Inferring Asset Volatility from Equity in the Merton Model
Summary
The document asks how to estimate a company’s asset volatility from its observed equity value and stock volatility using the Merton structural credit model. In this framework, equity is treated as a call option on the firm’s assets, with promised debt repayment acting as the strike at the debt maturity. The firm’s asset value is represented as the combined value of debt and equity, and the option’s sensitivity links equity volatility to asset volatility.
The response clarifies that book equity and market equity are different, and describes the option calculator inputs conceptually: current asset value, asset volatility, debt amount and maturity, with market equity and equity volatility informing the calibration. The questioner’s lecture procedure is to obtain option delta and use it to infer asset volatility, but the exchange gives no completed calculation or validation. The result depends on model assumptions, including the debt representation, maturity, and the stated no-dividend assumption; book financial statement values alone do not determine market asset value or volatility.
Key ideas
- The Merton model views a firm’s equity as a call option on its assets.
- The promised debt payment and its maturity define the option’s strike and expiry in the structural model.
- Asset value is represented as the combined value of equity and debt.
- Equity volatility and option delta are used to relate observed stock volatility to latent asset volatility.
- Book values and market values can differ, and the exchange does not provide a completed volatility estimate.
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Full text
# How to find volatility of Asset given volatility of Stock in Merton model? # How to find volatility of Asset given volatility of Stock in Merton model? I encounter a problem in one of my project to find the 1 year, 2 year and 3 year Asset volatility. We are given 2015 Bell Canada's financial report and a software to do this. The financial report can be found here http://www.bce.ca/investors/financialperformance/annual where asset=47993M, liability=30664M equity=17329M, stock volatility= 15% According to my lecture notes, we should do the following steps: - input data from the financial report into software to find Delta for 1,3,5 year(bottom left section), assuming no dividend - using the formula σSS =σVV*(∂S/∂V), plug in value of delta into ∂S/∂V and find σV for 1, 3, 5 year. However, I do not know what data I should put into the software. In the lecture, I see my prof first uses # of share outstanding* share price=S(equity), and use equity debt ratio to find B(debt) then he put V=B+S(total asset) into the section 'stock price', the given stock volatility into 'volatility', Ke^(rt) (the future debt) into 'exercise price'. Then he compute delta. I don't quite understand why he did this. Can anyone explain to me what kind of data I should use(market value of equity/book value of equity, market value of asset/book value of asset, market value of debt/book value of debt)? I am really confused. Thank you very much. ## Answer by Steinwolfe (score 1, accepted) https://quant.stackexchange.com/a/25314 The delta of an option is the amount the option value will change according to the change in the underlying. The Book value of a company is typically it's assets minus liabilities. This can differ from market value (which is the share price * number of shares outstanding). The picture you provided looks like an option calculator, with inputs: Stock price, Volatility, Risk Free Rate, and Dividends (top left) Option details - time to expiry, strike, call/put (middle left) and outputs of option price, Greeks, and a graph on the right. Edit: I realise now you mean Merton model (credit risk), so... "Equity can be viewed as a European call option on the firm's assets"... and as inputs we need "current value of the company’s assets, the volatility of the company’s assets, the outstanding debt and the debt maturity." Basically from this paper: Let E = value of the firm's Equity, and A = value of assets, and D be debt. Then $E_T = max[A_T – D, 0]$ and we can view equity as a call option on the assets of the firm with strike price equal to the promised debt payment... I think this summary will be of most help to you... "$V(0) = D(0) + E(0)$" - as your professor did. Adding equity and debt to get the firm's asset value.
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