How do you price a derivative? The deep idea of Black, Scholes, and Merton is to
replicate it — to build a portfolio of the stock and a bond that reproduces the
option's payoff exactly, in every state of the world. Once you can manufacture the payoff, the
law of one price
does the rest: the option must cost whatever the replicating portfolio costs, or there is a free
lunch.
Consider a portfolio holding \Delta_t shares of the stock
S_t and a bond position worth B_t in the
risk-free account. Its value is
V_t = \Delta_t S_t + B_t.
Both holdings are allowed to change through time — we rebalance, buying and selling
shares as the price moves. The crucial discipline is self-financing: after the
initial investment, no cash is ever added or withdrawn. Every purchase of stock is funded by
selling bonds and vice versa. The portfolio's value changes only because prices change,
never because we top it up.
The self-financing condition, line by line
We want to say precisely what "value changes only from price moves" means as a stochastic
differential. Write the bond as a risk-free account accruing continuously at rate
r,
dB_t = r B_t\,dt,
which just says cash in the bank grows at the risk-free rate. Now differentiate the portfolio
value.
Step 1 — write the value and its general differential. From
V_t = \Delta_t S_t + B_t, a naïve product-rule differential would
track changes in both the holdings and the prices:
dV_t = \underbrace{\Delta_t\,dS_t + dB_t}_{\text{prices move}} \;+\; \underbrace{S_t\,d\Delta_t + (\text{rebalancing of the bond})}_{\text{holdings move}}.
Step 2 — impose self-financing. The defining requirement is that the
holdings-move bracket contributes nothing to the value: any cash freed by changing
\Delta_t is exactly absorbed by the bond, and conversely. Killing
that bracket leaves only the price-driven terms,
dV_t = \Delta_t\,dS_t + dB_t.
This is the self-financing condition in differential form: the value moves only
as the assets you already hold are repriced.
Step 3 — substitute the bond dynamics. Using
dB_t = r B_t\,dt,
dV_t = \Delta_t\,dS_t + r B_t\,dt.
The first term is your exposure to the stock — \Delta_t shares times
the price move dS_t — and the second is the risk-free interest your
bond holding earns. That is the complete law of motion of a self-financing stock-and-bond
portfolio.
Step 4 — replicate, then price by no-arbitrage. Suppose we can choose the
rebalancing rule \Delta_t (and the bond position) so that the
terminal value matches a target option payoff H in every state,
V_T = H = (S_T - K)^+ \quad \text{(say, for a call).}
Then the portfolio and the option have identical payoffs, so by the law of one price they must
have identical prices at every earlier time — in particular today:
\text{option price at } t = 0 \;=\; V_0.
If the option traded for anything other than V_0, you would buy the
cheaper of {option, replicating portfolio}, sell the dearer, and harvest a riskless profit — the
contradiction is exactly the arbitrage argument from before. So the price of any replicable
derivative is the cost of the self-financing portfolio that reproduces it.
Let a portfolio hold \Delta_t shares and a bond
B_t with dB_t = r B_t\,dt, value
V_t = \Delta_t S_t + B_t. Then:
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The portfolio is self-financing exactly when its value moves only through
prices: dV_t = \Delta_t\,dS_t + r B_t\,dt (no cash injected or
withdrawn).
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If a self-financing portfolio replicates an option payoff,
V_T = H in every state, then no-arbitrage forces the option's
price to equal the replicating cost V_0 at every time.
Which \Delta_t replicates? Intuitively, hold exactly as many shares
as the option's sensitivity to the stock,
\Delta_t = \frac{\partial V}{\partial S}(t, S_t),
the option's delta. Then a small move dS_t changes
the share leg by \Delta_t\,dS_t, matching the option's own
first-order move and cancelling its randomness — the position is instantaneously
hedged. Because delta itself drifts as S_t and time change,
the hedge must be continuously rebalanced: this is delta-hedging, and the
requirement of continuous trading is exactly what makes the Black–Scholes machinery a
continuous-time theory. Substituting \Delta = \partial V/\partial S
into the self-financing equation and matching terms with
Itô's lemma
is precisely how one derives the
Black–Scholes PDE.
Replication is a promise that a self-financing \Delta_t
exists for the payoff in hand — and that promise is not automatic. It is guaranteed by
the martingale representation theorem: in a market driven by a single Brownian
motion, every square-integrable payoff can be written as a constant plus a stochastic integral
against that Brownian motion,
H = \mathbb{E}[H] + \int_0^T \phi_t\,dW_t.
The integrand \phi_t is the hedge — read off, it tells you
how many shares to hold at each instant. A market in which every payoff is replicable this way
is called complete, and completeness is what makes the no-arbitrage price not
merely bounded but unique. The Black–Scholes market, with one stock and one Brownian
driver, is complete — which is why every European option there has a single fair price.