Inferring Expected Volatility from Equity Prices with Structural Models
Summary
The document asks whether a stock price can imply future volatility in the same way an option price does. The response explains that structural credit models can connect equity value to volatility because equity behaves like a claim on firm assets while debt and default boundaries shape the equity payoff. In that framework, volatility is an input to the model relating firm value, debt, maturity, and the stock price, so it can in principle be inferred from equity and related capital-structure information.
The response points to the relation between corporate bond spreads and option-implied volatility measures as consistent with a shared role for firm risk. It emphasizes substantial practical complications: debt and equity change through payouts, repurchases, issuance, and refinancing, while default costs and information asymmetry are difficult to estimate. It mentions simplified structural models with closed-form solutions, but gives no calibration procedure or evidence establishing a direct equivalence between volatility inferred from stocks and option implied volatility.
Key ideas
- Structural credit models can include volatility as an input to the relationship between firm value and equity price.
- Inferring volatility from equity requires assumptions about debt, maturity, and the default boundary.
- Capital-structure changes and uncertain default costs make practical estimation difficult.
- The document does not show that stock-implied and option-implied volatility are identical.
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Full text
# Answer by Mats Lind (score 2, accepted) # If there was a way to back out implied volatility (IV) from a stock, would it be the same as the IV backed out from an option on that same stock? I know that it is not possible to back out an IV for a stock, because the concept of IV is based on a model with underlying assumptions applied to pricing an option. I was thinking of why IV is forward looking, and I realised that it was because the price of the option reflects all the current available information that investors had at that moment and also their expectations of where the asset price was going. Therefore, the IV backed out would be an expectation of future risk. Therefore, by this logic, shouldn't a stock price reflect all the current available information that investors had at that moment and also their expectations of where the asset price was going? And if an IV could be backed out, shouldn't it reflect its future expected risk? ## Answer by Mats Lind (score 2, accepted) https://quant.stackexchange.com/a/77283 Yes, IV is indeed possible, at least in theory, to back out of stock prices. This lies, I would say, in the core of the so called structural bond models, which, as far as I know, started out with Robert C. Merton's work. It is much the same logic for options: Option price = f1(IV, strike, maturity, Stock Price) as it is for stocks: Stock price = f2(IV, default boundary = g1(default costs, total firm debt,...), maturity, firm value = g2(Bond price, Stock Price)) The fact that IV is a parameter in both the Stock-to-option-price relation and in the Bond-to-stock-price relation shows up in the positive correlation between corporate bond spread indices and IV-measures such as the VIX. However, in practice the approach is complicated by equity and debt values constantly changing with dividends, stock-buybacks, possible equity-issuance, new funding and redemptions. Assymetric information between stakeholders with equity being usually more informed and hard-to-predict default costs makes the whole firm capital structure quite a game to play. Nevertheless, there are simplifications that lead to closed-form solutions such as the Leland 1994 model.
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