Self-Financing Delta Hedging and Black–Scholes Replication

# black scholes replication


# black scholes replication












I have a particular question regarding the black scholes equation. I see that if you sell a european call at the black scholes price and immediately purchased delta shares using this money and borrow/deposit the rest at the risk free rate then by definition of the price function, the first order change in value of your stock and cash position will be exactly that of the call option. The thing I dont see is how I can extend this reasoning to say that even if theoretically the portfolio share holdings continuoulsy varied using delta, how am I guaranteed that the first order changes at times different from the start coincide using the previous portfolio earnings? Or in other words maybe: It seems like I am making an additional assumption about the portfolio being self financing for the replication to happen? I dont know if I am expressing my doubt correctly. Thank you in advance !

## Answer by QuantCalc.net (score 1, accepted)

https://quant.stackexchange.com/a/85365

The main question is: "How are we guaranteed the previous earnings are enough to fund the next delta?"

You are correct: the "bridge" between a single-step hedge and a continuous replication strategy is the Self-Financing Property.

In the Black-Scholes framework, we don't just assume the portfolio is self-financing; we require it to be. The Black-Scholes price is the specific value that allows the portfolio to fund its own rebalancing without any external cash injections or withdrawals.

- Defining the Portfolio

We construct a portfolio $V$ at time $t$ consisting of $\Delta_t$ shares of the underlying stock $S_t$ and a remaining amount in a cash bond $B_t$ (the "bank account").

\begin{equation} V_t = \Delta_t S_t + B_t \end{equation}

- The Total Change in Wealth

When we look at the change in the portfolio value over an infinitesimal time $dt$, we must account for both the change in asset prices and the change in the amount of assets held. Using the differential form:

\begin{equation} dV_t = (\Delta_t dS_t + dB_t) + (S_t d\Delta_t + d\text{Rebalance}) \end{equation}





- The Self-Financing Condition The "guarantee" you are looking for is the Self-Financing Assumption. We define the strategy such that the cost of changing the number of shares is exactly offset by the cash taken from or deposited into the bank account:

\begin{equation} S_t d\Delta_t + d\text{Rebalance} = 0 \end{equation}

When this condition holds, the change in the portfolio value depends only on the price movements of the underlying assets:

\begin{equation} dV_t = \Delta_t dS_t + r B_t dt \end{equation}

Substituting $B_t = V_t - \Delta_t S_t$ back into the equation:

\begin{equation}dV_t = \Delta_t dS_t + r(V_t - \Delta_t S_t)dt \end{equation}

- How the PDE "Forces" the Guarantee To replicate the option $C(S,t)$, we need $V_t = C(S,t)$ at all times. By applying Ito’s Lemma to the option price $C(S,t)$, we get:

\begin{equation}dC = \frac{\partial C}{\partial t}dt + \frac{\partial C}{\partial S}dS + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}dt \end{equation}

For the portfolio to stay "in sync" with the option continuously ($dV_t = dC$):

- We set $\Delta_t = \frac{\partial C}{\partial S}$ (the Delta) to match the $dS$ terms.

- The remaining terms (Theta and Gamma) must equal the interest earned on the portfolio ($r(V_t - \Delta_t S_t)dt$).

This equality only holds if $C$ satisfies the Black-Scholes Partial Differential Equation:

\begin{equation} \frac{\partial C}{\partial t} + rS\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} - rC = 0 \end{equation}

In the discrete case, we move from the world of continuous differentials (dt, dS) to finite time steps (Δt).

- The Discrete Portfolio Setup

At time $t_k$, the portfolio value $V_k$ is comprised of the stock position and the bond position: $ V_k = \Delta_k S_k + B_k $

- Evolution to $t_{k+1}$

At the next time step $t_{k+1}$, before any rebalancing occurs, the value of the portfolio is: $ V_{k+1}^{\text{pre}} = \Delta_k S_{k+1} + B_k e^{r \Delta t} $

- The Discrete Self-Financing Requirement

To adjust the position to the new required delta $\Delta_{k+1}$, we must satisfy the condition that the net cost of the trade is zero: $ \underbrace{(\Delta_{k+1} - \Delta_k) S_{k+1}}_{\text{Cost of new shares}} + \underbrace{(B_{k+1} - B_k e^{r \Delta t})}_{\text{Change in bond}} = 0 $ This ensures that $V_{k+1} = V_{k+1}^{\text{pre}}$. The wealth at the end of the step is simply the result of the previous step's investment decisions.

- The Tracking Error (Discrete vs. Continuous)

In discrete time, the portfolio $V$ does not perfectly track the option $C$. The variance of the hedging error is driven by the Gamma ($\Gamma = \frac{\partial^2 C}{\partial S^2}$): $ \text{Error} \approx \frac{1}{2} \frac{\partial^2 C}{\partial S^2} (\Delta S)^2 $ As the rebalancing frequency increases ($\Delta t \to 0$), the discrete self-financing portfolio converges to the continuous Black-Scholes replication.