Abstract
In the the last two chapters we discussed the pricing of call and put options in relation to the price of the underlying stock. In fact, the option pricing method has a much wider field of application: it allows us to evaluate the liabilities of a limited company (corporation) — that is, a company whose shareholders are protected by a limited liability clause — in relation to the price of the assets considered as underlying security. In some cases, it even allows us to determine the value of the assets in relation to certain variables affecting the profitability of the firm.
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Notes to Chapter 8
In practice, when there is a default and therefore bankruptcy, the firm is placed under the control of the court. Two cases are therefore possible: either (i) the court sets up, in agreement with the creditors and shareholders, a plan of reorganization. There is a settlement of the liabilities, a reinjection of funds either by new, or by the old shareholders (there is therefore not necessarily a change of shareholdership) and the firm continues to operate. In this case, the old shareholders do not necessarily lose everything; or (ii) the court decides on a plan of transfer of assets by sale at auction. The firm ceases to exist and the creditors are compensated in order of priority. The balance, if it is positive, is recovered by the shareholders.
In the case of an administrative error by the shareholders, these may be called as fill-in liability even in a limited company.
We are considering here the financial point of view and not the legal point of view, according to which the shareholders are already the owners of the firm and consequently have no need of a call option. All that interests us here is evaluating the shares, and the formal analogy between the shares and a call option is particularly useful in this respect.
To apply the Black-Scholes formula, we must assume that the value of the assets of the firm follows the usual diffusion process, namely dV =αVdt + σVdz.
This is a period rate of return including not only the flows produced by the assets, but also the losses and gains in value. It is not, for example, the internal rate of return. The period return on an asset and its volatility are not easily determined, since the assets are most often not subject to quotation. But the volatility of the assets may be reconstructed from the volatility of the liabilities.
These ‘covenants’ may also be evaluated by options theory. See [Black and Cox 1976] and [Mason and Bhattacharya 1981].
Until recently, in France, debtholders, and in particular bondholders, constituting the ‘bondholding class’, could only call a general meeting to oppose a right of veto on the decisions of the firm. Plowever, this possibility was very rarely used and the debtholders were badly protected against the shareholders’ decisions.
There is no analogy here with the behaviour of a holder of call options on a share offering the prospect of a dividend (see Section 6.4.4 and Sections 7.4.1 and 7.4.2). The option holder may exercise his or her right prematurely. In the present case, the shareholder has no possibility of compensating the debtholders by prematurely paying them the redemption price. The only possible course of action available to the owners of the company is provided by the limited liability clause, in the case where the assets are less than the coupon amount.
This takes us back to the situation considered in Section 8.1.1, where there is no intermediate payment to the benefit of the shareholders or the creditors. However, we must now keep separate the risk of default attached to each category of debt. For the priority creditors, there is no change from the situation analysed in Section 8.1.1: their debt is still guaranteed by all the assets of the firm. On the other hand, the risk borne by the non-priority creditors is greater, since they will only be repaid after the priority creditors have been completely paid off (but nevertheless before the shareholders).
The sale of a call option made simultaneously with the purchase of another call option on the same underlying asset but with a higher exercise price is known to operators as a vertical spread.
See in particular [Black and Cox 1976].
Game theory deals with formalizing conflict situations between rational agents.
This is only accurate for priority debt or when the firm’s debts are all of the same class. We mentioned in Section 8.1.4 that when the firm had issued more than one class of debt with different priorities, the subordinated debts could be more risky than the assets of the firm.
We have already encountered this result in Chapter 6 (Equation 3.4).
Expected return over and above the risk-free interest rate.
We speak here of the ‘legal algebra’ of financial engineering.
z is equal to the ratio of the number of shares issued at the time of complete conversion to the total number of shares after conversion (old shares + new shares): z = (mq)/n + rag), where n is the number of old shares, m the number of convertible bonds and q the number of shares issued per convertible bond in the case of conversion.
This example is adapted from [Cox and Rubinstein 1985], Chapter 7.
This level of volatility is excessive, of course, but it does simplify the calculations.
The reasons why firms choose to issue such securities rather than’ simply’ debt or ordinary shares have given rise to a whole theoretical literature known as signalling theory and agency theory. It is mainly a question of minimizing the potential conflicts between the various partners of the firm as well as overcoming the problems caused by the asymmetry of information (the difference of information between firm managers and investors).
[Cox and Rubinstein 1985] includes a detailed biography (up-to-date in 1985) of the application of options theory to the evaluation of the various financial securities issued by firms. The reader may also consult [Smith 1979].
The discussion on the subject of warrants which follows applies also to convertible bonds.
[Grouhy and Galai 1991a, b, c] also show that the presence of warrants in the liabilities of the balance sheet will affect the volatility of the equity funds. As the volatility is no longer constant, the Black-Scholes formula can no longer be applied to evaluate call options on the shares of such a firm.
