5 Fundamentals of Arbitrage Pricing

This chapter introduces the change of measure (or change of numeraire or martingale) method for valuing derivative securities. The method is illustrated in a finite-state model and then extended to more general (continuum of states) models.

The pricing and hedging results in this book are not tied to any particular currency. However, for specificity, the discussion will generally be in terms of dollars.

5.1 Fundamental Ideas

The fundamental concept underlying most finance theory, from Miller-Modigliani to the Black Scholes model, is linear pricing. The concept of linear pricing is essentially the same as asking what is the price of five apples and six oranges. Of course, this is the price of one apple times five plus the price of one orange times six. However, instead of pricing bundles of commodities today, we will find prices of dollars in different contigencies at times in the future.

An arbitrage opportunity is a trading strategy that produces nonnegative cash flows at all times in all contingencies with a strictly positive cash flow at some time in some contingency. If such an opportunity were to exist, traders would exploit it until prices adjust to eliminate this possibility. Although simple, this has powerful implications.

Consider a world where there are $J$ possible states of the world at timme $t$. No arbritrage then implies the following properties.

If $X$ and $Y$ are time $t$ random cash flows, the time 0 price today of the random cash flow $aX+bY$ is t $a$ times the time 0 price of $X$ plus $b$ times the time zero price of $Y$. If $P(\cdot)$ is a pricing function that gives us the price of a random cash flow today, then $P(aA+bB)=aP(A) + bP(B)$. In other words, the pricing operator is linear. This follows because if this fails to hold, there is an arbitrage assuming an agent can buy and sell any quantities at these prices. Because the pricing operator is linear, it must have the form $P(X) = \sum_{j=1}^J \pi_j X_j$ where $X_j$ si the random cash flow in state~$j$. We call $\pi_j$ the price of a dollar at time $t$ in state $j$ or a state price.
The price of any positive nonzero random cash flow at time $t$ is strictly positive. Again, this follows from no arbitrage (think how much of an asset paying a positive dividend you would want if it were free or even if you were paid to take the asset). This implies the state prices are strictly positive.
The pricing operator obeys time value of money: $1=\sum_{j=1}^J \pi_j e^{rt}$, where $r$ is the continuously compounded yield to maturity of a bond maturing at $t$. This is the present value rule, in other words, the present value of the future value $e^{rt}$ is one or, equivalently, the present value of a dollar paid at $t$ is $e^{-rt}$.
The pricing operator must be consistent with observable prices. The price $S_0$ of a traded asset with random cash flows at time $t$ is given by $S_0= \sum_{j=1}^J \pi_j S_j$ where $S_j$ is the random cash flow in state $j$.

Important Principle

In the absence of arbitrage opportunities, there exist positive state prices such that the price of any security is the sum across the states of the world of its payoff multiplied by the state price.

This conclusion generalizes to other models, including models in which the stock price takes a continuum of possible values, interpreting “sum” in that case as an integral. We discuss more general models later in this chapter.

We can think of any security as a portfolio of what are called Arrow securities (in recognition of the seminal work of Kenneth Arrow (Arrow 1964)). An Arrow security’s payoff is tied to a specific state of the world, paying $1 at time t if the state in question occurs and paying nothing otherwise. State prices are the prices of these Arrow securities, and, given these state prices, we can value any security. The state prices can be found by inverting the pricing relationship for prices of traded securities; in other words use traded securities to find the prices of securities that pay off a dollar in each time/state contingency. While in general this can be challenging, the procedure is easily illlustrated in the binomial model that we examine next.

Fundamental Ideas in a Simple Setting

In this section we consider the following very simple framework. There is a stock with price $S$ today (which we call date $0$). At the end of some period of time of length $t$, the stock price will take one of two values: either $S_u$ or $S_d$, where $S_u > S_d$. If the stock price equals $S_u$, we say we are in the up state of the world, and, if it equals $S_d$, we say we are in the down state. The stock does not pay a dividend prior to $t$. There is also a risk-free asset earning a continuously compounded rate of interest $r$.

We can relax the assumption that the stock does not pay a dividend prior to $t$. We do that by assuming we are enrolled in a dividend-reinvestment program, so we receive additional shares of the asset instead of receiving cash dividends. If a company does not operate a dividend-reinvestment program, then we can simulate one on our own by buying shares with cash dividends received. In our model, we start with one share at date $0$ with price $S_0$, and we consider the future value – either $S_u$ or $S_d$ – of owning more than one share at date $t$, because the number of shares grows through reinvestment of dividends. We call this future value a “dividend-reinvested asset price” because of this property, and we formulate valuation principles for such assets throughout the book.

Assume \[ \frac{S_u}{S} > \mathrm{e}^{rt} > \frac{S_d}{S}\;. \qquad(5.1)\] This condition means that the rate of return on the stock in the up state is greater than the risk-free rate, and the rate of return on the stock in the down state is less than the risk-free rate. If it were not true, there would be an arbitrage opportunity: if the rate of return on the stock were greater than the risk-free rate in both states, then one should buy an infinite amount of the stock on margin, and conversely if the rate of return on the stock were less than the risk-free rate in both states, then one should short an infinite amount of stock and put the proceeds in the risk-free asset. So what we are assuming is that there are no arbitrage opportunities in the market for the stock and risk-free asset.

