Black-Scholes-Merton
The Black-Scholes-Merton model, sometimes just called the Black-Scholes model, is a mathematical model of financial derivative markets from which the Black-Scholes formula can be derived. This formula estimates the prices of call and put options. Originally, it priced European options and was the first widely adopted mathematical formula for pricing options. Some credit this model for the significant increase in options trading, and name it a significant influence in modern financial pricing. Prior to the invention of this formula and model, options traders didn't all use a consistent mathematical way to value options, and empirical analysis has shown that price estimates produced by this formula are close to observed prices.
In their initial formulation of the model, Fischer Black and Myron Scholes (the economists who originally formulated the model) came up with a partial differential equation known as the Black-Scholes equation^{[1]}, and later Robert Merton published a mathematical understanding of their model, using stochastic calculus^{[2]} that helped to formulate what became known as the Black-Scholes-Merton formula. Both Myron Scholes and Robert Merton split the 1997 Nobel Prize in Economists, listing Fischer Black as a contributor, though he was ineligible for the prize as he had passed away before it was awarded.
Roughly, their model determines the price of an option by calculating the return an investor gets less the amount that investor has to pay, using log-normal distribution probabilities to account for volatility in the underlying asset. The log-normal distribution of returns used in the model is based on theories of Brownian motion, with asset prices exhibiting similar behavior to the organic movement in Brownian motion.
The formula helped to legitimize options trading, making it seem less like gambling and more like science. Today, the Black-Scholes-Merton formula is widely used, though in individually modified ways, by traders and investors, as it is the fundamental strategy of hedging to best control, or "eliminate", risks associated with volatility in the assets that underlie the option.
Contents
The Black-Scholes-Merton Formula
Again, the Black-Scholes-Merton formula is an estimate of the prices of European call and put options, with the core difference between American and European options being that European options can only be exercised on their one exercise date versus American call options that can be exercised any time up to that expiration date. It's also used only to determine prices of non-dividend paying assets.
The Black–Scholes-Merton formula of value for a European call option is (note: the formula for a European put option is similar)
\[C(S_0,t) = S_0N(d_1) - Ke^{-r(T-t)}N(d_2),\]
where
- \(S_0\) is the stock price;
- \(C(S_0,t)\) is the price of the call option as a formulation of the stock price and time;
- \(K\) is the exercise price;
- \((T-t)\) is the time to maturity, i.e. the exercise date \(T\), less the amount of time between now \(t\) and then. Generally, this is represented in years with one month equaling \(\frac{1}{12}\) or \(0.08\overline{3};\)
- \(N(d_1)\) and \(N(d_2)\) are cumulative distribution functions for a standard normal distribution with the following formulation:
\[\begin{align} d_1 &= \frac{\ln{\frac{S_0}{K}} + \big(r + \frac{\sigma^2}{2}\big)(T-t)}{\sigma\sqrt{T-t}} \\
d_2 &= d_1 - \sigma\sqrt{(T-t)} \\&= \frac{\ln{\frac{S_0}{K}} + \big(r - \frac{\sigma^2}{2}\big)(T-t)}{\sigma\sqrt{T-t}},\end{align} \]
where
- \(\sigma\) represents the underlying volatility (a standard deviation of log returns);
- \(r\) is the risk-free interest rate, i.e. the rate of return an investor could get on an investment assumed to be risk-free (like a T-bill).
On Volatility
Volatility, in the case of financial assets, is the measure of how much and how quickly the asset's price changes. It's a measure of uncertainty. If traders were certain that an asset was worth a certain amount, then they'd buy at that price and sell below it. They'd sit on it. But highly uncertain assets get traded at a wider range of prices. Implied volatility, what options use, is the value of the volatility of the underlying asset.
Prior to the Black-Scholes-Merton formula, investors had their own ways of estimating the price of options. These methods varied, but generally they incorporated some measure of implied volatility. Stocks with more volatility had a higher chance of having a very high value in the future, or a very low value. In the above graph, an OIL ETF and an ETF for Live Cattle (COW) are graphed, with the Oil ETF being a much more volatile asset (spiking up and down more). The price not only decreases more drastically (larger slope in the trend line), but on a daily basis the stock fluctuates more significantly (the price fluctuates from the trend line more drastically).
Because of the way options work, the buyer of a call only makes money if an asset is above the strike price. If it's below that price, they don't care how much below the strike it is, they have spent the same amount. But they do care how much above the strike price it is. As such, highly volatile assets (options with higher implied volatility) are more likely to make investors more money, and are more valuable.
