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An Intuitive Explanation of Black-Scholes

I explain the Black-Scholes formula using only basic probability
theory and calculus, with a focus on the big picture and intuition
over technical details.

Published

28 September 2024

The Black-Scholes formula is the crown jewel of quantitative finance.
The formula gives the fair price of a European-style option, and its
success can ultimately be measured by its impact on option markets.
Before the formula's publication in 1973 (Black & Scholes, 1973;
Merton, 1973), option markets were relatively small and illiquid, and
options were not traded in standardized contracts. But after the
formula's publication, option markets grew rapidly. The first
exchange to list standardized stock options, the Chicago Board
Options Exchange, was founded the same year that Black-Scholes was
published. And today, options are a highly liquid, mature, and global
asset class, with many different tenors, exercise rights, and
underlying assets.

The financial and mathematical theory underpinning Black-Scholes is
rich, and one could easily spend months learning the foundational
ideas: continuous-time martingales, Brownian motion, stochastic
integration, valuation through replication, and risk-neutrality to
name just a few key concepts. But properly contextualized, the
formula can be surprisingly inevitable. It can almost feel like a law
of nature rather than a financial model. My goal here is to justify
this claim.

To begin, let's setup the problem and then state the formula. Recall
that a call option (CCC) is a contract that gives the holder the
right but not obligation to buy the underlying asset (SSS) at an
agreed-upon strike price (KKK). A put option (PPP) is the right but
not obligation to sell the underlying short, but since calls and puts
are fungible through put-call parity, we will only concern ourselves
with call options in this post. If we can price one, we can price the
other. We say the holder exercises the option if they choose to buy
or sell the underlying. A European-style option can only be exercised
at a fixed time in the future, called expiry (TTT).

Clearly, the payoff at expiry of a European-style option is just a
piecewise linear ramp function,

payoff of CT=max[?][0,ST-K],(1) \text{payoff of $C_T$} = \max\left[0,
S_T - K\right], \tag{1} payoff of CT =max[0,ST -K],(1)

where STS_TST  denotes the value of the stock at expiry (black line,
Figure 111). The single most important characteristic of an option is
this asymmetric payoff. For a call option, our downside is limited,
but our upside is unlimited. In finance, this kind of asymmetric
behavior is called "convexity".

Given this, we might guess that the price of the call before expiry,
so CtC_tCt  where t<Tt \lt Tt<T, should look like a smooth
approximation of the ramp function (colored lines, Figure 111), at
least to a first approximation. Why? Imagine we hold a call before
expiry, but the underlying StS_tSt  is currently less than the strike
KKK. Our position is not worthless precisely because it's still
possible that the stock price will rise before expiry. So we should
still be able to sell the contract for more than zero. The closer the
stock StS_tSt  is to the strike KKK, the more we should be able to
re-sell our option for. Put differently, and this is the key point
here, the price of the option will change nonlinearly with the price
of the stock, and as time passes, the smooth approximation should
look more and more like the payoff function, because the option is
losing its optionality.

[black_scho]
Figure 1. The price of optionality (colored lines) should be always
above the payoff of an option (black line) because the option could
be worth something if the underlying price changes. More optionality
(higher underlying volatility, more time to expiry) should be priced
higher than less optionality. As time passes, an option loses its
optionality and converges towards the payoff at expiry.

This tension between time decay and convexity is the central dynamic
of an option, and the Black-Scholes formula encapsulates this tension
beautifully. According to the Black-Scholes model, the fair price of
a European-style call option is the following:

Ct=StPh(d1)-Ke-r(T-t)Ph(d2),d1:=1sT-t[log[?](St/K)+(r+(1/2)s2)(T-t)],d2:=
d1-sT-t.(2) \begin{aligned} C_t &= S_t \Phi(d_1) - K e^{-r(T-t)} \Phi
(d_2), \\ d_1 &:= \frac{1}{\sigma \sqrt{T-t}} \left[ \log(S_t/K) + \
left(r + (1/2) \sigma^2 \right) (T-t) \right], \\ d_2 &:= d_1 - \
sigma \sqrt{T-t}. \end{aligned} \tag{2} Ct d1 d2  =St Ph(d1 )-Ke-r(T-t
)Ph(d2 ),:=sT-t 1 [log(St /K)+(r+(1/2)s2)(T-t)],:=d1 -sT-t . (2)

Here, Ph(x)\Phi(x)Ph(x) is the cumulative distribution function (CDF)
of the standard normal distribution, s\sigmas is the standard
deviation or volatility of the underlying asset, and rrr is the
risk-free interest rate. Black-Scholes assumes that the volatility s\
sigmas and risk-free rate rrr are both constant.

At a high level, this formula is just the weighted difference between
the stock and strike prices. But the terms Ph(d1)\Phi(d_1)Ph(d1 ) and Ph
(d2)\Phi(d_2)Ph(d2 ) are not obviously interpretable. So can we make
headway here? Can we say something more precise without quickly
getting bogged down in mathematical details? Let's try.

