easy to decide which is the right probability distribution. Except for very few obvious cases, like the rolling of a die or the throwing of a coin, the probability distribution of a random variable is a matter of subjective judgement often assisted by some stochastic (or econometric) modelling of an underlying but unknown process that drives the emergence of this or that value for a random variable. Who could know with certainty with which probability the weather tomorrow will take on this or that form?

This brings us to a fundamental problem when working with random variables in finance. Although we reasonably assume that there exists a true distribution for the concepts we identify, we do not know it and must make an assumption-based approximation. In this book, we made it explicit in the various pricing approaches how they solve this dilemma by which assumptions, and the potential consequences of these, for the exactness of the pricing method.

10.2.3 Time and Its Notation

Intertemporal considerations are key to asset pricing, which turns around relating a series of future (and sometimes past) cash flow events to a current, or reference, date. Hence, it is important to clarify the time concepts and corresponding notation.

Throughout the book, we tried to limit our discussion to discrete random variables, closely linked with the concept of discrete time measurement. The reason is that these, first, are easier to understand, and second, capture the same concepts as continuous random variables.

Since pricing is always done from a relative perspective—that is, seeking to determine the asset price for a given reference date, around which all other relevant events are ordered and measured—we also model time in a relative way. Out of a series of dates t = {−t_T, …, −t₂, −t₁, 0, t₁, t₂, …, t_T} we denote by t = 0 as the reference or pricing date (we might be tempted to call this date “today”). This logic is illustrated in Figure 10.1.

Figure 10.1: Discrete time notational conventions.

In line with market conventions, we measure difference in time as the year fraction between two dates. Calendars have the unpleasant feature to be rather inhomogeneous (some months are 28, 29, 30, or 31 days in length), so the term year fraction is not a standardised concept. Rather, we must resort to a so-called day count convention that specifies the exact method to transform two dates into a year’s fraction. There exist quite a number of day count conventions that are in use in the markets.

The most straightforward one is the rule ACT/ACT, which counts the number of actual calendar days between two dates and divides it by the actual number of calendar days in the current year. A related method is ACT/365, which always divides the first term of ACT/ACT by 365, even if the current year has 366 days. As these methods could reach widespread usage only with the emergence of computers, quite a number of simpler methods have been designed and are still in use.

In this book, however, we abstract from specific day count conventions by denoting the year fraction between two dates t₁ and t₂ as Δ(t₁,t₂), whatever may be the day count convention to be used. In addition, with reference to the central date t = 0 we specify Δ(t) = Δ(0,t) making the formulation of certain formulas less heavy without any loss in precision.

10.2.4 Discount Factor and Future Value

Future values do not have the same relevance today that current values do. Think of a cake that you can eat tomorrow, but not today. Although you will still appreciate the outlook (if you like cakes), today’s utility will be diminished by the fact that you have to wait another day to eat the cake.

Similarly, you can say that about money (cash flow) you will earn tomorrow, but not today. In the context of investing, another aspect becomes relevant: Money you have today can be invested and will bring you additional income, whereas money you have available only in the future cannot obviously be invested today. We call this concept the opportunity cost of time, the cost you incur due to the fact that the money is only available in the future is equal to the lost income. …

Hence, we may say in general that from today’s perspective, future values are worth less today than they are tomorrow. The depreciation of future values to have their current day equivalent is captured by the concept of a discount factor.

We denote the discount factor that transforms a future cash flow x_t, t > 0 to get its value equivalent at date t = 0 as m_0,t ∀t. By construction m_0,t < 1. As there is no depreciation for current values we have m_0,0 = 1.

However, there is no general rule what the “right” discount factor should be. As you will see next, determining the discount factor is one of the key issues when pricing an asset. To make things worse, the uncertainty inherent in future cash flows often makes even the discount factor uncertain from the perspective of the investor and, hence, a random variable.

10.4.1 Utility Functions and Investor Preferences

The investor needs to make sure that he or she draws the maximum utility, or pleasure, from his or her inter-temporal decision. Obviously, with unknown future values of wealth, and hence, consumption, this is a difficult one.

