11.1.1 Risk-free asset

The risk-free asset is modeled by a (deterministic) process B = {B_k, k = 0, 1, 2, n − 1, n} with B₀ = 1 and where B_k ≥ B_{k − 1}, for each k = 1, 2, ..., n. If r is the constant continuously compounded (annual) interest rate, then

and, if r is the periodically compounded (annual) interest rate, i.e. compounded n times per year, then

In other words,

is independent of k: it is the constant one-period discount factor. Of course, units of h and r must coincide for everything to make sense.

11.1.2 Risky asset

The risky asset is modeled by a stochastic process S = {S_k, k = 0, 1, 2, …, n − 1, n}, where S₀ is a known (deterministic) quantity and

For simplicity, we assume that {U_k, k = 1, 2, …, n − 1, n} are independent and identically distributed random variables taking values in {u, d} with (real-world or actuarial) probabilities {p, 1 − p}. Once again, we assume that d < u and 0 < p < 1.

As before, in order to verify the no-arbitrage assumption, we must make sure that

(11.1.1)

when interest is compounded continuously, or

when interest is compounded at each period. In plain words, one dollar accumulated at the risk-free rate over each period does not earn systematically more (or less) than an equivalent investment in the risky asset.

It should be clear that in this multi-period setup, we restrict ourselves to the simplified version of the binomial tree model as discussed in Chapter 10. Consequently, the n-period binomial tree model is based upon five parameters (r, S₀, u, d, p) and it is recombining. Because the tree is recombining, the 2ⁿ possible trajectories lead to only n + 1 possible prices at time n.

The following illustration shows all possible trajectories of the risky asset price (time series) in a three-step binomial tree:

The sample space and the price of the risky asset

It is sometimes useful to describe the full probability model on which the binomial tree is constructed. For an n-period binomial model, one can choose its sample space Ω_n as follows:

It is then clear that each state of nature ω ∈ Ω_n, that is each scenario, corresponds to a full trajectory of the risky asset S (a path in the tree). For example, if n = 3 as in the previous illustration, then the possible paths and prices at maturity are

Example 11.1.1 Construction of a three-step binomial tree

Assume that, each week, a stock price can increase by 3% or decrease by 2%. If the initial stock price is 50, build the corresponding three-step binomial tree.

For each time point k = 1, 2, 3, the corresponding random variable S_k, modeling the stock price in k weeks from now, can take one of the following values:

where i = 0, 1, …, k. The corresponding tree is

where the highlighted path {50, 51.50, 50.47, 49.46} is a possible realization (time series) for the weekly stock price process S.

As illustrated by the previous trees (with n = 3), after k periods/steps there are k + 1 (different) nodes or, in other words, k + 1 different time-k prices. But overall, after k periods/steps, there are 2^k different price trajectories for the process S = {S_k, k = 0, 1, 2, …, n − 1, n}. One of these particular paths was illustrated in the preceding and the next trees: one upward movement followed by two downward movements.

At maturity (after n periods), the rationale is similar: there are n + 1 different prices for the risky asset but a total of 2ⁿ paths leading to those values. Such a distinction is fundamental when pricing path-dependent derivatives as opposed to simple vanilla derivatives (see Section 11.1.3).

11.1.2.1 Probability distribution

We now look at the probability distribution of the risky asset time-k price, where k = 1, 2, …, n. We can write

where the random variable I_k counts the number of upward movements in the trajectory after k periods. Mathematically, it is defined as

If there are i upward movements after k time steps, then

where i can be as little as 0 but at most equal to k (k up-moves in k trials).

It is clear that I_k follows a binomial distribution (with respect to ) with parameters (k, p):

which means that

and consequently

Example 11.1.2 Probability distribution in a three-step binomial tree

We follow example 11.1.1 and we add that the probability of an up-move is 58%. We want to determine the probability distribution of the random variables S₁, S₂ and S₃.

The probability distribution of S₁ is simply

Also, we find that

because there are two possible paths leading to a price of 50.47 (up & down, down & up). Moreover, . Finally,

where again the constant 3 comes from the three possible paths leading to the price 51.9841 (same reasoning applies to 49.4606).

11.1.3 Derivatives

We now introduce a derivative (a third asset) in our model which is an asset whose value V is derived from the risky asset S. Therefore, the derivative’s price is modeled by a stochastic process V = {V_k, k = 0, 1, …, n}, where once again V₀ is today’s value and V₁, V₂, …, V_n are (random) future values. It is assumed that the derivative matures at time n which means that V_n is a random variable representing the payoff.