State prices

It is a simple matter to solve for state prices in this model. We have two equations and two unknowns.
\[ 1 = \pi_u e^{rt} + \pi_d e^{rt} \] \[ S = \pi_u S_u + \pi_d S_d \]

Solving these equations gives $\pi_u = \frac{S-e^{-rt}S_d}{S_u - S_d}$ and $\pi_d = \frac{e^{-rt}S_u-S}{S_u - S_d}$. Notice the requirement that $\pi_s > 0$ is precisely the no arbitrage condition Equation 5.1 in the previous section.

A somewhat more enlightening description of the state prices is that $\pi_u$ is the price of an Arrow securtity that pays $1$ if the stock price goes up and nothing if the stock price goes down. We can solve for portfolios of the stock and risk-free asset that have these payoffs. Let $A$ be the number of shares of stock and $B$ be the amount borrowed or lent at time zero. Then this portfolio will have a payoff of 1 in the up state and 0 in the down state if \[A S_u + B e^{rt} = 1\] and \[A S_d + B e^{rt} =0\,.\]

Solving these for $A$ and $B$, we get $A= \frac{1}{S_u - S_d}$ and $B = -e^{-rt}\frac{S_d}{S_u - S_d}$. The time zero cost of this portfolio is $AS +B = \frac{S - e^{-rt} S_d}{S_u - S_d} = \pi_u$. A similar exercise reveals $\pi_d$ is the time 0 price of a portfoio that has a value of 0 in the up state and 1 in the down state.

Given the state prices, we can value any derivative security whose payoffs depend on the stock price and the risk free asset. For example, consider a European call option on the stock with maturity $t$ and strike $K$. A call option gives the owner the right (but not the obligation) to buy the stock at the fixed price called the strike. The value of this at the option maturity $t$ is the excess of the stock price over $K$ if the excess is positive and is zero otherwise, which we write as $C_u=\max(0,S_u-K)$ in the up state and $C_d=\max(0,S_d-K)$ in the down state. Therefore, the value of the call option today, $C$, is given by \[C = \pi_u C_u + \pi_d C_d\,.\]

Option Deltas and Replication

A more traditional way to derive option prices is by finding a portfolio that replicates the option payoff. By no arbitrage, the initial cost of the portfolio must be the option price. We now show this is consistent with the state price approach. Again consider a European call option on the stock with maturity $t$ and strike $K$.

The delta of the call option is defined to be the difference between the call values in the up and down states divided by the difference between the underlying values; that is, $\delta = (C_u-C_d)/(S_u-S_d)$. Multiplying by $S_u-S_d$ gives us $\delta(S_u-S_d) = C_u-C_d$ and rearranging yields $\delta S_u - C_u = \delta S_d-C_d$, which is critical to what follows. Consider purchasing $\delta$ shares of the stock at date $0$ and borrowing \[\mathrm{e}^{-rt}(\delta S_u-C_u) = \mathrm{e}^{-rt}(\delta S_d-C_d)\] dollars at date $0$. Then you will owe \[\delta S_u-C_u = \delta S_d-C_d\] dollars at date $t$, and hence the value of the portfolio at date $t$ in the up state will be \[\text{value of delta shares} - \text{dollars owed} = \delta S_u - (\delta S_u-C_u) = C_u\; ,\] and the value of the portfolio at date $t$ in the down state will be \[\text{value of delta shares} - \text{dollars owed} = \delta S_d - (\delta S_d-C_d) = C_d\;\;.\] Thus, this portfolio of buying delta shares and borrowing money (i.e., buying delta shares on margin) replicates the call option. Consequently, the value $C$ of the option at date $0$ must be the date–0 cost of the portfolio; i.e., \[ C = \text{cost of delta shares} - \text{dollars borrowed}\] \[ = \delta S - \mathrm{e}^{-rt}(\delta S_u-C_u)\,. \qquad(5.2)\]

Example

Suppose a stock price today is $S=100$, and suppose that the price at a future date $t$ will be either $S_u=110$ or $S_u=90$. The following figure depicts this situation.

{fig-align=“center”}

Consider a call option with a strike of $K=105$ that matures at $t$. The value of being able to buy a stock at $105$ and sell it for $110$ is $5$, and the value of being able to buy at $105$ and sell at $90$ is zero, so the value of the call option at date $t$ is as shown below. Our goal is to answer the question: What is the call worth at date $0$?

We calculate the option $\delta$ as described above: \[\frac{C_u - C_d}{S_u-S_d} = \frac{5-0}{110-90} = \frac{1}{4}\,.\] The following shows the value of $\delta$ shares at date $0$ and at date $t$.

The critical feature of this figure is that the difference between the up and down values at date $t$ is the same as the difference between the up and down values of the call; that is, it is $5$ dollars. This is guaranteed by our formula for $\delta$. The portfolio of $\delta$ shares is worth more than the call at date $t$, but if we owed $22.50$, then the value of the portfolio after paying back the loan would match the call value. At an interest rate of 5% (as a rate from date $0$ to date $t$ instead of continuous compounding, for simplicity), we could borrow $21.43$ and then owe $22.50$. This is depicted in the following.

