Black and Scholes option pricing model, from intuition to formula
The Black and Scholes equation has led to the most basic and most widely known and applied option pricing model. Its simplicity has allowed for a widespread implementation in the financial market industry, but also in some subjects of corporate finance (real option pricing for example). Even if it is also famous for its short-comings, BS has proven to be a robust way of computing vanilla European option prices (once we know its limitations).
Fischer Black and Myron Scholes offered a break-through option valuation model in a publication in 1973. It was the first ever closed-form formula that allowed to compute precisely the right price for a generic European stock option. By “closed-form” understand that the formula is sufficient in itself, and doesn’t require iterations or additional processing, simply input the right variables in the formula and you get the result.
I won’t pretend to be a math person, I am not. Instead of giving you a math oriented explanation of this model, I will try and give practical explanations and interpretations so that the formulas make sense to you. It is good to know a formula by heart if you need to refer to it, especially Black & Scholes which is such a landmark. But it is even better to understand why this formula makes sense and why it is written that way! I won’t have a simple explanation for everything below, and although I will put a lot of effort in proof-reading and validation of what I write, please let me know if you see what you would describe as an error, I would be very happy to correct and reference your input in the sources!
The Black & Scholes option pricing formula
Without further introduction (and I know most of you have no time to lose in useless digressions), behold the so called Black and Scholes option call and put pricing formula:
Call price
$$C(S_0,t)=N(d1)S_0 +N(d2)K e^{-r(T-t)} $$
Put - Call parity
$$P(S_0,t)=Ke^{-r(T-t)}-S_0+C(S,t)$$
Put price
$$P(S_0,t)=N(-d2)K e^{-r(T-t)}-N(-d1)S_0 $$
Where
$d1= 1/(σ√(T-t)) [ln(S_0/K)+(r+σ^2/2)(T-t)]$
$d2= 1/(σ√(T-t)) [ln(S_0/K)+(r-σ^2/2)(T-t)]$
$d2= d1- σ√{T-t}$
Here are the main parameters:
- σ is the (flat) volatility of the underlying. It represent the risk and uncertainty of the return distribution, it is used in d1 and d2.
- K is the strike price of the option, $S_0$ is the underlying price when the option is traded,
- t is the time at which we compute the option price, T is the time to maturity (t < T obviously),
- r is the (flat) risk-free interest rate,
- N() is the normal cumulative probability distribution,
- N(d1) is the probability of expiring in the money under the equivalent exponential martingale measures using the stock as numéraire and
- N(d2) is the probability of expiring in the money using the risk-free asset as numéraire.
In the rest of this article, we explain the assumptions behind this formula, and how we can interprete it. Specifically, what are the meanings of $N(d1) S_0$ and $N(d2) K e^{-r(T-t)}$, d1 and d2... Let's have a look!
Understanding how Black and Scholes works
The vanilla call option price can be decomposed into a portfolio of two simpler options
Here we will theoretically decompose a stock option as the sum of two more elemental options, to understand where the value of the option comes from.
When you hold an option, you can either do nothing at maturity, or you can exercise it and receive the stock against the strike price. Obviously you will exercise it only if you can sell that stock at a better price in the market. That is only if St > K where St is Market spot price at time t (maturity) and K is the strike price. So at expiry your profit on a call option is $MAX(0, St – K)$. Ok, you probably already knew that.
Now imagine two products:
- Product A: buyer gets the Stock if St > K at exercise date, he doesn’t have to pay the strike price for it. He gets the stock literally for free but can only have it if St > K.
- Product B: buyer gets cash amount K if St > K at exercise date.
Suppose we buy product A and sell product B, the payoff at maturity is the same as the call option:
- 0 if St <= K,
- St – K if St > K
A very important assumption of the Black and Scholes model (and of most financial models, really) is that the market is “arbitrage free”, meaning that two products with exactly the same payoff will have the same price. If we assume that this assumption holds, then [A - B] and a call option must have the same value, right?
The Black and Scholes formula is built on the previous example. The value of the vanilla call option is the value of A - B.
What do $S_0$ and $K e^{-r(T-t)}$ represent in the Black & Scholes formula?
