What is an option?

Is it optional to know what an option is? No!

In the context of finance, an option refers to a contract that gives its owner the right, but not the obligation, to buy or sell an underlying asset e.g. shares in a company or bushels of wheat at a predetermined price on or before a specific date.

Options are used for various purposes including risk management and speculation. Herein we derive the formula for pricing a European option.

Calls & Puts: Bets In-Favor & Bets Against

Options come in two flavors. A call is a bet in-favor of the underlying asset whereas a put essentially is a bet against underlying asset.

A call option gives the purchaser the right, but not the obligation to purchase an underlying security, an asset like a stock or commodity, at a specified price, known as the strike price, within a certain timeframe, known as the tenor. For this right, the purchaser pays an upfront cost, known as the premium, to the seller.

If the market value of the security exceeds the strike price, the option becomes profitable, also known as in-the-money. To see this, note that the purchaser can now buy the asset at the strike price and sell the asset at the higher market rate, directly profiting from the difference. Otherwise, why would you execute the contract? You will buy an asset for a higher price than the market will, losing money, hence the option is otherwise out-of-the-money.

A put option achieves the opposite effect by enabling its purchaser the right but not the obligation to sell the underlying asset at a specified price.

Europeans and Americans

Options in finance come in two types: European and American. Here’s a detailed comparison between them based on various features:

1. Exercise Timing: The key difference lies in when the holder can exercise their option rights (buy or sell):
   • European Options: These can only be exercised at maturity, i.e., exactly at a specific date predetermined on the contract’s expiration day. They do not allow exercising before this maturity date, which makes them simpler to price compared to American options but limits their use for certain strategies.
   • American Options: These can be exercised any time up until and including the option’s expiration date. This flexibility allows investors more control over whether or not to exercise an option based on market conditions, making them slightly more complex to price due to this added layer of decision-making.
2. Pricing: Given their differences in exercise timing, these options are priced differently using different mathematical models:
   • European Options: The most common pricing model used for European options is the Black-Scholes Model (BSM), which assumes constant volatility and risk-free interest rates over time. We delve into the BSM formula later in the post, so keep reading.
   • American Options: Pricing an American option generally involves numerical methods such as binomial trees, finite difference method or Monte Carlo simulation because of their flexibility in exercise timing. Each model has its own limitations and assumptions regarding the behavior of asset prices (volatility, return rates) and market conditions over time.
3. Applications: The choice between European and American options often comes down to specific investment strategies or needs of an investor due to their different exercise features:
   • European Options: These are typically used when a buyer is seeking protection (hedging) against losses in the underlying asset’s price movements. For instance, they might be employed by companies holding large amounts of stock for hedging purposes or by investors who don’t need to exercise early due to specific market views.
   • American Options: Their flexible nature often suits buyers looking to speculate on future price changes in the underlying asset over a longer horizon, as they can choose the optimal time to exercise based on changing market conditions or news announcements affecting the underlier’s value. However, their complexity and the need for advanced mathematical models make them more suitable for professional investors with deep trading expertise.

In summary, while both types of options grant a right to buy or sell an underlying asset at a predetermined price before maturity, European and American options vary significantly in terms of exercise timing, pricing methods, and their applications within investment strategies and risk management techniques.

How do we price one of these things?

The canonical formula for the European option was developed by three economists Fischer Black and Myron Scholes, and Robert C. Merton, known as the Black–Scholes–Merton (BSM) model.

Black-Scholes Model

\[C = S \cdot N(d_1) - K \cdot e^{-r \cdot \tau}N(d_2)\] \[d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)\tau}{\sigma\sqrt{\tau}} \quad \text{and} \quad d_2 = d_1 - \sigma\sqrt{\tau}\]

Required Inputs:

• S = Current stock price
• K = Option strike price
• r = Risk-free interest rate
• $\tau$ = Time remaining until option expiration, often expressed as T-t
• $\sigma$ = Volatility of the stock

Where N is the normal distribution function.

What are the four assumptions?

BSM requires one to make four reasonable assumptions about the market.

1. The stock price follows a geometric Brownian motion with constant volatility and drift rate. This means the changes in stock prices over time have an exponential distribution, exhibit continuous compounding, and that this process is not influenced by other factors (e.g., market events).

2. There are no dividends during the option’s life period. Dividends can reduce the price of a call or increase the price of a put because they decrease the stock price when paid out. By excluding them, we simplify calculations.

3. Markets are efficient, meaning that asset prices reflect all available information, and investors cannot profit from trading in an arbitrage opportunity (i.e., buying undervalued securities and selling overvalued ones).