If we subtract the previous figure from the figure showing the value of $\delta$ shares, we obtain the figure below. What it means is that if we invest $3.57$ of our own money, and borrow $21.43$, then we can buy $\delta$ shares, and, after paying back the loan with interest, our portfolio value will match the value of the call option. In other words, we can replicate the call option by buying $\delta$ shares of the stock on margin. Because we can create the call option value at date $t$ by investing $3.57$ of our own money at date $0$, the price of the call option at date $0$ should be $3.57$. If the price of the call option were different, then there would be an arbitrage opportunity. This is the calculation that is made in Equation 5.2.

State Prices

We now rewrite the option pricing Equation 5.2 in terms of state prices. By substituting for $\delta$ in Equation 5.2, we can rearrange Equation 5.2 as

\[ C = \frac{S-\mathrm{e}^{-rt}S_d}{S_u-S_d} \times C_u + \frac{\mathrm{e}^{-rt}S_u-S}{S_u-S_d}\times C_d\; . \qquad(5.3)\]

A little algebra also shows that \[ S = \frac{S-\mathrm{e}^{-rt}S_d}{S_u-S_d} \times S_u + \frac{\mathrm{e}^{-rt}S_u-S}{S_u-S_d}\times S_d\; , \qquad(5.4)\]

and \[ 1 = \frac{S-\mathrm{e}^{-rt}S_d}{S_u-S_d} \times \mathrm{e}^{rt}+ \frac{\mathrm{e}^{-rt}S_u-S}{S_u-S_d}\times \mathrm{e}^{rt}\;. \qquad(5.5)\]

It is convenient to denote the factors appearing in these equations as \[ \pi_u = \frac{S-\mathrm{e}^{-rt}S_d}{S_u-S_d} \quad \text{and} \quad \pi_d = \frac{\mathrm{e}^{-rt}S_u-S}{S_u-S_d}\;. \qquad(5.6)\]

The numbers $\pi_u$ and $\pi_d$ are called the state prices, for reasons that will be explained below.

With these definitions, we can write Equation 5.3–Equation 5.5 as

\[ C = \pi_uC_u + \pi_dC_d\;, \qquad(5.7)\]

\[ S = \pi_uS_u+\pi_dS_d\;, \qquad(5.8)\]

\[ 1 = \pi_u\mathrm{e}^{rt} + \pi_d \mathrm{e}^{rt}\;. \qquad(5.9)\]

These equations have the following interpretation: the value of a security today is its value in the up state times $\pi_u$ plus its value in the down state times $\pi_d$. This applies to Equation 5.9 by considering an investment of $1 today in the risk-free asset—it has value 1 today and will have value $\mathrm{e}^{rt}$ in both the up and down states at date $t$. Moreover, this same equation holds for any other derivative asset – for example, a put option – for the same $\pi_u$ and $\pi_d$ defined in Equation 5.6.

We can think of any security as a portfolio of what are called Arrow securities (in recognition of the seminal work of Kenneth Arrow (Arrow 1964)). In this model, one of the Arrow securities pays $1 at date $t$ if the up state occurs and the other pays $1 at date $t$ if the down state occurs. For example, the stock is equivalent to a portfolio consisting of $S_u$ units of the first Arrow security and $S_d$ units of the second, because the stock is worth $S_u$ dollars in the up state and $S_d$ dollars in the down state. Equation 5.7–Equation 5.9 show that $\pi_u$ is the price of the first Arrow security and $\pi_d$ is the price of the second. For example, the right-hand side of Equation 5.8 is the value of the stock at date $0$ viewed as a portfolio of Arrow securities when the Arrow securities have prices $\pi_u$ and $\pi_d$. Because the stock clearly is such a portfolio, its price today must equal the value of that portfolio, which is what Equation 5.8 asserts.

As mentioned before, the prices $\pi_u$ and $\pi_d$ of the Arrow securities are called the state prices, because they are the prices of receiving $1 in the two states of the world. The state prices should be positive, because the payoff of each Arrow security is nonnegative in both states and positive in one. A little algebra shows that the conditions $\pi_u>0$ and $\pi_d>0$ are exactly equivalent to our no-arbitrage assumption Equation 5.1.

5.2 Risk-Neutral Probability

We have our basic pricing formula \[ P(X)= \sum_{s=1}^S \pi_s X_s \] Let $p_s = \pi_s e^{rt}$. Then $\sum_{s=1}^S p_s = 1$ and $p_s>0$. Therefore, we can think of $p_s$ as probabilities. We refer to $p_s$ as risk neutral probabilities. Given this we can write \[P(X)= \sum_{s=1}^S p_s e^{-rt} X_s \] which can be thought of the expected discounted value of the future cash flows. The risk neutral probabilities are so-called since we discount by the risk-free rate. However, the risk neutral probabilities should not be confused with actual probabilities since they are really prices.

Unlike the Capital Asset Pricing Model, for example, there is no risk premium in the discount rate. This is the calculation we would do to price assets under the actual probabilities if investors were risk neutral (or for zero-beta assets). So, we can act as if investors are risk neutral by adjusting the probabilities.¹ Of course, we are not really assuming investors are risk neutral. We have simply embedded any risk premia in the probabilities.

