Finance: Comparing Option Price, Time to Maturity, and Strike Price

FINS3635 – ASSIGNMENT (a) Comparing Option price, time to maturity, and strike price:

1. Option price against Time to Maturity, For a given strike price:

i) APOL calls (K=40) and puts (K=40)

(ii) SBUX calls (K=40) and puts (K=40)

(iii) ABAT calls (K=5.00) and puts (K=5.00)

2. Option price against Strike Price, for a given maturity:

i) APOL calls and puts

ii) SBUX July 2011 calls and puts

iii) ABAT 2011 calls and puts:

(b) Observing the effect on option price due to changing strike price and time to maturity:

* Option price vs. time to maturity:

We observe that as time to maturity increases, so too does the respective price for both call and put options. This can be explained by the increasing time value characteristic of options. This asymmetry of option payoffs offers a higher probability of finishing in the money for options with longer maturities, whilst capping the losses on the premium paid.

There were some abnormalities, most prevalent in the ABAT put option, which may be due to illiquidity considering their small firm size.

* Option price vs. Strike Price:

We generally see an inverse relationship between call options price and their respective strike price, whilst noting a positive relationship with puts and their respective strike prices. This is explained by their payoff, where for calls is: ST – K, and for puts: K – ST. This shows that the higher the strike price of a call, the lower its payoff will be, hence a lower option price, and vice versa for put options.

We did observe an abnormality for the ABAT call option; however this once again was most likely due to its small size as a company, resulting in few trades during the period in which it was observed.

* It seems to be that as an option price is deeper in the money, its intrinsic value becomes more potent and strike price seems to affect option price to a larger degree. As it moves out of the money, time value increases, and the effect of time to maturity becomes more prominent.

(c) Differences observed across the three stocks:

* We note that the graphs regarding APOL and SBUX tend to be smoother compared to the ABAT stock. This is most likely due to a larger no. in trades, and higher liquidity, leading to a wider range of options being available on the market. Abnormalities observed within the former two stocks could possibly be isolated incidents, due to different trading volume, turnover, and other such factors.

* There was a seemingly greater level of abnormalities and “angular” characteristics in ABAT’s graphs. This is once again most likely due to fewer option series available and certain strike prices not being available on some days.

* Regarding put price over time, it seemed that APOL’s price seemed to move at a more dramatic rate compared to SBUX’s prices. This may be due to APOL’s put being out of the money at the time, therefore having no intrinsic value and price being totally dependent on its time value. However, SBUX put option was in the money at the time, and therefore its intrinsic value allows it to be affected less so by its time to maturity. The opposite seemed to hold true for the call options regarding these two stocks.

(d) Relationship between option prices and their underlying stock prices:

SBUX calls and put (Jan 2011, K=40)

Regression: * Call : y = 0.9364x – 31.2752, R2 = 0.7227 * Put: y = 0.1077x + 2.8082, R2 = 0.06372
Based on our understanding of options and stock price, we would expect a general increasing function of call on stock and a decreasing for put on stock. We see that a one dollar increase in stock price would increase call and put prices by 0.9364 and 0.1077 dollars respectively. However, noting the R-squared, we see that the model for put against stock price is statistically weak and therefore unreliable. This makes sense, since we would expect a rise in stock price to decrease put vale. This is most likely due to the fact that data was compromised in that some prices were not taken at the same maturity as others, as they were not available on certain dates. E.g. Jan13 was the longest to maturity to contract on some days and Jan20 on others. The call option model seems to show a good model in observing the relationship.

APOL call and put (K=40, JAN 2011)

For the APOL data, we strictly used the Jan 20, 2011 contracts, omitting any other Jan 2011 options as their prices would be unreliable to compare together. Once doing this, we see a more direct positive relationship with calls and stock prices, and a negative relationship with puts and their stock prices.
Regression:
* Call: y = 0.45599x – 12.9505, R-squared = 0.95537 * Put: y = -0.35854 + 19.1496, R-squared = 0.97083
When using strictly Jan 20 contracts, we see two very strong models in defining the relationship between options and stock prices. Both show the correct directional relationship that we would expect, with a 1 dollar increase in APOL stock to cause a 0.455 increase in a call, and a 0.359 decrease in put price.
ABAT call and put (K = 5, DEC 2011)

Put had a slight decreasing relationship with stock price, however we see that call price was not seemingly affected. This is most likely due to illiquidity because of ABAT’s small firm size.

Regression: * Call: y = 0.04597x + 0.0223, R-squared = 0.04597 * Put: y = -1.0901x + 5.3562, R-squared = 0.21865
We see that the regression model for the call option is relatively poor. R-squared is statistically very low, and the coefficient of stock price is also very small. Regression on put provides a better model; although it still only explains approx. 22% of all possible factors of put price, it does manage to show the expected relationship of a 1 dollar increase in ABAT stock leading to a 1.0901 decrease in its respective put price.

(e) With expected rises in coffee bean prices, this could prove dangerous for coffee businesses that cannot eat up or pass on the increased prices to their customers. Fortunately, it seems that Starbucks has a clear advantage in coming out ahead of its competitors with the highest gross profit margin so far. It is also expecting to go through a massive expansion in outlets located most notably in China, tripling its number of outlets in the country by 2015. We would thus expect a relatively modest to moderate decrease in stock price in the future, with potential rise further in time.
We form the following option strategies based on this assumption:
Strip:
Positions | Costs | Final Payoff | Profit/Loss | Long 1 JUL 30 calls | $5.47 | max{ 0, ST -30} | max{ 0, ST -30} - $5.47 | Long 2 JUL 30 put | $0.38 | 2 * Max{0, 30 – ST} | 2 * Max{0, 30 – ST} - $0.38 | Total | $5.85 | ST – 30 if ST > 3060 - 2ST if ST < 30 | ST – 35.85 if ST > 3054.15 - 2 ST if ST < 30 |

Profit/Loss on expiration date: P/L

27.08 30 35.85 ST

| | | | | | | |
Bull Spread using Calls: We use 2 call options with the lower strike being close to the underlying stock price and the higher strike being slightly out of the money Positions | Costs | Final Payoff | Profit/Loss | Short 1 May 36 Put | -$0.59 | -Max{0, 36 – ST} | -Max{0, 36 – ST} + $0.59 | Long 1 May 40 Put | $3.76 | Max{0, 40 - ST} | Max{0, 40 - ST} -$3.76 | Total | $3.17 | 4 if ST < 36 40 - ST if 36 < ST < 40 0 if ST > 40 | $0.83 if ST < 36 36.83 - ST if 36 < ST < 40$3.17 if ST > 40 |

Payoff at expiration date:
P/L

0.83

36 40 ST
-3.17