In formal language we use utility functions to summarize a consumer’s behaviour by mapping the space of possible consumption bundles C to the real line R. The mapping is ordered in such a way that a higher value on the real line stands for a consumption bundle c, which is preferred by an individual to a consumption bundle c′ with a lower value on the real line. Formally, this is

Working with actual consumption bundles (i.e., with bundles of goods) that may be consumed is realistic, but very unhandy in practice. We therefore prefer to refer to monetary consumption levels; that is, the monetary cost of goods being consumed.

It can be shown that if the preference ordering is complete, reflexive, transitive, and continuous—in short, representing a rationally behaving individual—it can be represented by a continuous utility function.

Figure 10.3 shows the classical form of a stylised continuous utility function over monetary consumption levels c, u(c). It has the property that its first derivative is positive and the second derivative negative:

• : The marginal utility of any consumption level is always positive, reflecting a desire for more consumption: People always want to consume more. The marginal utility is the infinitesimal (very small) change in utility due to a change in consumption.
• : The marginal increase in utility is decreasing or constant, reflecting the declining marginal value of additional consumption. The last bite is never as satisfying as the first.

Figure 10.3: Concave utility function, u(c), over monetary consumption levels c.

Functions with this property are called concave, hence we speak of a concave utility function.⁵ An important property of utility functions is that any monotonic transformation of such a function represents the same preferences. Formally, if u(c) represents some preferences c ≻= c′ and f:R → R is a monotonic function (i.e., has no breaks or steps in it), then f(u(c)) will represent exactly the same preferences since f(u(c)) ≥ f(u(c′)) if and only if u(c) ≥ u(c′).

In the context of uncertainty, when the utility over uncertain future consumption levels is measured, it can be shown that the expected utility over various possible monetary consumption levels must be in additive form. Consider the following situations:

• 1. In the first situation, you consume c¹ with probability π and c² with the (remaining) probability 1 – π: (π → c¹, (1 − π)→ c²).
• 2. In the second situation, you consume c¹ with probability θ and c² with the (remaining) probability 1 − θ : (θ → c¹, (1 − θ)→ c²).

It can be shown⁶ that there exists a continuous utility function u that describes the consumer’s preferences; that is (π → c¹, (1 − π)→ c²) ≻ (θ → c¹, (1 − θ) → c²) if and only if u(π → c¹, (1 − π) → c²) > (θ → c¹, (1 − θ) → c²)

This utility function u( ) is not unique; any monotonic transform would do as well.

Under some additional hypotheses, we can find a monotonic transformation of the utility function that has the very convenient expected utility property:

The expected utility property says that the utility of a lottery is the expectation of the utility from its prizes. We can compute the utility of any lottery by taking the utility that would result from each outcome, multiplying that utility times the probability of occurrence of that outcome, and then summing over the outcomes. Utility is additively separable over the outcomes and linear in the probabilities.

10.4.2 Risk Aversion and Risk Neutrality

Utility functions can be used to represent the attitude of investors toward risk. The expected utility property applies in particular in the case that uncertainty is with respect to different amounts of money; that is, cash flows an individual may receive under different circumstances (“states of nature”).

It is hence possible to describe an individual’s behaviour who is faced with uncertainty concerning future monetary outcomes, if we only know this particular representation of his or her utility function for monetary consumption levels. For example, to compute the consumer’s expected utility in the situation (π → c¹, (1 − π)→ c²), we just need to look at the corresponding expected utility πu(c¹) + (1 − π)u(c²).

This construction is illustrated in Figure 10.4 with π = 0.5. Notice that in this example the individual prefers to get the expected value rather than being exposed to the uncertain situation (“lottery”). That means that for the investor, the utility of the lottery u(π → c¹, (1 − π) → c²) is less than the utility of the expected value of the lottery, πu(c¹) + (1 − π)u(c²). Such a behaviour is called risk aversion.⁷

Figure 10.4: Expected utility of a lottery. The expected utility of the lottery is 0.5u(c¹) + 0.5u(c²). The utility of the expected value of the gamble is u(0.5c¹ + 0.5c²). In this case, the utility of the expected value is higher than the expected utility of the lottery, so the individual is risk averse.

