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Erschienen in:

2023 | OriginalPaper | Buchkapitel

9. Derivatives and Corporate Finance

verfasst von : Kuo-Ping Chang

Erschienen in: Corporate Finance: A Systematic Approach

Verlag: Springer Nature Singapore

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Abstract

This chapter discusses how forward and futures contracts and options are related to corporate finance.

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Vorheriges Kapitel Capital Structure in a Perfect Market

Nächstes Kapitel Real Options

Nur mit Berechtigung zugänglich

E.g., Fama and Jensen (1983, p. 328) argue that “the residual risk—the risk of the difference between stochastic inflows of resources and promised payments to agents—is borne by those who contract for the rights to net cash flows. We call these agents the residual claimants or residual risk bearers”.

See also the Robin Hood story in Chap. 2.

This is an example of ‘expectation matters’, i.e., our expectations for the future will determine our current behavior.

Strictly speaking, the Modigliani–Miller first proposition is: In a complete market with no transaction costs and no arbitrage, the market value of the firm is independent of its capital structure.

The literature of derivatives (e.g., Hull (2018, p. 269) and Merton (1973b, p. 183)) incorrectly states the upper and lower bounds of the put options as: $\frac{K}{1+r}\ge p\ge Max\left[\frac{K}{1+r}-{S}_{0}, 0\right]$ and $K\ge P\ge Max\left[K-{S}_{0}, 0\right]$. Hull (2018, p. 270) and Merton (1973b, p. 183) erroneously argue that “the maximum pay-off to a European put is the exercise price, $K$, which occurs if the underlying asset price ${S}_{0}$ is zero”. This argument is wrong because ${S}_{0}=0$ if and only if people believe ${S}_{t}=0, \forall t>0$. Since ${S}_{t}=0, \forall t>0$, is not a random variable, all options do not exist.

The Gordan theory is a corollary of Farkas theory:

Let $\boldsymbol{A}$ be a $m \times n$ matrix and $\boldsymbol{c}\in {R}^{n}$ be a vector. Then, exactly one of the following systems has a solution:

System 1: $\boldsymbol{A}\boldsymbol{x}\ge 0$ and ${\boldsymbol{c}}^{t}\boldsymbol{x}<0$ for some $\boldsymbol{x}\in {R}^{n};$

System 2: ${\boldsymbol{A}}^{t}\boldsymbol{y}=\boldsymbol{c}$ and $\boldsymbol{y}\ge 0$ for some $\boldsymbol{y}\in {R}^{m}$.

The Farkas theory is a corollary of Separating Hyperplane Theory:

Let S be a nonempty, closed convex set in ${R}^{n}$ and $\boldsymbol{y}\notin S$. Then, there exists a nonzero vector $\boldsymbol{z}\in {R}^{n}$ and a scalar $\alpha$ such that ${\boldsymbol{z}}^{t}\boldsymbol{y}<\alpha$ and ${\boldsymbol{z}}^{t}\boldsymbol{x}\ge \alpha$ for each $\boldsymbol{x}\in S$.

${{\varvec{A}}}^{t}{\varvec{p}}=0$ And ${\varvec{p}}$ is a non-zero vector imply the rank of ${{\varvec{A}}}^{t}$, $R({{\varvec{A}}}^{t})$, is less than $m$. Unique solution for (${p}_{1},...,{p}_{m}$) and ${\sum }_{i=1}^{m}{p}_{i}=1$ imply $R({{\varvec{A}}}^{t})=m-1$.

Chang (2017) has shown that because an asset’s current price (e.g., ${S}_{0}=\$\mathrm{8,000}$) is determined by people’s expectation of the asset’s future possible payoffs and their probabilities, the probabilities of the Gordan theory derived from ${S}_{0}$ ($\mathrm{e}.\mathrm{g}., \pi \mathrm{ and }1-\pi$ in Eq. (9.15)) are the actual world (not the risk-neutral world) probabilities.

The Black–Scholes-Merton option pricing model is:$c = S_{0} \cdot N\left( {d_{1} } \right) - K \cdot e^{ - rT} \cdot N\left( {d_{2} } \right)$; $p = Ke^{ - rT} \cdot \left[ {1 - N\left( {d_{2} } \right)} \right] - S_{0} \left[ {1 - N\left( {d_{1} } \right)} \right]$,

where ${d}_{1}=\frac{\mathit{ln}(\frac{{S}_{0}}{K})+(r+\frac{{\sigma }^{2}}{2})T}{\sigma \sqrt{T}}$, ${d}_{2}={d}_{1}-\sigma \sqrt{T}$.

The Dupire formula.

Chang (2015, pp. 49-51) has shown that when both u and d change, and let (${S}_{0}u-{S}_{0}d$) be the range, the sign of $\frac{\partial c}{\partial ({S}_{0}u-{S}_{0}d)}=\frac{\partial p}{\partial ({S}_{0}u-{S}_{0}d)}$ could be positive or negative. The Black-Scholes-Merton option pricing model, on the other hand, has: $\frac{\partial c}{\partial \sigma }=\frac{\partial p}{\partial \sigma }>0$, where $\sigma$ is the volatility. Ross (1993, p. 470) and Chang (2014) have shown that with complete market, no transaction costs and no arbitrage, the Black-Scholes-Merton option pricing model has the restriction: $r=\mu +\frac{1}{2}{\sigma }^{2}$.

Under certainty, the example which uses Eq. (2.4) in Chapter 2 also shows the same result: ${r}_{S}={r}_{WACC}={r}_{B}$.

In the Black–Scholes-Merton option pricing model, the p-index is:

$\frac{p}{K} = \frac{p}{{e^{\delta T} S_{0} }} = e^{ - rT} \cdot \left[ {1 - N\left( {d_{2} } \right)} \right] - e^{ - \delta T} \cdot \left[ {1 - N\left( {d_{1} } \right)} \right]$, where $d_{1} = \frac{{\ln \left( {e^{ - \delta T} } \right) + \left( {r + \frac{{\sigma^{2} }}{2}} \right)T}}{\sigma \sqrt T }$ and $d_{2} = d_{1} - \sigma \sqrt T$.

Because $\frac{\partial c}{\partial \sigma }=\frac{\partial p}{\partial \sigma }>0$, higher $\sigma$ means higher risk of the asset’s providing at least $\delta$ rate of return.

This is financial diversification irrelevancy, i.e., it does not add or decrease value.

Also, let the maximum value of the debt at $t=T$ be: $K={S}_{0}^{D}\left(1+{r}_{D}^{max}\right)$ where ${S}_{0}^{D}$ is the market value of the debt at $t=0$. Then the debt’s call option is: $c=0$ and the debt’s put-call parity is: $0=\frac{0}{{S}_{0}^{D}(1+{r}_{D}^{max})}=\left[\frac{1}{1+{r}_{D}^{max}}-\frac{1}{1+r}\right]+\frac{p}{K}$.