We will consider two types of derivatives:

• simple (vanilla) options, whose final payoff V_n is some function g of the final asset price, i.e. V_n = g(S_n). Examples include standard call and put options, binary and gap options, etc.;
• exotic or path-dependent options, whose final payoff V_n is some function g of the entire trajectory, i.e. V_n = g(S₁, S₂, …S_n). Examples include Asian, lookback and barrier options.

Again, when dealing with call (resp. put) options, we will use C (resp. P) instead of V to represent the option’s price.

11.1.4 Labelling the nodes

The notation of the form v^uu, x^d or s^ud (with superscripts) was appropriate for a one-period or a two-period binomial tree to emphasize on the path taken by the price of the risky asset. Unfortunately, for n large, this notation is not convenient.

Therefore, we will use a more efficient notation relying on the fact that the underlying tree is recombining. The idea is that a node in the tree is uniquely identified by:

1. time;
2. number of upward movements needed to reach this node.

Therefore, for k = 1, 2, …, n and j = 0, 1, …, k, the pair (k, j) uniquely identifies the node at time k reached by a total of j upward movements (and therefore k − j downward movements) after k steps. This notation will be useful to simplify and formalize some, but not all, computations and eventually for coding the binomial tree model with a programming language.

The following tree illustrates how each node is identified in a three-step binomial tree. We see that, for each time point k, nodes are numbered j = 0, 1, …, k, from bottom to top.

To code the tree in a spreadsheet, or in an array-oriented programming language, we can also organize the tree as a table/matrix as illustrated next, where j corresponds to rows and k to columns. Note that nodes are numbered j = 0, 1, …, k from top to bottom which is much more convenient for referencing array cells.

As a first application of this notation, we can define S_k( j) as the realization/value of the random variable S_k corresponding to the j-th node. For a given path, in which we observe j upward movements from time 0 to time k, we define: for each j = 0, 1, …, k,

Simple vanilla options are easier to handle than path-dependent options as there is no need to analyze all possible paths: only the final asset price matters. Therefore, we define V_k( j) as the realization/value of the random variable V_k corresponding to the j-th node.

We have the following relationship at maturity: for each j = 0, 1, …, n,

Our goal, in the next sections, is to determine the derivative’s price for all times and nodes prior to maturity. In other words, we will compute V_k( j), for all k = 0, 1, …, n − 1 and j = 0, 1, …, k.

Moreover, when dealing with call (resp. put) options, we will use C (resp. P) instead of V to represent the option price. In particular, for a call option with exercise price K and maturing after n time steps, we have

whereas for a similar put option, we have

The evolution of the risky asset price and the corresponding derivative price is depicted in the following tree over three periods:

Example 11.1.3 Call option payoff in a general tree

The initial stock price is 32 and it may go up by 6% or down by 4% every period. Let us represent the evolution of the stock price by a three-step binomial tree along with the payoff of an at-the-money call option.

We have S₀ = 32 and S_k = 32 × 1.06 ^j × 0.96^{k − j}, for j = 0, 1, …, k and k = 1, 2, 3. Moreover, the payoff of the call option is

Those values are illustrated in the following tree:

11.1.5 Path-dependent payoffs

Exotic or path-dependent options such as Asian or lookback options cannot be treated as easily in the n-step binomial tree. One has to be very careful with the current notation (labeling of the nodes) because it focuses on the risky asset price at any given node, not taking into account the different paths reaching that node. Unless we rely on advanced techniques,¹ the only choice we have when computing the final payoff of an exotic option is to consider each possible path separately. This is illustrated in the following example.

Example 11.1.4 Payoff of a fixed-strike Asian call option

Using the context of example 11.1.3, we now introduce a fixed-strike Asian call option with strike price K = 31.50. Determine the behavior of the random variable V₃.

Asian options are path-dependent derivatives so we have to be careful and consider all eight possible payoffs (eight possible trajectories) in the three-step binomial tree of example 11.1.3. For this option, we cannot use the algorithmic notation V_k( j). This Asian option has a payoff given by whose realizations/values are computed in the next table. Note that the random variable is based on an arithmetic average.

For example, the path corresponding to udu is highlighted in the following tree:

For this path, the computation of the average goes as follows:

So, the payoff in this scenario is given by

Computations are done the same way for other trajectories.

