Question

14. 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 S0, the stock’s volatility σ, and the risk-free rate of return r : c = c(T, K, S0, σ, r). All the variables but one are “observable,” because an investor can quickly observe T, K, S0, 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, S0, r and c, what volatility σ should the stock have in order for the Black-Scholes formula to be correct. This is called the implied volatility, σBS. Some calculus, shows that σBS 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. With trial and error, find a one-percentage point range which contains the implied volatility. So your answer should be of the form “14% ≤ σBS ≤ 15%”

Answer 1

19% ≤ σBS ≤ 20%

А В 1 Input Data 2 Stock Price now (P) 3 Exercise Price of Option (EX) 4 Number of periods to Exercise in years (t) 5 Compounded Risk-Free Interest Rate (rf) 6 Standard Deviation (annualized o) 40 40 0.333333333 3.00% 19.58% 7 8 9 Output Data 10 Present Value of Exercise Price (PV(EX)) 11 *t.5 39.6020 0.1130 12 d1 0.1450 13 d2 0.0320 14 Delta N(d1) Normal Cumulative Density Function 15 Bank Loan N(d2) PV(EX) 0.5576 20.3061 16 17 Intrinsic Value of Call 18 Intrinsic Value of Put 0.0000 0.0000 19 20 Time Value of Call 1.9996 21 Time Value of Put 1.6016 22 1.9996 23 Total Value of Call 24 Total Value of Put 1.6016 25.

А В C 1 Input Data 2 Stock Price now (P) 3 Exercise Price of Option (EX) 4 Number of periods to Exercise in years (t) 5 Compounded Risk-Free Interest Rate (rf) 6 Standard Deviation (annualized 40 40 4/12 0.03 0.195750196774911 7 8 9 Output Data 10 Present Value of Exercise Price (PV(EX)) 11 *t.5 =++C3*EXP ( -C5*C4) =+C6 *C4^0,5 12 d1 +(LN(C2/C3)+(C5+C6*C6/2)*C4)(C6*C440.5) 13 d2 =+C12-C11 =NORMDIST (C12,0,1,TRUE) =NORMDIST ( C13,0,1,TRUE)*C10 14 Delta N(d1) Normal Cumulative Density Function 15 Bank Loan N(d2)*PV(EX) 16 17 Intrinsic Value of Call MAX(C2-C3,0) 18 Intrinsic Value of Put MAX(C3-C2,0) 19 20 Time Value of Call 21 Time Value of Put -C23-C17 -C24-C18 22 - +C14*C2- C15 23 Total Value of Call 24 Total Value of Put -+C23+C10-C2