A.1. Additional information on the single period CRR model

In Chapter 5, we saw that the random financial flows, that result from a call may be expressed as follows:

where:

• – is the value of the support at maturity;
• – K is the exercise price.

The flows are random. To evaluate an option at the point in time t = 0, we can adopt two points of view:

• – from the point of view of an eventual buyer of an option, we will examine their updated expectation of gain. In concrete terms, this is a simple question: what sum, C₀, are we ready to invest to obtain a (random) gain that is, in expectation, at least equal to the premium paid to commit to an option?
• – from the writer’s point of view, we can ask what amount to set the premium, C₀, in order to create a hedging portfolio that will make it possible to cover any payment that may need to be made in favor of the holder if the option is in the money at t = T.

The CRR model uses the strategy of using a hedging portfolio and, thus, takes the writer’s point of view. Nonetheless, as we saw in Chapter 5, the estimated value of C₀ is the same whether considered from the holder’s point of view or the writer’s point of view. If this were not so, no option would ever be signed.

A.2. Hedging portfolio

The writer of a call takes on a large risk of maximal loss. This is because if the price of the support increases sharply, they will suffer a loss equal to the difference between the price at maturity, S_T , and the strike price, K. Let us recall that a loss for the writer represents a gain for the buyer as options make up a zero-sum game. We consider the limit case where the writer puts in place a perfect hedge, which means that the total flow of their operation will be zero. They will make no loss but, to balance this, they will also realize no gain! The concept of a perfect hedging portfolio is, in fact, a very powerful analytical tool that makes it possible to establish a formal link between the price dynamic of the support and the price dynamic of the derivative product. Let us consider an option written on a date t = 0 and with a maturity date t = T, with the interval [0, T ] being divided into n periods. At t = 0, the writer will (1) buy a certain quantity of the support, S₀; (2) write a certain number of calls. The principle behind the hedge is simple: if the support increases in value, the writer will “be exercised” upon maturity. However, they will be able to cover this as their portfolio will have increased in value as it contains a certain proportion of the support. The hedge ratio, h, represents the ratio between the quantity of the support and the number of calls. The writer must, thus, resolve two problems: what ratio, h, must they choose to create their hedging portfolio? And what premium, C₀, must they ask for in order to put in place their hedging strategy?

A.2.1. Determining the hedge ratio – h

To answer this question, we set a simple rule: we create a portfolio whose final value will be the same, regardless of whether the support has gained or lost value. While several types of portfolios may be created, we will consider here a portfolio P = {+S; −hC}, where S represents one unit of the support, h is the hedge ratio and C is a call. The “+” sign signifies that we hold one unit of S (we take the long position), while the “−” sign means that we have written h C calls (we have taken the short position). , the value of the portfolio at maturity, can be expressed as follows:

C_up is the value of the call at maturity if the price of the support has increased, multiplied by a factor u > 1. C_down is the value of the call if the price of the support has dropped, multiplied by a factor d ϵ]0; [1. The value of the call is compared to the financial flows it produces.

From this system, we can deduce the following equation:

Hence:

[A.1]

A.2.2. Determining the premium C₀

The hedging portfolio is risk-free and, thus, its return over the period must be equal to r, the risk-free interest rate that prevails in the economy. On the date 0, we buy one unit of the support that costs S₀, but we sell h calls, resulting in hC₀, C₀ being the amount of the premium. The amount invested through the heding portfolio is thus equal to (S₀ − hC₀); this sum, invested with a rate of interest r, must be equal to the final value of the portfolio in case of an increase or a decrease. We thus obtain the equation:

[A.2]

We introduce a value q:

r is a rate of interest, and thus:

• – (1 + r) > d. If this were not the case, all investors would buy S₀ because even in case of a drop, it would give greater returns than the risk-free interest. However, no risky asset can always, in all circumstances, bring in more than the risk-free rate of interest;
• – (1 + r) < u. If this were not the case, no investor would buy S₀. In the case of a favorable change, a risky asset must necessarily yield a greater return than a risk-free rate of interest.