Technically, and in the case of the Black-Scholes-Merton model, implied volatility is the annualized standard deviation of the return on the asset, and is expressed as a decimal percentage. This will be explained more below. But in the B-S-M formula, \(\sigma\) is both a measure of implied volatility and the standard deviation. This is because, in a measure of possible returns for an asset, highly volatile assets will have a wider standard deviation than less volatile ones. The graph below shows the periodic daily returns for the previous Cow and Oil ETF, essentially a count of how many days the price changed \(+1\%\), or \(0\%\), or \(-2\%\), etc. Because the Cow ETF is a less volatile stock, the graph of its normal distribution is narrower, and the standard deviation is lower at ~\( 3.7\); expressed as a percentage that's \(13.8\%\) from the mean, meaning that the price of the ETF on any given day (because this is a graph of periodic daily returns) is highly unlikely to be \(\pm13.8\%\) more than the mean. One standard deviation in a normal distribution is \(68.2\%\), and the mean was \(\$26.85\), so the expected prices for \(68.2\%\) of the days were \(\$23.13 \le p \le \$30.56\). And indeed, compared to this, the Oil ETF's graph is much wider; in fact it goes beyond the current \(x\)-axis of this graph (there is a wider graph of this same ETF over this same time period in the section below). It has a standard deviation of \(~7.5\), or \(55.4\%\). The Oil ETF had a mean of \(\$13.58\), so for \(68.2\%\) of the day's prices were \( \$6.03 \le p \le \$21.01\), a much wider range of prices (both absolutely and relatively).
High-level Explanation of the Black-Scholes-Merton Formula
Overall: Intuitively, and roughly, the Black-Scholes-Merton formula subtracts \(Ke^{-r(T-t)}N(d_2)\), the exercise price discounted back to present value times the probability that the option is above the strike price at maturity, from \(S_0N(d_1)\), the stock price today times a probability that is \(0\) if the stock is below the strike price but is some probability representing the stock's value if it's above the strike price. Roughly, it's an investor's return, minus the cost of the option.
Discounting to Present Value: The \(e^{r(T-t)}\) portion of the formulation is simply a calculation of the present value of that strike price. It compounds the risk-free interest rate over the period between now (when the calculation is done) and the future expiration date. This is done because the price of this option should reflect that alternative risk-free choice an investor has. If an investor could put some money in a risk-free T-Bill and get a \(2\%\) return over one year, then an option needs to generate an additional return above and beyond that \(2\%\) to justify the increased risk associated with it.
Probability: Those probability weightings, \(N(d_1)\) and \(N(d_2)\), come from a normal probability distribution curve. If an investor graphed the periodic daily returns (the returns for this option each day) the resulting graph would be a normal distribution, a bell-shaped curve, like the one for the Oil ETF to the right. Just as the historical prices were normally distributed, the B-S-M model assumes that future prices will be normally distributed. Therefore \(N(d_1)\) is, roughly, looking for \(N(z\text{-score})\), the area under the bell curve up to some \(z\)-score, or the probability that the future price will be above the strike price on the expiration date. The standard notation for \(z\)-score is
\[z\text{-score} = \frac{x - \mu}{\sigma}.\]
The \(N(d_1)\) and \(N(d_2)\) functions are simply calculations of area on the curve. For instance, a calculation of \(N(d_2)\) for this Oil ETF is represented in the image to the right. If the curve is the normal distribution of all probabilities for the option, then \(N(d_2)\) is the percentage of probabilities that the option will expire in the money.
Example + Problem
Given the complexity of the model, it's always good to see it in action:
Smart investors calculate the price of an option for themselves before they buy. If you have the chance to buy a European call option with the following parameters, what cost should you pay less than to make it worth it?
- stock price: $50
- strike price: $45
- time to expiration: 80 days
- risk-free interest rate: 2%
- implied volatility: 30%
In other words, using the B-S-M formula, what should the cost of this call option be?