4. The risk-free interest rate is constant throughout the option’s life. This assumption simplifies calculations by assuming that there are no investment opportunities with guaranteed returns over time.

Further exploration of the difference between the ideal world of BSM and flesh-and-blood markets is discussed in the Volatility Smile, a seminal book in finance. We couch discussion of the nuances of the assumptions themselves for a future post.

A Detour to Japan

To follow the derivation that follows, you will need to learn about the world of stochastic calculus. If you’ve taken a first course in calculus, you already know the chain rule: given a smooth function $f(g(x))$, its derivative is $f’(g(x)) \cdot g’(x)$. You also know that when $dx$ is infinitesimally small, $(dx)^2$ is negligible and we throw it away. These two facts — the chain rule and the negligibility of squared infinitesimals — are the bedrock of deterministic calculus.

But stock prices are not smooth, deterministic curves. They jitter randomly at every instant. To handle functions of these jittery processes, we need a modified version of the chain rule — and it turns out that $(dx)^2$ can no longer be thrown away. This is the world of stochastic calculus.

This field was pioneered by Japanese mathematician Kiyosi Itô, who discovered an essential result in stochastic calculus that provides the mathematical foundation for analyzing continuous-time stochastic differential equations (SDEs).

Here are its key components:

1. Stochastic Processes: In finance, many quantities—such as stock prices or interest rates—are modeled as stochastic processes rather than deterministic ones due to their inherent randomness and unpredictability over time. Stochastic differential equations (SDEs) are used to describe these dynamic systems mathematically. In a stochastic system, each movement from one state to the next is governed by a probability, in our case, for the financial markets, the probability is sampled from a normal distribution. The canonical name for such a motion is a Brownian motion from physics or martingale from economics.

2. Itô’s Lemma: Itô’s lemma is a powerful tool for finding the differential of functions of stochastic processes that satisfy an SDE, particularly when applied in financial modeling and derivative pricing. In essence, Ito’s lemma enables us to understand how certain observable quantities (like option prices or portfolio values) change over time under volatility and random market conditions.

3. Functional Derivative: At its core, the lemma provides a formula for the differential of a function composed with a stochastic process defined by an SDE. For instance, if we have a function f(t, X(t)), where t denotes time and X(t) represents a stochastic process following an SDE, Ito’s Lemma tells us how df changes in relation to dt (time increment) and dX (increment of the random variable).

4. Formula: The formula for Itô’s Lemma is given by: \(df(t, X_t) = \left( \frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x} + \frac{1}{2}\sigma^2 \frac{\partial^2 f}{\partial x^2} \right) dt + \sigma \frac{\partial f}{\partial x} dW_t\) where $dX_t = \mu dt + \sigma dW_t$.

A Sketch of the Proof

To see why this looks different from the standard chain rule, consider the Taylor expansion of $f(t+dt, X+dX)$: \(df = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial x}dX + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}(dX)^2 + \frac{1}{2}\frac{\partial^2 f}{\partial t^2}(dt)^2 + \frac{\partial^2 f}{\partial x \partial t}dX dt + \dots\) In deterministic calculus, we drop everything of order higher than $dt$. But here, $dX = \mu dt + \sigma dW_t$. When we calculate $(dX)^2$: \((dX)^2 = (\mu dt + \sigma dW_t)^2 = \mu^2 (dt)^2 + 2\mu\sigma dt dW_t + \sigma^2 (dW_t)^2\) Using the heuristic rules of stochastic calculus:

1. $dt \cdot dt \to 0$
2. $dt \cdot dW_t \to 0$
3. $dW_t \cdot dW_t \to dt$

The third rule is the “Magic of Itô.” It arises because the variance of Brownian motion over $dt$ is exactly $dt$. Thus, the $(dX)^2$ term contributes a $\sigma^2 dt$ component, which is of the same order as $dt$ and cannot be ignored. Substituting $dX$ and $(dX)^2 = \sigma^2 dt$ back into the Taylor expansion yields the lemma.

The first term, $\frac{\partial f}{\partial t}\,dt$, captures the explicit time dependence of $f$.
The second term, $\frac{\partial f}{\partial x}\,dX_t$, reflects the sensitivity of $f$ to changes in $X_t$.
The third term, $\frac{1}{2}\frac{\partial^2 f}{\partial x^2}\,(dX_t)^2$, accounts for the quadratic variation of the stochastic process—this is a key feature of Itô calculus that distinguishes it from ordinary calculus. In particular, when $X_t$ follows a diffusion process (e.g., $dX_t = \mu\,dt + \sigma\,dW_t$), the term $(dX_t)^2$ contributes a non-vanishing $O(dt)$ component (e.g., $\sigma^2\,dt$), which must be retained.