If an individual consumer is risk averse, the line drawn between any two points of the graph of his or her utility function must lie below the function. This is equivalent to the mathematical definition of a concave function. It implies that an agent prefers averages to extremes; this is equivalent to risk aversion. Hence, concavity of the expected utility function is equivalent to risk aversion.

It is often convenient to have a measure of risk aversion. Intuitively, the more concave the expected utility function, the more risk averse the consumer. Thus, we might think we could measure risk aversion by the second derivative of the expected utility function that measures the curvature, or concavity of the function. However, this definition is not invariant to changes in the expected utility function: if we multiply the expected utility function by 2, the consumer’s behaviour doesn’t change, but our proposed measure of risk aversion does.⁸ However, if we normalize the second derivative by dividing by the first, we get a reasonable measure, known as the Arrow-Pratt measure of (absolute) risk aversion:

An important special case is the risk-neutral individual: Risk-neutrality implies that the consumer is indifferent between a sure expected value and the expected utility of it. Formally, this is expressed by a linearly increasing utility function:

The implication of this situation is shown in Figure 10.5. A risk neutral individual is indifferent on whether he or she gets the expected value of a lottery or the lottery itself.

Figure 10.5: Risk neutrality.

10.4.3 The Investor’s Decision

With the utility function toolkit at hand, we can now formulate the investor’s decision problem at date t = 0.

Using the formulations for c₀ in Equation (10-4) and in Equation (10-5), we model the investor by the expected utility function defined over current and future values of consumption:

From Equation (10-4) we know that the future consumption is random as it depends on the future cash flow . Discounting the future by β, 0 ≤ β ≤ 1_t captures impatience, and β is called the subjective discount factor. Note that we use the expectation generator E₀[] to account for uncertain states and their probabilities. In Section 10.5 we will make this formulation more explicit.

Because we do not want to enter a structural discussion of the Central Pricing Equation,⁹ it will suffice to make the standard concavity assumptions about the utility function; that is, the direct utility function u( ) is increasing, reflecting a desire for more consumption, and concave, reflecting the declining marginal value of additional consumption units.

In line with Section 10.4.2, this formalism captures our investor’s impatience and his or her aversion to risk. Both assumptions are enough to derive an optimum characterisation of the investor’s inter-temporal decision.

Our objective is to find the value at time t = 0 of the single future cash flow , t > 0. This future cash flow structure consists of the future price of the instrument at date t, and (possibly) an additional income flow ; that is, .

Assuming that the investor can freely buy or sell as much of the payoff as he or she wishes, how much will he or she buy or sell? The problem is to find the optimal number of assets n* that maximises the inter-temporal utility, Equation (10-6), under the constraint that this investment will decrease the current consumption level in t = 0, c₀. Formally this is

Substituting the constraints into the objective function, and setting the derivative with respect to n equal to zero,¹⁰ we obtain the first-order condition for an optimal consumption and portfolio choice,

The investor continues to buy or sell the instrument until the marginal loss equals the marginal gain. The investor buys more or less of the instrument until this first-order condition holds. Equation (10-6) expresses the condition for an optimum:

• p₀ u′(c₀) is the marginal loss in utility if the investor buys another unit of the instrument
• is the marginal expected increase in (discounted) utility the investor obtains from the extra payoff at t

The marginal utility loss of consuming a little less today and buying a little more of the instrument should equal the marginal utility gain of consuming a little more of the asset’s payoff in the future.

If the price and payoff of the asset do not satisfy this relation, the investor will adapt the amount of the asset he or she holds (i.e., buy more or less of the asset) and, hence, implicitly adjust the amount of consumption.

As c₀ is known, we can say that u′(c₀) = E₀[u′(c₀)] and therefore write

It follows that the asset’s price should equal the expected discounted value of the asset’s payoff, using the investor’s marginal utility to discount the payoff.

