The assignment of financial derivatives course assignments, assignment of hedging program assignments, assignment of assignments for normality test
requires two documents, one Excel file and one word document. The text is mainly used to explain how each topic is done
(1) Data collection and arrangement (10)
1 Download the daily data of S&P500 from January 2, 2016 to December 31, 2016 on yahoo finance. We take Adj Close as our data
2 Processing data:
adjust the first data of S&P500 to 100, and adjust the other data to relative prices, that is: we get the data.
From then on we regard St as our original data, and according to this data for analysis.
3 We assume that St is a non-interest bearing and well-liquid individual stock price trajectory. It is assumed to obey the Black-Scholes model, i.e., under the true probability measure:
(note that here, under the true measure, the parameters are not important, we only care about the volatility parameter:
Also, note that we assume that the unit of a year is 1, a job Day is 1/250, week is 1/50)
Suppose we can use the sample variance as a point estimate of variance, design a method to estimate the volatility of S&P500
4 pairs for normality test (design your own method, such as QQ plot)
(2) Binary tree model (20)
1 Assuming that the interest rate is r=0.01, build an N-step CRR binary tree on the Excel sheet for the S&P500 data for the whole year of 2017.
Note that at time T=1, under the risk-neutral measure of the CRR tree, after taking the logarithm, the variance should be the same as the normal distribution of the BS model
(what are the U and D of the CRR tree?)
2 Use this CRR tree to price European call options for ATMs signed from January 2, 2017, with an expiration date of December 31, 2017 (one year).
3 On the binary tree of the above N steps, calculate the price of the European-style and American-style put options of the ATM
4 Using the BS formula, accurately calculate the price of the European call option and European put option with K=100. And gives the price range for American put options.
5 The stock price of the CRR tree, after taking the logarithm, follows a binomial distribution at time T. Consult the literature for an analytical formula for this binomial distribution. Using this analytical formula, the BS analytical solution of the ATM European put and the numerical solution of the CRR tree are plotted on the same graph, with the changing trend of the binary tree steps increasing.
(The X-axis is the number of steps in the binary tree, and the Y-axis is the option price. Draw two lines, one is the BS price, and the other is the price of the numerical solution of the CRR tree)
(3) Hedging (10)
1 Find the trajectories of the nodes corresponding to our S&P500 data on the N-step binary tree in question (2). For example (udu....)
(Of course, the data cannot be exactly equal to the price on this node, use the rounding method to select the nearest node)
2 Assuming that the trajectory of the S&P500 really satisfies the trajectory of this node, perform delta hedging on the European call option of an ATM
3 In fact, the trajectory of the S&P500 must be different from the relative node. At each node time, we use the delta calculated by the black-scholes formula for delta hedging
(we assume that the whole process is self-financing, and the hedging error is determined by the final deviation from the hedging target)
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