Deriving the Black–Scholes Delta Hedge with the Pricing PDE
Summary
The document asks how to express the bank-account holding in a self-financing hedge for a derivative priced as a function of time and the underlying stock. Applying Itô’s formula gives a stock position equal to the derivative’s sensitivity to the stock price. The remaining drift term can be related to the cash position by using the Black–Scholes pricing equation.
The intended conclusion is that the cash holding equals the derivative value minus the stock position’s market value, divided by the bank account value. The document itself presents the Itô expansion and proposed holdings, but does not show the PDE substitution that connects them. Its setup assumes a Black–Scholes market and a derivative price function satisfying the corresponding pricing PDE; it is a theoretical replication argument, not evidence about hedge performance under market frictions or model error.
Key ideas
- The hedge is formed from positions in the underlying stock and the bank account.
- Itô’s formula identifies the stock holding as the derivative’s delta.
- The Black–Scholes PDE links the derivative’s time and curvature terms to its value and delta.
- The cash position follows from matching portfolio value to derivative value after setting the stock holding.
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Full text
# Delta hedge for derivative in Black-Scholes market
# Delta hedge for derivative in Black-Scholes market
Consider a derivative in the Black-Scholes market with the price formula $\Pi_t = F(t,S_t)$. I want to find a self-financing portfolio consisting of the stock and the bank account that hedges the derivative.
The value process of the hedging portfolio is given by $ V_t = h^S_t S_t + h^B_t B_t.$ The dynamics of the value process for the hedging portfolio is now given by (according to the condition of self-financing) $$ dV_t = h^S_t dS_t + h^B_t dB_t.$$
If we apply Itô's formula to the price process, $\Pi_t = F(t, S_t)$, we get that $$ d\Pi_t = \left(\frac{\partial}{\partial t} F(t, S_t) + \frac{\sigma^2 S_t^2}{2} \frac{\partial^2}{\partial S_t^2} F(t, S_t)\right) \frac{1}{rB_t} \, dB_t + \frac{\partial}{\partial S_t} F(t, S_t) dS_t $$ If we choose $$ h^B_t = \left(\frac{\partial}{\partial t} F(t, S_t) + \frac{\sigma^2 S_t^2}{2} \frac{\partial^2}{\partial S_t^2} F(t, S_t)\right) \frac{1}{rB_t} \, , $$ $$ h^S_t = \frac{\partial}{\partial S_t} F(t, S_t), $$
the dynamics of $V$ and $\Pi$ are the same. How can I simplify this so that I get $$ h^B_t = \frac{F(t,S_t)-h^S_tS_t}{B_t} = \frac{F(t,S_t)-S_t \frac{\partial}{\partial S_t}F(t,S_t)}{B_t}, $$ $$ h^S_t = \frac{\partial}{\partial S_t} F(t, S_t), $$
I am supposed to use that $F$ satisfies the Black-Scholes PDE. But I don't understand how $F$ satisfies the Black-Scholes PDE. Any help or tips would be greatly appreciated.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.