Why doesn’t $(dX_t)^2$ vanish like it does in ordinary calculus? The key is the Brownian motion $W_t$ that drives the randomness. Each step of $W_t$ over a time interval $\Delta t$ has standard deviation $\sqrt{\Delta t}$, so when you square it you get something of order $\Delta t$ — not $(\Delta t)^2$. Summing infinitely many of these squared increments gives a quantity proportional to $t$, not zero. This is called the quadratic variation of Brownian motion, and it is what forces us to keep the third term in Itô’s Lemma.

• Partial Derivatives: These measure how the function $f(t, X_t)$ changes with respect to its independent variables. Specifically:
  • $\frac{\partial f}{\partial t}$ captures the sensitivity to time $t$ (with $X_t$ held fixed),
  • $\frac{\partial f}{\partial x}$ and $\frac{\partial^2 f}{\partial x^2}$ capture the first- and second-order sensitivities to the stochastic process $X_t$ (with time held fixed).
• Quadratic Variation Term: The term $\frac{1}{2}\frac{\partial^2 f}{\partial x^2}(dX_t)^2$ arises from the quadratic variation of the stochastic process $X_t$. Unlike in ordinary calculus, $(dX_t)^2$ does not vanish; for Itô processes driven by Brownian motion, it contributes a deterministic $O(dt)$ component (e.g., if $dX_t = \mu\,dt + \sigma\,dW_t$, then $(dX_t)^2 = \sigma^2\,dt$). This term accounts for the cumulative effect of random fluctuations and is essential for correctly modeling dynamics in stochastic calculus.

In financial mathematics, Itô’s lemma plays an instrumental role by allowing analysts and traders to derive the dynamic pricing models (like Black-Scholes formula) that govern modern derivative markets—essentially helping us understand how prices evolve in response to market conditions and risk factors over time.

Now that we have Itô’s Lemma in hand, we can set up the BSM economy and see it in action.

The BSM Economy

To understand BSM, you will need to learn some stochastic calculus — which, having taken our detour to Japan, you now have.

There are two assets: a risky stock $S$ and riskless bond $B$. These assets are driven by the SDEs

\[\begin{aligned} dS_t &= \mu S_t dt + \sigma S_t dW_t \\ dB_t &= r_t B_t dt \end{aligned}\]

The time zero value of the bond is $B_0 = 1$ and that of the stock is $S_0$. By Itô’s Lemma the value $V_t$ of a derivative written on the stock follows the diffusion

\[\begin{aligned} dV_t &= \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} dS + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} (dS)^2 \quad \text{(3)} \\ &= \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} dS + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} \sigma^2 S^2 dt \\ &= \left( \frac{\partial V}{\partial t} + \mu S_t \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} \right) dt + \left( \sigma S_t \frac{\partial V}{\partial S} \right) dW_t. \end{aligned}\]

Notice how the Itô correction term $\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} dt$ appears naturally — this is exactly the third term from Itô’s Lemma applied to $V(S_t, t)$ with $(dS_t)^2 = \sigma^2 S_t^2 dt$.

Risk-Neutral Pricing and the Equivalent Martingale Measure

Before we can price the option, we need one more deep idea: the Equivalent Martingale Measure (EMM).

A martingale is a stochastic process whose expected future value, given all information available today, equals its current value. Formally, $M_t$ is a martingale if $E[M_T \mid \mathcal{F}_t] = M_t$. Think of it as a “fair game” — on average, you neither win nor lose.

The principle behind risk-neutral pricing is that if the market is complete and free of arbitrage, we can find a probability measure $\mathbb{Q}$ — the EMM — under which the discounted stock price $\tilde{S}_t = S_t / B_t$ becomes a martingale. Under this measure, every asset earns the risk-free rate on average, so we don’t need to know the stock’s actual drift $\mu$ to price derivatives. We simply take expectations under $\mathbb{Q}$ and discount at $r$.

Formalizing the Measure Change: Girsanov’s Theorem

Girsanov’s Theorem provides the mathematical bridge to “neutralize” the drift of the stock price. Suppose we have a Brownian motion $W_t$ under the physical measure $\mathbb{P}$. We want to move to a measure $\mathbb{Q}$ where the stock price has a drift equal to the risk-free rate $r$.