Notice the expected return using the risk neutral probabilities for any investement is the risk free rate: \[\frac{\sum_{s=1}^S p_s X_s}{P(X)} = e^{rt}\], so it is important to realize that the risk neutral probabilities are distinct from the actual probabilities used in portfolio theory.

Risk-Neutral Probability in the Binomial Model

To apply Key Principle $\ref{principle:stateprices}$ in the most convenient way, we manipulate the state prices so we can interpret the sums on the right-hand sides of Equation 5.7–Equation 5.9 in terms of expectations. The expectation (or mean) of a random variable is of course its probability-weighted average value.

Set $p_u = \pi_u\mathrm{e}^{rt}$ for the up state and $p_d = \pi_d\mathrm{e}^{rt}$ for the down state. Equation 5.7–Equation 5.9 can be written as

\[ C = \mathrm{e}^{-rt}[p_uC_u+p_dC_d]\;, \qquad(5.10)\]

\[ S = \mathrm{e}^{-rt}[p_uS_u+p_dS_d]\;, \qquad(5.11)\]

\[ 1= p_u+p_d\;. \qquad(5.12)\]

The numbers $p_u$ and $p_d$ are called the risk-neutral probabilities of the up and down states. The numbers are both positive (because the state prices are positive under our no-arbitrage assumption) and Equation 5.12 states that they sum to one, so it is indeed sensible to consider them as (artificial) probabilities. Equation 5.10 and Equation 5.11 state that the value of a security today is its expected value at date $t$ discounted at the risk-free rate, when we take the expectation using the risk-neutral probabilities. Thus, these are present value formulas.

It follows from Equation 5.11 that the expected return of the stock under the risk-neutral probabilities is the risk-free return; that is, \[\frac{p_uS_u + p_dS_d}{S} = \mathrm{e}^{rt}\,.\] This is a general fact: expected returns relative to the risk-neutral probabilities equal the risk-free return.

5.3 Martingale Pricing

A martingale is a stochastic process for which the expected value of tomorrow’s value is today’s value. In the context of our model discounted values of non-dividend paying trading strategies are martingales, that is if $X(0)$ is today’s value of a portfolio with payoffs random $X$ at time $t$, \[ \frac{X(0))}{R(0))} = \sum_{s=1}^S p_s \frac{X_s}{e^{r t}} = \mathbb{E}^R\left[\frac{X}{R_t}\right] \] where $R(0) = 1$, the initial investment in the risk free asset. In this sense, the discounted value today is the expected discounted discounted value in one period, where the expectation uses the risk neutral probabilities.

’’

Martingales

Equation 5.10 and Equation 5.11 can be written in an equivalent form, which, though somewhat less intuitive, generalizes more readily. First, let’s introduce some notation for the price of the risk-free asset. Considering an investment of $1 today that grows to $\mathrm{e}^{rt}$ at date $t$, it is sensible to take the price today to be $R=1$ and the price in the up and down states at date $t$ to be $R_u=R_d=\mathrm{e}^{rt}$.² In terms of this notation, Equation 5.10–Equation 5.12 can be written as:

\[ \frac{C}{R} = p_u \frac{C_u}{R_u} + p_d \frac{C_d}{R_d}\;, \qquad(5.13)\]

\[ \frac{S}{R} = p_u \frac{S_u}{R_u} + p_d \frac{S_d}{R_d}\;, \qquad(5.14)\]

\[ 1=p_u+p_d\;. \qquad(5.15)\]

Equation 5.13 and Equation 5.14 each state that the price of a security today divided by the price of the risk-free asset equals the expected future value of the same ratio, when we take expectations using the risk-neutral probabilities. In other words, the mean of the date–$t$ value of the ratio is equal to the ratio today.
A variable that changes randomly over time with the expected future value being always equal to the current value is called a martingale. Thus, we have shown that the ratio of the stock price to the price of the risk-free asset is a martingale when using the risk-neutral probabilities.

The ratio of one price to another is the value of the first (numerator) asset when we are using the second (denominator) asset as the numeraire. The term numeraire means a unit of measurement. For example, the ratio $C/R$ is the value of the call when we use the risk-free asset as the unit of measurement: it is the number of units of the risk-free asset for which one call option can be exchanged (to see this, note that $C/S$ shares is worth $C/R \times R = C$ dollars, so $C/R$ units of the risk-free asset is worth the same as one call). Thus, we have shown that asset prices using the risk-free asset as numeraire are martingales relative to the risk-neutral probabilities.

5.4 Change of Numeraire

Our choice of the risk free asset to derive risk neutral probabilities was somewhat arbitrary. In fact any strictly positive non-dividend paying trading strategy can be used as a numeraire. In other words instead of quoting the price of apples and oranges in dollars, we can quote prices in numbers of oranges. One apple is equal to the price of one apple in dollars divided by the price of oranges in dollars and one dollar is equal to one divided by the price of oranges. In our model, instead of measuring payoffs in units of an investment in the risk free asset, we can measure payoffs in different units. We explain this as follows.

