Abstract
Although various option price properties have been derived under the Black-Scholes-Merton and binomial option pricing models, to date only a few model-free option price properties have been developed in the literature. The literature also has not much say about the relationship between derivatives and corporate finance. In this chapter, I use simple arbitrage argument to derive a dozen of model-free option price properties. In addition to deriving the Greeks under model-free framework, it is found that first, in contrast to the traditional view, a European call (put) option for a non-dividend-paying asset can also be a European call (put) option for any other non-dividend-paying asset, and every non-dividend-paying asset is also both a European call option and a European put option for any other non-dividend-paying asset. Second, in some cases the time value of the European put option can be negative, and adjust the exercise price of an option can decrease or even erase the time value of the option. Third, the Modigliani-Miller capital structure irrelevancy proposition is a corollary of the put-call parity. Fourth, each of the firm’s resource is both a European call option and a European put option, and each resource is a stock plus a forward contract.
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Notes
- 1.
If all investors believe that at \( t \in [T/2, \, T] \), \( S_{t} \;\le\; \it{\$} 105 \), then \( S_{T/2} = \it{\$} 105 \) cannot sustain.
- 2.
At \( t = T \), if \( S_{T}\; <\; K \), then the equityholders will not pay K, and the debtholders will have the firm \( S_{T} \).
- 3.
Suppose that a person owns both the equity and debt of the firm. Then, pay more or pay less to the debtholder, i.e., the changes of K, will not affect the market value of the firm \( S_{0} \).
- 4.
Suppose that at \( t = T \), a riskless debt will pay K, and a risky debt (if possible) will also pay K. Then, if at \( t = 0 \), r increases to \( r^{\prime } \), we will have \( \left( {\frac{K}{{1 + r{\prime }}} - \frac{K}{1 + r}} \right) - \left\{ {\left( {\frac{K}{{1 + r{\prime }}} - {p^{\prime }}} \right) - \left( {\frac{K}{1 + r} - p} \right)} \right\} = p^{\prime } - p < 0 \). This is because at \( t = T \) if bad economic situations happen, the riskless debt will still be paying K, but the risky debt will be paying less than K, and rising r will make the riskless debtholders lose more money.
- 5.
See also Property 5.5.4 in Chap. 5.
- 6.
Here c is also the value of a forward contract on the non-dividend-paying underlying asset. Suppose there already exists a forward contract with K as its delivery price. Then, the value of this forward contract f at the current time, i.e., at \( t = 0 \), is \( f = (F_{0} - K)/(1 + r) \), where \( F_{0} \) is the forward price if both parties negotiated at the current time. Combine this equation with \( F_{0} = S_{0} (1 + r) \), the relationship between the forward price and the spot price, we get \( f = S_{0} - K/(1 + r) \).
- 7.
In the Black-Scholes option pricing model, the European call option price is: \( c = S_{0} \cdot \phi (d_{1} ) - K \cdot \text{e}^{ - rT} \cdot \phi (d_{2} )\quad {\text{where}}\quad d_{1} = \frac{{ \ln (\frac{{S_{0} }}{K}) + (r + \frac{{\sigma^{2} }}{2})T}}{\sigma \sqrt T },\;d_{2} = d_{1} - \sigma \sqrt T . \)
If \( K = \it{\$} 10.2 \), then \( S_{0} \) needs to be infinite to make \( c = S_{0} - K \cdot \text{e}^{ - rT} \).
- 8.
Since American call and put options can be exercised on or before the expiration date, their time values must be non-negative.
- 9.
For the multi-period put-call parity, \( c + \frac{K}{{(1 + m)^{T} }} = S_{0} + p \) where \( t = 0, \, 1, \, 2,\ldots, \, T \), and m is the risk-free interest rate in each period. If K is very small relative to \( S_{0} \) and T is short so that at \( t = T \) almost surely \( S_{T} > K \), then \( p = 0 \), and the European call option price is: \( c = S_{0} - \frac{K}{{(1 + m)^{T} }} \), and \( \frac{\partial c}{\partial T} > 0 \). If K is very big relative to \( S_{0} \) and T is short so that at \( t = T \) almost surely \( S_{T} < K \), then \( c = 0 \), and the European put option price is: \( p = \frac{K}{{(1 + m)^{T} }} - S_{0} \), and \( \frac{\partial p}{\partial T} < 0 \).
- 10.
If the equityholder also suffers losses, the firm will not move to the more uncertain project. See also Example 5.4 in Chap. 5.
- 11.
Since each resource is a stock plus a forward contract, it will be meaningless to say that only the equityholder (shareholder) is the owner of the firm, and other resource providers are not. Also, when the firm’s product causes damages to its customers, it will be unfair to ask only the equityholder needs to compensate the customers.
References
Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81:637–654
Knight F (1933) Risk, uncertainty and profit. Reprinted by the University of Chicago Press, Chicago, 1971
Merton R (1973) Theory of rational option pricing. Bell J Econ Manage Sci 4:141–183
Miller M (1988) The Modigliani-Miller propositions: after thirty years. J Econ Perspect 2:99–120
Modigliani F, Miller M (1958) The cost of capital, corporation finance and the theory of investment. Am Econ Rev 48:261–297
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Chang, KP. (2015). Derivatives and the Theory of the Firm. In: The Ownership of the Firm, Corporate Finance, and Derivatives. SpringerBriefs in Finance. Springer, Singapore. https://doi.org/10.1007/978-981-287-353-8_4
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