Question

You observe a premium of $44.00 for a call option on Birdwell Enterprises common stock, which is currently selling for $44. The strike E price on the call option is $44. The option has four months to maturity. The stock pays no dividends. The current risk-free interest rate is 3.00%. What is the implied volatility of the stock? (Round your answer to the nearest whole percent.) Implied volatility %

Answer 1

Generally, option premium quoted for one option contract and each option contract represents 100 share of underlying stocks.

In above case,

Option premium for one contract = $44

Option premium for one share = 44/100 = $0.44

Stock Price (P) = $44

Strike Price (X) = $44

Maturity in years = 4/12 = 0.3333

Risk free rate = 3%

Implied volatility = $\sigma$

We can compute the Implied volatility of stock using black-scholes equation but back calculation of equation to find implied volatility is very Complex. Thus, we can use Goal Seek function in excel to compute the implied volatility.

Please refer to below spreadsheet for calculation and answer. Cell reference also provided.

I c F G X $44.00 0.33 1471% $44 3.00% Goal Seek ? X E15 Using the Black Scholes formula: 0.44 Set cell: To value: By changing cell: D OK $D$4 (d1) 4.2487 N(d1) 1.0000 Cancel (d2) -4.2463 N(D2) 0.0000 Value of Call option 13 14 15 Vc T $44.00

ДА В DE F LG H I J K X $44.00 0.33 1% $44 3.00% Goal Seek Status ? X Goal Seeking with Cell E15 found a solution. Step Using the Black Scholes formula: Pause Target value: 0.44 Current value: $0.44 (d1) 1.8577 N(1) 0.9684 LOK Cancel (d2) 1.8524 Nd2) 0.9680 Value of Call option Vc Vc = $0.44

Cell reference -

F G X 44 44 =4/12 0.00933695065097976 PRE 0.03 Using the Black Scholes formula: (d1) =(LN(D2/G2)+(G3+D4^2/2)*D3)/(D4*SQRT(D3)) N(01) ENORMSDIST(E8) (d2) =E8-D4*SQRT(D3) N(D2) ENORMSDIST(E10) Value of Call option =D2*G8-(G2*G10)/EXP(G3*D3)

Thus, Implied volatility = 1%

Please note : If we assume that option premium provided in problem is for one share instead of one contract then, implied volatility would be 1484%.

Hope this will help, please do comment if you need any further explanation. Your feedback would be highly appreciated.