Finance 527: Lecture 31, Options V3 [John Nofsinger]: This is the third video for the options topic. And the final topic is option pricing is what we re gonna talk about. So what is the price of an option? Now what we ve noticed so far is that say for a call option, the price has something to do with the intrinsic value or the profit that s already embedded in the fact if it s an in-the-money option. That intrinsic value is zero if it s an out-of-the-money option. But even if it s an out-of-the-money option and the intrinsic value is zero, there s still a price that you have to pay to buy it. So that part of the value comes from what we call time value. And it s more than just time. The actual factors that contribute to the price or this time value-well the whole price-obviously the underlying stock price, right. For call option, as the stock price goes higher, you get more and more in-the-money the call price goes higher. Also time until expiration. The longer you have, the more chances you can think of that the price could get up and to you know a high price and you would make a lot of money. It s harder to see that that would happen if you re option expires tomorrow. But if it expires in 6 months, you see there s a lot better chance it s gonna happen. So that s factored into this time value. Also the volatility. If the underlying stock price goes you know very volatile, then there s lots of chances that it s gonna be high at some point and you can sell the option or exercise the option for a lot of value. If the volatility is very low, then it s a less chance that you re gonna make a lot of money, right. The size of the dividend if the stock pays one. And the interest rates in the economy are the others. Options V3 John Nofsinger Option pricing Factors contributing value of an option o Price of the underlying stock o Time until expiration o Volatility of underlying stock price o Cash dividend o Prevailing interest rate Intrinsic value: difference between an in-the-money option s strike price and current market price Time value: speculative value Call price=intrinsic value + time value [John Nofsinger]: This diagram shows us the call price at various relationships of the stock price as it goes up or down and what the actual call option price should be. This is the exercise price. So if the stock price is at the exercise price, it s an at-the-money option. And so there s no 1
internal fundamental value profit inside of it. So the entire option value is what we call this time value because there is no fundamental value. The fundamental portion starts here at exercise price, and if the stock goes higher, then it s in the money. And there is an embedded fundamental type value that the higher the stock goes, the higher the intrinsic value. Still if the stock price was here, right, the call options price would be here, you know be right there over here on the left. This portion of it is still time value, and then this portion of it is intrinsic value. So the intrinsic value-time value I should say-is largest around the exercise price. If it gets way out of the money, you can see that the time value sort of disappears. And also if it s way in the money. The call price tends to converge to its intrinsic value. And for these reasons, most of all the options that are trading and that are interesting to people tend to be around the exercise price. People rarely buy and sell deep out of the money options or deep in the money options or deep out of the money options. Figure 18.5 Call Prices Converge toward Intrinsic Value as Option Expiration Draws Near [Image of graph of stock price vs. call option value] [John Nofsinger]: This is the famous Black-Scholes Option Pricing formula. Scholes won a Nobel Prize in economics for this. I believe Black actually passed away by the time that happened. So he did not get to participate if I remember correctly. So here s how it works. The call price or the price of a call option. We focus on the call. We ll get to the put later. So here it is. Has to do with of course where the stock price is if that makes sense. And the relationship between the stock price and the strike price. So the minus the strike price there that makes a lot of sense. The strike price isn t of course known, and it s you re gonna have to pay that in the future, right. When you call this at maturity-this is assumed you re calling this at maturity-and so we actually want to find the present value of the strike price. And that s what we re doing here. We re finding the present value of it. Most people I think, most students are more aware of present value in terms of to find a present value, you would do one plus the interest rate to the t power. Right for t years. And that would be a way to find the present value of the strike price. This is a different form of that. It s called the continuous time present value function. So that s where that comes from. Alright so that part makes sense. What are these? These are basically distributions. This is the normal distribution of some variable. The normal cumulative distribution of a different variable. And the idea here is what s the distributions? What s the likelihood of this stock price going real high and getting a lot of value here? Well let s find out where that comes from. So here s the formula and you can see things like time, right. The longer the time, the higher the cumulative density function is gonna be. Also the volatility of that underlying stock. That s the variance of the returns there. The higher the volatility, the higher the d1, which would be the higher cumulative density function that would go in there. So if you had d1 and d2, you could put them into-you could look it up in the tables or spreadsheets and stuff. 2