Technical difficulties

Understanding how to derive Equation 222 from first principles
requires a complex set of mathematical and financial ideas. Perhaps
the trickiest part for most people is the use of stochastic calculus
(Ito, 1944; Ito, 1951; Bru & Yor, 2002). Stochastic calculus is
required because we assume the underlying stock price follows this
stochastic differential equation (SDE), which is a geometric Brownian
motion:

dStSt=mdt+s2dZt.(3) \frac{\text{d} S_t}{S_t} = \mu \text{d}t + \sigma
^2 \text{d} Z_t. \tag{3} St dSt  =mdt+s2dZt .(3)

Here, m\mum is the drift of the stock, and s\sigmas is the volatility
of the stock. This is the same variable in as Equation 222. Finally, 
dZt\text{d}Z_tdZt  is an infinitesimal change in a Brownian motion.

The important but subtle point here is that Equation 333 is
mathematically imprecise to most people, even those with strong
technical backgrounds. In standard calculus, we cannot take the
derivative of a random function. Informally, it breaks the required
assumption that the function is smooth enough and can therefore be
locally approximated by a tangent line. Thus, the notation dZt\text
{d} Z_tdZt  has no meaning in standard calculus. So it appears that
to even understand the generative model above, one must understand
stochastic calculus.

As an aside, it's worth mentioning that this technical difficulty is
made even more subtle in the original paper (Black & Scholes, 1973)
because it's only a technical detail. The main content of the paper
is the derivation of the famous Black-Scholes partial differential
equation (PDE), and this is done by reasoning about the dynamics of
an option. But since an option's price is a function of its
underlying stock's price, describing its dynamics via a PDE requires
partial derivative such as

[?]Ct[?]St.(4) \frac{\partial C_t}{\partial S_t}. \tag{4} [?]St [?]Ct  .(4)

And of course, computing this term requires stochastic calculus since
StS_tSt  is a random function. So without stochastic calculus, any
derivation of the Black-Scholes PDE is high-level at best. My
approach in this post is to circumvent the PDE entirely and to just
make sense of Equation 222 directly. The PDE is beautiful--and perhaps
I'll write a separate post about it in the future--but we can still
make progress by just thinking about the generative model of the
stock price in simple, probabilistic terms.

Lognormally distributed prices

So let's side-step the challenge of stochastic calculus implicit in
Equation 333. The key point is this: a differential equation is an
equation that expresses the relationship between a function and its
derivatives. And a solution here is the closed-form expression of 
StS_tSt  such that we could plug StS_tSt  and its derivative(s) into
Equation 333 and it would hold. A standard result is that the
solution to the SDE in Equation 333 is the following definition of 
StS_tSt :

St=S0exp[?]{(m-12s2)t+sZt},Zt~N(0,t).(5) S_t = S_0 \exp\left\{ \left( \
mu - \frac{1}{2} \sigma^2 \right) t + \sigma Z_t \right\}, \quad Z_t
\sim \mathcal{N}(0, t). \tag{5} St =S0 exp{(m-21 s2)t+sZt },Zt ~N(0,t
).(5)

If you want more detail, read about solving the SDE of geometric
Brownian motion. But the main idea for us is that the Black-Scholes
modeling assumption can be expressed in a form that does not require
stochastic calculus. Equation 333 can be reframed as the assumption
that stock prices are lognormally distributed or, put differently,
that log returns are normally distributed:

log[?]St-log[?]S0~N(mt-12s2t,s2t).(6) \log S_t - \log S_0 \sim \mathcal
{N}\left(\mu t - \frac{1}{2} \sigma^2 t, \sigma^2 t \right). \tag{6} 
logSt -logS0 ~N(mt-21 s2t,s2t).(6)

So as a pedagogical trick, let's just assume Equations 555 and 666
rather than Equation 333. This is our new starting point.

For some intuition for the leap between the two, recall that the
derivative of log[?]x\log xlogx is 1xdx\frac{1}{x}\text{d}xx1 dx, and
this looks a bit like the left-hand side of Equation 333. So think of
the left-hand side of Equation 333 as a log return. And think of the
right-hand side of Equation 333 as a normally distributed random
variable, since it is an incremental (additive) change in Brownian
motion plus a constant drift. The extra term, -(1/2)s2t-(1/2) \sigma^
2 t-(1/2)s2t, can only be properly understood with stochastic
calculus--it's from the quadratic variation of Brownian motion--, but
it provides no real intuition for us here, and we will take it as a
given.

Let's look at some examples. In Figure 222, I have plotted the stock
price over varying drifts m\mum and volatilities s\sigmas. This looks
like random noise with drift because that's precisely what it is. The
normal distribution is as random as it gets, in the sense that the
Central Limit Theorem states that the appropriately scaled sum of
independent random variables converges to the normal distribution.
Things that aren't initially normal can still become normal over
time.