It can be shown that the call option price is somewhere in-between 0 and $S_T - K$. It is easy to be convinced of this. On the one hand, it is not possible for the option price to be negative, because an option is in essence a right for its buyer, but not an obligation. It can only provide value to the option buyer. If it doesn't, the option won't be used, so its value in this case is 0. On the other hand, if the call option is very much in the money, that is if $K << S_0$, then the value of the option at any point in time t until maturity will most probably be very close to $S_t - K$ where $S_t$ is the underlying price at time t.
By no means the option can be worth more at maturity than $S_T - K$, where $S_T$ is the price of the underlying at time T (maturity), whatever its level of moneyness. At time t before maturity, $S_T$ will usually be replaced by the forward value of the underlying at maturity $F_t$ so that the option can't be worth more than the present value at time t of $F_t - K$. When the option is traded, we don't know $S_T$, so we can replace it by the future price $F_0 = S_0 e^{r T}$, as we just described. Note that the present value of $F_0 - K$ is $S_0 - K e^{-rT}$.
So far, we still can't really value the option, we can only say that :
$$ 0 ≤ C_0 ≤ S_0 - K e^{-rT}$$
(with $C_0$ the price of the vanilla call option when it is traded)
To go further, we need to price more precisely products A and B described in the previous paragraph. And for this we need to understand better the probabilities for options A and B to end-up in the money.
The problem with "real world" probabilities
How can we price A and B? The absolute rule in finance is that the fundamental value of an asset is the discounted value of its expected payoff (meaning the probability weighted average payoffs in all possible future states of the world). However, investors are assumed in B&S to be risk averse.
Risk aversion means that the investors will be willing to pay more for a product which price doesn't variate much compared to a product with the same expected payoff but that is more variable. This is actually rather intuitive: imagine products I and J with the same expected future value of 100:
- I pays 200 EUR with a 50% probability, 0 otherwise,
- J pays 100 EUR with a 100% probability.
You understand that these two products have the same probability-weighted average value (100 EUR in both cases) but in practice, a risk averse investor will choose product J as it is less risky.
The assumptions that the investors are risk-averse is very important in the Black and Scholes framework. This implies that the price of a product isn't exactly its expected payoff (at least in real world probabilities). It is rather the following:
Value for risk-averse investors of any risky financial product: Value = DiscountFactor × E(Payoff)–X , where X is an adjustment for uncertainty. A product with very uncertain cash flows (understand very variable, scattered) will have a higher X.
This formula is not extremely powerful and certainly not fit for derivative pricing, as the X value is hard to compute and has to be calibrated for each market, and even for investors who have different levels of risk aversion. To get around this problem, financial research has come up with the concept of risk neutral measure or equivalent Martingale measure.
Risk neutral measure
This concept consists in defining a probability measure that is completely theoretical and instead of reflecting only real world probabilities of payoffs, also reflects the risk aversion of investors for improbable extreme payoffs.
With such a theoretical probability measure (let it be Q) we would have the following formula for the value of a financial product: $Value = DiscountFactor × Q_{Expected}(Payoff)$ where $Q_{Expected} (Payoff)$ designates the expected value of payoffs in the probability measure Q.
To get things clearer, P ("real world" probability measure) and Q (theoretical risk-neutral probability measure) are similar but Q includes some sort of correction for investors' risk aversion.
Real world probability |
$Value = DiscountFactor × E_P(Payoff) – X$ |
Risk neutral probability |
$Value = DiscountFactor × E_Q(Payoff)$ |
It is important to note that we don’t really care about finding out what the actual level of the risk-neutral probability is. It is more a theoretical notion that is used in the Black and Scholes framework (and many other financial models too). The name of that measure (“risk-neutral”) must not be misleading. Risk-neutral doesn’t imply investors are neutral to risk. On the contrary! That risk-neutral measure allows weighting the expected future value of a financial product in such a way that aversion to risk isn’t an additional factor in the pricing, but is included in the probability, and then makes the pricing neutral to the specific risk aversion of the investor.
The difference between the real-world probability and the risk-neutral probability is described as the Radon-Nicodym derivative of the risk-neutral measure.
