Option: Option and Stock Price Essay

Introduction 2
Arbitrage, Delta neutral Strategy 3
1) Defined the Mispriced option 3
2) Risk neutral portfolio 4
3) Transaction cost 4
4) Summary of the strategy 5
5) Adjustments for the strategy 5
6) Close out the position 6
Further developed strategy 7
1) Develop strategy 7
2) Close out of the position 9
References: 11

Introduction
Options are the derivatives of the underlying assets. Its value depends on the underlying assets. Like a stock, an option has its fair value and will be mispriced which gives us the arbitrage opportunity. In this report, we discuss how to find a mispriced option and using the real transaction data to develop risk neutral strategies.
When coming to the strategies, we adjust the strategy every week to make it stays delta neutral and compare the profit/loss on different closing date. We also introduce other Greeks in order to make the strategy improved. In the end, we compare the profit/loss two strategies and analysis the reason of the result.

Arbitrage, Delta neutral Strategy
1) Defined the Mispriced option
Options are derivatives of the underlying assets, their value will be influenced by the current stock price, the exercise price, the time to maturity, interest rate and the most important, the volatility, which indicates how much the stock will change. According to the c, the option’s value is positive related to the volatility and the implied volatility which measures the market expectation that how much the stock will change is calculated by setting the current option price to the model price. If our expectation of the stock’s volatility different the implied one, then we can say that the option is mispriced.
We choose the AMP Ltd stock and the options derivate from it. We find from Financial Review website, on the 11th September, 2009, the information for AMP stock and options are as follow:
AMP Ltd Last Sale Price $6.40

ASX Code/Series
Ex Price
Fair Value
Last Sale
Weekly Vol 000's
Open Int
Implied Volatility
Delta
Annual % Return
Jun 10
6.50
0.66
0.13
-
48
30.60
0.60
13.00
Jun 10
7.46
0.32
0.22
-
121
29.80
0.39
6.40
Jun 10
6.98
0.47
0.25
-
216
30.20
0.49
9.20

To find its theoretical volatility, we assume the volatility that prevailed over the recent past will continue to the future and then take a sample of continuously compounding returns on the stock over the recent 60 days.
Continuously Compounding Return =In ( )
Then we computer the standard deviation of the continuously compounding return and converted it into annualized.

We find that the theoretical volatility is 33.71%, the implied volatility of option we want to invest (expires in June 10) is 30.20%, which is lower than our expectation, as a result, so we believe it is underpriced.
2) Risk neutral portfolio
Delta indicates the change of call price due to the small change of stock price. The call option has a delta of 0.49 which means if the stock price decreases $1. The call price will decrease $0.49 when it is correctly priced. However, as we regard the option as underpriced, theoretical, if the stock price decreases $1, the call price will drop less than 49 cents in order to correct its mispricing. On the contrast, if the stock price increases $1, the call price will increase more than 49 cents to make it closer to its fair value. So we can long the option and short the delta amount of the stock to make an arbitrage, delta-neutral strategy.
Arbitrary Strategy
a) Long 1 call option which expires in Jun 10 (1000 units)
b) Short 490 shares of AMP stock

3) Transaction cost
Transaction cost occurs whenever long or short these financial assets. Doing the strategy above, we will encounter a series of transaction costs, including contract commission, brokerage cost and Australian clearing house fees.ACH charges a transaction fee of $1.43 per share option contract, including GST and the contract commission is $1.88 per option contract .The brokerage fee is different by choosing different broker and

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