[drift]
Figure 2. Thousands of random samples from geometric Brownian motion
with an initial stock price S0=100S_0 = 100S0 =100 and varying drifts
and volatilities.

Now if log[?]St\log S_tlogSt  is normally distributed, then StS_tSt  is
lognormally distributed. So as time ttt passes, the stock price StS_t
St  is always lognormally distributed with a variance that increases
linearly with time and thus a volatility that increases with the
square root of time. To visualize this assumption, I've plotted the
appropriate lognormal distribution at various time slices along with
a many empirical samples (Figure 333). In my mind, this figure
captures the geometric essence of Equations 333 and 555.
Black-Scholes assumes that the stock price is unpredictable modulo
the drift, and that "infinitesimal change in Brownian motion",
mathematically vague for the uninitiated, amounts to an uncertainty
about the stock price that grows with the square root of time.

[lognormal]
Figure 3. Thousands of random samples from geometric Brownian motion
with an initial stock price S0=100S_0 = 100S0 =100, drift m=0.2\mu =
0.2m=0.2, and volatility s=0.15\sigma = 0.15s=0.15. The lognormal
distribution at various time points (Equation 666) is plotted in
dashed black lines.

Of course, we could make the same figure but using log returns. In
that case, the lines indicating the lognormal distributions in Figure
333 would change to indicate normal distributions. In either case,
the generative model of Black-Scholes is that prices and therefore
returns are unpredictable.

Now pause. We're going to make a subtle but critical tweak to our
assumptions so far. We're going to replace the stock-specific drift m
\mum with the risk-free interest rate rrr. Concretely, rather than
assuming Equation 666, we're going to assume a stock's log returns
follow this normal distribution:

log[?]St-log[?]S0~N(rt-12s2t,s2t).(7) \log S_t - \log S_0 \sim \mathcal
{N}\left(r t - \frac{1}{2} \sigma^2 t, \sigma^2 t \right). \tag{7} lo
gSt -logS0 ~N(rt-21 s2t,s2t).(7)

We haven't discussed rrr in detail yet, but this is just the an
idealized number representing the time-value of money. To a first
order, think of rrr as the interest rate from a very secure or
reliable asset, such as a short-term US government bond. It is the
risk-free rate and thus the lower-bound on what you can earn without
risk.

Of course, different stocks might be modeled with different drifts.
The drift of a blue-chip company might not be the same as the drift
of a penny stock. So in essence, this assumption is a claim that
Black-Scholes is making: the drift of the stock doesn't matter when
pricing an option! This is a surprising and deep claim. Let's
understand it.

Risk neutral world

In the original paper, Black and Scholes assume that the market has
no arbitrage. Here, an "arbitrage" is an opportunity to make a
risk-free profit starting at zero wealth. In other words,
Black-Scholes assumes the market is perfectly efficient, and the
formula (Equation 222) represents the fair price of the option in
this idealized world.

Now when I first learned this stuff, I found this assumption
confusing, because I thought it was analogous to assuming "no
friction" in a high-school physics problem. In physics, we might
simplify the world by assuming that a box slides on plane without
friction. This makes calculations easier for students, but the
consequence is that predictions are systematically wrong. At some
point, we have to add friction back into our model to make it
realistic.

But friction is a poor analogy here, and I propose a different one:
assuming no arbitrage is analogous to assuming no wind or air
resistence when modeling projectile motion. This assumption also
simplifies calculations, but really it helps us understand the
essence of the phenomenon: the parabolic arc of projectile motion.
Later, we can add air resistance or wind depending on our particular
circumstances, but the underlying parabolic arc is a kind of platonic
ideal. It is the signal without the noise.

This is the sense in which Black-Scholes assumes no arbitrage. The
assumption is not an implausible claim that real financial markets
are perfectly efficient. Rather, assuming no arbitrage is assuming a
perfectly coherent, noise-free system where prices are consistent and
make sense relative to each other. So this isn't about simplifying
calculations. It's about finding the platonic price. And thus it's
not an assumption that we remove in the future to get a more
realistic price. Quite the opposite! Adding arbitrage would make the
problem impossible to solve because prices would then be
inconsistent. There would be no universally agreed-upon market price.

Now that we understand this, let's retrace the main line of argument
of the original paper (Black & Scholes, 1973), but let's do so in a
simpler context. Imagine we sold a hypothetical derivative contract:
a redeemable certificate on a stock, which can be exercised only at
time TTT. An investor pays us R0R_0R0 , the fair price of the
redeemable at inception, and in return we give them a certificate
that is redeemable for the value of the stock at time TTT. Our goal
is to solve for RtR_tRt , the fair price of the redeemable at any
given moment in time.

As dealers, the risk to us is that the price of the stock goes up. So
what could we do? Well, the moment we sell a redeemable, we could buy
the underlying stock. Now we are perfectly hedged. We don't care if
the stock goes up or down in value, because we own the stock.
Whenever a customer comes to redeem the value of the stock, we simply
sell the appropriate share, and we're done. We neither make nor lose
money on average, and thus the price is fair.

