Arbitrage and Valuation of Different Contracts

Abstract

In this chapter, I first derive the Arbitrage Theorem, and use the theorem to show that, in a complete market with no transaction costs and no arbitrage, all securities or assets are derivatives for each other, and they are dependent on each other. It also shows a capital structure irrelevancy proposition: changes in firms’ debt-equity ratios will not affect equityholders’ wealth (welfare), and equityholders’ preferences toward risk (or variance) are irrelevant. When the firm moves from a more certain project to a more uncertain one, the time-0 price of equity will increase, but the time-0 prices of fixed-income assets will decrease. Different labor contractual arrangements will not affect the time-0 price of labor input. When the firm moves from a more certain project to a more uncertain one, the time-0 price of labor input will increase if it is under the share or the mixed contract.

Appendices

Appendix A: Incomplete Market

In incomplete markets , securities may not be replicated, but with no arbitrage (i.e., System 2 of Theorem 5.4 has a solution), they are still priced by the same (which may not be unique) risk neutral probability measure. For example, assume that only two securities (one of them is a money market with interest rate $r = 0.25$) exist in a non-arbitrage, one-period, five states of nature world:

$$ \left\{ {\begin{array}{*{20}l} {\text{Money Market (Security 1):}} \hfill & {S_{0}^{1} = \frac{1}{1 + 0.25}(p_{1} \times 1.25 + p_{2} \times 1.25 + p_{3} \times 1.25 + p_{4} \times 1.25 + p_{5} \times 1.25)} \hfill \\ {\text{Security 2:}} \hfill & {S_{0}^{2} = \frac{1}{1 + 0.25}(p_{1} \times 10 + p_{2} \times 8 + p_{3} \times 4 + p_{4} \times 2 + p_{5} \times 1)} \hfill \\ {} \hfill & {p_{1} ,\;p_{2} ,\;p_{3} ,\;p_{4} ,\;p_{5} \ge 0;\quad \sum\limits_{i = 1}^{5} {p{}_{i} = 1} } \hfill \\ \end{array} } \right. $$

Suppose that there is a new security: Security 3 whose time-1 payoff is: ${\mathbf{c}} = \left[ {\begin{array}{*{20}c} {12.8} \hfill \\ {10.4} \hfill \\ {5.6} \hfill \\ {3.2} \hfill \\ 2 \hfill \\ \end{array} } \right]$. Because c lies in the subspace spanned by ${\mathbf{a}} = \left[ {\begin{array}{*{20}c} {1.25} \hfill \\ {1.25} \hfill \\ {1.25} \hfill \\ {1.25} \hfill \\ {1.25} \hfill \\ \end{array} } \right]$ and ${\mathbf{b}} = \left[ {\begin{array}{*{20}c} {10} \hfill \\ 8 \hfill \\ 4 \hfill \\ 2 \hfill \\ 1 \hfill \\ \end{array} } \right]$ (i.e., ${\mathbf{c}} \in S = \{ \alpha {\mathbf{a}} + \beta {\mathbf{b}}:\;\alpha ,\;\beta \in R^{1} \}$), the time-1 payoff of Security 3 can be replicated by those of Securities 1 and 2:

$$ \left[ {\begin{array}{*{20}c} {12.8} \hfill \\ {10.4} \hfill \\ {5.6} \hfill \\ {3.2} \hfill \\ 2 \hfill \\ \end{array} } \right] = (0.64)\left[ {\begin{array}{*{20}c} {1.25} \hfill \\ {1.25} \hfill \\ {1.25} \hfill \\ {1.25} \hfill \\ {1.25} \hfill \\ \end{array} } \right] + (1.2)\left[ {\begin{array}{*{20}c} {10} \hfill \\ 8 \hfill \\ 4 \hfill \\ 2 \hfill \\ 1 \hfill \\ \end{array} } \right] $$

