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日期：2024-09-30 08:24

Assignment 2

Due on September 29

1. Given a Call option .

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    = 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

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    = 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

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    = 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)

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    = 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 (    = 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    of daily percentage changes   .

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(%)

regression to give the estimated    and   .