The key insight of Black-Scholes is to realize that our perfectly
hedged portfolio, short a redeemable and long a stock, is risk-less.
And thus, it must grow or drift at the risk-free rate! Formally, we
can say that the value of our portfolio at expiry TTT is

ST-RT=erT(S0-R0).(8) S_T - R_T = e^{rT} (S_0 - R_0). \tag{8} ST -RT =
erT(S0 -R0 ).(8)

This is remarkable because STS_TST  is random! But the left-hand side
is not an expectation, because we are always hedged! Do you see the
trick? This is the big idea of the original paper. We've neutralized
the randomness in StS_tSt  by assuming we can perfectly hedge it out!

Now a terminal condition is that our redeemable is worth the value of
the stock at expiry, so ST=RTS_T = R_TST =RT . This is fair in the
sense that neither we nor the investor makes more money than the
other. Under this condition, we can write Equation 888 as

erTR0=erTS0.(9) e^{rT} R_0 = e^{rT} S_0. \tag{9} erTR0 =erTS0 .(9)

But if our portfolio is perfectly hedged and riskless, then clearly 
R0R_0R0  must be a riskless derivative. And so it must drift at the
risk-free rate, giving us

RT=erTS0.(10) R_T = e^{rT} S_0. \tag{10} RT =erTS0 .(10)

And of course, we can replace big TTT with little ttt, in general.
Doesn't this make sense? The fair price of the redeemable is simply
the price of the stock at contract inception, adjusted for the
time-value of money! Again, despite StS_tSt  being a random process,
we have no expectations. We have neutralized randomness through
perfect hedging in a world without arbitrage.

Now let's extend this line of reasoning to options. The challenge
here is that an option has convexity. Its price changes nonlinearly
with changes in the underlying. So unlike with the redeemable, we
would not want to buy exactly one share of stock for one option
contract. Instead, at each moment, we want precisely the amount of
stock such that, if the stock price moved an infinitesimally small
amount, our hedge would move an infinitesimally small amount that
perfectly matched the price of the option. This sounds like a
derivative from calculus because it is! The amount we want to hedge
at an instant in time is simply the derivative of the call price with
respect to the stock price, called the delta:

delta:=Dt=[?]Ct[?]St.(11) \text{delta} := \Delta_t = \frac{\partial C_t}
{\partial S_t}. \tag{11} delta:=Dt =[?]St [?]Ct  .(11)

So if the stock changes by one dollar, our option price changes by
approximately Dt\Delta_tDt  dollars.

Now imagine a world without arbitrage or transaction fees and with
continous trading. In this world, we (an options dealer) can always
be perfectly delta hedged. Our portfolio is always just

DtSt-Ct.(12) \Delta_t S_t - C_t. \tag{12} Dt St -Ct .(12)

In the original paper, the authors repeat the argument made above for
redeemables but in the context of options. The calculations become
more complicated because working with Dt\Delta_tDt  requires
stochastic calculus, since StS_tSt  is a random variable. But the
argument is essentially the same. We assume a world without arbitrage
and then just model the dynamics of a perfectly hedged and thus
risk-less portfolio. We use these dynamics (Black-Scholes PDE) and a
terminal condition to solve for the fair price CtC_tCt .

That's the original argument. It's an argument about no arbitrage and
perfect hedging. But Black-Scholes feels like a law of nature because
it's the solution to a noise-free system and because it can be
derived in many ways. Another way to derive Black-Scholes is by
modeling a portfolio which perfectly replicates the price of a call
at each moment. This idea of valuation through replication is
far-reaching in finance. To quote Emanuel Derman (Derman, 2002):

    If you want to know the value of a security, use the price of
    another security that's as similar to it as possible. All the
    rest is modeling.

The discrete-time version of this argument is the binomial
options-pricing model. Yet another way is to connect the idea of no
arbitrage to the idea of risk-neutrality. This relationship is called
the fundamental theorem of asset pricing, and it's the relationship
we want to explore here.

(As an aside, in my mind, one of the confusing things about many
presentations of Black-Scholes is that presenters will seemlessly
switch between these different arguments, without acknowledging that
it is non-obvious that they are equivalent and without acknowledging
which line of argument Black, Scholes, and Merton used.)

