Advantages of Continuous Time-Space Options Pricing
- Question 1: Option Price Inputs and Limits
I’ve asked a few questions about option pricing below. I don’t really intend for you to bring out your calculator top answer them (unless I made some mistakes) because I’ve also provided solutions below. Instead, let me know about any questions or comments that my questions and solutions might provoke.
Let’s look at the Black-Scholes model a little differently now. Basically, I’d like for you to start to see how the Black-Scholes Model works on a “grand scale.” Let’s use an exaggerated scenario to trivialize the model, then start adding back some realism a little bit at a time. Don’t do any calculations using the Black-Scholes model for Question 1 below; just use your intuition for the first question.
1. Suppose that we have a one-year call (T=1) on an underlying stock with the stock currently trading at 50. Interest rates are zero (rf = 0), and the underlying stock return variance is zero. The exercise price of the option is zero (X = 0). What is the value of this call option on the stock?
2. Same as Question 1, but we’ll increase the variance of underlying stock returns to .01 (standard deviation is now .1) and the exercise price to one X = 1, both still pretty small values. Based on the Black-Scholes model, shat is the call worth now? Show your calculations.
3. Same as Question 2, but we’ll increase the exercise price to 49.
4. Same as Question 3, but we’ll increase the underlying stock return standard deviation to .5.
5. Same as Question 4, but we’ll increase the riskless return rate to 0.1.
Here are my answers to these questions. Feel free to respond with additional questions.
1. If you use the Black-Scholes model, the arithmetic will look a bit ugly as we attempt to avoid trying to divide by zero. However, with both the riskless rate and the variance of underlying stock returns equal to zero, it’s pretty obvious that the stock will still be worth 50 in one year. We can exercise our call option to pay zero for stock that is worth 50, which makes the option have a future value of 50. Since the riskless rate equals zero, the call option is worth 50 today. So, what does this mean? The option gives you the stock for free – if you want it, you don’t mind waiting a year for the stock since the riskless return is zero, and you know with certainty that you’ll get a $50 share for nothing. So, the coupon that allows you to get a free share of stock is worth as much as the stock.
2. I deliberately kept the underlying stock return variance and exercise price small, but non-zero so that you could do some arithmetic here. Basically, you’ll see that both d1 and d2 values are very large, depending on how you round your calculations. For me, d1 worked out to be about 39.17023 and d2 worked out to be about 39.07023. But, the main issue here is that both N(d1) and N(d2) worked out to be so close to 1 that we can call both 1. With both N(d1) and N(d2) equal to 1, this means that the Black-Scholes model simplifies to 50 – 1e-0×1 = 50-1 = 49. So, what does this mean? The variance is so small that we’ll still almost certainly exercise the option and the stock will end up being worth close to 50-1. But, since the option requires that we pay 1 to buy the stock worth really close to 50, the option is worth about 49.
3. d1 and d2 values are now quite a bit smaller, with d1 about 0.252027 and d2 about 0.152027. N(d1) is now .59949 and N(d2) is .560417. The second means that there is now only a 56.0417% chance that we will exercise the option. In any case, the Black-Scholes model simplifies to 50 × .59949 – 49e-0×1 × .560417 = 2.514053. So, what does this mean? There’s only a little better than 50/50 chance that we will exercise this option, which will be worth zero if we don’t. So, the option is worth a little over 2-1/2.
4. d1 is now about 0.290405 and d2 about -0.20959. N(d1) is now .614247 and N(d2) is .416992. The second means that there is now only a 41.6992% chance that we will exercise the option. In any case, the Black-Scholes model simplifies to 50 × .614247 – 49e-0×1 × .416992 = 10.27974. So, what does this mean? Because of the significantly increased variance in underlying stock returns, the expected value of the option is much higher if we exercise it. However, the option is still worth zero if we don’t exercise it. So, we have greater prospects on the upside without worse prospects on the downside. With the higher underlying stock return variance, the option is worth a lot more at 9.27937.
5. d1 is now about 0.490405 and d2 about -0.00959. N(d1) is now .688076 and N(d2) is .496172. The second means that there is now only a 49.6172% chance that we will exercise the option. In any case, the Black-Scholes model simplifies to 50 × .688076 – 49e-.10×1 × .496172 = 12.40501. So, what does this mean? The increase in the riskless return rate increases the value of the option because we now can earn interest on our exercise money for a year before exercising the option.
- Question 2: Option Pricing and Corporate Securities
Explain how the shares of stock in a modern corporation might be considered to be options on the firm’s assets?
- Question 3: Minimum Option Value
What is the smallest value that a stock option could have? Now, give some thought to how this might motivate corporate managers who are compensated with stock options.
- Question 4: Advantages of Discrete Time-Space Option Pricing
Let’s say that you are a professional options trader. Under what circumstances in an actual trading environment would you prefer to value options and construct option hedges with a discrete model such as the Binomial Model rather than a continuous model such as Black-Scholes?
- Question 5: Advantages of Continuous Time-Space Options Pricing
Following up on my last question, let’s say that you are a professional options trader. Under what circumstances in an actual trading environment would you prefer to value options and construct option hedges with a continuous time-space model such as Black-Scholes rather than a discrete time-space framework such as the Binomial Model?
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