# Merton Model

## What is the 'Merton Model'

The Merton model is an analysis model â€“ named after economist Robert C. Merton â€“ that is used to assess the credit risk of a company's debt. Analysts at brokerage firms and investors utilize the Merton model to understand how capable a company is at meeting financial obligations, servicing its debt and weighing the general possibility that the company will go into credit default. This model was later built out by Fischer Black and Myron Scholes to develop the Black-Scholes pricing model.

## BREAKING DOWN 'Merton Model'

Loan officers and stocks analysts utilize the Merton model to analyze a corporation's risk of credit default. This model allows for easier valuation of the company and also helps analysts determine if the company will be able to retain solvency by analyzing maturity dates and debt totals.

## The Black-Scholes Model and the Merton Model

Merton purchased his first stock at age 10 and eventually went to MIT for graduate work. There, he developed and published groundbreaking and precedent-setting ideas to be utilized in the financial world.

Black and Scholes, during Mertonâ€™s time at MIT, developed a critical insight that by hedging an option, systematic risk is removed. Merton then developed a derivative showing that hedging an option would remove all risk. In their 1973 paper, "The Pricing of Options and Corporate Liabilities," Black and Scholes included Merton's report, which explained the derivative of the formula. Merton changed the name of the formula to the Black-Scholes model because he felt it would be pretentious to name something after oneself.

## Understanding the Model

The Merton (or Black-Scholes) model calculates theoretical pricing of European put and call options without considering dividends paid out during the life of the option. The model can, however, be adapted to consider these dividends by calculating the ex-dividend date value of underlying stocks.

The Merton model makes basic assumptions: all options are European and are exercised only at the time of expiration; no dividends are paid out; market movements are unpredictable (efficient markets); no commissions; underlying stocks' volatility and risk-free rate are constant; returns on underlying stocks are regularly distributed. Variables that were taken into consideration in the formula include options strike price, present underlying price, risk-free interest rates and the amount of time before expiration.

The formula for the model is: C = SN(d1)-Ke(-rt)N(d2), where C = Theoretical call premium, S = Current stock price, t = time, K = option striking price, r = risk free interest rate, N = Cumulative standard normal distribution, e = exponential term (2.7183), d1 = ( ln(S/K) + (r + (s2/2))t ) / s_t, d2 = d1 - s_t, s = standard deviation of stock returns. Consider a company's shares sell for \$210.59, stock price volatility is 14.04%, the interest rate is 0.2175%, the strike price is \$205 and expiration time is four days. With the given values, the theoretical call option is negative 8.13.