You may have noticed that Equation (10-7) is actually very close to the Central Pricing Equation. Indeed, if we define the stochastic discount factor

Then, we again get the basic pricing formula, Equation (10-7):

From Equation (10-9) we see that the discount factor is a function of a random variable , is a random variable itself, and hence follows a distribution. From Equation (10-4) we know that is a function of and so it is . This means that and are correlated. is hence called stochastic (another word for random) because it is not known with certainty at time t = 0. Because of Equation (10-8), also often is called the marginal rate of substitution, the rate at which the investor is willing to substitute consumption at time t for consumption at time t = 0. net.

10.4.4 Multiple Periods

So far, we have analysed the investor’s decision in a two-date (or two-period) context. We pointed out that this setting captures a wide range of practical situations, in particular if we interpret . Using this, we can now formally show that the central pricing equation holds for any two periods of a multi-period model.

Think of the future cash flow , consisting of the future price and, possibly, an income cash flow , all random variables, unknown at date t = 0. This formulation allows us to think of multi-period pricing problems in just two-period contexts.

In situations, however, where a set of future cash flows is prescheduled and hence known at date t = 0, we will want to relate a price to an entire cash flow stream, rather than just to one dividend and next period’s price.¹² Using the consumption-based approach, we extend Equation (10-5) to a multi-period utility function,

Note that, as β⁰ = 1, for T = 1, this is exactly the two-period utility function of Equation (10-4). As with the two-period model, the investor’s first-order condition gives the pricing formula:

which for T = 1 is equivalent to the two-period Central Pricing Equation. An interesting formulation of Equation (10-11) is obtained when using incrementally to derive a series of future cash flows.

We can write p₀ as

Acknowledging and extending the logic for , , … etc., this is equivalent to the formulation in Equation (10-10).

10.5.1 States of Nature and Contingent Claims

We start with formalising the concept of an ex ante unknown “state of nature at date t.” By ex ante we mean from the point of view of any date prior to t.

The “state of nature at date at date t” concept satisfies the definition for a random variable in Section 2.2. We denote this random variable as with a domain set Ω = {ω¹, ω², …, ω^Ω} containing Ω mutually exclusive states of nature that may occur at the future date t. A particular realisation of is ω_t, being one of the elements in Ω; that is, ω_t ∈ Ω. The exact value ω_t is not revealed until date t, hence the time-specific subscript.

In its earliest formulations, contingent claim analysis considered states representing physical conditions of the environment; for example, “rain” or “shine.” But the notion of a state soon became and can be more broadly interpreted as representing other forms of uncertainty in financial markets, such as “issuer default,” “underlying price above/below strike price,” and so on.

The set-up is depicted in Figure 10-6 where, at date t = 0, the future state is unknown and represented by the random variable . At a date t > 0 a specific value from the domain Ω emerges.

Figure 10.6: Two-period (date) example of the random variable “state of nature at date t,” . At date t = 0, its value is unknown, but one of three possible states, either , , or , is revealed at date t.

Given , a contingent claim is defined as a contractual claim on a unit monetary cash flow (e.g., one Euro or one Great Britain Pound). These claims are traded and agreed upon at date t = 0, prior to the realization of state . It can be shown¹⁴ that this trading results in a competitive equilibrium and hence the resulting outcome is stable. At date t, trading stops, the state of nature, ω_t, is revealed, and only those contingent claims that are conditional on the realized state, , are paid out.

In the case where a certain state of nature ω_t materialises at date t > 0, the corresponding contingent claim pays one monetary unit, and zero otherwise. Formally, this is expressed as

Due to their contractual nature, contingent claims may be interpreted as elementary securities. Elementary securities generate a cash flow only in one single state of nature. Contingent claims are elementary securities with the special feature that their cash flow is a unitary value. We will use this important property in the next section.

In his 1953 article, Arrow assumed that there exist precisely as many contingent claims as there are possible states of nature, with states being mutually exclusive¹⁵ so that only one of all possible states can materialise at date t.

