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Unformatted text preview: Notes 8 Finance 5108 Learning Objective 1. Understand the application of Delta and gamma in option position management 2. Be able to explain delta hedging and its applications 3. Calculate the delta of a portfolio of options 4. Understand the means of calculating the theta of an option and its relationship to the gamma and delta of options 5. Be able to calculate the Vega of an option 6. One of the most interesting aspects of options is how a financial institution that has written an option will be able to hedge the risks. As can be seen by looking at the Black-Scholes formula, the price of an option is dependent on certain parameters. The price of an option of a given strike price and expiration depends on the price of the underlying asset, the time to expiration, the volatility and the risk free rate. If a change in these parameters affect the value of the option them is imperative to have an estimate of the change in the value of the option given a change in these parameter. This could lead to a better understanding of the risks. The first thing that we will consider is the change in the price underlying asset and the change in the price of the option and try to understand its effect on the hedging of the option. Let us first assume that a firm has decided to write 200 (20,000 share underlying) options with a strike price of 100 and an expiration of 6 months on a stock that is currently trading at $100. The volatility of the underlying stock 20% and the risk free rate is 3%. The Black-Scholes price of the option is $6.37 or the premium that was collected is $6.37 (20000) = $127,400. Hedging the position 1. Naked write - This entails just writing the option and not doing anything to hedge. The loss possible is limitless 2. Covered write - Purchase 20,000 shares of the stock when the option is sold. This strategy limits the upside potential and mitigates the loss by the amount of the premium In theory these would on average result in losses equal to the value of the option. But on any one transaction there could be virtually unlimited losses. Another approach would to set up a program of buying the stock when it rises above the strike price of the option and selling it when it falls. This could generate substantial transaction costs. This leads us to Delta Hedging Delta Hedging The delta of an option measures it sensitivity to small changes in the price of the underlying asset. The delta of an option will be between 0 and 1. Deep in the money options will have an delta of one and deep out of the money options have a delta near zero. The delta of a put option is negative.zero....
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