And you could find the cumulative density function for the normal distribution and put those I there. Black-Scholes Option Pricing model Call price= value of upside potential opportunity cost of invested funds CC = SS[NN(dd 1 )] XX ee rrrr [NN(dd 2)] Where C: current price of a call option S: current market price of the underlying stock X: exercise price r: risk free rate t: time until expiration N(d1) and N(d2) : cumulative density functions for d1 and d2 dd 1 = ln SS XX +(rr+0.5σσ2 )tt σσ tt dd 2 = dd 1 σσ tt [John Nofsinger]: Now this is something that a lot of people aren t actually going to calculate out by hand like this. But I just wanted to give you a flavor of what s going on here. Let s say a stock trades for 50. You want to call option at 55 so that s out of the money. Right the right to buy at 55 isn t very valuable if the stock s only trading at 50 anyway. Here s the risk free rate. Volatile stock of 40%, a lot of time until it expires. What s the call price? So first we have to find d1 and d2. So we put all those numbers in here. We find d1. Once we find d1, we can find d2. We could look up in the table the cumulative normal density function and we can get these numbers. And so now we re ready to put them into the actual call price function. And here s our 50 dollars in there and the present value of the strike price and all of that and we end up with 4 dollars and 30 cents. So even though this option has basically-well it has no fundamental value cause it s out of the money-it still trades for a hefty four dollars and 30 cents because we have lots of time and quite a bit of volatility for that option to jump up there. Example Current stock price: 50 exercise price: 55 risk free rate: 6.25% Time to expiration: 6 months volatility: 40% what is the call price? Solution dd 1 = ln 50 55 + 0.0625 + 0.5 xx 0.42 xx 0.5 0.4 0.5 dd 2 = 0.0851 0.4 0.5 3
= 0.0953+0.0713 = 0.0851 = 0.3679 0.2828 N(d1)=0.4661 N(d2)+0.3564 Call price = SS[NN(dd 1 )] XX ee rrrr [NN(dd 2)] = 50[0.4661] ee 55 (0.0625)(0.5) [0.3564] = $4.30 [John Nofsinger]: Now what about the put option? Well recall from the previous video that calls and puts through the actual stock itself are related. That is if you have a protective put strategy where you own the stock price-you own the stock-and you buy a put. You re payoff strategy ends up looking a lot like a call option. Right so that s how they re related. And so through that, they ve created this put price. First you call the call price. And then you ll find-you ll want to find the difference-between the present value of the strike price and where the stock price is right now. So you can find the put option price here. For that same example that I just did, the put price was in the money and so it would be 7.61. Put call parity Relationship between the price of a put option and the price of a call option on the same underlying equity XX pppppp pppppppppp = SS + CC eerrrr Using the same values before, 55 pppppp pppppppppp = 50 + 4.30 = $7.61 ee (0.0625)(0.5) [John Nofsinger]: This is the alphabet soup of options. You can see it s the actual Greek alphabet soup of options. So why do option prices change? Well obviously if the underlying stock price changes then the options price is gonna change too. What s the sensitivity? That is if the stock price changes one dollar, say it goes up one dollar, what does the call option price do? Does it go up a whole dollar? If it goes up 95 cents, does it go up a dollar ten? Delta tells us the sensitivity. So you know is it you know a hundred percent? Is it 80 percent? Whatever it is so that s what delta is. And that s really important when you want to use options to hedge. If you have a large a hundred million dollar portfolio, you re worried about stocks going down. You can get into say the S&P 500, S&P 100. You could buy you know the protective put. How many put options do you buy? Well you buy enough so that if you want 100 % right, then you would buy enough and the delta-the hedge ratio or delta-helps tell us how much we need. Other reasons that the option is going to change in value-well this delta actually depends a lot on where the stock price is. So if the stock price is at 50 that s gonna have a different delta than if the stock price is at 60 right. So 4