The price of any non dividend paying asset or dividend reinvested trading strategy with strictly positive value is \[Y(0) = \sum_{s=1}^S \pi_s Y_s \] Rearranging, we can define probabilities $\text{prob}^Y_s = \frac{\pi_s}{Y(0)}$. Then we have $\sum_{s=1}^S \text{prob}^Y_s =1$ and we can write the price of the risk free asset and any traded security as \[1 = Y(0) \sum_{s=1}^S \text{prob}^Y_s \frac{e^{rt}}{Y_s} = Y(0) \mathbb{E}^Y\left[\frac{e^{rt}}{Y} \right]\] \[S(0) = Y(0) \sum_{s=1}^S \text{prob}^Y \frac{S_s}{Y_s} = Y(0) \mathbb{E}^Y\left[\frac{S}{Y}\right] \] Notice $\frac{S}{Y}$ is a martingale using the probabilities $\text{prob}_s^Y$. The ratio $\frac{S}{Y}$ is the number of shares of $Y$ that security $S$ is worth. Therefore, using the probabilities $p_s^Y$, the value of the number of shares of $Y$ an investment is worth follows a martingale.

Important Principle

If there are no arbitrage opportunities, then for each dividend-reinvested asset, there exists a probability measure such that the ratio of any other dividend-reinvested asset price to the first (numeraire) asset price is a martingale.

An assignment of probabilities to events is called a probability measure, or simply a measure (because it measures the events, in a sense). Thus, we have described two different probability measures: one using the risk-free asset as numeraire and one using the stock as numeraire. The probability measure using the risk-free asset as numeraire defined in the previous section is universally called the risk-neutral probability. When we use one of these probabilities, we commonly say that we are changing numeraires.

We have applied this statement to the risk-free asset, which pays dividends (interest). However, the price $R_s=\mathrm{e}^{rt}$ for all $s=1,\dots,S$, includes the interest, so no interest has been withdrawn—the interest has been reinvested—prior to the maturity $t$ of the option. This is what we mean by a dividend-reinvested asset. In general, we apply the formulas developed in this and the following section to dividend-paying assets by considering the portfolios in which dividends are reinvested.

Other Numeraires in the Binomial Model

Note that the risk-neutral probabilities are the state prices multiplied by the gross return on the risk-free asset. Analogously, define numbers $\text{prob}^S_u = \pi_uS_u/S$ and $\text{prob}^S_d = \pi_dS_d/S$. Substituting for $\pi_u$ and $\pi_d$ in Equation 5.7–Equation 5.9 and continuing to use the notation $R$ for the price of the risk-free asset, we obtain

\[ \frac{C}{S} = \text{prob}^S_u \frac{C_u}{S_u} + \text{prob}^S_d \frac{C_d}{S_d}\;, \qquad(5.16)\]

\[ 1 = \text{prob}^S_u + \text{prob}^S_d\;, \qquad(5.17)\]

\[ \frac{R}{S} = \text{prob}^S_u \frac{R_u}{S_u} + \text{prob}^S_d \frac{R_d}{S_d}\;. \qquad(5.18)\]

Equation 5.17 establishes that we can view the $\text{prob}^S$’s as probabilities (like the risk-neutral probabilities, they are positive because the state prices are positive). Equation 5.16 and Equation 5.18 both state that the ratio of a security price to the price of the stock is a martingale when we use the $\text{prob}^S$’s’ as probabilities. Thus, asset prices using the stock as numeraire are martingales when we use the $\text{prob}^S$ probabilities.

Practical Implementation

For this exposition, it was convenient to first calculate the state prices and then calculate the various probabilities. However, that is not the most efficient way to proceed in most applications. In a typical application, we would view the prices of the stock and risk-free asset in the various states of the world as given, and we would be attempting to compute the value of the call option. Note that t Equation 5.7–Equation 5.9, Equation 5.13–Equation 5.15, and Equation 5.16–Equation 5.18 are all equivalent. In each case we would consider that there are three unknowns—the value $C$ of the call option and either two state prices or two probabilities. In each case the state prices or probabilities can be computed from the last two equations in the set of three equations and then the call value $C$ can be computed from the first equation in the set. All three sets of equations produce the same call value.

In fact, as we will see, it is not even necessary to calculate the probabilities. The fact that ratios of dividend-reinvested asset prices to the numeraire asset price are martingales tell us enough about the probabilities to calculate derivative values without having to calculate the probabilities themselves.

5.5 Stochastic Discount Factor

Up until now, no mention of the true proabilities has been made. However, another pricing formula that is commonly encountered is to use a stochastic discount factor. Denote the actual probability of state $s$ as $\text{prob}_s$.If we define $m_s = \frac{\pi_s}{\text{prob}_s}$, we can write \[ P(X) = \sum_{s=1}^S \text{prob}_s m_s X_s = \mathbb{E}\left[m X\right] \].

We call the random variable $m$ the stochastic discount factor. We can calculate an asset value as the expectation of its future value discounted (multiplied) by the the stochastic factor $m$.