From this double constraint, it can be deduced that 0 < q < 1. By introducing q in [A.2], we obtain:

[A.3]

C₀ thus represents the value of the premium that the writer asks for in order to agree to engaging in an options contract. Formula [A.3] calls for a few basic observations:

1. 1) p is called the “physical” probability, such that and such that . We can see that formula [A.3], which makes it possible to evaluate C₀, does not bring in p, but brings in a value q, which depends on u and d and, therefore, on the amplitude of the rise or drop in value. In other words, the pricing of an option depends on the volatility, not the physical probability;
2. 2) q can be interpreted as a probability. Equation [A.3] compares pricing to calculating an expectation. We say that the probability q is a risk-neutral probability as a risk-neutral operator could use this value to calculate the expected, updated terminal flows. To establish this result, let us consider ρ, the expected return from the asset S₀. We also posit the condition that this return must be equal to the rate of interest r. We look for a probability value, p^∗, such that:

[A.4]

By elaborating and simplifying, we can easily obtain the value of p^∗ fulfilling this condition:

[A.5]

1. 3) we can write a very general formula for evaluating assets with the random flows :

[A.6]

where E_q represents the expectation operator using the probability q, n is the number of periods and r_n is the rate of interest for one period.

A.3. Elements on the CRR model with n periods

We now consider a financial process that is carried out over a time interval [0, ; T ]; this interval is divided into n equal steps. At each step, the asset gains in value if it is multiplied by a value u > 1, or it loses value if it is multiplied by a value d, with 0 < d < 1. The initial value of the asset, S, is denoted by S₀, knowing that S₀ > 0. The recombinant binomial tree given in Figure A.1 represents the initial steps in the evolution of the price of the asset S.

At the end of n periods, the support attains the value S_T :

The CRR model thus makes it possible to determine the set of possible values that the support can take upon maturity. Once this first step is accomplished, we study the terminal flows resulting from a call with the exercise price, K. These terminal flows can then be expressed more simply as:

To estimate C₀, the value of the call at the time 0, we introduce a perfect hedging portfolio. In the case of an n-periods model, an essential point must be highlighted: it must be possible for the hedging strategy to be self-financed, which signifies that the monetary flow must be positive or null for the writer. This property must be verified at each step if the writer has to adjust their hedging portfolio between 0 and T. In order to resolve the problem of how to create the hedging portfolio, we proceed in two successive steps. We first determine h and then evaluate C₀. Formally, we can write, for instance, that the hedging portfolio takes the form P_n = {C_n; h_nS_n} , with C_n being the value of a risk-free asset in the period n. P_n is the value of the portfolio at t = n. For example, at the end of the first period, the equality must be respected, being the value of the portfolio if the price of the support has increased, while is the value of the portfolio if the price has fallen.

This property will be illustrated using the simplest case, where the exercise price, K, is such that uS₀ > K > dS₀. If S₁ = uS₀ > K, the holder exercises their option and the writer must pay a flow that is equal to uS₀ − E. As a counterpart, the support within the portfolio increases in value and makes it possible to generate a plus value that is equal to h(uS₀ − S₀). If S₁ = dS₀ < K, the holder does not exercise their option but the support included in the portfolio loses value equivalent to dS₀ − S₀. The condition makes it possible to determine h. This process then continues between t = 1 and t = 2; if the hedging portfolio has to be adjusted, we must buy (or sell) a certain quantity of the underlying support to sell (or buy) a certain quantity of the derivative product. The process through which the price of the support changes then takes place and any eventual adjustments to the portfolio are carried out. Finally, the premium C₀ must be high enough to make it possible to create an initial hedging portfolio that can then be adjusted with every price change. This dynamically adjusted portfolio must make it possible to cover the random flows to be paid, and this holds good regardless of the price of the support at maturity. The formula that allows us to determine the theoretical value of the premium C₀ is very close to the formula used for the single-period model:

where E_q denotes the mathematical expectation operator, using the probability q, and r_n is the rate of interest for a period when the interval [0,T] is divided into n periods¹. This equation translates a simple principle: the premium C₀ for an option is equal to the mathematical expectation of financial flows (updated on the date 0) that the option may produce on the date of maturity, T, knowing that this expectation is computed using a risk-neutral probability.

This method – which was initially developed for European options – can be extended to the evaluation of any assets that result in random financial flows. Two major hypotheses are brought in. The first concerns the law of random process, while the second is related to the market balance. As concerns the law of random process, it is assumed that we know, in each period, the possible increase – measured by u – and the possible decrease – measured by d. This makes it possible to determine the set of possible values of , the terminal value of the support and, thus, the flows that are produced by the derivative product built on S. The law of random process is given by the parameters u and d; we also know the value of the risk-free rate of interest that prevails in the economy and can, therefore, compute the risk-neutral probability q = (1 + r − d)/(u − d) that we can use to compute the expectation of the final random flows. The second major hypothesis consists of assuming that the market is complete. This means that there is no opportunity for arbitrage. The prices are equilibrium prices.