The time-0 price of Security 3 is: $S_{0}^{3} = (0.64)\;S_{0}^{1} + (1.2)\;S_{0}^{2}$, and with no arbitrage, the three securities are priced by the same risk neutral probability measure:

$$ \left\{ {\begin{array}{*{20}l} {\text{Money Market (Security 1):}} \hfill & {S_{0}^{1} = \frac{1}{1 + 0.25}(p_{1} \times 1.25 + p_{2} \times 1.25 + p_{3} \times 1.25 + p_{4} \times 1.25 + p_{5} \times 1.25)} \hfill \\ {\text{Security 2:}} \hfill & {S_{0}^{2} = \frac{1}{1 + 0.25}(p_{1} \times 10 + p_{2} \times 8 + p_{3} \times 4 + p_{4} \times 2 + p_{5} \times 1)} \hfill \\ {\text{Security 3:}} \hfill & {\begin{array}{*{20}l} {S_{0}^{3} = \frac{1}{1 + 0.25}(p_{1} \times 12.8 + p_{2} \times 10.4 + p_{3} \times 5.6 + p_{4} \times 3.2 + p_{5} \times 2)} \hfill \\ { = (0.64)\;S_{0}^{1} + (1.2)\;S_{0}^{2} } \hfill \\ \end{array} } \hfill \\ {} \hfill & {p_{1} ,\;p_{2} ,\;p_{3} ,\;p_{4} ,\;p_{5} \ge 0;\quad \sum\limits_{i = 1}^{5} {p{}_{i} = 1} } \hfill \\ \end{array} } \right. $$

Suppose that the time-1 payoff of Security 3 is: ${\mathbf{c}}^{\prime } = \left[ {\begin{array}{*{20}c} {11.25} \hfill \\ {9.25} \hfill \\ {5.25} \hfill \\ {3.25} \hfill \\ 2 \hfill \\ \end{array} } \right]$. Because ${\mathbf{c}}^{\prime } \notin S = \{ \alpha \;{\mathbf{a}} + \beta \;{\mathbf{b}}:\;\alpha ,\;\beta \in R^{1} \}$, the time-1 payoff of Security 3 cannot be replicated by those of Securities 1 and 2. But with no arbitrage (i.e., System 2 of Theorem 5.4 has a solution), all the three securities will be priced by the same risk neutral probability measure:

$$ \left\{ {\begin{array}{*{20}l} {\text{Money Market (Security 1):}} \hfill & {S_{0}^{1} = \frac{1}{1 + 0.25}(p_{1} \times 1.25 + p_{2} \times 1.25 + p_{3} \times 1.25 + p_{4} \times 1.25 + p_{5} \times 1.25)} \hfill \\ {\text{Security 2:}} \hfill & {S_{0}^{2} = \frac{1}{1 + 0.25}(p_{1} \times 10 + p_{2} \times 8 + p_{3} \times 4 + p_{4} \times 2 + p_{5} \times 1)} \hfill \\ {\text{Security 3:}} \hfill & {S_{0}^{3\prime } = \frac{1}{1 + 0.25}(p_{1} \times 11.25 + p_{2} \times 9.25 + p_{3} \times 5.25 + p_{4} \times 3.25 + p_{5} \times 2)} \hfill \\ {} \hfill & {p_{1} ,\;p_{2} ,\;p_{3} ,\;p_{4} ,\;p_{5} \ge 0;\quad \sum\limits_{i = 1}^{5} {p{}_{i} = 1} } \hfill \\ \end{array} } \right. $$

where $p_{1} ,\;p_{2} ,\;p_{3} ,\;p_{4} ,\;p_{5}$ may not be unique.

Appendix B: Incomplete Market and Replication of Securities

In an incomplete market , after the firm changes its debt-equity ratio , the equityholders may not be able to create a home-made equity to replicate the time-1 payment of the old equity. For example, assume that only two securities exist (where one of them is a money market with risk-free interest rate $r = 0.25$) in a no-arbitrage, one-period, five states of nature world:

$$ \left\{ {\begin{array}{*{20}l} {\text{Money Market (Security 1):}} \hfill & {S_{0}^{1} = 1 = \frac{1}{1 + 0.25}(1.25 \times p_{1} + 1.25 \times p_{2} + 1.25 \times p_{3} + 1.25 \times p_{4} + 1.25 \times p_{5} )} \hfill \\ {\text{Security 2:}} \hfill & {S_{0}^{2} = 4 = \frac{1}{1 + 0.25}(10 \times p_{1} + 8 \times p_{2} + 4 \times p_{3} + 2 \times p_{4} + 1 \times p_{5} )} \hfill \\ {} \hfill & {p_{1} ,\;p_{2} ,\;p_{3} ,\;p_{4} ,\;p_{5} \ge 0;\quad \sum\limits_{i = 1}^{5} {p{}_{i} = 1} } \hfill \\ \end{array} } \right. $$

