The Black-Scholes model assumes, among other things, that the underlying asset f

Question

The Black-Scholes model assumes, among other things, that the underlying asset follows a Geometric Brownian Motion process given by: 4. (15 points). Using Ito's Lemma and by forming a portfolio which is a combination of a call option C and short "delta shares" of S, derive the Black- Scholes (BS) partial differential equation (PDE). Then, by taking derivatives and substituting show that the following functions are exact solutions of the BS PDE a. (5 points). C(S, t)AS b. (5 points). C(S,t) S-Ke-T c. (5 points). C(S, t)- e-(T-)

Explanation / Answer

Template - Black-Scholes Option Value Formulas Input Data Stock Price now (P) 15 Exercise Price of Option (EX) 14 Number of periods to Exercise in years (t) 0.5 Compounded Risk-Free Interest Rate (rf) 8.00% Standard Deviation (annualized s) 36.06% Square root of 13% Output Data Present Value of Exercise Price (PV(EX)) 13.4511 =++C5*EXP(-C7*C6) s*t^.5 0.2550 =+C8*C6^0.5 d1 0.5550 =++(LN(C4/C5)+(C7+C8*C8/2)*C6)/(C8*C6^0.5) d2 0.3000 =+C14-C13 Delta N(d1) Normal Cumulative Density Function 0.7105 =NORMDIST(C14,0,1,TRUE) Bank Loan N(d2)*PV(EX) 8.3117 =NORMDIST(C15,0,1,TRUE)*C12 Value of Call 2.3465 =+C16*C4-C17 Value of Put 0.7975 =+C19+C12-C4

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