the delta is changing and that s what gamma tells us. What is the rate of change of delta? Other reasons the call option changes in price is that the great sands of time keep moving, and that option that had 6 months to maturity will eventually have 5 months to maturity, 4 months to maturity. And the time value will start going down. So theta is a measure of that time value-the option price sensitivity to how that time is wasting away. The underlying stock of changes its level of inherent risk, right. It could change its volatility-become more volatile or less volatile. So that should change the option price as well, and Vega tells us about that. And lastly, interest rates in the economy can change while we own the option. And since that s a part of the factor, we could get the variable rho would tell us how much the option will change in price when interest rates are changing. So if you want to know you know that s just the alphabet soup of what all those are. Option risks Delta: the sensitivity of option value to a unit change in the underlying asset (hedge ratio) Gamma: the responsiveness of delta to unit changes in the value of the underlying asset Theta: the sensitivity of option value to change in time Vega: the sensitivity of option value to change in volatility Rho: the sensitivity of option value to changes in interest rate [John Nofsinger]: Now if you don t like calculating those things by hand-the Black-Scholes Option Pricing-and who does? There are plenty of option calculators on the web. This is one example. This is an old example that I did. I had a put in Wal-Mart, and it had come up with the current stock price of Wal-Mart. And I chose a strike price. I could even change this though-the price of Wal-Mart if I wanted to. When it expired. So this told me then oh it s only 8 days expiration. It told me the volatility of Wal-Mart and what the current interest rates are, when the dividend was going to be paid. And then out pops-you can see here the symbols for the option for this particular Wal-Mart call and put option-and it gave me what the option values are for the call and the put. And so we see here that it also gives us all of this Greek alphabet soup to tell us the rate of change and everything. Or we could say we could put in the option price-right we could put in the option price-and we could have it calculate the volatility. Right instead of having telling us what the volatility is and commuting the price, we could put in the price and then it would calculate the volatility. Such values are presented in CBOE Option Calculator (www.cboe.com) [Image of table of online calculator of option values of Wal-Mart call and put] 5
[John Nofsinger]: This is a current shot of that type of calculator. It is at the Chicago Board Options Exchange web page. And we see the-you could put in the symbols for whatever we re looking for: Wal-Mart or Microsoft-we could populate it. We could pick a different strike or whatever-a different expiration date. And it could populate that for us or we could reverse it and say we ll put in the option price and you tell us the volatility. So all of that is easily available at the Chicago Board Options Exchange tools is where the options calculator is. So you can do that instead of calculating Black-Scholes by hand, which seems like a pretty good idea to not do it by hand. [Image of screenshot of Options Calculator on the CBOE web page] [John Nofsinger]: The last issue that I d like to talk about here is the VIX. Again what if we for say the S&P 500, what if we entered into that calculator the S&P 500 and we put in the price of the call price, right. And we told it to calculate the volatility-the implied volatility is what we call it. That is since we know what interest rates are, we know when the option matures, we know what the underlying S&P 500 index is at, and we are taking for granted the price of the option that s being traded. We can have them tell us what the volatility is. So that s what this option is implying-the volatility of the S&P 500 is. We call that the VIX, and that has become a very popular measure. People often call it a fear index. As VIX goes way up, it indicates a lot of volatility in the market, which is associated with fear. So also it s very useful to see what s that volatility mean? So if the VIX is currently at 22, what that means is the S&P 500 index is expected to on a one month basis, it s expected to if you do the 22%, which is from right from here, divided by the square root of 12, you get 6.35. It says in the next month, the S&P 500 is expected to move by six and a third percent. That doesn t mean it s gonna go up six and a third or down. We don t know which direction. That s volatility. It s just expected to move that much in the next month. So you can see as volatility goes up from here, its expectations of the market moving even more. If the VIX goes down, we are expecting the markets to move less. So the VIX is a commonly reported statistic. It s just the implied volatility of the S&P 500. But people have taken a big interest into it more recently. So anyway that is the end of the third video for options. VIX If you know the price of the option, you can calculate the volatility that the price implies, called implied volatility The VIX is the scaled implied volatility of the S&P 500 Index option o Say the VIX is at 22.06 o Means the S&P500 Index is expected to move by 6.35% (=22% 12) in the next month 6
The VIX is known as the fear index and is a tradable product 7