The Black-Scholes world[edit]
The Black–Scholes model assumes that the market consists of at least one risky asset, usually called the stock, and one riskless asset, usually called the money market, cash, or bond.
Now we make assumptions on the assets (which explain their names):
(riskless rate) The rate of return on the riskless asset is constant and thus called the risk-free interest rate.
(random walk) The instantaneous log returns of the stock price is an infinitesimal random walk with drift; more precisely, it is a geometric Brownian motion, and we will assume its drift and volatility is constant (if they are time-varying, we can deduce a suitably modified Black–Scholes formula quite simply, as long as the volatility is not random).
The stock does not pay a dividend.[Notes 1]
Assumptions on the market:
There is no arbitrage opportunity (i.e., there is no way to make a riskless profit).
It is possible to borrow and lend any amount, even fractional, of cash at the riskless rate.
It is possible to buy and sell any amount, even fractional, of the stock (this includes short selling).
The above transactions do not incur any fees or costs (i.e., frictionless market).
With these assumptions holding, suppose there is a derivative security also trading in this market. We specify that this security will have a certain payoff at a specified date in the future, depending on the value(s) taken by the stock up to that date. It is a surprising fact that the derivative's price is completely determined at the current time, even though we do not know what path the stock price will take in the future. For the special case of a European call or put option, Black and Scholes showed that "it is possible to create a hedged position, consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock".[5] Their dynamic hedging strategy led to a partial differential equation which governed the price of the option. Its solution is given by the Black–Scholes formula.
Several of these assumptions of the original model have been removed in subsequent extensions of the model. Modern versions account for dynamic interest rates (Merton, 1976)[citation needed], transaction costs and taxes (Ingersoll, 1976)[citation needed], and dividend payout.[6]
Notation[edit]
Let
S, be the price of the stock, which will sometimes be a random variable and other times a constant (context should make this clear).
V(S, t), the price of a derivative as a function of time and stock price.
C(S, t) the price of a European call option and P(S, t) the price of a European put option.
K, the strike price of the option.
r, the annualized risk-free interest rate, continuously compounded (the force of interest).
\mu, the drift rate of S, annualized.
\sigma, the standard deviation of the stock's returns; this is the square root of the quadratic variation of the stock's log price process.
t, a time in years; we generally use: now=0, expiry=T.
\Pi, the value of a portfolio.
Finally we will use N(x) to denote the standard normal cumulative distribution function,
N(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x} e^{-\frac{z^2}{2}}\, dz.
N'(x) will denote the standard normal probability density function,
N'(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}
The Black–Scholes equation[edit]
Main article: Black–Scholes equation
Simulated geometric Brownian motions with parameters from market data
As above, the Black–Scholes equation is a partial differential equation, which describes the price of the option over time. The equation is:
\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0
The key financial insight behind the equation is that one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk".[citation needed] This hedge, in turn, implies that there is only one right price for the option, as returned by the Black–Scholes formula (see the next section).
Black-Scholes formula[edit]
A European call valued using the Black-Scholes pricing equation for varying asset price S and time-to-expiry T. In this particular example, the strike price is set to unity.
The Black–Scholes formula calculates the price of European put and call options. This price is consistent with the Black–Scholes equation as above; this follows since the formula can be obtained by solving the equation for the corresponding terminal and boundary conditions.
The value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is:
\begin{align}
C(S, t) &= N(d_1)S - N(d_2) Ke^{-r(T - t)} \\
d_1 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right] \\
d_2 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S}{K}\right) + \left(r - \frac{\sigma^2}{2}\right)(T - t)\right] \\
&= d_1 - \sigma\sqrt{T - t}
\end{align}
The price of a corresponding put option based on put–call parity is:
\begin{align}
P(S, t) &= Ke^{-r(T - t)} - S + C(S, t) \\
&= N(-d_2) Ke^{-r(T - t)} - N(-d_1) S
\end{align}\,
For both, as above:
N(\cdot) is the cumulative distribution function of the standard normal distribution
T - t is the time to maturity
S is the spot price of the underlying asset
K is the strike price
r is the risk free rate (annual rate, expressed in terms of continuous compounding)
\sigma is the volatility of returns of the underlying asset
Alternative formulation[edit]
Introducing some auxiliary variables allows the formula to be simplified and reformulated in a form that is often more convenient (this is a special case of the Black '76 formula):
\begin{align}
C(F, \tau) &= D \left( N(d_+) F - N(d_-) K \right) \\
d_\pm &=
\frac{1}{\sigma\sqrt{\tau}}\left[\ln\left(\frac{F}{K}\right) \pm \frac{1}{2}\sigma^2\tau\right] \\
d_\pm &= d_\mp \pm \sigma\sqrt{\tau}
\end{align}
The auxiliary variables are:
\tau = T - t is the time to expiry (remaining time, backwards time)
D = e^{-r\tau} is the discount factor
F = e^{r\tau} S = \frac{S}{D} is the forward price of the underlying asset, and S = DF
with d+ = d1 and d− = d2 to clarify notation.