Let's understand this connection between the original argument of no
arbitrage and the modern argument of risk-neutral pricing. Consider
this: in a Black-Scholes world, do we care about the drift of the
underlying stock? As with a redeemable, we are always perfectly and
continuously delta hedged. We have no price exposure. In this world,
all options dealers would perfectly hedge. And all options investors
would become risk-neutral, because it would not pay to take risk. So
one trick that makes our logic and calculations easier is to just
assume the drift of the stock is rrr! This is equivalent to assuming
the world has no arbitrage. And what's the fair price of an option in
this world? It's a time-discounted expected value:

Ct=e-r(T-t)EQ[max[?][0,St-K]].(13) C_t = e^{-r(T-t)} \mathbb{E}_{\
mathbb{Q}} \left[\max[0, S_t - K]\right]. \tag{13} Ct =e-r(T-t)EQ [
max[0,St -K]].(13)

Here, the subscript Q\mathbb{Q}Q is called the risk-neutral measure,
and this notation is used to make it explicit that the expectation is
computed in this imaginary world, not in the real risky world. In
this imaginary world, StS_tSt  is a martingale, which is a stochastic
process with an expected value that is always equal to the current
value. Intuitively and importantly, martingales have no drift. So in
Equation 131313, the only drift is from the discount factor e-r(T-t)e
^{-r(T-t)}e-r(T-t). The stock itself is a drift-less martingale.

So in one telling, we assume the world has no risk, and thus we can
work without expectations. Random processes can be forced to become
non-random. In another telling, we assume the world is risk-neutral,
and random processes become martingales. In both tellings, the only
meaningful drift is the risk-free rate.

There is a lot of theory one could get into here. For example, the
Girsanov theorem is a result from probability theory on how a
stochastic process changes under a change in measure (here, from the
true "physical" measure to the risk-neutral measure). And you might
read things like, "Under the risk-neutral measure, the stock price
after discounting by the risk-free rate follows a martingale". You
can easily get lost in the technical details. But in my mind, at a
high-level, the concept is fairly simple, if a bit non-obvious. In a
world in which all investors and traders can perfectly hedge their
risk, there is no risk premia; everyone is forced to become risk
neutral. And thus, the price of everything is its risk-neutral
expected value, or its value if there was no premia to risk.

And again, this simplifying assumption is not like assuming "no
friction". We do not need to re-add arbitrage or drift in order to
compute a more realistic option price later. Instead, this assumption
strips out all the noise, and CtC_tCt  represents a platonic price in
a coherent and consistent market.

Armed with this understanding, let's revisit the generative model for
a stock. In a risk-neutral world, the drift of every stock is the
same: it's just the risk-free rate rrr. This is the line of reasoning
which converts Equation 666 to Equation 777.

Contingent values

We're now ready to try to directly make sense of Equation 222. At
this point, we'll need to a bit of tedious algebra, but nothing in
this section requires more than basic probability. And the conceptual
work is mostly done. As promised, I've tried to side-step as much
stochastic calculus as possible in order to get to this point.

For easy access, let's repeat the main formula:

Ct=StPh(d1)-Ke-r(T-t)Ph(d2),d1:=1sT-t[log[?](St/K)+(r+(1/2)s2)(T-t)],d2:=
d1-sT-t.(14) \begin{aligned} C_t &= S_t \Phi(d_1) - K e^{-r(T-t)} \
Phi(d_2), \\ d_1 &:= \frac{1}{\sigma \sqrt{T-t}} \left[ \log(S_t/K) +
\left(r + (1/2) \sigma^2 \right) (T-t) \right], \\ d_2 &:= d_1 - \
sigma \sqrt{T-t}. \end{aligned} \tag{14} Ct d1 d2  =St Ph(d1 )-Ke-r(T-
t)Ph(d2 ),:=sT-t 1 [log(St /K)+(r+(1/2)s2)(T-t)],:=d1 -sT-t . (14)

Let's also repeat our modeling assumption (Equation 777) but in terms
of TTT and ttt:

log[?]ST-log[?]St~N(m,s2),m:=(r-12s2)t,s:=st,t:=T-t.(15) \begin{aligned}
\log S_T - \log S_t &\sim \mathcal{N} \left(m, s^2\right), \\ m &:= \
left( r - \frac{1}{2} \sigma^2 \right) \tau, \\ s &:= \sigma \sqrt{\
tau}, \\ \tau &:= T - t. \end{aligned} \tag{15} logST -logSt mst ~N(m
,s2),:=(r-21 s2)t,:=st ,:=T-t. (15)

As a final preliminary, two observations for notational clarity.
First, note that at time ttt, the price StS_tSt  is non-random. So we
can push that into the mean if we would like, giving us:

log[?]ST~N(log[?]St+m,s2).(16) \log S_T \sim \mathcal{N}\left(\log S_t +
m, s^2 \right). \tag{16} logST ~N(logSt +m,s2).(16)

Second, here I've used notation d1d_1d1  and d2d_2d2  to match what I
have commonly seen in the literature. But I think it's extremely
useful to rewrite d2d_2d2  as

d2=d1-st=log[?](St/K)+(r+(1/2)s2)t-s2tst=-log[?](K/St)-(r-(1/2)s2)tst=
-log[?](K/St)-ms.(17) \begin{aligned} d_2 &= d_1 - \sigma \sqrt{\tau} \
\ &= \frac{\log(S_t/K) + \left(r + (1/2) \sigma^2 \right) \tau - \
sigma^2 \tau}{\sigma \sqrt{\tau}} \\ &= -\frac{\log(K/S_t) - \left(r
- (1/2) \sigma^2 \right) \tau}{\sigma \sqrt{\tau}} \\ &= -\frac{\log
(K/S_t) - m}{s}. \end{aligned} \tag{17} d2  =d1 -st =st log(St /K)+(r
+(1/2)s2)t-s2t =-st log(K/St )-(r-(1/2)s2)t =-slog(K/St )-m . (17)

We'll use this momentarily.