While the formula is involved, this is essentially a matter of plugging in the given variables:
\[\begin{align} d_1 &= \frac{\ln{\frac{S_0}{K}} + \big(r + \frac{\sigma^2}{2}\big)(T-t)}{\sigma\sqrt{T-t}}\\\\ &=\frac{\ln{\frac{50}{45}} + \big(0.02 + \frac{.3^2}{2}\big)\big(\frac{80}{365}\big)}{0.3\sqrt{\frac{80}{365}}}\\\\ &=\frac{0.105+0.014}{0.140} \approx 0.851 \\\\\\ d_2 &= d_1 - \sigma\sqrt{T-t} \\ &= 0.851 - 0.3\sqrt{\frac{80}{365}} \approx 0.711. \end{align}\]
\(N(d_1)\) and \(N(d_2)\) can be found by looking at a \(z\)-score table:
\[\begin{align} N(d_1) &= 0.8023 \\ N(d_2) &= 0.7611\\\\ \Rightarrow C(S_0,t) &= S_0N(d_1) - Ke^{-r(T-t)}N(d_2)\\ &=($50\times 0.8023) - \left($45\times e^{\big(-0.02\times \frac{80}{365}\big)}\times 0.7611\right) \\ &= $40.12 - $34.10 \\ &= $6.02.\ _\square \end{align}\]
Note: In the case of this stock, there is a probability of \(\approx80\%\) that represents the expected value at expiration, which is multiplied by \(\$50\) to yield a \(\$40.12\) return. The strike price is \(\$45\) and its present value is \(\$44.80,\) which is multiplied by \(\approx76\%,\) probability of the call option expiring in the money, to yield \(\$34.10\).
Suppose you're an investor and are curious what the market thinks the implied volatility of the S&P 500 is today. You know a few things:
- You can assume that every other investor is using Black-Scholes-Merton formula for pricing.
- You look a common ETF of the S&P 500, the SPY spider.
- Today it's priced at $216.
- A European call option has a strike price of $210.
- To expire, there are 30 days left from today.
- The risk-free interest rate is 1.8%.
- The market is pricing this European call option at $7.93.
What is the implied volatility?
(All answers are truncated.)
Note: Using Excel's "Goal Seek" may be helpful.
Hedging to "Eliminate" Risk
Once an asset is priced, the key idea is to hedge the option by buying and selling the underlying asset in just the right way so as to "eliminate risk." This is referred to as delta hedging or dynamic hedging. The idea is to maintain a zero option Greeks — delta, where delta is the sensitivity of an option to changes in the price of the underlying assets. This is a fairly complex form of hedging, and is principally performed by large investment institutions (investment banks, hedge funds, private equity funds, etc.). The formal calculation for delta is \[\Delta = \frac{\partial V}{\partial S}.\] That is the first derivative of the value of the option over the first derivative of the value of the underlying asset. As such, the basic strategy of delta hedging is to buy or sell some of the underlying asset (the denominator) in responses to changes in the value of that asset; it is to keep \({\partial S}\) static, even though the value of that asset change regularly.
One important note is that the risk eliminated here is not the risk that the underlying asset will go down in value; this is not to prevent normal negative returns from assets simply not performing, but to eliminate the more sharp shifts in price not correlated to changes in underlying value — the tail ends of that periodic daily return graph above where in a single day the price of an asset can change significantly, and then correct back to the original price in following days. Also, risk is never really "eliminated." That is the goal, but there are always risks, like default risk, that are harder to control for.
Criticisms of the Black-Scholes-Merton Model
Nassim Nicholas Taleb, famous for his 2007 bestselling book "Black Swan" which discussed unpredictable events in financial markets, along with Espen Gaarder Haug has criticized the Black-Scholes-Merton model, saying that it is "fragile to jumps and tail events" and can only handle "mild randomness."^{[3]} This is one of a few known challenges to the model:
- Fragility to "tail-risk" or other extreme randomness: In general, returns do not absolutely follow a normal distribution. The \(p\)-value on the Anderson-Darling normality test is 0.000 when applied to S&P returns, showing that market returns are leptokurtic (having greater kurtosis, or more concentrated about the mean with fat tails).^{[4]}
- The structure of B-S-M doesn't reflect present realities: The B-S-M model assumes a market using European call options when most options traded today are American call options that can be sold at any point. It also does not allow for dividends, something that is commonly found in options.
- Assumption of a risk-free interest rate: A theoretical calculation of risk-free rates is hard to come up with and, in practice, investors use proxies like the long-term yield on the US Treasury coupon bonds (generally 10-year bonds). However, this assumes that US Treasury bonds are "risk-free" when a more accurate statement would be that they're what the market assumes the least risky investment vehicles.
- Assumption of costless trading: Trading generally comes with exchange fees, the costs to buy or sell stocks and options, and the cost of time; the time it takes for the order to go through may result in changes to the price on the market. These costs can be managed, but are not included in the model.
