Need help with my Statistics question – I’m studying for my class.
In the following exercise, please refer to the FE535_SPY_OPT_2019-01-14.csv data file. This data was collected on January 14th, 2019 at 10 PM from Yahoo Finance covering all the options last traded on the SPY ETF at that time.
Each observation corresponds to a different contract, i.e. maturity, exercise, and type (call or put).
In addition to the data, you are given the following information:
Treat the Jan 14th, 2019 as the current date
– where the spot price on Jan 14th, 2019 was $256.27
The risk-free rate is 2%
Treat the SPY volatility to be equal to 10%
The maturity date is April 18th, 2019
Put-Call Parity (10 Points)
1. In the class, we talked about put-call parity. Given the data, provide a plot, where the y-axis denotes the left hand side (LHS) of the parity and the x-axis denotes the right hand side (RHS) of the parity (6 Points). In addition, how do both sides compare? (4 Points)
Black and Scholes: Market vs. Theory (10 Points)
2. The prices from the data denote the price that traders are willing to buy (respectively sell). Moreover, these prices reflect traders’ views on the options prices and, hence, the underlying asset, i.e. the SPY. On the other hand, option pricing models, such as the Black-Scholes Model (BSM), are theory based. To bridge between theory and reality, you need to price each of the 44 call and put options using the BSM. This should result in 88 prices, 44 for the calls and 44 for the puts. Note that the BSM computes the price for a European call option. To compute the put price, you need to refer to the put-call parity.
As a summary, you are asked to
(a) Plot the BSM prices versus the market prices along with a 45-degree line.
(b) Regress the market prices on the BSM prices. Report the mean-squared error (MSE) and the R2 from the regression.
Black and Scholes: Implied Volatility (10 Points)
3. The previous part assumes that ? = 10%. Traders, on the other hand, may share different views about the market volatility. To see this, assume that the BSM is the true pricing model that traders use to evaluate different options. As a result, differences between market and model prices could arise due to different perception of the market volatility by traders. Your task is, thus, to deduce the volatility perceived by the market, which is known as the implied volatility. As a summary,
• Report the implied volatility – this should correspond to a single number (5 Points)
• Report a similar summary from the previous part, however, using the implied volatility (5 Points)Black and Scholes: Implied Volatility (10 Points)
Option Pricing via Monte Carlo (20 Points)
Consider the same assumptions and parameters from the previous question. Assume that the asset price follows a GBM with drift r and volatility ?. Your task is to price the following options using MC:
• A European Call option with K = 265 (5 Points)
• B Asian call option with a strike price of K = 265 (5 Points)
• C Knock-Out Barrier Option with exercise price of K = 265 and knock-out price of 255 (5
Points)
Finally, how do you justify the difference in prices among the three options? (5 Points)
Assignment about Option Pricing via Empirical Investigation and Monte Carlo
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