As we saw in the previous example, pricing exotic options in an n-period binomial tree can be tedious, especially when n is large. Therefore, unless stated otherwise, the rest of this chapter will focus on simple vanilla options, even if most of the upcoming material could apply to path-dependent options with some adaptation to the notation.

11.2.1 Trading strategies/portfolios

We now look at the concept of trading or investment strategies, also called portfolios, of which replicating strategies are a sub-group. Let us first define by Δ_k (resp. Θ_k), the number of units of the risky asset (resp. risk-free asset) that we hold during the k-th period, that is from time k − 1 to time k or during the time interval [k − 1, k). It is important to note that both Δ_k and Θ_k are determined at time k − 1, therefore using the information about the tree available up to that time point. This is illustrated on the following timeline:

The evolution of the number of units held in the risk-free asset and in the risky asset is represented by the corresponding (stochastic) processes Θ = {Θ_k, k = 1, 2, …, n} and Δ = {Δ_k, k = 1, 2, …, n}. Note that the time-index starts at k = 1, since Θ₁ and Δ₁ are the quantities held in the portfolio during the first period (from time 0 to time 1). A trading strategy is given by (Θ, Δ).

An investment strategy (Θ, Δ) is said to be static if Θ_k and Δ_k are constant over time, that is

This is also known as a buy-and-hold strategy. Otherwise, when the strategy (Θ, Δ) is updated periodically as asset prices evolve, the strategy is said to be dynamic.

Example 11.2.1 Static and dynamic investment strategies

We know from Chapter 3 that to replicate a forward contract we need to hold one unit of stock and borrow the present value of K. Therefore, using the previously defined notation, this trading strategy is given by

and

This is a static strategy.

An example of a dynamic investment strategy is when we update our portfolio to maintain a fixed proportion of, say, 60%–40% (of the portfolio value) in stocks and bonds. Suppose a stock and a bond trade at $100 each and we want/need to invest $10,000. Therefore, at time 0, we choose Δ₁ = 60 and Θ₁ = 40, for a total investment of

After one period, if for example the stock price is worth 150 and the bond 110, then the portfolio will be worth 60 × 150 + 40 × 110 = 13400. However, the position in the stock is worth 9000, which corresponds to 9000/13400 = 67.16% of the portfolio value at time 1. To maintain, our 60–40 strategy, we need to sell some shares of stock and buy more bonds, so that we ultimately set

and

In other words, we sell 60 − 53.6 shares of stock and buy 48.73 − 40 units of the bond. Those are the quantities set up at time 1 and prevailing during the second period. Because the portfolio has been rebalanced following the information observed at time 1, we say the investment strategy is dynamic.

Similar trades would need to be performed in the future, depending on the values of the basic assets at those times.

11.2.2 Portfolio value process

Let us now define by Π_k the value of a trading strategy at time k (before rebalancing). Thus, for each k = 1, 2, …, n, set

(11.2.1)

This portfolio value is computed before rebalancing takes place. In particular, Π_n = Θ_nB_n + Δ_nS_n is the value of the portfolio at maturity, a time where no rebalancing is needed.

At time k, based upon asset prices B_k and S_k, we will choose (Θ_{k + 1}, Δ_{k + 1}) for the upcoming period, i.e. the (k + 1)-th period (from time k to time k + 1). This is the rebalancing procedure.

To complete the definition of the portfolio value process, started in equation (11.2.1), we define the portfolio initial value as follows:

It is the value of the portfolio at time 0 (inception).

The notation for trading strategies and its corresponding portfolio value is complicated by the fact that the portfolio quantities Θ_k and Δ_k are set up at time k − 1 and remain unchanged during the whole k-th period. They affect the portfolio value at both time k − 1 (after rebalancing) and time k (before rebalancing).

Moreover, we can see that the portfolio value at time k, that is Π_k, depends on the risky asset price, so it is also a random variable. Therefore,

is a stochastic process and it is often called the portfolio value process.

Example 11.2.2 Portfolio value process of the 60–40 portfolio

We continue example 11.2.1 where we had S₀ = B₀ = 100. Recall that to hold 60% in stocks and 40% in bonds, we need to set Δ₁ = 60 and Θ₁ = 40 initially, so that

One period later, i.e. at time 1, in the scenario that S₁ = 150 and B₁ = 110, the portfolio would be worth

before any rebalancing takes place.

To maintain the 60–40 proportion (of the new portfolio value of 13, 400), we need to update our portfolio quantities. Based upon the information available at time 1 (in the given scenario), we set Δ₂ = 53.6 and Θ₂ = 48.73 for the second period (from time 1 to time 2).

