Note that the Black-Scholes formula gives the price of European call c given the

Question

Note that the Black-Scholes formula gives the price of European call c given the time to expiration T, the strike price K, the stock's spot price S_0, the stock's volatility sigma, and the risk-free rate of return r : c = c(T, K, S_0, sigma, r). All the variables but one are "observable," because an investor can quickly observe T, K, S_0, r. The stock volatility, however, is not observable. Rather it relies on the choice of models the investor uses. The price of the option, c, if traded, is observable. So we can flip the problem around. Given observables T, K, S_0, r and c, what volatility sigma should the stock have in order for the Black-Scholes formula to be correct. This is called the implied volatility, sigma_BS. Some calculus, shows that sigma_BS exists and is unique. The current spot price is $40, the expected rate of return of the stock is 8%, the risk-free rate is 3%. A European call option on the stock with strike price $40 expiring in 4 months is currently trading for $2. Estimate by trial and error the implied volatility of the stock.

Explanation / Answer

he Black-Scholes option pricing model provides a closed-form pricing formula BS()BS() for a European-exercise option with price PP. There is no closed-form inverse for it, but because it has a closed-form vega (volatility derivative) ()(), and the derivative is nonnegative, we can use the Newton-Raphson formula with confidence.

Essentially, we choose a starting value 00 say from yoonkwon's post. Then, we iterate

n+1=nBS(n)P(n

The answer comes out to be 30.85%

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