Answer to Question #156906 in Financial Math for Saso

Question

1. The Moon company currently sells for 100 $. The annual stock price volatility is %10 and risk- free interest rate %8, the price of a call on a company’s stock with strike price 200 $ and time period 2 months. Find the stock option price with Black and Scholes Model. If option market value 320 $ what is the option strategy?

2. The Moon company currently (t1) sells for 100 $ . The annual stock price volatility is %10 and risk- free interest rate %8, the price of a call on a company’s stock with strike price 120 $ and time period 2 months. Changes in parameters in period t2 are as shown in the table.

t =2

Price =110

volatility =%8

period =3 months

Risk free interest rate= %5

a) What is the option price changes if delta value 0.05?

b) What is the Gamma value of the stock?

c) What is the Vega value of the stock?

d) What is the option price changes if theta value 0.04?

e) What is the Rho value of stock?

Answer 1

Answer:

Question 1.

The question is based on valuation of option by use of Black and Scholes model.

Formula as:

C₀=SN(d₁)-Ke^-rtN(d₂)

d₁=$ln \lparen {S \over K} \rparen + \lparen r+ { σ^2 \over 2} \rparen t \over σ \sqrt{t}$

d₂= d₁- σ_-$\sqrt{t}$

Hedge Ratio (Delta) = N(d₁)

Current value of Option is $ 320, which is higher than fundamental value of option as calculated.

Investor should create short position in call.

Question 2

a) If the delta value is 0.05 and the stock price increases by $10, then the value of the call option should increase by 10*0.05 = $0.5.

b) Gamma = $sigma \over S$

sigma decreases by 2% when share price increases by $10

Gamma = $-0.02 \over 10$ = -0.002

c) Vega =$call \over sigma$ = $0.5 \over 0.02$ = 25

d) Theta = $Call \over Time$

Assumption: Theta here is given as a yearly rate

Change in call = 0.04 * $2 \over 12$ = $0.00667

e) Rho value of stock= $Change in call \over risk-free rate of interest$ = $0.00667 \over 0.08$ = 0.083375