In 1973 Black and Scholes produced their paper on option pricing and the valuation of corporate liabilities. There was, however, a third member of the group that produced this landmark work – Robert Merton whose contribution to modern finance is just as significant.
The major applications of option pricing and the Black Scholes model have been in the analysis of conventional derivative securities, equity options, currency options and so forth. However, the OPM is of much wider significance. It gives us a simple method of establishing the value of a future opportunity rather than one we can exercise immediately. The Black Scholes model does have a number of limiting assumptions but in principal it can be used in any area of decision making where our choices can be deferred. In an important sense it supersets conventional valuation models that assume that the choice to buy, to sell or to hold, or the decision to go or not go ahead, is being made immediately.
So, what Black Scholes and Merton realised was that option pricing gave an economic basis for a more general theory of value. In figure 1 below we show how value is derived.
Share options derive their value from the equities against which they are written. Equities in their turn derive their value from the underlying assets of the business and finally a firm’s assets derive their value from the cash flows against which they are written. The first type of option is the easiest to understand, the last is what we know as a ‘real option’.
But how does the limited liability option work? Limited liability, in principle at least, gives the equity investors the right to walk away from the business if the value of the assets of the business falls below the value of its outstanding liabilities. What the lenders have done is to effectively write an option wherby the equity investors, on the maturity of the debt can purchase the firm's assets for the value of the outstanding liabilities. So, if the firm prospers and the value of its assets rise and remain above the value of the outstanding liabilities then the equity investors claim the difference. If it goes the other way they can hand back the keys to the business and walk away.
In the figure below, the nature of this call option is made clear. Without limited liability the value of the firm to the owner falls to a loss in the event that the value of the assets drops below the liabilities. In effect, with unlimited liability, the equity investor has entered into a forward contract to buy the assets from the lender at the value of the outstanding liability, no matter what the value of those assets might be. However, in the limited liability firm, the loss is limited to the amount the investors paid for their shares – the equivalent of their call option premium.
In option pricing we know that the value of the option has two parts: the intrinsic value which is the difference between the value of the underlying and the exercise price and the time value. Time value is driven by five variables: the value of the underlying and the exercise price, the volatility of the returns on the underlying, the time to exercise and the risk free rate. When looking at the value of the firm those five variables controlling the value of the shares in the equity investors’ hands are:
Value of the underlying = present value of the cash flows generated by the firm’s assets
Exercise price = the value of the firm’s outstanding liabilities
Time to exercise = the term to maturity on those liabilities
Volatility = continuously generated rate of return on the firm’s assets (the asset volatility)
Risk free rate of return over the term of the liabilities.
Suppose a firm had a value of its outstanding assets of $110bn and its liabilities were also valued at $110bn we would immediately infer that the equity value of the business was zero and presumably its share price would also be zero. Not so. Let us assume that the volatility of the assets was 10% per annum (we will return to how this is measured later), the liabilities had a term to maturity of one year and the risk free rate was 5%. Applying the Black Scholes model we come up with d1 and d2 values of 0.55 and 0.45 respectively and a call option value of $7.49bn.
How can this be? The answer is that when the company is ‘at the money’ as in this case the shareholders have the right to enjoy the benefit if the risk of the business works in their favour but can ignore the loss if it goes against them. They have an asymmetric claim on the future value of the business and it is this asymmetry in their claim which gives them the value. Now in practice there are many issues which cloud the picture – most of the assumptions of the Black Scholes model are violated to a greater or lesser degree – but what the option pricing approach is forcing us to do is to focus on the economic value which accrues to limited liability and to identify the crucial variables which drive that value.
The time value of an option is at its maximum when it is ‘at the money’, it diminishes as the option moves deeper into the money or, indeed, in the other direction, deep out of the money. In option pricing we measure proximity to the critical threshold of being at the money through what is referred to as the options ‘gamma’. So a high gamma firm where the time value of the firm is at a maximum and its intrinsic value is close to zero. Conversely, the low gamma firm is the firm which is deep in, or out of the money.
But, what are the high gamma firms? These are the firms that are in their start up phase and have been largely financed by debt (private equity buyouts, management buy outs etc.) or indeed firms going in the other direction towards default. Indeed the example given above are numbers very close to those of Northern Rock, the first British Bank to become a victim of the credit crunch taken under public control in December 2008.
One key point to note is that no matter what type of firm we are dealing with – the equity investors in the high gamma firm will have a high appetite for risk. For a given intrinsic value, time value is positively associated with the volatility of the future returns generated by the assets of the business. With firms where the level of debt is low in proportion to the assets of the business (i.e., their market gearing is low) then the time value of their call option will be of little or no significance. What will matter is the intrinsic value of the business represented by the present value of its future cash flows, the magnitude of which is inversely related to the riskiness of those assets.
This notion that the equity investors in the high gamma firm, where time value dominates, will be risk preferring throws some light on the problems of the banks. Take the Northern Rock case above and with the conservative assumption of 10% volatility workout what the capital requirement should be to effectively eliminate the perverse incentive effect. Now take, HSBC or Barclays where the volatility of the bank’s assets are somewhat higher. A 12% capital adequacy ratio is wholly inadequate to prevent the shareholders banging on the door saying, ‘hey you lot, go for broke and guess what here is a huge incentive package if you do our bidding’.