# 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:

1. input data from the financial report into software to find Delta for 1,3,5 year(bottom left section), assuming no dividend

2. 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. • What software is this? I'm in APM466 too. Also do we have to take into account current assets and liabilities or is just assets and liabilities ok? (Current = 1 year) – RYR Apr 8 '16 at 21:39
• www-2.rotman.utoronto.ca/~hull/software I think we are asked to use this software to find out asset volatility given stock volatility – usagi drop Apr 8 '16 at 22:13
• This is what I did: enter stock price as assets, enter strike price as debt, enter time to expiry as 1/3/5, Risk free rate as Treasury rate, 0 dividends. Then, test different values of Volatility until the value you entered equals the value of 0.15*S/V * delta. That value is the implied asset volatility – RYR Apr 8 '16 at 22:44

## 1 Answer

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.

• I think in Merton model, we treat the corporate like an underlying asset, the exercise price = Book value of debt*e^(rt), which is the future debt payment, and option price= Market value of equity. That is all I know. Our purpose is to find Delta using the software, which is the derivative of equity W.R.T asset. The point I got confused is that my prof inputed market value of corporate into the 'stock price' but I don' t know why. Beside, I don't know how to find the market value of asset as well :( – usagi drop Apr 8 '16 at 22:33
• Updated.. I didn't study this, sorry, but I hope the link is helpful to your understanding. – Steinwolfe Apr 9 '16 at 3:19