- Gap risk: Also the model assumes that trading occurs continuously, unlike reality, where markets shut down for the night and then can reopen at significantly different prices to reflect new information.
Empirically, significant pricing discrepancies between B-S-M and reality have occurred more often than if returns were log-normal. But the B-S-M model continues to be used. It is simple, easy to determine, and can be adjusted for various inadequacies.
One of the more common criticisms of the B-S-M model is the existence of a volatility smile. The Black-Scholes-Merton pricing model suggests a constant volatility and log-normal distributions of returns, where, in reality, implied volatility varies widely. Options whose strike price are said to be "deep-in-the-money" or "out-of-the-money," i.e. whose strike price is further away from the assumed underlying asset price, command higher prices than a flat volatility would suggest — their implied volatility is higher.
Discussion
The notation is not standard mathematical notation but is the standard forms used in the finance industry.
- What is called a normal distribution is not a normal distribution; rather, it is the cumulative distribution function of a log-normal distribution. The use of an underlying normal distribution with a mean of 0 and a standard deviation of 1 is assumed and seldom mentioned. \[\]
- The use of the log-normal distribution is because the compound interest, which is a power law, is being modeled. Taking the logs of the growth factors makes the growth factors nearly linear and the distribution nearly normal. The values of \(mu\) and \(\sigma\) are the expected growth factor (interest rate) and the expected standard deviation (volatility) for one time period. Therefore, values close to 0 are expected.
- Continuous functions are used to model discrete functions to simplify the computations without warning, e.g. dividends and interest computed continually and not periodically. This fact is not mentioned in the discussion. Mathematicians do this also, but they generally mention the practice.
- What is being modeled is a random one-dimensional walk or martingale. Since a binomial distribution models a normal distribution over a large number of trials, e.g. the changes in prices over a year's time, this modeling of the normal distribution is a reasonable approximation.
The use of the logarithm function requires that the argument be a positive real to avoid infinities and complex numbers:
\[\text{Log Normal Distribution}\big[\mu ,\sigma \big]= \text{Transformed Distribution}\big[\exp (x),x \approx \text{Normal Distribution}[\mu ,\sigma ]\big]\]
\(\text{PDF}[\text{NormalDistribution}[\mu ,\sigma ]] \Rightarrow x\to \frac{e^{-\frac{(x-\mu )^2}{2 \sigma ^2}}}{\sqrt{2 \pi } \sigma }\)
\(\text{PDF}[\text{LogNormalDistribution}[\mu ,\sigma ]] \Rightarrow\) \(x\to\) \( \begin{array}{cc} \frac{e^{-\frac{(\log (x)-\mu )^2}{2 \sigma ^2}}}{x \sqrt{2 \pi } \sigma } & x>0 \\ 0 & \text{True} \\ \end{array} \)
\(\text{Mean}[\text{LogNormalDistribution}[\mu t,\sigma t]] = \text{Expectation}[x,x\overset{\sim}{\sim}\text{LogNormalDistribution}[\mu t,\sigma t]] = e^{\frac{\sigma ^2 t^2}{2}+\mu t}\)
Using material from Wolfram Mathematica 12 Expectation documentation: \[\text{expectedPrice}[\text{currentPrice},\mu,\sigma,t]= \\ \text{currentPrice}\times\text{Mean}[\text{LognormalDistribution}[\mu\,t,\sigma\,t]] \Rightarrow \\ \text{currentPrice}\!\times\mathbb{e}^{\mu t+\frac{{\sigma}^2}{2}\,t}\]
"Assuming an investor can invest money in a stock with dividend yield \(q\) for a year at a continuously compounded yearly rate \(r\) risk-free, the risk-neutral pricing condition requires:"
With the \(\text{currentPrice}\) factored out of both sides of the equation and the increase of value caused by risk-free interest rate less the effective interest rate from the dividend yield, assuming that both rates are compounded continuously:
\[\text{expectedPrice}\left(\text{currentPrice},\mu ,\sigma ,t+1\right)=\mathbb{e}^{r-q}\,\text{expectedPrice}\left(\text{currentPrice},\mu ,\sigma ,t\right)\]
\[\mathbb{e}^{\mu\,(t+1)+\frac{1}{2} \sigma^2 (t+1)}=\mathbb{e}^{-q+r+\mu \,t+\frac{\sigma ^2 t}{2}}\]
Solving for \(\mu\) over all positive time gives \(\mu=\frac{1}{2} \left(-2 q+2 r-\sigma ^2\right)\).