A further decomposition of $m$ is useful. Notice that \[ 1 = \mathbb{E}\left[m e^{rt}\right] \] so in a sense to be made precise later, $m e^{rt}$ is a martingale when we identify the time 0 value of $m$ as 1. Therefore, we can write the price of any asset with time $t$ value $X$ as \[ P(X) = \mathbb{E}\left[m e^{rt} \frac{X}{e^{rt}}\right] = \sum_{s=1}^S \text{prob}_s m_s e^{rt} \frac{X_s}{e^{rt}} =\mathbb{E}^R\left[\frac{X}{e^{rt}}\right]\] Therefore, the risk neutral probabilities are \[p_s = \text{prob}_s m_s e^{rt} \] and the random variable $m e^{rt}$ changes the probability measure from the true probability measure to the risk neutral measure: for any random variable $Z$, $\mathbb{E}[m e^{rt} Z] = \mathbb{E}^R[Z]$.

Stochastic Discount Factor in the Binomial Model

We now consider yet another reformulation of the pricing relations Equation 5.7–Equation 5.9. This formulation generalizes more easily to pricing when there are a continuum of states.
Let $\text{prob}_u$ denote the actual probability of the up state and $\text{prob}_d$ denote the probability of the down state. These probabilities are irrelevant for pricing derivatives in the two-state up-and-down model, but we use them to write the pricing relations Equation 5.7–Equation 5.9 as expectations with respect to the actual probabilities. To do this, we can define \[\begin{align*} m_u &= \frac{\pi_u}{\text{prob}_u}\; ,\\ m_d &= \frac{\pi_d}{\text{prob}_d}\;. \end{align*}\] Then Equation 5.7–Equation 5.9 can be written as

\[ C = \text{prob}_um_uC_u + \text{prob}_dm_dC_d\;, \qquad(5.19)\]

\[ S = \text{prob}_um_uS_u + \text{prob}_dm_dS_d\;, \qquad(5.20)\]

\[ R = \text{prob}_um_uR_u + \text{prob}_dm_dR_d\;. \qquad(5.21)\]

The right-hand sides are expectations with respect to the actual probabilities. For example, the right-hand side of Equation 5.19 is the expectation of the random variable that equals $m_uC_u$ in the up state and $m_dC_d$ in the down state. The risk-neutral probabilities can be calculated from $m_u$ and $m_d$ as $p_u=\text{prob}_um_uR_u/R$ and $p_d=\text{prob}_dm_dR_d/R$. Likewise, the probabilities using the stock as the numeraire can be calculated from $m_u$ and $m_d$ as $\text{prob}^S_u=\text{prob}_um_uS_u/S$ and $\text{prob}^S_d=\text{prob}_dm_dS_d/S$.

The random variable $m$ the stochastic discount factor. Equation 5.20–Equation 5.21 show that we can calculate an asset value as the expectation of its future value discounted (multiplied) by the the stochastic factor $m$.

5.6 More General Models

Now, we drop the assumption that there are a finite number of possible future prices of the stock and allow a general distribution with potentially a continuum of possible values. Denote the future price as $S_t$. Our principle regarding the stochastic discount factor developed in the preceding section can in general be expressed as:³

Important Principle

If there are no arbitrage opportunities, then there exists for each date $t$ a strictly positive random variable $m_t$, called a stochastic discount factor, such that the date–$0$ value of any dividend-reinvested asset with price $P$ is \[ P_0 = \mathbb{E}[m_tP_t]\;. \qquad(5.22)\]

Here, $\mathbb{E}[m_tS_t]$ denotes the expectation of the random variable $m_tS_t$. In a two-state model (or in any model with only a finite number of states of the world), the concept of an expectation is clear: it is just a weighted average of outcomes, the weights being the probabilities. In the two-state model, the right-hand side of Equation 5.20 is the same as the right-hand side of Equation 5.22.⁴

To convert from state prices to probabilities corresponding to different numeraires, we follow the same procedure as at the end of the previous section: we multiply together (i) the probability of the state, (ii) the value of $m_t$ in the state, and (iii) the gross return of the numeraire in the state. If there is a continuum of states, then the actual probability of any individual state is typically zero, so this multiplication produces a zero probability. However, we can nevertheless add up these probabilities to define the probability of any event $A$, an event being a set of states of the world. To do this, let $1_A$ denote the random variable that takes the value 1 when $A$ is true and which is zero otherwise. Then the probability of $A$ using $S$ as the numeraire is defined as \[ \mathbb{E}\left[1_Am_t\frac{S_t}{S_0}\right]\;. \qquad(5.23)\]

This makes sense as a probability because it is nonnegative and because, if $A$ is the set of all states of the world, then its probability is $\mathbb{E}[m_tS_t/S_0]$, which equals one by virtue of Equation 5.22. From Equation 5.23 for the probability of any event $A$, it can be shown that the expectation of any random variable $X$ using $S$ as the numeraire is \[ \mathbb{E}\left[Xm_t\frac{S_t}{S_0}\right]\;. \qquad(5.24)\]

Different numeraires lead to different probability measures and hence to different expectations. To keep this straight, we use the numeraire as a superscript on the expectation symbol: for example, $\mathbb{E}^S$ denotes expectation with respect to the probability measure that corresponds to $S$ being the numeraire. Also, we use the symbol $\text{prob}^S(A)$ to denote the probability of an event $A$ when we use $S$ as the numeraire. So, Equation 5.23 and Equation 5.24 are written as

\[ \text{prob}^S(A) = \mathbb{E}\left[1_Am_t\frac{S_t}{S_0}\right]\;, \qquad(5.25)\]

\[ \mathbb{E}^S[X] = \mathbb{E}\left[Xm_t\frac{S_t}{S_0}\right]\;. \qquad(5.26)\]

