Introduction to Financial Mathematics
Choose a publicly trade company that offers a dividend and let XYZ be its common stock. One of the stocks listed among the Standard and Poors 100 (S&P100) is a good choice as these are heavily trade with widely published data. Use US treasury T-Bill discount rate to estimate riskless returns.
Analyze contingent claims on XYZ in these three ways:
1. Find the implied volatility sigma(K,T) of XYZ using options at various near the money strike prices K and near future expiry dates T. Justify your choice of your options. Use both CRR and Black Scholes pricing models adn compare the results. Also compare your results to published implied volatilities and comment on your diffrences. Plot the implied volatility surface sigma(K,T) for your results.
Choose a time to expiry T that includes at least two dividend payments expected to equal the most recent. Use the CRR model for American style options with dividend to price Call and Put options with expiry T at various near the money strike prices. What volatility should be used? How do your prices compare with the market prices for these options? How would your prices change if the second dividend was 20
Choose a time to expiry T that contains no expected dividends. Construct an implied binomial tree from the spot price and at least five near the money Call option premiums on XYZ with expiry T. Use it to price five near the money Put options on XYZ with the same expiry T. Compare your results with the market prices of those Puts and also with the values given by the Call-Put parity formula.
Answer:
To find the implied volatility sigma(K,T) of XYZ using options at various near-the-money strike prices K and near-future expiry dates T, we can use a variety of options pricing models. For this example, we will use the Cox-Ross-Rubinstein (CRR) binomial model, the Black-Scholes model, and the implied volatility surface (IVS).
For the CRR model, we can use the following formula to calculate the implied volatility:
Sigma(K,T) = Square Root [ln(S/K) + (r-d + (sigma^2)/2)T] / sigma*sqrt(T)
Where S is the stock price, K is the strike price, r is the US Treasury T-Bill discount rate, d is the dividend rate of the stock, sigma is the volatility of the stock, and T is the time to expiration.
For the Black-Scholes model, we can use the following formula to calculate the implied volatility:
Sigma(K,T) = Square Root [ln(S/K) + (r-d + (sigma^2)/2)T] / sigma*sqrt(T)
Where S is the stock price, K is the strike price, r is the US Treasury T-Bill discount rate, d is the dividend rate of the stock, sigma is the volatility of the stock, and T is the time to expiration.
For the implied volatility surface (IVS), we can use a variety of methods to calculate the implied volatility. The most common method is to use a two-dimensional interpolation of the implied volatilities from the near-the-money options at different expiry dates.
Once we have calculated the implied volatilities for each of the near-the-money options, we can plot the implied volatility surface sigma(K,T) for our results. This will give us a visual representation of the implied volatility for our chosen stock.
We can then compare our results to published implied volatilities and comment on any differences. Generally, the implied volatility from the CRR and Black-Scholes models should be similar. However, there may be some discrepancies due to market conditions, liquidity, and other factors.
Finally, we can compare our results to published implied volatilities and comment on any differences. Generally, the published implied volatilities should be more accurate as they are based on real-time market data. If our results differ significantly from the published implied volatilities, we should investigate further to determine why.</p
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