Girsanov tells us that if we define the Radon-Nikodym derivative process: \(\frac{d\mathbb{Q}}{d\mathbb{P}} = \exp\left( -\int_0^T \theta_t dW_t - \frac{1}{2} \int_0^T \theta_t^2 dt \right)\) where $\theta_t = \frac{\mu - r}{\sigma}$ is the Market Price of Risk, then the process: \(W_t^{\mathbb{Q}} = W_t + \int_0^t \theta_s ds\) is a Brownian motion under the new measure $\mathbb{Q}$. Substituting $dW_t = dW_t^{\mathbb{Q}} - \theta_t dt$ into the stock’s SDE: \(\begin{aligned} dS_t &= \mu S_t dt + \sigma S_t (dW_t^{\mathbb{Q}} - \theta_t dt) \\ &= \mu S_t dt + \sigma S_t dW_t^{\mathbb{Q}} - (\mu - r) S_t dt \\ &= r S_t dt + \sigma S_t dW_t^{\mathbb{Q}}. \end{aligned}\) This formally proves that the drift $\mu$ disappears, replaced by $r$, allowing us to price derivatives without knowing the actual expected return on the stock.

\[d\tilde{S}_t = \frac{\partial \tilde{S}}{\partial B} dB_t + \frac{\partial \tilde{S}}{\partial S} dS_t \tag{15}\]

since all terms involving the second-order derivatives are zero. Expand Equation (15) to obtain

\[\begin{align} d\tilde{S}_t &= -\frac{S_t}{B_t^2} dB_t + \frac{1}{B_t} dS_t \tag{16} \\ &= -\frac{S_t}{B_t^2} (r_t B_t dt) + \frac{1}{B_t} \left( r_t S_t dt + \sigma S_t dW_t^{\mathbb{Q}} \right) \\ &= \sigma \tilde{S}_t dW_t^{\mathbb{Q}}. \end{align}\]

The solution to the SDE (16) is

\[\tilde{S}_t = \tilde{S}_0 \exp\left( -\tfrac{1}{2} \sigma^2 t + \sigma W_t^{\mathbb{Q}} \right).\]

The drift terms have cancelled, confirming that $\tilde{S}_t$ is indeed a martingale under $\mathbb{Q}$.

• Filtration: The filtration ${\mathcal{F}t}{t \geq 0}$ represents the accumulation of information about market events up to time $t$. It is a mathematical structure that models how our knowledge of asset price dynamics evolves over time, incorporating new data, observable trends, and other relevant information available in the financial market.

• Conditional Expectation: The conditional expectation $\mathbb{E}[S_T \mid \mathcal{F}_t]$ is the best estimate (in the mean-square sense) of the future asset price $S_T$, given all information available up to time $t$, encoded in the sigma-algebra $\mathcal{F}_t$. This includes historical price paths, current market conditions, and any other observable information that may affect the asset’s future behavior.

With the EMM in hand, we can now state the fundamental pricing equation. Under a constant interest rate $r$ the time-$t$ price of a European call option on a non-dividend paying stock when its spot price is $S_t$ and with strike $K$ and time to maturity $\tau = T - t$ is

\[C(S_t, K, T) = e^{-r\tau} E^{\mathbb{Q}}\left[(S_T - K)^+ \mid \mathcal{F}_t\right] \tag{17}\]

which can be evaluated to produce Equation (1), reproduced here for convenience

\[C(S_t, K, T) = S_t\Phi(d_1) - Ke^{-r\tau}\Phi(d_2)\]

where

\[d_1 = \frac{\log\frac{S_t}{K} + \left(r + \frac{\sigma^2}{2}\right)\tau}{\sigma\sqrt{\tau}}\]

and \(\begin{align} d_2 &= d_1 - \sigma\sqrt{\tau}\\ &= \frac{\log\frac{S_t}{K} + \left(r - \frac{\sigma^2}{2}\right)\tau}{\sigma\sqrt{\tau}}. \end{align}\)

To evaluate Equation (17) and arrive at the BSM formula, we will need two tools: the lognormal distribution and a technique we call “completing the square.” We take a detour to develop these tools before returning to the derivation.

The Lognormal PDF and CDF

In this Note we make extensive use of the fact that if a random variable $Y \in \mathbb{R}$ follows the normal distribution with mean $\mu$ and variance $\sigma^2$, then $X = e^Y$ follows the lognormal distribution with mean

\[E[X] = e^{\mu + \frac{1}{2}\sigma^2} \tag{4}\]

and variance

\[Var[X] = \left(e^{\sigma^2} - 1\right) e^{2\mu + \sigma^2}. \tag{5}\]

The pdf for $X$ is

\[dF_X(x) = \frac{1}{\sigma x\sqrt{2\pi}} \exp\left(-\frac{1}{2}\left(\frac{\ln x - \mu}{\sigma}\right)^2\right) \tag{6}\]

and the cdf is

\[F_X(x) = \Phi\left(\frac{\ln x - \mu}{\sigma}\right) \tag{7}\]

where $\Phi(y) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^y e^{-\frac{1}{2}t^2} dt$ is the standard normal cdf.

