One area of concern with the binomial model and our solutions is that options have negative vegas (i.e., as we increase the underlying’s volatility in our model, option prices fall).

Challenge When considering the option greeks, we saw that, while the BSOPM had explicit formulas for all of the greeks, the BINOM only had a formula for one: the delta. In this challenge, we will compare the two and see that the BINOM approximation for shorter expirations is actually quite similar to the BSOPM delta! Myron and Stephen are each pricing a three-day option. Myron uses the BSOPM while Stephen uses a three-period BINOM with the CRR solutions (see equation sheet). They agree that annualized volatility of the stock’s log returns 65 percent for a stock whose spot price is $30.50. The current annualized continuously compounded risk-free rate is 5 percent. What is the percentage difference between Myron’s (BSOPM) delta estimate and Stephen’s (BINOM) delta estimate for the $31-strike put? Enter your answers as a percentage, rounded to the nearest 0.001%. For example, for 0.123456, enter 12.346. Enter your answer as a positive number.

Challenge When considering the option greeks, we saw that, while the BSOPM had explicit formulas for all of the greeks, the BINOM only had a formula for one: the delta. In this challenge, we will compare the two and see that the BINOM approximation for shorter expirations is actually quite similar to the BSOPM delta! Myron and Stephen are each pricing a three-day option. Myron uses the BSOPM while Stephen uses a three-period BINOM with the CRR solutions (see equation sheet). They agree that annualized volatility of the stock’s log returns 70 percent for a stock whose spot price is $45.50. The current annualized continuously compounded risk-free rate is 5 percent. What is the percentage difference between Myron’s (BSOPM) delta estimate and Stephen’s (BINOM) delta estimate for the $46-strike put? Enter your answers as a percentage, rounded to the nearest 0.001%. For example, for 0.123456, enter 12.346. Enter your answer as a positive number.

Challenge When considering the option greeks, we saw that, while the BSOPM had explicit formulas for all of the greeks, the BINOM only had a formula for one: the delta. In this challenge, we will compare the two and see that the BINOM approximation for shorter expirations is actually quite similar to the BSOPM delta! Myron and Stephen are each pricing a three-day option. Myron uses the BSOPM while Stephen uses a three-period BINOM with the CRR solutions (see equation sheet). They agree that annualized volatility of the stock’s log returns 70 percent for a stock whose spot price is $26.50. The current annualized continuously compounded risk-free rate is 5 percent. What is the percentage difference between Myron’s (BSOPM) delta estimate and Stephen’s (BINOM) delta estimate for the $27-strike call? Enter your answers as a percentage, rounded to the nearest 0.001%. For example, for 0.123456, enter 12.346. Enter your answer as a positive number.

In the context of options trading, what does “pricing” an option involve?

Challenge When considering the option greeks, we saw that, while the BSOPM had explicit formulas for all of the greeks, the BINOM only had a formula for one: the delta. In this challenge, we will compare the two and see that the BINOM approximation for shorter expirations is actually quite similar to the BSOPM delta! Myron and Stephen are each pricing a three-day option. Myron uses the BSOPM while Stephen uses a three-period BINOM with the CRR solutions (see equation sheet). They agree that annualized volatility of the stock’s log returns 80 percent for a stock whose spot price is $29.00. The current annualized continuously compounded risk-free rate is 5 percent. What is the percentage difference between Myron’s (BSOPM) delta estimate and Stephen’s (BINOM) delta estimate for the $29-strike call? Enter your answers as a percentage, rounded to the nearest 0.001%. For example, for 0.123456, enter 12.346. Enter your answer as a positive number.

Challenge Consider an option trader than wants to avoid time decay. So, they want to find an option position that neither suffers from time decay nor appreciate over time. They limited their search for a position in an option that is 10% in-the-money (so, K = 0.9*St or 1.1*St, depending on the type of option). Assume the BSOPM is a correct model of the stock’s price evolution. If the risk-free rate is currently 10.00 percent per year, continuously compounded, and the trader is only interested in options that have 126 days until expiration, what must annualized volatility of the underlying’s log-returns be to meet all the parameters of their trade? Enter your answer as a percentage, rounded to the nearest 0.01%. For example, for 0.12345, enter, 12.35.  

Challenge Consider an option trader than wants to avoid time decay. So, they want to find an option position that neither suffers from time decay nor appreciate over time. They limited their search for a position in an option that is 10% in-the-money (so, K = 0.9*St or 1.1*St, depending on the type of option). Assume the BSOPM is a correct model of the stock’s price evolution. If the risk-free rate is currently 8.00 percent per year, continuously compounded, and the trader is only interested in options that have 126 days until expiration, what must annualized volatility of the underlying’s log-returns be to meet all the parameters of their trade? Enter your answer as a percentage, rounded to the nearest 0.01%. For example, for 0.12345, enter, 12.35.  

Suppose a trader buys a call and shorts a put with the strike price and expiration. What must happen in the future?

Challenge Consider an option trader than wants to avoid time decay. So, they want to find an option position that neither suffers from time decay nor appreciate over time. They limited their search for a position in an option that is 10% in-the-money (so, K = 0.9*St or 1.1*St, depending on the type of option). Assume the BSOPM is a correct model of the stock’s price evolution. If the risk-free rate is currently 5.00 percent per year, continuously compounded, and the trader is only interested in options that have 126 days until expiration, what must annualized volatility of the underlying’s log-returns be to meet all the parameters of their trade? Enter your answer as a percentage, rounded to the nearest 0.01%. For example, for 0.12345, enter, 12.35.