In order for the market to be arbitrage free, one and only one risk-neutral measure must exist.
How to use Risk-Neutral pricing in practice?
Here we will have to make large cuts in demonstrations as the numerical considerations are out of the scope here.
Lets consider the option that gives cash amount K if it expires in the money or nothing otherwise (product B in the example above). It is quite straight forward to use the risk neutral measure Q to define $Price of B = e^{-r T} E_Q[ K × 1_{S_T > K } ]$ where $1_{S_T > K }$ is the indicative function that is equal to 1 if $S_T > K$ and 0 otherwise. It can be shown in a development using Ito Lemma that $E_Q[1_{S_T > K }] = P_Q(S_T > K) $, and that:
$$ P_Q(S_T > K) = P_Q( (r - {σ^2}/2) T + σ W_T > ln (K/S_0) )$$ where $W_T$ is a Weiner process.
$$ = P_Q( W_T > {ln (K/S_0) - (r - {σ^2}/2) T } / σ )$$
$$ = P_Q( {W_T}/√T > {ln (K/S_0) - (r - {σ^2}/2) T } / {σ √T} )$$
Let G ∼ N(0,1) and $d2 = {ln (S_0/K) + (r - {σ^2}/2) T }/{σ √T} = - {ln (K/S_0) - (r - {σ^2}/2) T } / {σ √T} $
Note that G and -G have the same distribution (the standard normal distribution is symetric), then we have:
$$ P_Q(S_T > K) = P_Q(-G > -d2) = P_Q(G < d2) = N(d2) $$
We got slightly off topic with this small and partial demonstation! You only have to see that the expected value of the cash option can be computed using a cumulative standard normal distribution applied on $d2 = {ln (S_0/K) + (r - {σ^2}/2) T }/{σ √T} $.
What do N(d1) and N(d2) represent in the Black and Scholes formula?
N() is the normal cumulative probability distribution, and N(d1) represents the probability of expiring in the money under the equivalent exponential martingale measures with the stock as numéraire and N(d2) is the same type of probability with the risk-free asset as numéraire. The numeraire refers to the unit of measure taken to label the payoff and infer the risk-neutral measure.
In the case of the Cash option (product B described at the beginning), its payoff is constant, so it is appropriate to take the risk-free asset as numeraire, and use the risk free-rate as a discount factor. However in the case of the pure equity option (product A described at the beginning), the risk-free asset isn't appropriate as numeraire, as the payoff isn't constant but rather depends on $S_T$, the underlying value at maturity. $S_T$ is very hard to guess, we prefer to have a payoff that yields either 0 or a fixed number of units of numeraire. This is why we use the underlying as the numeraire in the risk-neutral pricing of the pure equity option. Using this numeraire, we will receive either 0 or 1 unit of numeraire at maturity.
The explanation above exhibits a change of numéraire, which is a widely used technic in derivative pricing models, but we won’t dig into the details here.
If we have a look into d2 that is assumed to be normally distributed under the risk-neutral measure, $d2 = {ln (S_0/K) + (r - {σ^2}/2) T }/{σ √T} $, here are some observations that we can make:
(1) ${ln (S_0/K)$ is clearly a measure of moneyness at time 0,
(2) $(r - {σ^2}/2) T$ refers to the drift of the underlying, that follows an îto Process,
(3) ${σ √T}$ represents the volatility of the underlying over the whole life of the option up to maturity where σ is the annualized maturity.
To conclude
It was a long article but it looks like we barely scratched the surface of the topics surrounding Black and Scholes. I really advise that you have a look into the other articles of the section of the blog dedicated to Black and Scholes and option pricing! I will add more articles on other topics and try to zoom on concepts like numéraire, risk-neutral pricing, Ito Lemma, or Wiener processes.
References
The Wikipedia articles on these subjects are a great read:
- https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model
- https://en.wikipedia.org/wiki/Num%C3%A9raire
- https://en.wikipedia.org/wiki/It%C3%B4_calculus
Many other documents exist on the internet and quality is usually great!
Please contact me if you have remarks on this article, I will be happy to improve this article and even quote your contribution in the references!