"Consider a call option to buy this stock a year from now, at a fixed price \(\mathcal{K}\). The value of such an option is:"
\[\text{callOptVal}(\text{expectedPrice},\mathcal{K})=\max ((\text{expectedPrice}-\mathcal{K},0)\]
This is because a call option is worthless if an immediate profit can not be made.
"[C]onsider a put option to sell this stock a year from now, at a fixed price \(\mathcal{K}\). The value of such an option is:"
\[\text{putOptVal}(\text{expectedPrice},\mathcal{K})=\max (\mathcal{K}-\text{expectedPrice},0)\]
This is because a put option is worthless if an immediate profit can not be made.
In the formulas below, all parameters are positive real, \(\mu\) is as computed above and the distribution is as in the argument to the Mean function above:
The call option price is \(\mathbb{e}^{-r\,t} \text{Expectation}[\text{callOptVal}(f\,\text{currentPrice},\mathcal{K}]),f\overset{\sim}{\sim} \text{LognormalDistribution}(\mu,\sigma,t)]\).
The put option price is \(\mathbb{e}^{-r\,t} \text{Expectation}[\text{putOptVal}(f\,\text{currentPrice},\mathcal{K}]),f\overset{\sim}{\sim} \text{LognormalDistribution}(\mu,\sigma,t)]\).
\[\text{BlackScholesCallOptionPrice}(\text{currentPrice},\mathcal{K},r,q,\sigma,t)= \\ \frac{1}{2} \mathbb{e}^{-r t} \left(\text{currentPrice}\times\mathbb{e}^{t (r-q)} \left(\text{erf}\left(\frac{-2 \log (\mathcal{K})+t \left(-2 q+2 r+\sigma ^2\right)+2 \log (\text{currentPrice})}{2 \sqrt{2} \sigma \sqrt{t}}\right)+1\right)- \\ \mathcal{K}\,\text{erfc}\left(\frac{2 \log (\mathcal{K})+t \left(2 q-2 r+\sigma ^2\right)-2 \log (\text{currentPrice})}{2 \sqrt{2} \sigma \sqrt{t}}\right)\right)\]
\[\text{BlackScholesCallOptionPrice}(\text{currentPrice},\mathcal{K},r,q,\sigma,t)= \\ \frac{1}{2} \mathbb{e}^{-r t} \left(\mathcal{K}\,\text{erf}\left(\frac{t \left(2 q-2 r+\sigma ^2\right)+2 \log \left(\frac{\mathcal{K}}{\text{currentPrice}}\right)}{2 \sqrt{2} \sigma \sqrt{t}}\right)- \\ \text{currentPrice}\times\mathbb{e}^{t (r-q)} \text{erfc}\left(\frac{-2 \log (\mathcal{K})+t \left(-2 q+2 r+\sigma ^2\right)+2 \log (\text{currentPrice})}{2 \sqrt{2} \sigma \sqrt{t}}\right)+\mathcal{K}\right)\]
The function \(\text{erf}(z)=\frac{2}{\sqrt{\pi }} \int_0^z e^{-t^2} \, dt\) and \(\text{erfc}(z)=1-\text{erf}(z)\). Both functions assume a normal distribution with a mean of 0 and a standard deviation of 1. This means that the arguments must be normalized to a a mean of 0 and a standard deviation of 1 which has been done in the call and option value formulae above.
\[\text{CDF}[\text{NormalDistribution}[0,1],x] = \frac{1}{2} \text{erfc}\left(-\frac{x}{\sqrt{2}}\right)\] \[\text{erfc}(-z)=1+\text{erf}(z)\]
References
- Black, F., & Scholes, M. The Pricing of Options and Corporate Liabilities. Retrieved November 1, 2016, from http://www.jstor.org/stable/1831029
- Merton, R. Theory of Rational Option Pricing. Retrieved November 1, 2016, from http://www.jstor.org/stable/3003143
- Haug, E., & Taleb, N. Why We Have Never Used the Black–Scholes–Merton Option Pricing Formula. Retrieved November 9th 2016, from http://polymer.bu.edu/hes/rp-haug08.pdf
- Hurvich, C. SOME DRAWBACKS OF BLACK-SCHOLES. Retrieved November 9th 2016, from http://people.stern.nyu.edu/churvich/Forecasting/Handouts/Scholes.pdf
- Brianegge, . Volatility Smile. Retrieved November 9th 2016, from https://en.wikipedia.org/wiki/File:Volatility_smile.svg