Fundamental Pricing Formula

Our key result in the two-state up-and-down example considered earlier was that the ratio of the price of any dividend-reinvested asset to the price of the numeraire asset is not expected to change when we use the probability measure corresponding to the numeraire. We now demonstrate the same result in this more general model. Recall that $t$ denotes an arbitrary but fixed date at which we have defined the probabilities using $S$ as the numeraire in Equation 5.23. At each date $s<t$, let $\mathbb{E}^P_s$ denote the expectation given information at time $s$ and using a dividend-reinvested asset price $P$ as the numeraire (we continue to write the expectation at date $0$ without a subscript; i.e., $\mathbb{E}^S$ has the same meaning as $\mathbb{E}^S_0$). Let $Y$ denote the price of another dividend-reinvested asset. We will show that \[ \frac{Y_s}{P_s} = \mathbb{E}^P_s \left[\frac{Y_t}{P_t}\right]\;. \qquad(5.27)\]

Thus, the expected future (date–$t$) value of the ratio $Y/P$ always equals the current (date–$s$) value when we change probability measures using $P$ as the numeraire. As discussed in the preceding section, the mathematical term for a random variable whose expected future value always equals its current value is martingale. Thus, we can express Equation 5.27 as stating that the ratio $Y/P$ is a martingale when we compute expectations using the probability measure that corresponds to $S$ being the numeraire.

The usefulness of Equation 5.27 is that it gives us a formula for the asset price $Y_s$ at any time $s$—and recall that this formula holds for every dividend-reinvested asset. The formula is obtained from Equation 5.27 by multiplying through by $P_s$:

Important Principle

In the absence of arbitrage opportunities, prices $P$ and $Y$ of dividend-reinvested assets satisfy, for all $s<t$, \[ Y_s = P_s\mathbb{E}^P_s \left[\frac{Y_t}{P_t}\right]\;. \qquad(5.28)\]

We call Equation 5.28 the fundamental pricing formula. It is at the heart of modern pricing of derivative securities. It is a present value relation: the value at time $s$ of the asset with price $Y$ is the expectation, under the appropriate probability measure, of its value $Y_t$ at time $t$ discounted by the (possibly random) factor $P_s/P_t$.

For example, assume the risk free rate is constant. Letting $P_s = R_s$ denote the value $\mathrm{e}^{rs}$ of the risk-free asset and using it as the numeraire, Equation 5.28 becomes \[ Y_s = \mathrm{e}^{rs}\\E^R_s\left[\frac{Y_t}{\mathrm{e}^{rt}}\right] = \mathrm{e}^{-r(t-s)}\mathbb{\mathbb{E}}^R_s [Y_t]\;, \qquad(5.29)\]

which means that the value $Y_s$ is the expected value of $Y_t$ discounted at the risk-free rate for the remaining time $t-s$, when the expectation is computed under the risk-neutral probability.

Notice that this implies $m_t e^{rt}$ is a martingale. This tells us how to change probabilities as follows: for any time $t$ random variable⁵, $Z$, \[E[m_t e^{rt} Z] = E^R[Z] \] Similar remarks also apply to any dividend reinvested trading strategy, $X$, $m_t X_t$ is a martingale. Then \[E[m_t X_t Z] = X(0) E^X[Z]\].

We end this section with a proof of Equation 5.27.⁶

Theory Extra

Consider any time $s<t$ and any event $A$ that is distinguishable by time $s$. Consider the trading strategy of buying one share of the asset with price $Y$ at time $s$ when $A$ has happened and financing this purchase by short selling $Y_s/P_s$ shares of the asset with price $P$. Each share of this asset that you short brings in $P_s$ dollars, so shorting $Y_s/P_s$ shares brings in $Y_s$ dollars, exactly enough to purchase the desired share of the first asset. Hold this portfolio until time $t$ and then liquidate it. Liquidating it generates \[1_A\left(Y_t-\frac{Y_s}{P_s}P_t\right)\] dollars. The multiplication by the random variable $1_A$ is because we only implement this strategy when $A$ occurs (i.e., when $1_A=1$). Consider the security that pays this number of dollars at time $t$. Because we obtained it with a trading strategy that required no investment at any time, its price at time $0$ must be $0$. We already observed that we can represent the price in terms of state prices, so we conclude that \[\mathbb{E}\left[m_t1_A\left(Y_t-\frac{Y_s}{P_s}P_t\right)\right] = 0\;\;.\] When we divide by $P_0$, this still equals zero. Factoring $P_t$ outside the parentheses gives \[\mathbb{E}\left[1_A\frac{P_t}{P_0}m_t\left(\frac{Y_t}{P_t}-\frac{Y_s}{P_s}\right)\right] = 0\;\;.\] We see from Equation 5.26 for expectations using $P$ as the numeraire that we can write this as \[\mathbb{E}^P\left[1_A\left(\frac{Y_t}{P_t}-\frac{Y_s}{P_s}\right)\right]=0\;.\] This is true for any event $A$ distinguishable at time $s$, so the expectation of $Y_t/P_t-Y_s/P_s$ must be zero given any information at time $s$ when we use $P$ as the numeraire; i.e., \[\mathbb{E}^P_s\left[\frac{Y_t}{P_t}-\frac{Y_s}{P_s}\right]=0\; ,\] or, equivalently \[\mathbb{E}^P_s\left[\frac{Y_t}{P_t}\right] = \frac{Y_s}{P_s}\;.\]

