Question

In: Finance

Consider Apple Inc. as the underlying asset, use its daily adjusted clsing prices from August 14,...

Consider Apple Inc. as the underlying asset, use its daily adjusted clsing prices from August 14, 2017 to August 14, 2018 as historical data.

1. Estimate the daily standard deviation of the returns of this stock.

2. Deduce the yearly standard deviation

Consider the yearly standard deviation as the volatility of the stock and use the rate r=0.02 as annual risk-free rate. Assume you want to build a portfolio of options containing one call option with strike K1=180, and one put option with strike K2=225. Let C1(t,x) denote the call option pricing function. Let P2(t,x) denote the put option pricing function. let the maturity T=12 months. Using the adjusted closing price of August 15, 2017 as the intial stock price.

3. Compute the option prices C1, P2 on that date

4. Compute the Delta of this portfolio

5. Compute the Gamme of this portfolio

Solutions

Expert Solution

The data is collected from 14th August 2017 to 14th August 2018 in the Excel Sheet.

First calculate the log return and then calculate the daily standard deviation by using Excel formula Stdev.S.

Log return formula is= ln (current price/ previous price)

1. The daily standard Deviation of the returns of the stock is = 1.35 %

2. The yearly standard deviation is- 1.35% * sqrt (250)= 21% (lets assume 250 trading days in a year)

on 15th August the stock price was $161.6

Rate of interest = 0.02

Call Strike price = 180

Put strike price= 225

Time to maturity= 12 months or 365 days

By using Black & Scholes model,

C= S*N(d1) - K1e^(-r*t) N(d2)

P= K2e^(-r*t)N(-d2) - S*N(-d1)

C= call option price

P= put option price

S= underlying stock price = 161.6

K1= call strike price = 180

K2= put strike price = 225

r= interest rate 2 %

t= time to maturity

annual volatility= 21%

N(d1)= Normal distribution of d1

N(d2)= Normal distribution od d2

d1= [ln(S/K) - (r +( annual volatility^2)/2)*t] / (annual volatility * sqare root (t))

d2= d1 - annual volatility *sqrt (t)

Delta = Change in premium / Change in undelysing stock's price

Gamma= Change in Delta / Change in Underlying stock price

3. Call Option price (at K1= 180) will be= $7.337 Delta = 0.359 Gamma= 0.011

Put option price (at K2= 225) will be= $61.631 Delta = -0.923 Gamma- 0.004

4. Therefore, the portfolio delta will be- .359 + (-.923)= -0.564

jeff jeffy answered 2 years ago
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