The Lognormal Conditional Expected Value

The expected value of $X$ conditional on $X > x$ is $L_X(K) = E[X \mid X > x]$. For the lognormal distribution this is, using Equation (6)

\[L_X(K) = \int_K^{\infty} \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{\ln x - \mu}{\sigma}\right)^2} dx.\]

Make the change of variable $y = \ln x$ so that $x = e^y$, $dx = e^y dy$ and the Jacobian is $e^y$. Hence we have

\[L_X(K) = \int_{\ln K}^{\infty} \frac{e^y}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{y-\mu}{\sigma}\right)^2} dy. \tag{8}\]

Combining terms and completing the square, the exponent is

\[-\frac{1}{2\sigma^2}\left(y^2 - 2y\mu + \mu^2 - 2\sigma^2 y\right) = -\frac{1}{2\sigma^2}\left(y - \left(\mu + \sigma^2\right)\right)^2 + \mu + \frac{1}{2}\sigma^2.\]

Equation (8) becomes

\[L_X(K) = \exp\left(\mu + \frac{1}{2}\sigma^2\right) \frac{1}{\sigma} \int_{\ln K}^{\infty} \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{1}{2}\left(\frac{y - \left(\mu + \sigma^2\right)}{\sigma}\right)^2\right) dy. \tag{9}\]

Consider the random variable $X$ with pdf $f_X(x)$ and cdf $F_X(x)$, and the scale-location transformation $Y = \sigma X + \mu$. It is easy to show that the Jacobian is $\frac{1}{\sigma}$, that the pdf for $Y$ is $f_Y(y) = \frac{1}{\sigma} f_X\left(\frac{y-\mu}{\sigma}\right)$ and that the cdf is $F_Y(y) = F_X\left(\frac{y-\mu}{\sigma}\right)$. Hence, the integral in Equation (9) involves the scale-location transformation of the standard normal cdf. Using the fact that $\Phi(-x) = 1 - \Phi(x)$ this implies that

\[L_X(K) = \exp\left(\mu + \frac{\sigma^2}{2}\right) \Phi\left(\frac{-\ln K + \mu + \sigma^2}{\sigma}\right). \tag{10}\]

Now we have all the tools we need. Let us return to the derivation.

Deriving From Integration

The European call price $C(S_t, K, T)$ is the discounted time-$t$ expected value of $(S_T - K)^+$ under the EMM $\mathbb{Q}$ and when interest rates are constant. Hence from Equation (17) we have

\[\begin{align} C(S_t, K, T) &= e^{-r\tau} E^{\mathbb{Q}}\left[(S_T - K)^+ \mid \mathcal{F}_t\right] \tag{18}\\ &= e^{-r\tau} \int_K^{\infty} (S_T - K) dF(S_T) \\ &= e^{-r\tau} \int_K^{\infty} S_T dF(S_T) - e^{-r\tau} K \int_K^{\infty} dF(S_T). \end{align}\]

To evaluate the two integrals, we make use of the result derived above that under $\mathbb{Q}$ and at time $t$ the terminal stock price $S_T$ follows the lognormal distribution with mean $\ln S_t + \left(r - \frac{\sigma^2}{2}\right)\tau$ and variance $\sigma^2\tau$, where $\tau = T - t$ is the time to maturity. The first integral in the last line of Equation (18) uses the conditional expectation of $S_T$ given that $S_T > K$

\(\int_K^{\infty} S_T dF(S_T) = E^{\mathbb{Q}}[S_T \mid S_T > K]\) \(= L_{S_T}(K).\)

This conditional expectation is, from Equation (10) \(\begin{align} L_{S_T}(K) &= \exp\left(\ln S_t + \left(r - \frac{\sigma^2}{2}\right)\tau + \frac{\sigma^2\tau}{2}\right)\\ &\quad \times\Phi\left(\frac{-\ln K + \ln S_t + \left(r - \frac{\sigma^2}{2}\right)\tau + \sigma^2\tau}{\sigma\sqrt{\tau}}\right)\\ &= S_t e^{r\tau} \Phi(d_1), \end{align}\) so the first integral in the last line of Equation (18) is

\[S_t\Phi(d_1). \tag{19}\]

Using Equation (7), the second integral in the last line of (18) can be written

\(\begin{align} e^{-r\tau} K \int_K^{\infty} dF(S_T) &= e^{-r\tau} K [1 - F(K)] \tag{20}\\ &= e^{-r\tau} K \left[1 - \Phi\left(\frac{\ln K - \ln S_t - \left(r - \frac{\sigma^2}{2}\right)\tau}{\sigma\sqrt{\tau}}\right)\right]\\ &= e^{-r\tau} K [1 - \Phi(-d_2)]\\ &= e^{-r\tau} K\Phi(d_2). \end{align}\) Combining the terms in Equations (19) and (20) leads to the expression (1) for the European call price.

