Options as distributions
See the payoff, its density, and the Black–Scholes formula together
A call pays in high-price states. A put pays in low-price states. Their densities show which observations the contracts emphasize, including how much weight each observation receives. Here we draw those densities continuously, using the normal model.
Throughout, prices are in units of the reference asset \(Y\), so \(Y(t)=1\). The underlying is \(X(t)\), and a strike \(K\) is fixed in those units. We return to currency prices below.
Three densities on one axis
Let \(z\) be the standardized normal observation at maturity \(T\), and set \(a=\sigma\sqrt T\). The market and asset assign it different normal laws:
\[ \begin{aligned} f^Y(z)&=\varphi(z),\\ f^X(z)&=\varphi(z-a), \end{aligned} \]
where \(\varphi\) is the standard normal density. Thus \(z\) has mean zero under \(\mathbb P^Y\) and mean \(a\) under \(\mathbb P^X\), with variance one under both.
Dividing the densities determines the terminal price:
\[ \frac{X(T)}{X(0)} =\frac{f^X(z)}{f^Y(z)} =\exp\!\left(az-\frac{a^2}{2}\right). \]
The observation is normal; the price is lognormal. This is the geometric Brownian motion model from the normal-model page, not a model in which the price itself is normally distributed.
Now choose an option payoff \(h(X(T))\). Its price and density are
\[ C(0)=\mathbb E^Y[h(X(T))], \]
\[ f^C(z)=\frac{h(X(T))}{C(0)}\,f^Y(z). \]
Multiply the market density by the payoff, then divide by its price. For a positive finite price, the result has area one. A call has no density below the exercise threshold; above it, larger payoffs receive more weight.
Draw the option’s view
Start with a call at strike 100. The blue and green curves are the two normal densities; the purple curve is the option’s induced density. Switch to a digital: its flat payoff above the strike conditions the market density on that region, without the call’s extra weighting toward larger gains.
The interactive density explorer requires JavaScript. With \(X(0)=K=100\), \(\sigma=0.25\) and \(T=1\), the call price is approximately \(9.948\). Its density is proportional to \(\bigl(100\varphi(z-0.25)-100\varphi(z)\bigr)^+\).
The density plot uses the normal observation \(z\) throughout. The separate payoff plot uses the terminal price \(X(T)\). Changing coordinates would change density heights; keeping a common observation axis makes the likelihood ratios directly visible.
\(\mathbb P^C\) describes the exposure built into one contract. It is not automatically the investor’s subjective forecast \(\mathbb P^M\). It also differs from recovering the underlying’s market distribution from a whole surface of option prices.
A call price is a difference of two probabilities
Let \(A=\{X(T)>K\}\). The call pays the underlying on \(A\), less the strike payment on the same event. Since \(X(T)f^Y=X(0)f^X\),
\[ \begin{aligned} C(0) &=X(0)\mathbb P^X(A)\\ &\quad-K\mathbb P^Y(A). \end{aligned} \]
The underlying payment is valued using the asset measure; the strike payment is valued using the reference measure. This identity holds before choosing any particular distribution.
Its density has an equally direct expression:
\[ f^C(z) =\frac{\bigl(X(0)f^X(z)-Kf^Y(z)\bigr)^+}{C(0)}. \]
The numerator is the positive part of the difference between two scaled densities. Its integral is the call price.
Black–Scholes from the normal tails
In the normal model, exercise means
\[ z>\frac{\log(K/X(0))}{a}+\frac a2. \]
Write \(\Phi\) for the standard normal cumulative distribution function, and define
\[ \begin{aligned} d(2)&=\frac{\log(X(0)/K)}{a}-\frac a2,\\ d(1)&=d(2)+a. \end{aligned} \]
Under \(\mathbb P^Y\), the observation has mean zero; under \(\mathbb P^X\), its mean is shifted by \(a\). The exercise probabilities are therefore
\[ \begin{aligned} \mathbb P^Y(A)&=\Phi(d(2)),\\ \mathbb P^X(A)&=\Phi(d(1)). \end{aligned} \]
Substitute them into the two-measure price identity:
\[ \boxed{ \begin{aligned} C(0)&=X(0)\Phi(d(1))\\ &\quad-K\Phi(d(2)). \end{aligned}} \]
This is the Black–Scholes call formula in reference-asset units. The two normal probabilities have financial roles: one values delivery of the underlying, the other values payment of the strike. Neither probability is an investor’s personal forecast unless that investor chooses the corresponding measure.
For a put, use the complementary exercise region:
\[ \begin{aligned} P(0)&=K\Phi(-d(2))\\ &\quad-X(0)\Phi(-d(1)). \end{aligned} \]
Subtracting yields put–call parity,
\[ C(0)-P(0)=X(0)-K. \]
Let the reference asset be a bank account \(B(t)=e^{rt}\) with constant rate \(r\) and \(B(0)=1\). For a non-dividend-paying stock with currency price \(S(t)\), we have \(X(t)=S(t)/B(t)\). A currency strike \(K^{\mathrm{cash}}\) at maturity becomes \(K=K^{\mathrm{cash}}e^{-rT}\) in reference units.
Consequently,
\[ \begin{aligned} C^{\mathrm{cash}}(0) &=S(0)\Phi(d(1))\\ &\quad-K^{\mathrm{cash}}e^{-rT}\Phi(d(2)), \end{aligned} \]
where
\[ \begin{aligned} d(1)&=\frac{\log(S(0)/K^{\mathrm{cash}})+rT}{\sigma\sqrt T}\\ &\quad+\frac{\sigma\sqrt T}{2},\\ d(2)&=d(1)-\sigma\sqrt T. \end{aligned} \]
The discount factor appears when converting the currency strike into reference units. It was already accounted for by the choice of units above.
Other payoffs, and a reference-asset reserve
A digital call pays one unit on \(A\). Its price is \(\mathbb P^Y(A)=\Phi(d(2))\), and its induced distribution is \(\mathbb P^Y\) conditioned on \(A\).
A straddle combines a call and a put. Its density is the mixture of their densities with weights \(C(0)/(C(0)+P(0))\) and \(P(0)/(C(0)+P(0))\). The weights are fractions of the initial cost, even though the position holds one unit of each contract.
Calls, puts and digitals pay zero on entire regions. If an investor assigns positive probability to a zero-payoff region, committing all wealth to that option gives expected log performance \(-\infty\). The option density is still perfectly valid.
To retain a positive payoff floor, keep a fraction \(b\) of the budget in \(Y\) and allocate \(1-b\) to the option. The portfolio density is
\[ f^V(z)=b\,f^Y(z)+(1-b)f^C(z), \]
and its gross payoff is at least \(b\). The explorer’s reserve control draws this mixture in gold. It is the same cost-weighted construction used in portfolio mixtures.
How could trading the underlying reproduce one of these payoffs? The discrete-hedging page compares a target density with the density implemented by rebalancing at finitely many dates.
Based on Chapter 4 of Jan Vecer’s Principles of Portfolio Choice: An Information-Theoretic, Likelihood-Based Perspective: “Option Payoffs,” the call and put induced distributions, the two-measure price identity, and “Black–Scholes Formula from State Prices.” The normal driver is the standardized cumulative return used there. Chapter 5 develops the same likelihood ratio as an additive normal model.