Replication & Self-Financing Portfolios

How do you price a derivative? Here is the deep idea of Black, Scholes, and Merton, and it fits in one sentence: to price something, build it. Suppose you can assemble a portfolio of just the stock and cash, and — rebalancing as prices move, but never adding or removing a penny — steer it so that at expiry it holds exactly the option's payoff in every possible scenario. Then the option and the portfolio are the same asset wearing different clothes, and the law of one price does the rest: the option must cost today whatever the portfolio costs today, or there is a free lunch. This is replication, and it is the engine under every no-arbitrage price in this course — the Black–Scholes formula is, at heart, the running cost of a construction project.

Concretely, 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 The crucial discipline is self-financing: after the initial investment, no cash is ever added or withdrawn.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. Without that discipline, "replication" would be a cheat: anyone can end up holding the payoff if they're allowed to wire in fresh money along the way.

Worked example: replication in a two-scenario world

Before continuous time, solve a market small enough to do by hand. Today the stock is S_0 = 100. Tomorrow it will be worth either S_u = 120 or S_d = 80 — just two scenarios. The bank pays zero interest (a bond worth 1 today is worth 1 tomorrow; a nonzero rate only sprinkles discount factors on the same argument). We want to price a call with strike K = 100, which pays C_u = 20 in the up scenario and C_d = 0 in the down one.

Step 1 — set up the replicating portfolio. Buy \Delta shares and put B in the bank (negative B means borrowing). To replicate, the portfolio must match the option's payoff in both scenarios — two equations, two unknowns:

\begin{aligned} 120\,\Delta + B &= 20 \quad &\text{(up)} \\ 80\,\Delta + B &= 0 \quad &\text{(down)} \end{aligned}

Step 2 — solve. Subtract the second equation from the first: 40\,\Delta = 20, so \Delta = \tfrac{1}{2}. Substitute back: 80 \cdot \tfrac{1}{2} + B = 0, so B = -40. The recipe: buy half a share, borrow 40.

Step 3 — check it really replicates. Up: \tfrac{1}{2}\cdot 120 - 40 = 20 ✓. Down: \tfrac{1}{2}\cdot 80 - 40 = 0 ✓. The portfolio pays exactly what the call pays, whichever way the coin lands.

Step 4 — read off the price. Building the portfolio today costs

V_0 = \Delta S_0 + B = \tfrac{1}{2}\cdot 100 - 40 = 10,

so the call is worth 10. Not "about 10", not "10 if the market is rational about probabilities" — exactly 10, enforced by arbitrage. If it traded at 12, you would sell the call (collect 12), spend 10 building the portfolio, pocket 2 today, and at expiry hand over the portfolio's payoff — which is the call's payoff to the penny. Risk-free profit; the mispricing cannot survive.

Step 5 — notice what's missing. At no point did we use the probability of the up-move. Whether the stock rises with chance 90% or 10%, the replication equations — and hence the price — are identical, because matching the payoff in every state makes the odds of each state irrelevant. This is the great surprise of derivative pricing, and it foreshadows why the Black–Scholes formula contains the volatility but not the stock's expected return. In general, the share holding that matches the spread of payoffs to the spread of prices is

\Delta = \frac{C_u - C_d}{S_u - S_d} = \frac{20 - 0}{120 - 80} = \frac{1}{2},

the discrete ancestor of the continuous-time delta \partial V/\partial S you'll meet below.

The self-financing condition, line by line

Now let time flow continuously. We want to say precisely what "value changes only from price moves" means as a stochastic differential. In the literature you will often see the holdings written \varphi_t (shares) and \psi_t (bond units), with V_t = \varphi_t S_t + \psi_t B_t and the self-financing condition dV_t = \varphi_t\,dS_t + \psi_t\,dB_t; we keep the equivalent \Delta_t-and-bond-value notation. 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. To feel what the condition permits, watch one rebalance up close. Suppose at some instant you hold 20 shares at £40 and decide to lighten up: you sell 5 shares, receiving £200, and immediately park the £200 in bonds. Your share leg drops by £200, your bond leg rises by £200 — the portfolio's value is unchanged at the instant of the trade. Rebalancing pays for itself. That is all self-financing demands: trades shuffle value between the legs, they never create or destroy it.