More stochastic calculus and fanciful maths

Having derived BSM through direct integration, we now turn to a second derivation that arrives at the same formula by a completely different route — the change of numeraire.

Change of Numeraire

The principle behind pricing by arbitrage is that if the market is complete we can find a portfolio that replicates the derivative at all times, and we can find an equivalent martingale measure (EMM) $\mathbb{N}$ such that the discounted stock price is a martingale. Moreover, the EMM $\mathbb{N}$ determines the unique numeraire $N_t$ that discounts the stock price. The time-$t$ value $V(S_t, t)$ of the derivative with payoff $V(S_T, T)$ at time $T$ discounted by the numeraire $N_t$ is

\[V(S_t, t) = N_t E^{\mathbb{N}}\left[\frac{V(S_T, T)}{N_T} \mid \mathcal{F}_t\right]. \tag{21}\]

In the derivation of the previous section, the bond $B_t = e^{rt}$ serves as the numeraire, and since $r$ is deterministic we can take $N_T = e^{rT}$ out of the expectation and with $V(S_T, T) = (S_T - K)^+$ we can write

\[V(S_t, t) = e^{-r(T-t)} E^{\mathbb{N}}\left[(S_T - K)^+ \mid \mathcal{F}_t\right]\]

which is Equation (17) for the call price.

What is a Change of Numeraire?

The change of numeraire is a technique in financial mathematics that helps simplify the pricing of derivatives, such as options. The numeraire is essentially the “unit of value” that we use to measure other assets. By changing this unit of measurement, we can sometimes make complicated pricing problems easier to solve.

1. What is a Numeraire? A numeraire can be thought of as a reference asset against which all other assets are measured. In typical pricing problems, we often use a risk-free bond as the numeraire because it grows at a predictable rate, making it easier to handle in calculations. However, we can choose other assets as the numeraire, depending on the scenario.

2. Why Change the Numeraire? In finance, different assets may behave differently under various market conditions. By switching the asset used as the numeraire, we can sometimes transform a complex pricing problem into one that’s easier to calculate. This is particularly useful when pricing exotic options or handling currencies.

3. Martingale Property A key concept in pricing derivatives is the martingale property. This means that, under the right conditions (no arbitrage opportunities), the discounted value of an asset evolves in such a way that its expected future value is equal to its current price. When we change the numeraire, we also change the “measure” (the mathematical lens) we use to view future prices, but the martingale property still holds under this new measure.

4. Using a Bond as the Numeraire When we use a risk-free bond (which grows at the risk-free rate) as the numeraire, the pricing becomes relatively simple. For example, the price of a European call option (which pays off if the underlying asset’s price is higher than a set strike price) can be calculated by discounting the expected payoff of the option by the bond’s growth rate.

5. Application to Option Pricing In option pricing, changing the numeraire can be particularly useful. For instance, when pricing a European call option, we typically use the risk-free bond as the numeraire, leading to the familiar Black-Scholes pricing formula. This formula essentially discounts the future expected payoff of the option (based on the stock price exceeding the strike price) by the risk-free rate over time.

Summary

In short, the change of numeraire is a method that simplifies pricing derivatives by changing the reference asset (the numeraire) used to measure value. When we do this, we switch to a new probability measure but retain certain properties, like the fact that discounted asset prices behave in a predictable way (the martingale property). For standard derivatives like European call options, using the risk-free bond as the numeraire makes the pricing straightforward and leads to well-known formulas like Black-Scholes-Merton.

