Consider a portfolio X that holds stocks and cash, such that
X = qS + CM
where
X = portfolio value
q = quantity of stock
S = stock price
C = quantity of cash
M = price of cash
We’ll let this portfolio be self-financing. A self-financing portfolio is one where there is no exogenous infusion or withdrawal of money. Therefore, the purchase of a new asset must be financed by the sale of an old one.
We’ll start with zero cash and a lot of credit. This means we can lend and borrow an arbitrary amount of money at the risk-free rate and the stock purchases and sales are directly funded from or to the cash account.
For instance, if we buy 100 shares at $20 stock, the cash account goes down by $2000. If initial X = $0, after buying the shares, q = 100, S = $20, C = -2000, M = $1. Therefore, X becomes 100*$20 – 2000*$1 = 0
Suppose we rebalance this portfolio at the end of each trading period (e.g. each trading day) by exchanging bonds for stock of equal value at the prevailing market price.
Let
Xtpre = Portfolio value before rebalancing at time t
Xtpost = Portfolio value after rebalancing at time t
St = Stock price at time t
(Stock price is assumed to be constant during the rebalancing, which occurs over a very short duration.)
\begin{aligned}
X_t^{post} - X_t^{pre} &= (S_tq_t + M_tC_t) - (S_tq_{t-1} + M_tC_{t-1})\\
&= S_t(q_t - q_{t-1}) + M_t(C_t - C_{t-1})\\
&= S_{t-1}(q_t - q_{t-1}) + M_{t-1}(C_t - C_{t-1}) + (S_t-S_{t-1})(q_t - q_{t-1}) + (M_t - M_{t-1})(C_t - C_{t-1})\\
&= Sdq + MdC + dSdq + dMdC
\end{aligned}As this portfolio is self-financing, rebalancing does not change the portfolio value. Therefore,
Sdq + MdC + dSdq + dMdC = 0
Now, let’s consider another self-financing portfolio. This time, we’ll consider one with three assets: an underlying security (e.g. stock), a derivative (e.g. option) and cash (e.g. bond).
We’ll start with an initial portfolio value of zero. We’re going to buy and hold the derivative and rebalance our stock position to hedge the portfolio.
\pi = V + qS + CM
where
𝜋 = portfolio value
V = option value
q = quantity of stock
S = stock price
C = quantity of cash
M = value of cash
Since the portfolio is self-financing, d𝜋 = 0 and Sdq + MdC + dSdq + dMdC = 0.
In continuous time (using product rule for Ito processes),
\begin{aligned}
d\pi &= dV + d(qS) + d(CM)\\
&= dV + (Sdq + qdS +dqdS) + (MdC +CdM + dCdM)\\
&= dV + qdS + CdM + (Sdq + MdC +dSdq + dCdM)\\
&= dV + qdS + CdM + 0\\
&= dV + qdS + CdM\\
&= dV + qdS + rCMdt\;\;(dM = rMdt\text{, where r is the risk-free interest rate)}\\
&= dV + qdS + r(\pi - V - dS)dt
\end{aligned}Using Ito’s lemma,
dV = \left(\frac{\partial V}{\partial t} + \frac{(\sigma S)^2}{2}\frac{\partial^2 V}{\partial S^2} \right)dt + \left(\frac{\partial V}{\partial S} \right)dSTherefore,
\begin{aligned}
d\pi &= \left(\frac{\partial V}{\partial t} + \frac{(\sigma S)^2}{2}\frac{\partial^2 V}{\partial S^2} \right)dt + \left(\frac{\partial V}{\partial S} + q \right)dS + r(\pi - V - qS)dt
\end{aligned}To get rid of the stochastic term dS, let
q = -\frac{\partial V}{\partial S}Substitute 𝜋 = 0, d𝜋 = 0 and q=−∂S/∂V into the equation above. We get
\begin{aligned}
d\pi &= \left(\frac{\partial V}{\partial t} + \frac{(\sigma S)^2}{2}\frac{\partial^2 V}{\partial S^2} -rV + rS\frac{\partial V}{\partial S} \right)dt \\
&= 0
\end{aligned}which gives us the Black-Scholes equation:
\begin{aligned}
\frac{\partial V}{\partial t} + \frac{(\sigma S)^2}{2}\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0
\end{aligned}This equation tells us that the price of an option depends on the volatility (σ) of the stock and the risk-free interest rate (r), but does not depend on the drift (μ) of the stock.
Other parameters such as the strike price, expiration date, type (call/put/exotic) will be set by boundary conditions when solving the equation.
Here’s an excellent video that explains how we use rebalancing to dynamically hedge our option position:
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