Financial Derivatives

Choice of options for analysis

In order to conduct the analysis data is collected for the American-style (OEX) and European-style (XEO) options. The different kind of options is to be selected for the purpose of analysis. The list of American and European options selected for the purpose of analysis is presented in table underneath. The data is extracted from Chicago Board Options Exchange.

American and European options selected with same maturity and strike price

The table presented above segregates data collected into three parts: At-the-Money (ATM), Out-of-the-money (OTM) and In-the-Money (ITM) options. The call option provides right to buy security at some specified time in future. The put option on other hand provides right to sell security at some specified time in future. In case of In-the-Money option the intrinsic value is associated with option. However, no intrinsic value is associated with at-the-money or out-of-the-money option (Cont and Kokholm, 2013). The prices paid for in-the-money options are higher due to adequate level of intrinsic value. It can be said that the prices paid is the premium charged for high level of intrinsic value.

The data is collected in pairs of American and European options. It is ensured that different pair of options selected is with same maturity date and strike price. Moreover, the options are categorized into in-the-money, at-the-money and out-of-the-money options on the basis of S&P 100 index. It is the underlying assets for options dealt on Central board of exchange. The data described above indicates the prices for American and European options so as to facilitate comparison.

Put-call parity for European options

Where,
C is call premium
P is put premium
X is the strike price of both call and put
R is the annual interest rate
T is time in years
So is initial price of underlying assets

Put-call parity for XEO1517D920 and XEO1517P920

In case 920 are taken as current price based on assumption that it is at-the-money option, following value will be estimated.

In both the above case, value of left hand and right hand is approximately equal to each other. This in turn indicates put-call parity exists between options thereby not providing arbitrage opportunity to investors.

Put-call parity for XEO1520C840 and XEO1520O840

The values on both side differs marginally which indicates that put-call parity holds true for the pair of options. It can be said that option for taking arbitrage benefit does not exist between the pair of options.

Put-call parity for American options

In case of American options the pull-call parity cannot be adopted due to fact that options can be exercised prior to expiry date (Goard and Mazur, 2013). However, the put call parity can be applied for American options only when it is exercised on maturity date. On the basis of assumption that the all the American options will be exercised on date of maturity, the put-call parity is applied below.

Put-call parity for OEX1517D920and OEX1517P920

In case 920 are taken as current price based on assumption that it is at-the-money option, following value will be estimated.

The value estimated above indicates that American option if exercised on expiry date does not provide opportunity for arbitrage. This is due to reason that values on both the sides are approximately equivalent to each other.

Put-call parity for XEO1520C840 and XEO1520O840

The marginal difference in values on both the sides indicates that if the American option is exercised on date of maturity, put-call parity holds true. The paid of options does not provide arbitrage opportunity.

Difference between prices of American and European options

The prices at which options are available in the market depend upon time to maturity and strike price. However, the American and European options with similar strike price and maturity have different prices. The difference in level of prices charged is estimated and interpreted below.

Difference in prices for American and European options

The table above indicates that prices of American options are most of the times higher than European Options. This is due to reason that the American options gives right to eservice option on the any date before maturity. The flexibility involved in case of American option results in increasing amount of premium charged by seller. In case of At-the-money call option, the prices for European option are higher than American option. This is due to reason that in case of at-the-money option the prices are highly closer than market price (Lian and Zhu, 2013). They are also expected to rise in future and European options are highly tradable in nature. This in turn makes them of worth comparatively higher than the American options. In case of out-of-money options premium is paid for the time remaining till maturity. There is a marginal difference in American and European options due to high flexibility associated with American option. In case of American options, investor can book profits by exercising options anytime before the maturity. In case of in-the-money call option the prices of American option is higher due to higher chances of increasing profits. The prices for at-the-money and out-of-the-money put options are higher for American options. This is due to additional rights and flexibility associated with the options. In-the-money put option is priced marginally higher for European option (Birt, Rankin and Song, 2013). Moreover, it is seen that out-of-the-money options is highly priced due to high risk associated with them. It can be said higher the risk associated more is the prices charged. However, the options buyer does not assume loss since they have right to exercise options.

Step 2 Binomial model

The different pricing models are developed so as to value and price the options appropriately. These models include binomial model, Black and schools model and so on. The binomial and Black Scholes model is described and applied for some of the options and index levels underneath.

Comparison between index level prices based on binomial tree and observed prices

It is essential for investors to decide the right time for exercising its options in case of American options. The binomial tree of index level will be similar for call and put option. However, option prices changes with change in type of option that is call or put option. In addition, the binomial tree helps in supporting decision related to point of time when exercising option looks attractive in nature (Levy and Stockwell, 2013). As per the binomial tree for index levels an investor can take benefit by exercising its options on date of maturity. In all the three cases changes are significant in step 3. It is suggested for investors to hold the options till maturity date and exercise the same on date of maturity. It should be ensured that call or put option when exercised generate sufficient level of profits. Besides, investor always possesses right to not exercise the options into consideration. Moreover, continuous observation of values is necessary to identify appropriate level for exercising the option. It is through continuous observation that investors are able to understand fluctuation so as to take benefit of exercising the call/put options entered into. The call and put prices for two of the options selected in the report is estimated through binomial model. The prices estimated through construction of binomial tree are presented underneath.

Black Scholes pricing model

Black Scholes model is the pricing model that emphasizes on pricing the option without considering dividend yield on stocks (Yan, 2013). It helps in estimating numerical values of options so as to judge whether it is undervalued or overvalued. The Black Scholes model is mostly adapted to European options. It is essential to assume limited risk on the basis of volatility index (May and Stiglitz, 2013). The volatility index can be modified so as to identify level at which prices estimated can equate the market price of underlying assets.

Conclusion

The report proposed herewith deals with evaluation of various American and European options. It helps in understanding manner in which options can be priced appropriately. The investor can be minimized through investment in options since it provides rights to investors.

Reference

• Benth, F. E. And Benth, J. S., 2013. Modeling and pricing in financial markets for weather derivatives. World Scientific.
• Birt, J., Rankin, M., and Song, C. L., 2013. Derivatives use and financial instrument disclosure in the extractives industry. Accounting & Finance. 53(1).pp. 55-83.
• Cont, R. And Kokholm, T. , 2013. A consistent pricing model for index options and volatility derivatives. Mathematical Finance. 23(2).pp.248-274.
• Goard, J., and Mazur, M., 2013. Stochastic volatility models and the pricing of VIX options. Mathematical Finance.23(3).pp.439-458.
• Hirsa, A. And Neftci, S. N., 2013. An introduction to the mathematics of financial derivatives. Academic Press.
• Levy, M., and Stockwell, G. , 2013. CALL dimensions: Options and issues in computer-assisted language learning. Routledge.
• Lian, G. H. And Zhu, S. P., 2013. Pricing VIX options with stochastic volatility and random jumps. Decisions in Economics and Finance.36(1).pp 71-88.
• May, R., and Stiglitz, J. ,2013. Complex derivatives. Nature Physics. 9(3).pp. 123-125.
• Yan, H., 2013, March. Adaptive wavelet precise integration method for nonlinear Black-Scholes model based on variational iteration method. In Abstract and Applied Analysis . Hindawi Publishing Corporation.
• Online
• CBOE quotes: OEX data, 2015. [Online]. Available through:< http://www.cboe.com/delayedquote/quotetable.aspx >. [Accessed on 9th March 2015].
• CBOE quotes: XEO data, 2015. [Online]. Available through:< http://www.cboe.com/delayedquote/quotetable.aspx >. [Accessed on 9th March 2015].

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