(B1)

where the risk neutral probability can be ${\mathbf{p}}^{\prime } = \left[ {\begin{array}{*{20}c} {0.2} \hfill \\ {0.2} \hfill \\ {0.2} \hfill \\ {0.2} \hfill \\ {0.2} \hfill \\ \end{array} } \right]$ or ${\mathbf{p}}^{{\prime \prime }} = \left[ {\begin{array}{*{20}c} {7/90} \hfill \\ {1/5} \hfill \\ {3/5} \hfill \\ {1/10} \hfill \\ {1/45} \hfill \\ \end{array} } \right]$ or others.

Assume that Security 2 is an all-equity firm and it plans to issue a riskless debt, e.g., a debtholder pays $0.8 at time 0 and obtains $1 at time 1:

$$ {\text{By}}\;{\mathbf{p}}^{\prime } ,\;{\text{Security}}\;2{:}\quad \left\{ {\begin{array}{*{20}l} {E_{0}^{{\prime }} = 3.2 = \frac{1}{1 + 0.25}(9 \times 0.2 + 7 \times 0.2 + 3 \times 0.2 + 1 \times 0.2 + 0 \times 0.2)} \hfill \\ {D_{0}^{{\prime }} = 0.8 = \frac{1}{1 + 0.25}(1 \times 0.2 + 1 \times 0.2 + 1 \times 0.2 + 1 \times 0.2 + 1 \times 0.2)} \hfill \\ \end{array} } \right. $$

or,

$$ {\text{by}}\;{\mathbf{p}}^{{\prime \prime }} ,\;{\text{Security}}\;2{:}\quad \left\{ {\begin{array}{*{20}l} {E_{0}^{{\prime \prime }} = 3.2 = \frac{1}{1 + 0.25}\left( {9 \times \frac{7}{90} + 7 \times \frac{1}{5} + 3 \times \frac{3}{5} + 1 \times \frac{1}{10} + 0 \times \frac{1}{45}} \right)} \hfill \\ {D_{0}^{{\prime \prime }} = 0.8 = \frac{1}{1 + 0.25}\left( {1 \times \frac{7}{90} + 1 \times \frac{1}{5} + 1 \times \frac{3}{5} + 1 \times \frac{1}{10} + 1 \times \frac{1}{45}} \right)} \hfill \\ \end{array} } \right. $$

That is, recapitalization through issuing riskless debt does not change the market value of the firm (i.e., the time-0 price of Security 2 is always $4), and the time-0 prices of equity and debt are independent of the risk neutral probability measure used. Also, after the firm issues riskless debt, the equityholder can always create a home-made equity by combining the new equity ($E_{0}^{{\prime }}$ or $E_{0}^{{\prime \prime }}$) with investing $0.8 in the money market, which will give exactly the same time-1 payment of the old equity: ${\mathbf{b}} = \left[ {\begin{array}{*{20}c} {10} \hfill \\ 8 \hfill \\ 4 \hfill \\ 2 \hfill \\ 1 \hfill \\ \end{array} } \right]$.