The reader interested in these questions is referred to the articles by [Emmanuel 1983], [Gonstantinides 1984] and [Gonstantinides and Rosenthal 1984], which showed, for example, (i) that a monopolist who held all the warrants could have an advantage in exercising the warrants one after another rather than simultaneously and (ii) that if the warrants were shared between many holders acting competitively, they may be led to undertake actions which would be unfavourable to them as a group, that is, which would bring them lower revenues than they could obtain if they co-operated.
It is important to realize that this critique does not so much concern the fundamental concept of the net present value of investments as the method of calculating this NPV.
We speak here of physical options to distinguish them from those exchanged on the financial markets.
For a numerical example, see [Brealey and Myers 1988], pages 496 to 498. For a more theoretical and complete treatment, refer to [McDonald and Siegel 1986]. They show that, in their case, it only becomes optimal to invest from the moment when the gross present value of the project is roughly double the cost. [Brennan and Schwartz 1985], [Trigeorgis and Mason 1987] and [Paddock et al. 1988] also deal with this issue.
A project which we may envisage being abandoned is more flexible than one for which this is impossible such as, for example, an investment undertaken for reasons of safety. This flexibility contributes to the value of the project.
For a detailed treatment, see [Myers and Majd 1983].
See [McDonald and Siegel 1985].
A compound option is an option whose underlying security is itself an option (cf. Chapter 7).
Remember that it is more a question of developing a method of calculation which allows us to take account of flexibility than of questioning the concept of net present value.
For other examples of the application of options theory to investment projects, read the surveys by [Mason and Merton 1985] and [Pindyck 1991].
This defines the rate of extraction of the output q(S,Q,t) as well as three critical values of the price of the output: S 1*(Q,t), which is the price of the output at which the mine is closed, S2*(Q, t), which is the price of the output at which the mine is reopened and S 0*(Q,t), which is the price of the output for which the mine is abandoned. Abandonment is permanent, whereas closure may only be temporary. In this latter case, maintenance costs must be borne. As it is costly to close and then reopen the mine, the value of the mine will be different according to whether it is closed (j = 0) or open (j = 1).
By convenience yield we mean the service flow which is received by the holder of the ore but which would not be received by the holder of a forward contract on the ore. For example, the holder of the ore may choose the location where the goods are stored and the date on which the stock will be liquidated. Brennan and Schwartz introduce the assumption that the convenience yield is proportional to the price of the ore: C(S,t) = cS. This service flow thus corresponds to a continuous dividend at the rate c from which the holder of the ore benefits. It must therefore be included in the equation of dH. (How a dividend paid by the underlying stock is taken into account has already been dealt with in Section 7.3.).
In the Brennan and Schwartz model, this flow K is equal to q(S − A) − M(l − j) − λ jH− T, where A(q,Q,t) is the average monetary cost of production of the output q at the time t and when the stock of ore in the mine is Q; M(t) is the fixed cost after tax of maintaining the mine when it is closed; λj(j = 0,1) is the proportional rate of tax when the mine is closed and opened and T(q, Q, S, t) is the total tax on income levied on the mine when it is in operation. Brennan and Schwartz also introduce the hypothesis that T(q, Q, S, t) = t 1 qS + maxt 2 q[S(1−t 1)−A], 0.
It will be noted that Equation 4.1 is very similar to Equation 3.2 of Chapter 7, which is not surprising since we followed the same reasoning as before. Two additional terms, qHQ and it, appear, however, reflecting the distinctive character of the problem here. Q being the stock of ore contained in the mine, the first term represents the impact of the variation of the stock on the value of the mine (observe that dQ/dt = −q), whilst the second represents the instantaneous cash flows produced by the operation of the mine (c/note 36 above).
The optimization of stochastic processes is outside the scope of this book. A brief presentation of the procedure to follow is given in the appendix to this chapter.
The value of the closure option is represented by the vertical distance separating the curve v from the segment of the broken line.
See note 34 above for the definition of S 1* and S2*.
This formulation assumes that the construction of the mine is instantaneous. The above model may be modified to take account of delays in construction. See on this subject the example in [Siegel et al 1988].
The reader will of course remember that it is also insufficient for the price of the underlying asset to be greater than the exercise price for it to be optimal to exercise the option.
Research and development may be modelled as a chain of interdependent options. The option of abandonment assumes a very special significance in this context.
The optimization of stochastic processes is beyond the scope of this book. The interested reader may consult [Merton 1971], Theorem 1, [Fleming and Rishel 1975], Chapter 6, and [Cox et al. 1978], Lemma 1. As far as we are directly concerned, we need only verify that the solution of this maximization problem gives us V Q=S − A, of which the intuitive interpretation is immediate: at the optimum, the (marginal) value of one additional unit of the ore in the mine, V Q, must be equal to the (marginal) value of one unit mined and sold, S−A.
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Dumas, B., Allaz, B. (1996). Evaluation of the liabilities and assets of a limited company; financial engineering. In: Financial Securities. The Current Issues in Finance Series. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7116-6_8
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