Black Scholes Under a Different Numeraire

In this section we show that we can use the stock price $S_t$ as the numeraire and recover the Black-Scholes call price. We start with the stock price process in Equation (14) under the measure $\mathbb{Q}$ and with a constant interest rate

\[dS_t = rS_t dt + \sigma S_t dW_t^{\mathbb{Q}}. \tag{22}\]

The relative bond price is defined as $\tilde{B} = \frac{B}{S}$ and by Itô’s Lemma follows the process

\[d\tilde{B}_t = \sigma^2 \tilde{B}_t dt - \sigma \tilde{B}_t dW_t^{\mathbb{Q}}.\]

The measure $\mathbb{Q}$ turns $\tilde{S} = \frac{S}{B}$ into a martingale, but not $\tilde{B}$. The measure $\mathbb{P}$ that turns $\tilde{B}$ into a martingale is

\[W_t^{\mathbb{P}} = W_t^{\mathbb{Q}} - \sigma t \tag{23}\]

so that

\[d\tilde{B}_t = -\sigma \tilde{B}_t dW_t^{\mathbb{P}}\]

is a martingale under $\mathbb{P}$. The value of the European call is determined by using $N_t = S_t$ as the numeraire along with the payoff function $V(S_T, T) = (S_T - K)^+$ in the valuation Equation (21) \(\begin{align} V(S_t, t) &= S_t E^{\mathbb{P}}\left[\frac{(S_T - K)^+}{S_T} \mid \mathcal{F}_t\right] \tag{24}\\ &= S_t E^{\mathbb{P}}\left[(1 - KZ_T) \mid \mathcal{F}_t\right] \end{align}\) where $Z_t = \frac{1}{S_t}$. To evaluate $V(S_t, t)$ we need the distribution for $Z_T$. The process for $Z = \frac{1}{S}$ is obtained using Itô’s Lemma on $S_t$ in Equation (22) and the change of measure in Equation (23)

\[\begin{align} dZ_t &= \left(-r + \sigma^2\right) Z_t dt - \sigma Z_t dW_t^{\mathbb{Q}}\\ &= -rZ_t dt - \sigma Z_t dW_t^{\mathbb{P}}. \end{align}\]

To find the solution for $Z_t$ we define $Y_t = \ln Z_t$ and apply Itô’s Lemma again, to produce

\[dY_t = -\left(r + \frac{\sigma^2}{2}\right) dt - \sigma dW_t^{\mathbb{P}}. \tag{25}\]

We integrate Equation (25) to produce the solution

\[Y_T - Y_t = -\left(r + \frac{\sigma^2}{2}\right)(T - t) - \sigma(W_T^P - W_t^P),\]

so that $Z_T$ has the solution

\[Z_T = e^{\ln Z_t - (r + \frac{\sigma^2}{2})(T-t) - \sigma(W_T^P - W_t^P)}.\]

Now, since $W_T^P - W_t^P$ is identical in distribution to $W_\tau^P$, where $\tau = T - t$ is the time to maturity, and since $W_\tau^P$ follows the normal distribution with zero mean and variance $\sigma^2 \tau$, the exponent in Equation (26)

\[\ln Z_t - \left(r + \frac{\sigma^2}{2}\right)(T - t) - \sigma(W_T^P - W_t^P),\]

follows the normal distribution with mean

\[u = \ln Z_t - \left(r + \frac{\sigma^2}{2}\right) \tau = -\ln S_t - \left(r + \frac{\sigma^2}{2}\right) \tau\]

and variance $v = \sigma^2 \tau$. This implies that $Z_T$ follows the lognormal distribution with mean $e^{u + v/2}$ and variance $(e^v - 1)e^{2u + v}$. Note that $(1 - K Z_T)^+$ in the expectation of Equation (24) is non-zero when $Z_T < \frac{1}{K}$. Hence we can write this expectation as the two integrals

\[E^P[(1 - K Z_T) \mid F_t] = \int_{-\infty}^k dF_{Z_T} - K \int_{-\infty}^k Z_T dF_{Z_T}\] \[= I_1 - I_2.\]

where $F_{Z_T}$ is the cdf of $Z_T$ defined in Equation (7). The first integral in Equation (27) is

\[I_1 = F_{Z_T} \left( \frac{1}{K} \right) = \Phi \left( \frac{ \ln \frac{1}{K} - u }{ \sqrt{v} } \right)\] \[= \Phi \left( \frac{ -\ln K + \ln S_t + \left( r + \frac{\sigma^2}{2} \right) \tau }{ \sigma \sqrt{\tau} } \right)\] \[= \Phi(d_1).\]

