Assignment 2
Due on September 29th 18:00.
1. Given a Call option with K = $45, r = 0.03, σ = 0.2, T = 1 year.
a) Use the RMFI software to plot a graph of delta on the stock price series. Explain this graph - sign, monotonicity, and the curvature.
b) Suppose the current underlying asset price is $40. Use the RMFI software to plot a graph of vega of the call option with the volatility as x axis. Explain this graph – sign and monotonicity.
2. Suppose the stock price today is 50 USD, its volatility is 0.25 and the interest rate is 0.5 percent. How many stocks we need to construct a delta-neutral portfolio, if
a) we sold 1000 put options with strike price 55 USD and expiration in 1 year,
b) we sold 1000 call options with strike price 50 USD and expiration in 1/4 year,
c) we bought 1000 put options with strike price 30 USD and expiration in 1/2 year,
d) we bought 1000 call options with strike price 50 USD and expiration in 1 month?
3. Suppose an existing long option position is delta-neutral, but has a gamma of 800.Also assume that there exists a traded option with a delta of 0.5 and a gamma of 1.25. In order to maintain the position gamma-neutral and delta-neutral, what is the appropriate strategy to implement?
4. The following tables record the cost of delta-hedging with different rebalancing frequencies. Fill in the blanks in parts a) – d).
a) Hedging cost when rebalanced every week and the option closes out-of-money. Suppose you are in the short position of 100,000 European put options with r = 5%, σ = 20%, t = 8 weeks, S0 = 40 , K = 45 . What is the cost of delta-hedging? (Assume there are 52 trading weeks per year, cost of hedging is free of interest charge.)
|
Week
|
Stock price
|
Delta
|
Shares
Purchased
|
Cost of Shares Purchased
($000)
|
Cumulative
Cash Outflow ($000)
|
|
0
|
40
|
|
|
|
|
|
1
|
42.75
|
|
|
|
|
|
2
|
45.5
|
|
|
|
|
|
3
|
48.25
|
|
|
|
|
|
4
|
47
|
|
|
|
|
|
5
|
49.75
|
|
|
|
|
|
6
|
52.5
|
|
|
|
|
|
7
|
50.25
|
|
|
|
|
|
8
|
51
|
|
|
|
|
b) Hedging cost when rebalanced every month and the option closes out-of-money. Suppose you are in the short position of 100,000 European put options with r = 5%, σ = 20%, t = 2 months (8 weeks), S0 = 40 , K = 45 . What is the cost of delta-hedging?
|
Week
|
Stock price
|
Delta
|
Shares
Purchased
|
Cost of Shares Purchased
($000)
|
Cumulative
Cash Outflow ($000)
|
|
0
|
40
|
|
|
|
|
|
4
|
47
|
|
|
|
|
|
8
|
51
|
|
|
|
|
c) Hedging cost when rebalanced every week and the option closes in-the-money. Suppose you are in the short position of 100,000 European put options with r = 5%, σ = 20%, t = 8 weeks, S0 = 40 , K = 45 . What is the cost of delta-hedging?
|
Week
|
Stock price
|
Delta
|
Shares
Purchased
|
Cost of Shares Purchased
($000)
|
Cumulative
Cash Outflow ($000)
|
|
0
|
40
|
|
|
|
|
|
1
|
45
|
|
|
|
|
|
2
|
50
|
|
|
|
|
|
3
|
55
|
|
|
|
|
|
4
|
50
|
|
|
|
|
|
5
|
45
|
|
|
|
|
|
6
|
40
|
|
|
|
|
|
7
|
35
|
|
|
|
|
|
8
|
30
|
|
|
|
|
d) Hedging cost when rebalanced every month and the option closes in-the-money. Suppose you are in the short position of 100,000 European put options with r = 5%, σ = 20%, t = 2 months (8 weeks), S0 = 40 , K = 45 . What is the cost of delta-hedging?
|
Week
|
Stock price
|
Delta
|
Shares
Purchased
|
Cost of Shares Purchased
($000)
|
Cumulative
Cash Outflow ($000)
|
|
0
|
40
|
|
|
|
|
|
4
|
50
|
|
|
|
|
|
8
|
30
|
|
|
|
|
e) Calculate the fair price of the 100,000 European put options ( r = 5%, σ = 20%, t = 8 weeks, S0 = 40 , K = 45 ) with the Black-Scholes-Merton formula using the RMFI software. What can you conclude from the comparison of hedging costs between a) and b)?
What if you compare the hedging cost between c) and d)? Comparing the option price with the hedging cost in a) and c), what conclusion can you make?
5. Based on the data in the attached csv file (AAPL.csv), let’s conduct the test of normality.
a) Calculate the standard deviation σ of daily percentage changes r.
b) Fill in the following table with the percentage of returns whose absolute size is greater than one, two, …, five standard deviations (S.D.). What’s your observation from the table?
|
|
Real World (%)
|
Normal Model(%)
|
|
> 1 σ
|
|
31.73
|
|
> 2 σ
|
|
4.55
|
|
> 3 σ
|
|
0.27
|
|
> 4 σ
|
|
0.01
|
|
> 5 σ
|
|
0.00
|
c) Define v = σ/|. Fill in the following table.
|
x
|
ln x
|
prob(v > x)
|
ln(prob(v > x))
|
|
1
|
|
|
|
|
2
|
|
|
|
|
3
|
|
|
|
|
4
|
|
|
|
|
5
|
|
|
|
d) Assuming v follows the power law, namely, prob(v > x) = Kx −α . Use the linear regression to give the estimated K and α .