This generic pricing method gives the theoretical value of the asset that generates random flows and is an invaluable indicator that allows us to answer the following question: what sum, C₀, is reasonable to invest at t = 0 in order to obtain a random flux at t = T? Symmetrically, this expectation gives an important indicator that makes it possible to answer the question: what sum C₀ is reasonable to demand at t = 0 in order to commit to paying a random flow at t = T?

A.4. Risk-neutral probability: some additional insight

The concept of risk-neutral probability is difficult to comprehend. However, we can provide a simple, intuitive approach by considering two investment supports:

• – the first support is a risk-free asset bought at the value 100 at t = 0 and whose value will be 104 at t = T . This means that there does exist a risk-free rate of interest, r;
• – the second support is an asset bought at the value 100 at t = 0 and whose value will be 105, with a probability q or 95 with a probability of (1 − q) at t = T.

We apply the formula to compute q by using the values u = 1.05, d = 0.95 and r = 0.04. We obtain q = 0.9 and we can easily verify that the expectation of gain from the second support is equal to 104. A risk-neutral operator – that is, an operator who only reasons through an expectation of gain – will be equally inclined to invest in both supports as they both have the same expectation. In this example, which is numerically simple, we can see that the parameters used are the terminal values resulting from the random financial instrument (95 and 100) and the risk-free rate of return (r = 0.04); these parameters make it possible to determine the risk-neutral probability without the necessity of knowing the physical probability of an increase, p, or that of a decrease, (1 − p), in the real random asset.

The key to the reasoning is as follows: in a no arbitrage market, all investment supports are perfectly known, which means that we know all the flows that result from these supports. If a support resulted in a higher return than others, then investors would buy it massively and this would cause an increase in its price and, consequently, reduce its returns to such an extent that all expectations of return would become equal. The reasoning is carried out as follows: we assume that the market is equilibrated and that operators are risk-neutral. Risk-neutral operators use risk-neutral probability (tautology). The terminal flows resulting from the investment supports are known as their current prices. We can, thus, determine q by considering that the present prices – which are equilibrium prices – are the expectations computed with q for the terminal flows. Knowing q will then make it possible to price all investment supports that lead to random flows.

A.4.2. Equivalence relationship between the balance price and the risk-neutral probability

A risk-neutral probability q is such that, for any asset j:

where:

• – is the price of the asset j at the present instant;
• – F^j represents the random terminal flows that result from j;
• – r_m denotes the risk-free rate of interest valid for the duration of the update;
• – E_q is the expectation operator with the risk-neutral probability q.

We can accept the following two properties:

• – in a no arbitrage situation (NA), a risk-neutral probability exists;
• – if the market is complete, this probability is unique.

For a pure asset e_i, the equilibrium price is given by the relationship that is valid for any asset:

Moreover, e_i pays a single unit in the state of world ω = i, which is denoted by ω_i and is zero in other states of the world. The probability that ω = ω_i is denoted by Pr(ω = ω_i). We compute the expectation under q of the flows generated by e_i:

In the case of a complete market, we can, thus, establish the link between the price of pure assets and the risk-neutral probability. If, at the instant 0, we create a portfolio that contains one unit of each pure asset, then we must invest a sum S₀ that is equal to . By depositing this sum S₀ at the risk-free rate of interest r_m, we will obtain a terminal flow that is equal to one, because one (and only one) state of the world will be realized and because the pure assets will pay a flow equal to one, while all other pure assets will result in a null flow. Formally, S₀(1 + r_m) = 1 thus, . Hence:

Moreover:

Finally:

[A.7]

Equation [A.7] expresses an equality between the probability of the state of the world and the relative price of the pure asset that will pay a flow equal to one, if this state of world is the one that occurs. In equation [A.7], we establish the ratio between the price of the asset, e_i, and the price of the portfolio that contains a unit of each pure asset. This price ratio is equal to the probability Pr(ω = ω_i). It must be noted that in order for this relationship to be valid, the market must be complete. Two fundamental observations are as follows:

• – this relationship establishes an equivalence between the market prices, , and the risk-neutral probability;
• – the prices, , of the pure assets, e_i, are equilibrium prices, that is, they are the product of a perfect and free functioning of the markets.

Let us recall that the existence of a unique risk-neutral probability, q, is equivalent to the existence of an equilibrium and an absence of arbitrage opportunities, i.e. a complete market situation.