Question

UW DUCupei tau. equipment and CLISSOLD INDUSTRIES OPTIONS You are currently working for Clissold Industries. The company, which went puoi dustries. The company, which went public five years ago, engages in the design, production, and distribution of lighting equipment specialty products worldwide. Because of recent events. Mal Clissold, the company prest dent, is concerned about the company's risk, so he asks for your input your discussion with Mal, you explain that the CAPM proposes that the market terminant of its expected return. Even though Mal agrees with this, he argues that his portfolio consists entirely of Clissold Indust Olio consists entirely of Clissold Industry stock anu options, so he is concerned with the total risk or standard deviation of the company stock. Furthermore, even though he has calculated the standard deviation of the compa ny s stock for the past five vears he would like an estimate of the stock's volatility moving forward. Mal states that you can find the estimated volatility of the stock for future periods by calculating the implied standard deviation of option contracts on the company stock. When you examine the factors that affect the price of an option, all of the factors except the standard deviation of the stock are directly observable in the market. Mal states that because you can observe all of the option factors except the standard deviation, you can solve the Black-Scholes model and find the implied standard de To help you find the implied standard deviation of the company's stock, Mal has provided you with the following option prices on four call options that expire in six months. The risk-free rate is 4 percent and the current stock price is $53. Strike Price $50 55 Option Price $12.78 10.14 7.99 5.81 60 65 1. How many different volatilities would you expect to see for the stock? 2. Unfortunately, solving for the implied standard deviation is not as easy as Mal suggests. In fact, there is no direct solution for the standard deviation of the stock even if we

Answer 1

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Answer:

1) From the given casse study, we find there are four possible volatilites that can be implied by four different call options.

2)

J K L M N O P Q H Set Solver as: Set Objective: $I$16=18.73 By changing variable cells: $1$7 Solver Parameters Set Objective: $I$16 To: Max Min Value Of: 18.73 so By Changing Variable Cells: $I$7 4% Subject to the Constraints: Add Chang 5 BSM Call Option Pricing Model X=65 X=70 X=75 X=80 6 current StockPrice 68 68 68 68 7 annual return volatility sigma 90.35% 84.24% 69.05% 57.86% 8 effective annual risk-free rater 4% 4% 4% 9 Exercise Price 65 70 75 10 Time to Maturity(yrs) T(years) 0.5 0.5 0.5 0.5 11 Dividend Yield 0 0 0 12 d1 d1=[In(S0/X)+(r-q+.5*sigma-2)*T]/[sigma*sqrt(T) 0.42136 0.28274 0.08443 -0.14376|| 13 d2 d2=[In(so/X)+(r-q-.5*sigma-2)*T]/[sigma*sqrt(T) -0.2175 -0.31292 -0.40384 -0.55291 14 N(1) Nd1)=NORM.S.DIST(d1, TRUE) 0.66326 0.61131 0.53364 0.44285 15 N(d2) N(02)=NORM.S.DIST(d2, TRUE) 0.41391 0.37717 0.34316 0.29016| 16 call option Price c=S0*exp(-9T)*N(d1)-x*exp(-rT)*N(02) 18.73 15.69 11.06 7.36 17 Delet Reset Load/Sa V Make Unconstrained variables Non-Negative Select a Solving Method: GRG Nonlinear Optior Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that non-smooth. 20 21 22 Help solved K BetaCalpvnpv-sales AcPayable NPV Am,gmleasing variance in states bondpricce sd, mean Sheet2 bsm < M M1npv-simple Circular References 2 Ready 200 140% 0

J16 fac =J6*EXP(-J11*J10) *J14-19*EXP(-18*J10)*115 H Solver Parameters KL Set Solver as for X=80; Set Objective:$L$16=7.36 Set Solver as for X=70; Set Objective:$J$16=15.69 By changing variable cells:$J$7 1 Set Solver as for X=75; 2 Set Objective:$K$16=11.06 By changing variable 3 cells: $K$7 Set Objective: $J$16 To: By changing variable cells:$L$7| Max O Min Value Of: 15.69 By Changing Variable Cells: $J$7 Subject to the Constraints: 4% Char Dele 5 BSM Call Option Pricing Model X=65 X=70 X=75 X=80 6 current StockPrice so | 68 68 68 68 7 annual return volatility sigma 90.35% 84.24% 69.05% 57.86% 8 effective annual risk-free rate 4% 4% 4% 9 Exercise Price X 65 70 75 80 10 Time to Maturity yrs) T(years) 0.5 0.5 0.5 0.5 11 Dividend Yield 0 0 0 0 12 d1 d1=[In(so/X)+(r-q+.5*sigma-2)*T]/[sigma*sqrt(T) 0.42136 0.28274 0.084429 -0.14376 13 d2 d2=[In(So/X)+(r-q-.5*sigma^2) *T]/[sigma*sqrt(T)] -0.2175 -0.31292 -0.40384 -0.55291 14 N(d1) N(d1)=NORM.S.DIST(d1, TRUE) 0.66326 0.61131 0.533642 0.44285 15 N(d2) N(D2)=NORM.S.DIST(d2,TRUE) 0.41391 0.37717 0.343164 0.29016 16 call option Price C=SO*exp(-qT)*N(d1)-x*exp(-rT)*N(02) 18.73 15.690 11.06 7.36 la Rese Load/ Make Unconstrained variables Non-Negative Select a Solving Method: GRG Nonlinear Opti Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the L engine for linear Solver Problems, and select the Evolutionary engine for Solver problems th non-smooth. 17 19 Help 20 K 21 M Ready BetaCalpvnpv-sales ACPayable NPV Amgmleasing variance in statesbondpricce sd, meanSheet2 bsm MM1 npv-simple Circular References 2 BOJ 140% 0- 4

3)

No all of the implied volatilities for the options are not the same.Possible reasons are the different prices of the call options ,generally higher is the price higher is the  implied volatilities and also different exercise prices which influence prices which in turn influence the implied volatilities.

4)

VIX is a volatility index and it measures the markets expectation of 30 day volatility.The index is calculated using the implied volatilties of wide range of S&P500 index options ,the volatilty caculated for both calls and puts is implied is forward looking.