Now at a high-level, my claim is that we can think of CtC_tCt  as
decomposable into two terms, one representing what we make (stock
price) and one representing what we pay (strike price), both
contingent on the call ending in-the-money (ST>KS_T \gt KST >K). This
idea is not original to me; it is from (Nielsen, 1992). We have:

C1=contigent value of stock={STif ST>K,0else,C2=
contigent value of strike={-Kif ST>K,0else.(18) \begin{aligned} C_1 &
= \text{contigent value of stock} &&= \begin{cases} S_T & \text{if
$S_T \gt K$,} \\ 0 & \text{else,} \end{cases} \\ \\ C_2 &= \text
{contigent value of strike} &&= \begin{cases} -K & \text{if $S_T \gt
K$,} \\ 0 & \text{else.} \end{cases} \end{aligned} \tag{18} C1 C2  =
contigent value of stock=contigent value of strike  ={ST 0 if ST >K
,else, ={-K0 if ST >K,else.  (18)

Furthermore, both of these terms will have a clear, simple,
probabilistic interpretation that will directly map onto Equation 
141414. Let's see this.

First, what is E[C2]\mathbb{E}[C_2]E[C2 ]? This is an expectation,
and the value of the claim is zero when the call is out-of-the-money
(when ST<KS_T \lt KST <K). By definition:

E[C2]=0xP(ST<K)+(-K)xP(ST>K)=-KxP(ST>K).(19) \begin{aligned} \mathbb
{E}[C_2] &= 0 \times \mathbb{P}(S_T \lt K) + (-K) \times \mathbb{P}
(S_T \gt K) \\ &= -K \times \mathbb{P}(S_T \gt K). \end{aligned} \tag
{19} E[C2 ] =0xP(ST <K)+(-K)xP(ST >K)=-KxP(ST >K). (19)

And it is easy to see that P(ST>K)\mathbb{P}(S_T \gt K)P(ST >K) is
equal to Ph(d2)\Phi(d_2)Ph(d2 )!

P(ST>K)=P(log[?](ST/St)>log[?](K/St))=P(log[?](ST/St)-ms>log[?](K/St)-ms)=P
(ZT>-d2)=1-P(ZT<-d2)=P(ZT<d2)=Ph(d2).(20) \begin{aligned} \mathbb{P}
(S_T \gt K) &= \mathbb{P}(\log(S_T / S_t) \gt \log(K / S_t)) \\ &= \
mathbb{P} \left( \frac{\log(S_T/S_t) - m}{s} \gt \frac{\log(K/S_t) -
m}{s} \right) \\ &= \mathbb{P} \left( Z_T \gt -d_2 \right) \\ &= 1 -
\mathbb{P}(Z_T \lt -d_2) \\ &= \mathbb{P}(Z_T \lt d_2) \\ &= \Phi
(d_2). \end{aligned} \tag{20} P(ST >K) =P(log(ST /St )>log(K/St ))=P(
slog(ST /St )-m >slog(K/St )-m )=P(ZT >-d2 )=1-P(ZT <-d2 )=P(ZT <d2 )
=Ph(d2 ). (20)

Here, ZTZ_TZT  denotes the zzz-score of the log move from StS_tSt  to
STS_TST , i.e.

ZT:=log[?](ST/St)-ms~N(0,1).(21) Z_T := \frac{\log(S_T/S_t) - m}{s} \
sim \mathcal{N}(0, 1). \tag{21} ZT :=slog(ST /St )-m ~N(0,1).(21)

So in words, we can see that Ph(d2)\Phi(d_2)Ph(d2 ) is just the
probability that the call ends up in-the-money, or the probability
that ST>KS_T \gt KST >K. All those extra variables embedded in d2d_2
d2  just represent normalizing the log move from StS_tSt  to STS_TST 
, such that we can represent the equation using the CDF of the
standard normal rather than the CDF of STS_TST . We could, if we
wanted to, represent all of this using the CDF of the un-standardized
lognormal distribution. But it's cleaner and conventional to work in
a standardized space.