5.7 Multiple Cash Flows

Our valuation formulae work for dividend paying assets or trading strategies that make periodic payments. We simply sum up the value of the individual cash flows. If a stock pays dividends $D_t$ up to and including time $T$ and the stock price at time $T$ is $S(T)$ then the stock price is given by \[S(0)= \mathbb{E}\left[\sum_{t=1}^T m_t D_t + M_T S(T)\right] \]

5.8 Further Discussion

How do the probabilities of different events change when we change probability measures? Which events become more likely and which become less likely? These are natural questions to ask. Let’s consider the risk-neutral probability assuming the risk free rate is constant. Equation 5.29 can be rearranged as \[\frac{\mathbb{E}^{R}_s[Y_t]}{Y_s} = \mathrm{e}^{r(t-s)}\,.\] This means that the expected return on on the dividend-reinvested asset with price $Y$ is the risk-free rate. Stocks generally have positive risk premia, meaning they are expected to return more than the risk-free rate. Hence, when we shift from the actual probabilities to the risk-neutral probability, we are decreasing expected returns. How does this happen? It is because we increase the probabilities of bad events (low stock returns) and decrease the probabilities of good events (high stock returns).

In Example $\ref{example:binomial}$, it is easy to calculate that the risk-neutral probabilities are $p_u=0.75$ and $p_d=0.25$. To value the call option, we did not need to specifythe actual probabilities, but if the expected return of the stock is higher than the risk-free return – that is, if the stock has a positive risk premium – then it must be the case that $p_u > 0.75$. For example, if the expected return is 8%, then $p_u=0.9$. Thus, if the stock has a positive risk premium, then the risk-neutral probability of the stock falling is higher than the actual probability.

The usual way to value risky assets is to discount expected future values at a risk-adjusted rate that is higher than the risk-free rate. What we have shown is that we can instead value risky assets by adjusting the probabilities and discounting at the risk-free rate. Instead of increasing the discount rate relative to the risk-free rate, we increase the probabilities of bad events. These are alternative ways of valuing risky assets.

5.9 Exercises

Exercise 5.1 Create a Python function in which the user inputs $S$, $S_d$, $S_u$, $K$, $r$ and $t$. Check that the no-arbitrage condition Equation 5.1 is satisfied. Compute the value of a call option in each of the following ways:

Compute the delta and use Equation 5.2.
Compute the state prices and use Equation 5.7.
Compute the risk-neutral probabilities and use Equation 5.10.
Compute the probabilities using the stock as numeraire and use Equation 5.16.

Verify that all of these methods produce the same answer.

Exercise 5.2 In a two-state model, a put option is equivalent to $\delta_p$ shares of the stock, where $\delta_p = (P_u-P_d)/(S_u-S_d)$ (this will be negative, meaning a short position) and some money invested in the risk-free asset. Derive the amount of money $x$ that should be invested in the risk-free asset to replicate the put option. The value of the put at date $0$ must be $x+\delta_pS$.

Exercise 5.3 Using the result of the previous exercise, repeat Exercise 5.1} for a put option.

Arrow, K. J. 1964. “The Role of Securities in the Optimal Allocation of Risk Bearing.” Review of Economic Studies 31: 91–96.

Cox, J., and S. Ross. 1976. “The Valuation of Options for Alternative Stochastic Processes.” Journal of Financial Economics 3: 145–66.

Geman, H., N. El Karoui, and J.-C. Rochet. 1995. “Changes of Numeraire, Changes of Probability Measure and Option Pricing.” Journal of Applied Probability 32: 443–58.

Harrison, J. M., and D. Kreps. 1979. “Martingales and Arbitrage in Multiperiod Securities Markets.” Journal of Economic Theory 20: 381–408.

This fundamental idea is due to Cox and Ross (Cox and Ross 1976).↩︎
All of the equations appearing below are also true if instead we take $R=\mathrm{e}^{-rt}$ and $R_u=R_d=1$.↩︎
We have proven this in the two-state model, but we will not prove it in general. As is standard in the literature, we simply adopt it as an assumption. A general proof is in fact difficult and requires a definition of no arbitrage that is considerably more complicated than the simple assumption Equation 5.1 that is sufficient in the two-state up-and-down model.↩︎
In general the expectation (or mean) of a random variable is an intuitive concept, and an intuitive understanding is sufficient for this book, so we will not give a formal definition. It should be understood that we are assuming implicitly, whenever necessary, that the expectation exists (which is not always the case). In this regard, it is useful to note in passing that a product of two random variables $XY$ has a finite mean whenever $X$ and $Y$ have finite variances.↩︎
In other words a random variable whose value is known at time $t$↩︎
The proof is due to Harrison and Kreps (Harrison and Kreps 1979). See also Geman, El Karoui and Rochet (Geman, El Karoui, and Rochet 1995). We omit here technical assumptions regarding the existence of expectations.↩︎