Given put-call parity, which is expressed in these terms as:
C - P = D(F - K) = S - D K
the price of a put option is:
P(F, \tau) = D \left[ N(-d_-) K - N(-d_+) F \right]
Interpretation[edit]
The Black–Scholes formula can be interpreted fairly handily, with the main subtlety the interpretation of the N(d_\pm) (and a fortiori d_\pm) terms, particularly d_+ and why there are two different terms.[7]
The formula can be interpreted by first decomposing a call option into the difference of two binary options: an asset-or-nothing call minus a cash-or-nothing call (long an asset-or-nothing call, short a cash-or-nothing call). A call option exchanges cash for an asset at expiry, while an asset-or-nothing call just yields the asset (with no cash in exchange) and a cash-or-nothing call just yields cash (with no asset in exchange). The Black–Scholes formula is a difference of two terms, and these two terms equal the value of the binary call options. These binary options are much less frequently traded than vanilla call options, but are easier to analyze.
Thus the formula:
C = D \left[ N(d_+) F - N(d_-) K \right]
breaks up as:
C = D N(d_+) F - D N(d_-) K
where D N(d_+) F is the present value of an asset-or-nothing call and D N(d_-) K is the present value of a cash-or-nothing call. The D factor is for discounting, because the expiration date is in future, and removing it changes present value to future value (value at expiry). Thus N(d_+) ~ F is the future value of an asset-or-nothing call and N(d_-) ~ K is the future value of a cash-or-nothing call. In risk-neutral terms, these are the expected value of the asset and the expected value of the cash in the risk-neutral measure.
The naive, and not quite correct, interpretation of these terms is that N(d_+) F is the probability of the option expiring in the money N(d_+), times the value of the underlying at expiry F, while N(d_-) K is the probability of the option expiring in the money N(d_-), times the value of the cash at expiry K. This is obviously incorrect, as either both binaries expire in the money or both expire out of the money (either cash is exchanged for asset or it is not), but the probabilities N(d_+) and N(d_-) are not equal. In fact, d_\pm can be interpreted as measures of moneyness (in standard deviations) and N(d_\pm) as probabilities of expiring ITM (percent moneyness), in the respective numéraire, as discussed below. Simply put, the interpretation of the cash option, N(d_-) K, is correct, as the value of the cash is independent of movements of the underlying, and thus can be interpreted as a simple product of "probability times value", while the N(d_+) F is more complicated, as the probability of expiring in the money and the value of the asset at expiry are not independent.[7] More precisely, the value of the asset at expiry is variable in terms of cash, but is constant in terms of the asset itself (a fixed quantity of the asset), and thus these quantities are independent if one changes numéraire to the asset rather than cash.
If one uses spot S instead of forward F, in d_\pm instead of the \frac{1}{2}\sigma^2 term there is \left(r \pm \frac{1}{2}\sigma^2\right)\tau, which can be interpreted as a drift factor (in the risk-neutral measure for appropriate numéraire). The use of d− for moneyness rather than the standardized moneyness m = \frac{1}{\sigma\sqrt{\tau}}\ln\left(\frac{F}{K}\right) – in other words, the reason for the \frac{1}{2}\sigma^2 factor – is due to the difference between the median and mean of the log-normal distribution; it is the same factor as in Itō's lemma applied to geometric Brownian motion. In addition, another way to see that the naive interpretation is incorrect is that replacing N(d+) by N(d−) in the formula yields a negative value for out-of-the-money call options.[7]:6
In detail, the terms N(d_1), N(d_2) are the probabilities of the option expiring in-the-money under the equivalent exponential martingale probability measure (numéraire=stock) and the equivalent martingale probability measure (numéraire=risk free asset), respectively.[7] The risk neutral probability density for the stock price S_T \in (0, \infty) is
p(S, T) = \frac{N^\prime [d_2(S_T)]}{S_T \sigma\sqrt{T}}
where d_2 = d_2(K) is defined as above.
Specifically, N(d_2) is the probability that the call will be exercised provided one assumes that the asset drift is the risk-free rate. N(d_1), however, does not lend itself to a simple probability interpretation. SN(d_1) is correctly interpreted as the present value, using the risk-free interest rate, of the expected asset price at expiration, given that the asset price at expiration is above the exercise price.[8] For related discussion – and graphical representation – see section "Interpretation" under Datar–Mathews method for real option valuation.
The equivalent martingale probability measure is also called the risk-neutral probability measure. Note that both of these are probabilities in a measure theoretic sense, and neither of these is the true probability of expiring in-the-money under the real probability measure. To calculate the probability under the real ("physical") probability measure, additional information is required—the drift term in the physical measure, or equivalently, the market price of risk.
Derivations[edit]
See also: Martingale pricing
A standard derivation for solving the Black–Scholes PDE is given in the article Black-Scholes equation.
The Feynman-Kac formula says that the solution to this type of PDE, when discounted appropriately, is actually a martingale. Thus the option price is the expected value of the discounted payoff of the option. Computing the option price via this expectation is the risk neutrality approach and can be done without knowledge of PDEs.[7] Note the expectation of the option payoff is not done under the real world probability measure, but an artificial risk-neutral measure, which differs from the real world measure. For the underlying logic see section "risk neutral valuation" under Rational pricing as well as section "Derivatives pricing: the Q world" under Mathematical finance; for detail, once again, see Hull.[9]:307–309