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. Contrast a strategy that is not self-financing: a manager who wires in £10 of fresh cash per day has

dV_t = \Delta_t\,dS_t + r B_t\,dt + 10\,dt.

That extra 10\,dt is exogenous money, and it wrecks the pricing logic: a topped-up portfolio can be steered to any payoff whatsoever, so its terminal value tells you nothing about what the payoff is worth. Only when the funding term is zero does "what it ends with" pin down "what it costs".

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:

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.

Why the price is unique (and what breaks if it isn't)

The theorem above claims more than "a replicating cost is a fair price" — it claims the price is the only one. Here is the argument, and it is pure arbitrage. Suppose two self-financing portfolios V and \widetilde V both replicate the same payoff, V_T = \widetilde V_T = H in every state, but cost different amounts today, say V_0 < \widetilde V_0. Go long the cheap one and short the dear one. The combined position is again self-financing (a difference of self-financing strategies is self-financing — no cash crosses the boundary of either), it banks \widetilde V_0 - V_0 > 0 at inception, and at expiry it holds V_T - \widetilde V_T = 0: nothing owed, nothing owned. Free money today, zero obligation tomorrow — an arbitrage. So in an arbitrage-free market the replicating cost is unique, and the derivative's price is not a matter of opinion, expectation, or risk appetite: it is forced.

Notice how the two hypotheses divide the labour. Replication makes the terminal values agree; self-financing guarantees that agreement at T propagates backwards to every earlier time — because the only way two self-financing value processes with equal endpoints could differ mid-flight is by someone slipping money in or out, which the condition forbids. Drop either hypothesis and the conclusion collapses: without replication the payoffs differ somewhere and the comparison is meaningless; without self-financing the £10-a-day strategy "replicates" everything and prices nothing. What remains — and it is the business of the next page — is to actually find the replicating strategy for a call and compute V_0. The answer is a partial differential equation.

Replication is not just a proof device: it is the day job of every option desk. When a bank sells you a call it does not pray the market falls — it manufactures the payoff it just sold, running exactly the recipe on this page: hold \Delta_t shares against the book, finance them at the money-market rate, and rebalance as the delta drifts (in practice a few times a day, within tolerance bands, since each trade costs commission and spread). On this view an option's premium is its manufacturing cost — the raw materials are shares and borrowed cash, and the cost of the recipe turns out to depend on how violently the ingredients move: the volatility. Desks even speak of their "hedging P&L": if realised volatility comes in below what the premium assumed, the manufacturing ran cheap and the desk keeps the difference.

The dark twin of this story is 19 October 1987. Through the mid-1980s, "portfolio insurance" funds sold institutions a synthetic put — not a real option, but a replication strategy for one, run in reverse: as the market falls, a put's delta says sell more stock; as it rises, buy back. Fine for one fund. But by 1987 an estimated $60–100 billion followed the same rule, and the rule has a feedback loop built in: prices fall → every insurer's model orders selling → selling pushes prices lower → the models order more selling. On Black Monday the Dow fell 22.6% in a single day — still the worst one-day percentage loss in its history — with the insurers' program trades a major amplifier. The moral is not that replication is wrong; it is that replication assumes you can trade continuously at posted prices without moving them. When half the market runs the same self-financing strategy, that assumption devours itself.

Watch the portfolio shadow the option

Below, a random stock path S_t (geometric Brownian motion) runs from 0 to maturity T. A delta-hedged self-financing portfolio value V_t is rebalanced along the way so that at T it lands exactly on the call payoff (S_T - K)^+ — marked on the right against the dashed strike K. The portfolio never receives a cash injection; it just rides the price.

Three things to look for. First, V_0 starts above the intrinsic value (S_0 - K)^+ — that gap is the option's time value, the cost of the hedging campaign still to come. Second, V_t wiggles with the stock but more gently: the portfolio holds only \Delta_t < 1 shares, so it feels a damped copy of every stock move — that is the hedge working. Third, press Refresh a few times and watch the endings: on paths that finish in the money, V_T lands on S_T - K; on paths that die below the strike, it glides down to exactly 0. One self-financing recipe, every scenario covered — which is precisely the property that lets V_0 serve as the price.