To summarize, we have shown:

expected contingent value of strike=E[C2]=-KPh(d2).(22) \text{expected
contingent value of strike} = \mathbb{E}[C_2] = -K \Phi(d_2). \tag
{22} expected contingent value of strike=E[C2 ]=-KPh(d2 ).(22)

This represents the expected value we must pay to exercise a call
option, contingent on the option being exercised. Of course, this is
an expected value, but the Black-Scholes price is the price in
today's terms. So we need a discount factor, giving us:

E[C2] at time t=-e-r(T-t)KPh(d2).(23) \text{$\mathbb{E}[C_2]$ at time
$t$} = -e^{-r(T-t)} K \Phi(d_2). \tag{23} E[C2 ] at time t=-e-r(T-t)K
Ph(d2 ).(23)

Now for C1C_1C1 , we again have an expectation where the contingent
value is zero when the option ends out-of-the-money. This is a bit
more complicated than the derivation for C2C_2C2 , since STS_TST  is
random while KKK is fixed. By the law of total expectation, we have:

E[C1]=0xP(ST<K)+E[ST|ST>K]xP(ST>K),=E[ST|ST>K]P(ST>K).(24) \begin
{aligned} \mathbb{E}[C_1] &= 0 \times \mathbb{P}(S_T \lt K) + \mathbb
{E}[S_T \mid S_T \gt K] \times \mathbb{P}(S_T \gt K), \\ &= \mathbb
{E}[S_T \mid S_T \gt K] \mathbb{P}(S_T \gt K). \end{aligned} \tag{24} 
E[C1 ] =0xP(ST <K)+E[ST |ST >K]xP(ST >K),=E[ST |ST >K]P(ST >K). (24)

While this is a bit trickier, it's still relatively straightforward
to put into words. It is the weighted average contribution of the
expected stock price, conditional on the fact that we exercise the
call. Since STS_TST  is lognormally distributed (Equation 161616), we
can write this expectation in terms of a truncated lognormally
distributed random variable S^T\hat{S}_TS^T :

S^T={STif ST>K,0else.(25) \hat{S}_T = \begin{cases} S_T & \text{if
$S_T \gt K$}, \\ 0 & \text{else}. \end{cases} \tag{25} S^T ={ST 0 if 
ST >K,else. (25)

And coincidentally, I recently wrote a blog post on the expected
value of the truncated lognormal distribution! Using the parameters 
mmm and sss from Equation 151515 here and plugging them into Equation
151515 in that blog post, we can easily compute the expectation:

E[S^T]=E[ST|ST>K]=E[ST]Ph(s-log[?]K-(log[?]St+m)s)1-Ph(log[?]K-(log[?]St+m))s).
(26) \begin{aligned} \mathbb{E}[\hat{S}_T] &= \mathbb{E}[S_T \mid S_T
\gt K] \\ &= \frac{\mathbb{E}[S_T] \Phi\left( s - \frac{\log K - (\
log S_t + m)}{s} \right)}{1 - \Phi\left(\frac{\log K - (\log S_t +
m))}{s}\right)}. \end{aligned} \tag{26} E[S^T ] =E[ST |ST >K]=1-Ph(slo
gK-(logSt +m)) )E[ST ]Ph(s-slogK-(logSt +m) ) . (26)

And denominator is the probability that we exercise the call!

1-Ph(log[?]K-(log[?]St+m))s)=1-Ph(-d2)=Ph(d2)=P(ST>K).(27) 1 - \Phi\left(\
frac{\log K - (\log S_t + m))}{s}\right) = 1 - \Phi(-d_2) = \Phi(d_2)
= \mathbb{P}(S_T \gt K). \tag{27} 1-Ph(slogK-(logSt +m)) )=1-Ph(-d2 )=Ph
(d2 )=P(ST >K).(27)

Putting this together, we can see that we can rewrite Equation 242424
as:

E[C1]=E[ST|ST>K]P(ST>K)=E[ST]Ph(s-log[?](K/St)-m)s)=E[ST]Ph(s+d2)=E[ST]Ph
(d1).(28) \begin{aligned} \mathbb{E}[C_1] &= \mathbb{E}[S_T \mid S_T
\gt K] \mathbb{P}(S_T \gt K) \\ &= \mathbb{E}[S_T] \Phi\left( s - \
frac{\log(K/S_t) - m)}{s} \right) \\ &= \mathbb{E}[S_T] \Phi\left( s
+ d_2 \right) \\ &= \mathbb{E}[S_T] \Phi\left( d_1 \right). \end
{aligned} \tag{28} E[C1 ] =E[ST |ST >K]P(ST >K)=E[ST ]Ph(s-slog(K/St )
-m) )=E[ST ]Ph(s+d2 )=E[ST ]Ph(d1 ). (28)

This looks very promising, and we can solve this because we also know
the expected value of a lognormally distributed random variable. It's

E[ST]=exp[?]{log[?]St+m+12s2}=Stexp[?]{(r-12s2)t+12s2t}=Ster(T-t).(29) \
begin{aligned} \mathbb{E}[S_T] &= \exp\left\{ \log S_t + m + \frac{1}
{2} s^2 \right\} \\ &= S_t \exp\left\{ \left(r - \frac{1}{2} \sigma^2
\right) \tau + \frac{1}{2} \sigma^2 \tau \right\} \\ &= S_t e^{r
(T-t)}. \end{aligned} \tag{29} E[ST ] =exp{logSt +m+21 s2}=St exp{(r-
21 s2)t+21 s2t}=St er(T-t). (29)