In this setting, the Ω-dimensional payoff vectors of contingent claims, each claim paying off 1 monetary unit if and zero in all other states, are linearly independent. The matrix of claim-specific unitary cash flows may be represented by a Ω-dimensional identity matrix

The contingent claims in this particular setting are called Arrow-Debreu securities, .

10.5.2 Contingent Claims and Cash Flows

Contingent Claims Analysis further distinguishes between the unit monetary contingent claims, , and the actual cash flows that are generated if a given state is adopted. Each state ω corresponds to a specific cash flow, x_t(ω), out of a set of possible cash flows X. At date t = 0, the actual cash flow generated at t = 0 is unknown and hence modelled as a random variable .

The various realisations of may be mapped to states of nature . Denote this mapping as . The mapping from to , , is non-reflexive, meaning that for each ω in Ω one may exactly identify a specific x, but not the other way around. It is therefore possible that multiple states of nature result in the same payment! This is illustrated in Table 10.1, where two states, ω¹ and ω³, result in the same cash flow, x¹. Formally, this may be expressed as

Table 10.1: Exemplary mapping from to at date t > 0

State at date t		Cash flow at date t
ω1	→	x1
ω2	→	x2
ω3	→	x1

or, more generic

10.5.3 Contingent Claims and State Prices

From the point of view of the investor, the realisation of different states of nature does not have the same value, and hence the monetary unit resulting from the corresponding claims are not of equal subjective value. How is that?

The usefulness of a good often depends on the circumstances (or “state of nature”) in which it becomes available: An umbrella when it is raining has a different appeal than an umbrella when it is not raining. Another example is the amount of ice cream that can be consumed in hot weather and cold weather. Although the actual ice cream will cost the same, the utility drawn from its consumption is most likely much higher in hot than in cold weather. A more serious example involves health insurance: How much would a million Euros be worth to someone lying in a coma?

As the investor is interested primarily in the utility drawn from consumption levels, it is important to distinguish levels of consumption by the state of nature in which they may be consumed.

This difference in utility is captured by the state price concept, q₀(ω_t), measuring the value the investor attributes to receiving a unit monetary value in a specific state ω_t. It is also the price at date t = 0 of a contingent claim (Arrow-Debreu security), and hence also is called contingent claim price.

With this concept in mind, we can decompose assets promising an uncertain future cash flow into Ω elementary securities, each consisting of x(ω_t) Arrow-Debreu securities . Every such elementary security represents the cash flow received in a specific state and, as utility differs for each state, is separately priced. To see how this works, consider the example in Table 10.2.

Table 10.2: States of nature and corresponding cash flows

In this example, there are two states of nature, Ω = {ω¹, ω²}, and two possible cash flow values X_t = {x¹, x²} = {100, 5} that are contingent according the mapping set out in Table 10.2. As each Arrow-Debreu security yields exactly one Euro in the corresponding state of nature, the value for the investor of receiving the 100 Euros in bad weather is

and the value of receiving the five Euros in good weather is

x_t(ω) denotes the asset’s cash flow in state of nature . Using q₀(ω) and x_t(ω) the decomposition of the asset into a bundle of orthogonal¹⁶ contingent claims, each potentially resulting in a different cash flow, the asset’s price must then equal the value of the contingent claims of which it is a bundle,

10.5.4 Investor Decision under Uncertainty

How should we interpret the notion of a state price and the pricing formula, Equation (10-14), and how is this related to the Central Pricing Equation? Why is one state preferred over another and hence given a higher value by the investor? To find this out, we again take a look at the decision problem of an individual in the consumption-based context, this time explicitly modelling the uncertainty across various states of nature.

From the contingent claim framework we now have an explicit formulation of the future consumption level, , as a function of the state of nature. Start with transforming the budget constraint for the future consumption level by using the mapping from to , .