Using the definition of $L_{Z_T}(x)$ in Equation (10), the second integral in Equation (27) is

\[I_2 = K \left[ \int_{-\infty}^{\infty} Z_T \, dF_{Z_T} - \int_{\frac{1}{K}}^{\infty} Z_T \, dF_{Z_T} \right]\] \[= K \left[ E^{\mathbb{P}} [Z_T] - L_{Z_T} \left( \frac{1}{K} \right) \right]\] \[= K \left[ e^{u + v/2} - e^{u + v/2} \Phi \left( \frac{ -\ln \frac{1}{K} + u + v }{ \sqrt{v} } \right) \right]\] \[= K e^{u + v/2} \left[ 1 - \Phi \left( \frac{ -\ln \frac{S_t}{K} - \left( r - \frac{\sigma^2}{2} \right) \tau }{ \sigma \sqrt{\tau} } \right) \right]\] \[= \frac{K}{S_t} e^{-r \tau} \Phi (d_2)\]

since $1 - \Phi (-d_2) = \Phi (d_2)$. Substitute the expressions for $I_1$ and $I_2$ from Equations (28) and (29) into the valuation Equation (24)

\[V(S_t, t) = S_t E^{\mathbb{P}} [(1 - K Z_T)^+ \mid \mathcal{F}_t]\] \[= S_t [I_1 - I_2]\] \[= S_t \Phi (d_1) - K e^{-r \tau} \Phi (d_2)\]

which is the Black-Scholes call price in Equation (1).

The Calculus Student’s Bridge: From Stocks to Heat

If you’ve taken a course in differential equations or physics, you’ve met the Heat Equation: \(\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}\) It describes how heat dissipates through a metal rod. If you start with a hot spot, it spreads out and flattens over time.

Black-Scholes is the Heat Equation in disguise.

1. The Black-Scholes PDE: The Replication Argument

How did Black and Scholes actually derive the formula? They didn’t start with expectations; they started with perfect hedging.

Consider a portfolio $\Pi$ consisting of one short option (value $V$) and $\Delta$ shares of the long stock (value $S$): \(\Pi = -V + \Delta S\) The change in this portfolio over an infinitesimal time $dt$ is: \(d\Pi = -dV + \Delta dS\) Using Itô’s Lemma to expand $dV$: \(d\Pi = -\left( \frac{\partial V}{\partial t} + \mu S \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \right) dt - \sigma S \frac{\partial V}{\partial S} dW_t + \Delta (\mu S dt + \sigma S dW_t)\) To eliminate the randomness (the $dW_t$ term), we choose $\Delta = \frac{\partial V}{\partial S}$. This is called Delta Hedging. The portfolio becomes risk-free: \(d\Pi = -\left( \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \right) dt\) Since the portfolio is now riskless, it must grow at the risk-free rate $r$ to avoid arbitrage ($d\Pi = r\Pi dt$). Substituting $\Pi = -V + \frac{\partial V}{\partial S} S$: \(-\left( \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \right) dt = r(-V + \frac{\partial V}{\partial S} S) dt\) Rearranging gives the celebrated Black-Scholes PDE: \(\frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV = 0\)

2. The Feynman-Kac Connection: Bridge from PDE to Expectation

Why does solving this PDE give the same answer as calculating an expectation? This is formalized by the Feynman-Kac Theorem.

The theorem states that the solution to a parabolic PDE of the form: \(\frac{\partial V}{\partial t} + \mu(S,t) \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2(S,t) \frac{\partial^2 V}{\partial S^2} - r V = 0\) subject to the terminal condition $V(S, T) = h(S)$, can be expressed as the conditional expectation: \(V(S_t, t) = e^{-r(T-t)} E^{\mathbb{Q}} [ h(S_T) \mid S_t ]\) under a measure where the process $S$ has drift $\mu(S,t)$.

In the Black-Scholes case, the drift is $\mu(S,t) = rS$. This provides the rigorous mathematical justification for switching between the Hedging Approach (the PDE) and the Risk-Neutral Approach (the expectation). They are two sides of the same coin.

3. Volatility as Diffusion

In physics, $\alpha$ represents how easily heat flows. In finance, $\sigma^2$ (Volatility) represents how “jittery” the stock is.

• High volatility means the probability “heat” spreads out quickly (a wider range of possible future prices).
• Low volatility means the probability stays concentrated.

3. Turning Finance into Physics

By performing a change of variables—specifically $x = \ln(S)$ (moving to log-space) and shifting time—you can transform the Black-Scholes PDE into the exact Heat Equation.

Why does this matter? It tells us that an option price isn’t just a random guess. It is the result of a diffusion process. The “price” of an option is essentially the “temperature” of the asset’s potential payoff, averaged out over all possible paths the stock could take, weighted by their probability.

When you see the $\Phi(d_1)$ and $\Phi(d_2)$ terms in the final formula, you are looking at the Cumulative Normal Distribution—which is the fundamental solution (the “Green’s function”) to the Heat Equation. The bell curve isn’t just a statistical assumption; it is the natural shape that heat (and probability) takes as it spreads through time.

Further reading

Excellent post on 4 ways to derive BSM Fabrice Douglas Rouah