Suppose that the time-1 payment of the debt is risky: ${\mathbf{d}}_{{\mathbf{1}}} = \left[ {\begin{array}{*{20}c} 3 \hfill \\ 3 \hfill \\ 3 \hfill \\ 2 \hfill \\ 1 \hfill \\ \end{array} } \right].$ Then the time-1 payment of the equity is: ${\mathbf{e}}_{{\mathbf{1}}} = \left[ {\begin{array}{*{20}c} 7 \hfill \\ 5 \hfill \\ 1 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{array} } \right]$. Since ${\mathbf{b}} \notin S = \{ \alpha {\mathbf{a}} + \beta {\mathbf{e}}_{{\mathbf{1}}} :\alpha ,\;\beta \in R^{1} \}$ where ${\mathbf{a}} = \left[ {\begin{array}{*{20}c} {1.25} \hfill \\ {1.25} \hfill \\ {1.25} \hfill \\ {1.25} \hfill \\ {1.25} \hfill \\ \end{array} } \right]$ and ${\mathbf{b}} = \left[ {\begin{array}{*{20}c} {10} \hfill \\ 8 \hfill \\ 4 \hfill \\ 2 \hfill \\ 1 \hfill \\ \end{array} } \right]$, b cannot be replicated by ${\mathbf{a}}$ and ${\mathbf{e}}_{{\mathbf{1}}}$. That is, the equityholder cannot combine the new equity ${\mathbf{e}}_{{\mathbf{1}}}$ with the money market to create a home-made equity to replicate the time-1 payoff of the old equity: ${\mathbf{b}} = \left[ {\begin{array}{*{20}c} {10} \hfill \\ 8 \hfill \\ 4 \hfill \\ 2 \hfill \\ 1 \hfill \\ \end{array} } \right]$. Also, with different risk neutral probability measures :

$$ \begin{aligned} & {\text{By}}\;{\mathbf{p}}^{{\prime }} ,\;{\text{Security}}\;2{:}\quad \left\{ {\begin{array}{*{20}l} {e_{0}^{{\prime }} = 2.08 = \frac{1}{1 + 0.25}(7 \times 0.2 + 5 \times 0.2 + 1 \times 0.2 + 0 \times 0.2 + 0 \times 0.2)} \hfill \\ {d_{0}^{{\prime }} = 1.92 = \frac{1}{1 + 0.25}(3 \times 0.2 + 3 \times 0.2 + 3 \times 0.2 + 2 \times 0.2 + 1 \times 0.2)} \hfill \\ \end{array} } \right. \\ & {\text{By}}\;{\mathbf{p}}^{{\prime \prime }} ,\;{\text{Security}}\;2{:}\quad \left\{ {\begin{array}{*{20}l} {e_{0}^{{\prime \prime }} = 1.71555 = \frac{1}{1 + 0.25}\left( {7 \times \frac{7}{90} + 5 \times \frac{1}{5} + 1 \times \frac{3}{5} + 0 \times \frac{1}{10} + 0 \times \frac{1}{45}} \right)} \hfill \\ {d_{0}^{{\prime \prime }} = 2.28444 = \frac{1}{1 + 0.25}\left( {3 \times \frac{7}{90} + 3 \times \frac{1}{5} + 3 \times \frac{3}{5} + 2 \times \frac{1}{10} + 1 \times \frac{1}{45}} \right)} \hfill \\ \end{array} } \right. \\ \end{aligned} $$

The time-0 price of Security 2 can be the sum of $e_{0}^{{\prime }}$ and $d_{0}^{{\prime \prime }}$ (which equals ${\$ 4} . 3 6 4 4 4> { \$ 4}$), and with no arbitrage,

$$ \left\{ {\begin{array}{*{20}l} {\text{Money Market (Security 1):}} \hfill & {S_{0}^{1} = 1 = \frac{1}{1 + 0.25}(1.25 \times p_{1} + 1.25 \times p_{2} + 1.25 \times p_{3} + 1.25 \times p_{4} + 1.25 \times p_{5} )} \hfill \\ {\text{Security 2:}} \hfill & {\left\{ {\begin{array}{*{20}l} {e_{0}^{{\prime }} = 2.08 = \frac{1}{1 + 0.25}(7 \times p_{1} + 5 \times p_{2} + 1 \times p_{3} + 0 \times p_{4} + 0 \times p_{5} )} \hfill \\ {d_{0}^{{\prime \prime }} = 2.28444 = \frac{1}{1 + 0.25}(3 \times p_{1} + 3 \times p_{2} + 3 \times p_{3} + 2 \times p_{4} + 1 \times p_{5} )} \hfill \\ \end{array} } \right.} \hfill \\ {} \hfill & {p_{1} ,\;p_{2} ,\;p_{3} ,\;p_{4} ,\;p_{5} \ge 0;\quad \sum\limits_{i = 1}^{5} {p{}_{i} = 1} } \hfill \\ \end{array} } \right. $$

