Question

Traders in major financial institutions use the Black-Scholes formula in a backward fashion to infer other traders' estimates of $\sigma$ from option prices. In fact, traders frequently quote sigmas to each other, rather than prices, to arrange trades. Suppose a call option on a stock that pays no dividenf for 6 months has a strike price of $35, a premium of $2.15, and time to myurity of 7 weeks. The current short-term T-bill rate is 7%, and the price of the underlying stock is $36.12. What is the implied volatility of the underlying security?

Answer 1

In a market in which options are traded actively, we can reasonably assume that the market price of the option is an accurate reflection of its true value. Thus, by setting the Black–Scholes–Merton price equal to the market price, we can work backwards to infer the volatility

Unfortunately, determining implied volatility is not a simple task. We cannot simply solve the Black–Scholes–Merton equation for the volatility. It is a complicated function with the volatility appearing several times, in some cases as ?²

c=S₀N(d1)?Xe^?rcTN(d2)

We compute the normal probabilities associated with values of d₁ and d₂ using the second equation above and insert these values into the formula as N(d₁) and N(d₂).

All the vlaues are inserted in the above formula and find ? (implied volatility)

X = strike price = $35

T = time to maturity = 7 weeks = 49/365 = .1342

r = risk free rate = 7%

S₀ = price of the underlying stock =$36.12

c = Premium = $2.15 (Price of the option which is found out by the above formula)

now we back track and set the option price = 2.15 and find the volatility using BSM model using goal seek function in excel or the basic method trial and error method.

So the volatility is .25158.