Now that both makes sense in a risk-neutral world, and it looks quite
promising. This means we have shown

E[C1]=er(T-t)StPh(d1).(30) \mathbb{E}[C_1] = e^{r (T-t)} S_t \Phi
(d_1). \tag{30} E[C1 ]=er(T-t)St Ph(d1 ).(30)

And once again, we need to discount this for the current time, giving
us:

E[C1] at time t=StPh(d1).(31) \text{$\mathbb{E}[C_1]$ at time $t$} =
S_t \Phi(d_1). \tag{31} E[C1 ] at time t=St Ph(d1 ).(31)

That's it. We're done. This is a simple, probabilistic interpretation
of the Black-Scholes formula. To summarize, it is the difference
between two contingent values:

Ct=e-r(T-t)[E[C1]+E[C2]]=StPh(d1)-e-r(T-t)KPh(d2).(32) \begin{aligned}
C_t &= e^{-r(T-t)} \left[ \mathbb{E}[C_1] + \mathbb{E}[C_2] \right] \
\ &= S_t \Phi(d_1) -e^{-r(T-t)} K \Phi(d_2). \end{aligned} \tag{32} 
Ct  =e-r(T-t)[E[C1 ]+E[C2 ]]=St Ph(d1 )-e-r(T-t)KPh(d2 ). (32)

In more words, the left term is the time-discounted and
weighted-average expected value of the stock, contingent on
exercising. And the right term is the time-discounted expected value
of the strike, again contingent on exercising. The main conceptual
hurdle was realizing why we needed to replace the stock-specific
drift m\mum with the risk-free rate rrr. But once we understood that,
we could assume that the underlying stock's log returns were normally
distributed as in Equation 151515. Everything else was just
computation.

[truncated_]
Figure 4. Thousands of random samples from geometric Brownian motion
as in Figure 333 but truncated at strike K=95K=95K=95. The truncated
lognormal distribution at various time points is plotted in dashed
black lines. The mean of these distributions (dashed purple line) is
the mean of the truncated lognormal. The term StPh(d1)S_t \Phi(d_1)St 
Ph(d1 ) is just this mean weighted by P(ST>K)\mathbb{P}(S_T \gt K)P(ST
 >K).

In my mind, this view of Black-Scholes finally helped me understand Ph
(d1)\Phi(d_1)Ph(d1 ). While Ph(d2)\Phi(d_2)Ph(d2 ) has a clean
interpretation--it's just the probability that the call is exercised
(Equation 202020)--I initially found Ph(d1)\Phi(d_1)Ph(d1 ) less clear.
But perhaps my confusion was due to my attempt to understand the term
in isolation. Really, I think it helps to think about StPh(d1)S_t \Phi
(d_1)St Ph(d1 ) as a single unit. This represents the weighted-average
present value of the stock, contingent on the option ending in the
money (Equations 282828 and 292929).

I think this has a nice geometric interpretation. We can think of the
contingent value of our stock as following a truncated lognormal
distribution (Figure 444). Here, StPh(d1)S_t \Phi(d_1)St Ph(d1 ) is the
expected value of this truncated lognormal random variable, weighted
by the probability that the call is exercised.

Conclusion

As a final sanity check, I've visualized the Black-Scholes price for
a call option across various times to expiry (Figure 555). Again, we
see the inherent tension between time-decay and convexity. As time
passes, an option loses value as it loses optionality, but it gains
value as it gains convexity. Ultimately, the Black-Scholes PDE models
this tension directly.

In this post, however, we side-stepped the PDE entirely in favor of
an intuitive understanding of risk-neutrality. In a world without
arbitrage, everyone can perfectly hedge their option positions, and
thus everyone is risk-neutral. In this world, all stock prices are
converted into martingales, stochastic process without drift or
memory. In this world, the fair price of an option is just its
time-discounted, risk-neutral expected value. And this discounted
expected value has a clean interpration: it is the difference between
two contingent values, the contingent value of the stock and the
contingent value of the strike.

[surface]
Figure 5. Visualization of the Black-Scholes formula for an initial
stock price S0=100S_0 = 100S0 =100, drift m=0.04\mu = 0.04m=0.04, and
volatility s=0.15\sigma = 0.15s=0.15 over hundreds of samples. At
each moment, the call gains or loses value, contingent on the stock
price StS_tSt  and the time to expiry T-tT-tT-t.

Hopefully this post justifies my original claim: Black-Scholes feels
inevitable, like a natural law. It makes some very simplifying
assumptions, such as constant volatility, but the model works
extremely well in practice. Many decades later, investors, traders,
and other market participants still use Black-Scholes daily. Much
like other simple models such as linear regression, Black-Scholes is
wrong but useful, interpretable, and general due to its simplicity.
And it is the foundation and reference point for more complex models.

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 6. Derman, E. (2002). The Boy's Guide to Pricing & Hedging.
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