Next, we adjust the utility function to account for uncertainty in a more explicit way. Denote as the probability as perceived by the investor at date t = 0 that state ω will emerge (i.e., ) at date t. The probabilities π₀(ω)are the investors’ subjective probabilities for the various states. Asset prices are set, after all, by investors’ demands for assets, and those demands are set by investors’ subjective evaluations of the probabilities of various events. Using Equation (10-15) we can formulate the expectation of the future utility as

The investor’s decision problem may then be written as

under the budget constraints

Note that we have proper probabilities for the various state of the lottery and this expectation shows how Equation (10-6) satisfies the expected utility criterion needed for working with additive utility functions in the context of uncertainty.

As in Chapter 4, we insert the restrictions into the objective function to get

The first-order condition for an optimum of this is:¹⁷

which is nothing else than a more detailed formulation of the Central Pricing Equation for the case that the uncertainty surrounding is described by the state of nature random variable .

Further differentiating Equation (10-16) with respect to x_t(ω) provides—in analogy to Equation (10-14)—a structural expression linking the state price q₀(ω) to the marginal utility of the investor:

This illustrates how the state price measures the value of the emergence of a state (and resulting cash flow) for the investor in terms of utility. It also brings along the interpretation of the stochastic discount factor as a set of contingent claim prices, scaled by probabilities. As a result of this interpretation, the combination of discount factor and probability is sometimes called state-price density:

This formulation also shows that the randomness in the discount factor comes from the uncertainty about the future state of nature.

Assuming smooth utility functions, the same logic as in Section 10.4 ensures that state prices q₀(ω_t) are proportional to each investor’s marginal rate of substitution between current and future consumption, that is

The investor’s first-order conditions say that the marginal rate of substitution between states at date t equals the relevant price ratio for the case that π₀(ω¹) = π₀(ω²),

m(ω¹)/m(ω²) gives the rate at which the investor can give up consumption in state 2 in return for consumption in state 1 through purchase and sales of contingent claims.

We learn that the discount factor m is the marginal rate of substitution between date—and state—contingent commodities. That is why it, like c(ω), is a random variable.

Figure 10.7 gives the economics behind this approach to asset pricing. The investor’s first-order conditions in a contingent claims market implies that the marginal rate of substitution equals the discount factor and the contingent claim price ratio.

Figure 10.7: Indifference curve and contingent claim prices.

10.5.5 Special Case I: The Risk-free Rate

An important concept to which we will have much recourse is the risk-free rate, which we can now define and characterise.

Think of an instrument that generates the same cash flow whatever the state of nature; that is, where . We call this instrument the risk-free instrument and the return it generates, the risk-free rate.

The risk-free rate is the ratio of future cash flow to current price, , for an instrument that guarantees (by all means of probability) the same cash flow in all possible states Ω; that is,

Accordingly, is called the net risk-free rate from date 0 to date t and stands for the corresponding gross risk-free rate.

10.5.6 Special Case II: Equivalent Martingale Measures

Another important concept that is fundamental for derivative pricing is the equivalent martingale measures. A martingale is a stochastic process such that at any date t = 0 its conditional expectation for date t > 0 coincides with its current value: . In discrete time (i.e., when time is measured by separable dates), a martingale is a process the value of which at any date is equal to its conditional expectation for one date later.

Price processes, p_t, are in general not martingales, meaning that empirically they usually exhibit a drift or trend. If we recognise that prices are equal to the discounted values of expected future cash flows, it can be demonstrated that in the absence of arbitrage there is a probability measure in which all price processes, appropriately discounted, become martingales. The law of iterated expectation then implies that price plus payoff processes are martingales.

To illustrate this, assume a continuity of dates starting with t = 0, that is, t = {0, 1, 2, 3, …, T}. If we use the risk-free rate from date 0 to t, , to discount future cash flows, we may formulate the Central Pricing Equation as

Note that this formulation presupposes the absence of arbitrage, a market-price generating concept that implies that the correct discount factor for market prices is the risk-free rate. We use this concept repeatedly in this book.

It is also important to note that this approach deviates from the consumption-based theory as it takes away the investor-specific inter-temporal preferences in its quest for statistically convenient, “investor-neutral” valuation approach. This has important consequences, when interpreting valuations that are done using the equivalent martingale measure approach.