Finally, at time 2, the portfolio value will be given by

which depends on the realization of the random variable S₂.

11.2.3 Self-financing strategies

A trading strategy (Θ, Δ) is said to be self-financing if, when it is rebalanced, no money is injected or withdrawn: the portfolio value, before and after rebalancing, is equal. Mathematically, this means that, at each time step k = 1, 2, …, n − 1,

(11.2.2)

where the left-hand side is in fact Π_k as given in equation (11.2.1). Thus, a self-financing strategy does not generate any cash flows (in or out) between inception and maturity. Only at maturity or inception does it require/allow money to be injected or withdrawn from it, making it comparable to the cash flows of most derivatives (premium at inception, payoff at maturity, no other cash flow). Note that equation (11.2.2) is an equality of random variables.

The profit or loss during the k-th period, for any investment portfolio, is given by

For a self-financing strategy, using the self-financing condition of equation (11.2.2), we get

(11.2.3)

This provides another interpretation of the self-financing condition: the variation in the portfolio value comes from changes in values from both assets within the portfolio. This interpretation is the one to have in mind when we consider continuous-time models.

Example 11.2.3 Self-financing condition in the 60–40 portfolio

We continue example 11.2.2 and want to verify that the investment strategy satisfies the self-financing condition at time 1, at least in the given scenario.

Indeed, at time 1, in the given scenario, we have

and

So, the self-financing condition is satisfied because the gain/loss of the portfolio corresponds to the sum of the gains/losses made on each asset. Alternatively, still in the given scenario, we have

In other words, before and after rebalancing, the portfolio is worth the same.

11.2.4 Replicating strategy

The main objective now is to determine the initial value V₀ of a derivative having final payoff V_n. The key idea is that we need to set up a (self-financing) trading strategy such that

in each possible scenario. Moreover, if the strategy is self-financing, which means it has no cash flows except at time 0 and time n (just like the derivative we want to replicate), the strategy is called a replicating strategy for this derivative.

By the no-arbitrage assumption, we can conclude that the portfolio process Π mimics the price of the derivative at each time point and in each possible scenario. Consequently, we have

for all k = 0, 1, …, n. In particular, the initial values are equal: V₀ = Π₀. The price of the derivative is equal to the cost of the replicating portfolio.

11.2.5 Backward recursive procedure

Recall that in what follows, we consider only simple options. Using the labeling of nodes presented in section 11.1.4, let

be the realized number of units of each asset an investor holds during the k-th period and in the scenario corresponding to the one-period sub-tree with root (k − 1, j). In other words, if at time k − 1 we have observed so far j upward movements, then during the next period (the k-th), we will hold Θ_k( j) units of the risk-free asset and Δ_k( j) units of the risky asset. Then, the time-(k − 1) value of the corresponding portfolio in this scenario is

(11.2.4)

Graphically, we have

In this case, we emphasize that, for each fixed k = 1, 2, …, n, the possible values for j are 0, 1, …, k − 1.

For k = 0, the only possible value for j is of course 0, which is in agreement with the fact that Θ₁ and Δ₁ are deterministic values, i.e. are fixed at time 0.

Example 11.2.4 Relating the notation from Chapters 9 and 10

In the one-step binomial tree, the investment strategy set up at inception to replicate the realizations v^u and v^d of a payoff V₁ was the pair (x, y). Using this chapter’s notation, we thus have

In the two-step binomial tree, the quantities (x^d, y^d) were chosen at time 1, in the scenario S₁ = s^d, to replicate the values v^du and v^dd, while the quantities (x^u, y^u) were chosen, in the scenario S₁ = s^u, to replicate v^uu and v^ud. With the new notation, in the scenario S₁ = s^d (i.e. no up-move so far), we have

and, in the scenario S₁ = s^u (i.e. one up-move so far), we have

The one-period sub-tree with root at node (k − 1, j) is given by:

In this tree, to replicate the value of the derivative, it should be clear that we need to solve the following system of equations:

where the only two unknowns at this point/stage are Θ_k( j) and Δ_k( j). As in Chapter 10, the solution is

which can be further simplified to obtain

(11.2.5)

Again, to avoid arbitrage opportunities, we must have

(11.2.6)

where Π_{k − 1}( j) is given in equation (11.2.4).

The full n-period binomial tree and replicating strategy for payoff V_n is obtained by pasting together all the one-period sub-trees, i.e. for each k = 1, …, n and j = 0, 1, …, k − 1.

