Estimating Equity Return from the Firm-as-Call Option Model

Summary

The document considers whether the required or expected return on equity can be inferred when equity is modeled as a call option on the firm’s assets, even though no corresponding option trades in the market. It presents a calculation that treats equity value as the call value and assumes the firm’s asset value grows at the equity’s continuously compounded return over a chosen horizon.

Under that assumption, the proposed return is the annualized logarithm of the ratio of current firm value to equity value. The response illustrates the calculation with specified inputs for firm value, debt face value, volatility, horizon, and the risk-free rate, and reports an equity call value and implied return for that example. The method depends on the assumed asset value growth relationship and the inputs to the option valuation, including volatility. The document gives no independent validation that this implied rate is a market-required return or an expected return under a particular probability model.

Key ideas

• The approach models equity as a call option on the value of the firm’s assets.
• It derives a continuously compounded equity return from the ratio of firm value to equity value over a chosen horizon.
• The option valuation requires inputs including debt face value, volatility, time to maturity, and the risk-free rate.
• The resulting rate is conditional on the assumed growth relationship and is not independently validated as a market-required return.

Tags

# Is it possible to calculate the equity required (or expected) return using Black-Scholes option pricing model?


# Is it possible to calculate the equity required (or expected) return using Black-Scholes option pricing model?












I know the method of calculating the equity value as a European call option (using Black-scholes formula). My question is: Is it possible to calculate the expected (or required) return of equity when we assume the equity to be a call option on the firms assets?It should be mentioned that no call or put option has been offered in the market.

## Answer by ZRH (score -1)

https://quant.stackexchange.com/a/43810

With $E=C(V,D)$, $V$ being firm value, and $D$ being face value of debt, and $r_{ROE}$ being continuously compounded Return on Equity, the following should hold

$E*e^{r_{ROE}T}=C(V,D)*e^{r_{ROE}T}=V$, and therefore:

$r_{ROE}=\frac{1}{T}ln(\frac{V}{C(V,D)})$

For the sake of making a numerical example, I use $V=100$, $D=80$, $\sigma=20\%$ (determined via index or peer firms with outstanding options), $T=10$, a risk-free rate of $r=1\%$, and get $C(V,D)=30.45$. $r_{ROE}=\frac{1}{10}ln(\frac{100}{30.45})=11.9\%$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.