where the risk neutral probability can be $\left[ {\begin{array}{*{20}c} {0.12127} \hfill \\ {0.24921} \hfill \\ {0.50507} \hfill \\ {0.10445} \hfill \\ {0.02} \hfill \\ \end{array} } \right]$ or $\left[ {\begin{array}{*{20}c} {0.10984} \hfill \\ {0.25635} \hfill \\ {0.54936} \hfill \\ {0.02445} \hfill \\ {0.06} \hfill \\ \end{array} } \right]$ or others. If this is the case, then with no arbitrage, the time-0 price of Security 2 in Eq. (B1) will be adjusted to $4.36444 in the first place (see also Litzenberger and Sosin 1977; Huang and Litzenberger 1988, pp. 128–129):

$$ \left\{ {\begin{array}{*{20}l} {\text{Money Market (Security 1):}} \hfill & {S_{0}^{1} = 1 = \frac{1}{1 + 0.25}(1.25 \times p_{1} + 1.25 \times p_{2} + 1.25 \times p_{3} + 1.25 \times p_{4} + 1.25 \times p_{5} )} \hfill \\ {\text{Security 2:}} \hfill & {S_{0}^{2} = 4.36444 = \frac{1}{1 + 0.25}(10 \times p_{1} + 8 \times p_{2} + 4 \times p_{3} + 2 \times p_{4} + 1 \times p_{5} )} \hfill \\ {} \hfill & {p_{1} ,\;p_{2} ,\;p_{3} ,\;p_{4} ,\;p_{5} \ge 0;\quad \sum\limits_{i = 1}^{5} {p{}_{i} = 1} } \hfill \\ \end{array} } \right. $$

(B1’)

where $p_{1} ,\,p_{2} ,\,p_{3} ,\,p_{4} ,\,p_{5}$ may not be unique.

Appendix C: More Uncertain Project and the Firm’s Value

In some cases, the time-0 price of a firm may decrease when the firm moves to a more uncertain project. For example, assume a firm exists in a no-arbitrage, one-period, two states of nature world (where the risk-free interest rate is: $r = 0.25$):

That is, the unique risk neutral probability for this world is: $\left[ {\begin{array}{*{20}c} {1/16} \hfill \\ {15/16} \hfill \\ \end{array} } \right]$, and

$$ {\text{Firm}}\;{\text{value}} = \left\{ {\begin{array}{*{20}l} {\text{Equity:}} \hfill & {400 = \frac{1}{1 + 0.25}\left( {\frac{1}{16} \times 875 + \frac{15}{16} \times 475} \right)} \hfill \\ {\text{Debt:}} \hfill & {100 = \frac{1}{1 + 0.25}\left( {\frac{1}{16} \times 125 + \frac{15}{16} \times 125} \right)} \hfill \\ \end{array} } \right. $$

(C1)

Suppose that the firm moves to a more uncertain project, and its time-1 payment is $\left[ {\begin{array}{*{20}c} {2000} \hfill \\ {100} \hfill \\ \end{array} } \right]$ instead of $\left[ {\begin{array}{*{20}c} {1000} \hfill \\ {600} \hfill \\ \end{array} } \right]$. Then, the time-0 prices of the whole firm, the equity, and the debt decrease:

$$ {\text{Firm}}\;{\text{value}} = \left\{ {\begin{array}{*{20}l} {\text{Equity:}} \hfill & {93.75 = \frac{1}{1 + 0.25}\left( {\frac{1}{16} \times 1875 + \frac{15}{16} \times 0} \right)} \hfill \\ {\text{Debt:}} \hfill & {81.25 = \frac{1}{1 + 0.25}\left( {\frac{1}{16} \times 125 + \frac{15}{16} \times 100} \right)} \hfill \\ \end{array}} \right. $$

(C2)

Since no one benefits, the firm will not move to this more uncertain project, and Eq. (C2) cannot hold.