11.2.6 Algorithm

Replication and valuation in the general n-step binomial tree are summarized by the following algorithm:

1. At maturity (time n), in each possible scenario, we must:

   1. compute the payoff for each j = 0, 1, …, n

      

   2. compute the corresponding (replicating) portfolio quantities

      

      as given in (11.2.5) for j = 0, 1, …, n − 1.

   3. compute the value of the replicating portfolio (at time n − 1):

      

   4. compute the value of the derivative:

      

2. Then, we work backward and recursively: for each time k = n − 1, n − 2, …, 1, and in each possible scenario, i.e. for each j = 0, 1, ..., k, we must:

   1. from the previous step, retrieve the derivative’s value V_k( j);

   2. compute the corresponding (replicating) portfolio quantities: for each j = 0, 1, …, k − 1, compute

      

      as given in (11.2.5).

   3. compute the value of the replicating portfolio (at time k − 1):

      

   4. compute the value of the derivative:

      

Note that in the very last step, i.e. for k = 1, we compute (Θ₁(0), Δ₁(0)) and then set V₀(0) = Π₀(0) or, written as before, V₀ = Π₀.

Example 11.2.5 Pricing a put option

Consider a European put option on a stock with spot price 50, maturity 2 years and exercise price 52. Assume the stock price evolves according to a two-period binomial tree, where each time step corresponds to 1 year and during which the price increases or decreases by 20%. In this model, the continuously compounded annual interest rate is 5%. Let us find the replicating portfolio of this option.

We have T = 2, K = 52, S₀ = 50, u = 1.2, d = 0.8, n = 2, h = 1 and r = 0.05. We have graphically:

Using the expressions in (11.2.5), we find the replicating strategies for the second period: in the sub-tree with root (1, 0), i.e. for S₁ = 40,

whereas, in the sub-tree with root (1, 1), i.e. for S₁ = 60,

So, the option price after 1 period (at time k = 1) is given by

During the first period, the portfolio quantities are given by

In conclusion, the initial price of the option is

and more generally we have obtained

Example 11.2.6 Replicating a GMMB in a binomial tree

A GMMB was issued by your insurance company several years ago. It has an annual premium of 2% with a guaranteed minimum payout of 102. As of today, the sub-account balance is $98.74 and the guarantee will apply 3 years from now. Given that the reference stock actually trades for 50 and increases by 7% or decreases by 5% every year, let us describe the replicating strategy covering the loss on the guarantee. Assume the risk-free rate is 3% (annually compounded) and use a binomial tree with annual time steps.

As we did in Chapter 10, replicating the loss on a variable annuity is similar to replicating options. Recall that the payoff is computed on the sub-account (after premiums and withdrawals), whereas replication is performed with the reference asset. Therefore, we will first build the binomial tree for the reference stock index and for the sub-account value. The tree for the reference stock is constructed using

for each j = 0, 1, 2, 3, with S₀ = 50, u = 1.07 and d = 0.95. The evolution of the reference asset is shown in the next table:

Then, the sub-account balance of the policyholder gets credited with the returns of the reference asset whereas premiums are withdrawn. Recursively, as we saw in Chapter 8, we have

which is equivalent to

knowing that A₀ = 98.74. Therefore, the values A_k( j) of the sub-account balance are:

The minimum guaranteed amount is G = 102 and hence the loss for the insurer is computed in the last column as max (102 − A₃, 0). Therefore, the insurance company must implement an investment strategy to replicate the payoff values {22.3213136, 12.2566374, 0.92063368, 0} in the final nodes of the tree. Indeed, when the sub-account balance is $79.68 upon maturity, the company needs to provide for an extra $22.32 in order for the policyholder to receive at least 102.

We will illustrate how to replicate in a single sub-tree. The whole replicating strategy and its cost are provided in the tables below.

For example, the one-period sub-tree with root (2, 0), i.e. at time 2 with no up-move so far, is given by

Using the formulas in (11.2.5), we get

and

Hence,

In order to cover a random loss of either $22.32 or $12.26, the insurance company must have $15.16 to short-sell 1.86 shares of stock and invest 93.34 at the risk-free rate.

Repeating the same procedure for each sub-tree, we can determine the replicating strategy. This is given in the next table.

Finally, the realizations Π_k( j) of the replicating portfolio value process are given in the last table. Note that this also corresponds to the no-arbitrage values of the insurer's loss.

11.5 Exercises