On Using Risk-Neutral Probabilities to Price Assets

This paper has used the Arbitrage Theorem under binomial case to show that in a complete market with no transaction costs and no arbitrage, for any asset, the current spot price is a function of the risk-free interest rate, the future possible prices and their probabilities. These probabilities are the actual world probabilities, not the so-called risk-neutral probabilities. The paper also proves that for the levered firm, (i) under riskless debt, increasing the debt-equity ratio increases the variance of the rate of return on equity and has no effect on the rate of return on debt; and (ii) under risky debt, increasing the debt-equity ratio increases the variance of the rate of return on debt but does not affect the probability density function of the rate of return on equity. With the actual world probabilities, it can be shown that changes in the debt-equity ratio do not affect the expected rate of return on the equity of the levered firm. These findings refute the Modigliani-Miller second proposition that the expected rate of return on the equity of the levered firm increases in proportion to the debt-equity ratio. With the actual world probabilities, it is also found that increasing the variance of the underlying asset price may either increase or decrease the option prices.