Acknowledge further that the series of cash flows generated by the instrument is

With the definition of the uncertain future cash flow at date t, , we can write the price of the instrument being discounted by risk-free rates as

From this formulation we know that

which, using risk free rates, is equivalent to

Hence we can write

implying, in turn, nothing else than . Equation (10-18) shows that the discounted price plus income process is a martingale. The terms on the left are the price processes, the terms on the right are the conditional expectations under the probability measure Q of the expected cash flows discounted with the risk-free rate (cash flow).

It is important to realise that, with equivalent martingale measures, we deviate from the pure Central Pricing Equation that acknowledges investor-specific discount factors and preferences. From a “true price” perspective, using would be justified only if there is no risk inherent to x_t (i.e., ), which is only true for very few instruments.

It is the absence of arbitrage principle that forces market prices to be consistent with Equation (10-18). Always keep in mind, however, that this price may not be the best one reflecting your own economic situation and preferences, but rather the one that you might expect to see in the markets as an equilibrium outcome.

The equivalent martingale measure E^Q[] is a mathematical construct. It illustrates that the concept of arbitrage depends only on the structure of the price and payoff processes and not on the actual probabilities. As we will see later, equivalent martingale measures simplify the computation of the pricing of derivatives in important ways.

The term “equivalent probability measure” comes from the statistical theory: Given a probability measure P the probability measure Q is said to be equivalent to P if both assign probability zero to the same events. An equivalent probability measure Q is an equivalent martingale measure if all price processes discounted will become martingales. More precisely, Q is an equivalent martingale measure if and only if the market value of any trading strategy is a martingale.

10.5.7 Special Case III: Risk-neutral Probabilities

The concept of equivalent martingale measure requires that the stochastic discount factor is equal to the risk-free rate . In this section we will have a closer look at this and the conditions under which this makes sense.

With the risk-free rate, the discount factor is state independent. It is common to assume that , that the subjective discount factor of the risk-neutral investor is equal to the risk-free rate.

Much of the difficulty in interpreting the Central Pricing Equation lies in the (unknown degree of) risk aversion of individual investors over time and between states of nature. How would the central pricing equation look if investors were not risk-averse?

Recall the definition of the state dependent discount factor from Section 10.5.4:

From the discussion of utility functions in Section 10.4.2 we know that if investors are risk-neutral, the first derivative of their utility function is constant for all consumption levels; that is, u′(c₀) = u′(c_t(ω)) ∀ω. As a consequence, u′(c₀)/u′(c_t(ω)) = 1 ∀ω and the discount factor becomes state-independent, equal to the subjective discount factor of a risk-neutral investor, independent of consumption levels:

Another useful relationship may be found if we take the right two expressions and rearrange them to have q₀(ω) = π(ω)β. As and, by its definition as a probability function we have

Consequently, with risk-neutral investors, the expected value of the discount factor is unique across all states and itself equal to the sum of state prices.

For the case of risk-neutral investors, we denote the discount factor as and the risk-neutral probability as . Using this and expanding it we find the definition of the risk-neutral probability in terms of discount factors

are positive, less than or equal to one, and sum to one, so they are a legitimate set of probabilities. Using and , assuming a risk-free asset, we can rewrite the asset pricing formula for risk-neutral investors as

which is the formulation we need to justify the equivalent martingale measure in the previous section; that is, . denotes the conditional expectation with respect to the modified probability distribution.

Bringing back the discount factor and inserting q₀(ω) = π₀(ω)m_0,t(ω) and into the definition of the risk-neutral probability (…) we obtain the transformation from actual (i.e., risk-averse) to risk-neutral probabilities:

We can also think of the discount factor m as the derivative or change of measure from the real probabilities π to the subjective probabilities π*. The risk-neutral probability representation of asset pricing is quite common, especially in derivative pricing where the results are often independent of risk adjustments.

If then we can write the two-dates Central Pricing Equation as