Delta Hedging and the Model-Based Black–Scholes Price
Summary
The document explains how a delta-hedging argument leads to the Black–Scholes partial differential equation. A portfolio holds one derivative and shorts an amount of stock equal to the derivative’s sensitivity to the stock price. Applying Itô’s formula removes the stock’s instantaneous random component from the portfolio change, allowing a no-arbitrage pricing equation to be derived.
It addresses a concern that the derivation may use the derivative’s actual market price, including bid–ask effects, as if that price were already known. The response clarifies that the price function in the derivation is the model’s price, determined by its assumptions and inputs; it does not explicitly take the traded derivative price as an input. Agreement with market quotes depends on how well the model represents reality. The explanation is theoretical and does not quantify transaction costs, imperfect hedging, or model error.
Key ideas
- Delta hedging combines a derivative with an offsetting stock position.
- Itô’s formula shows that the hedged portfolio’s instantaneous change has no stock-price randomness.
- The Black–Scholes equation follows from the hedged portfolio and a no-arbitrage condition.
- A model price is calculated from model assumptions rather than directly from the derivative’s market quote.
- The model’s fit to observed prices depends on the realism of its assumptions.
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Full text
# Delta hedging: theoretical value vs actual price
# Delta hedging: theoretical value vs actual price
One way to derive the Black-Scholes PDE is via the Delta-hedging argument:
Suppose that $V_t = V(t, S_t)$, for some function $V: [0,T] \times \mathbb{R} \to \mathbb{R}$. We construct a portfolio by buying one unit of the derivative and shorting $\frac{\partial V}{\partial S}(t, S_t)$ units of the underlying stock. Therefore, the porfolio has value $\Pi_t= V(t, S_t) - \frac{\partial V}{\partial S}(t, S_t) S_t$ and hence by the self-financing property and Ito's formula, $$ d \Pi_t = dV_t - \frac{\partial V}{\partial S}(t, S_t) \,dS_t = \bigg(\frac{\partial V}{\partial t}(t, S_t) + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}(t, S_t) \bigg) \,dt. $$ This allows us to derive the Black-Scholes PDE: $$\frac{\partial V}{\partial t} + rS \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} = r V.$$
However, I notice a strange thing in this argument:
> The Delta-hedging portfolio is constructed using one unit of the derivative, whose price function $V$ fluctuates according to the actual market price of that derivative, i.e. the compromised value following a bid-ask spread in trading. However, the objective of this argument is to find a PDE for $V$. Hence, this argument appears to assume that the theoretical value function (also denoted by $V$) is the same as the actual price function in the market. Have I mixed up anything in this argument? Any ideas?
## Answer by Dhruv Gupta (score 1)
https://quant.stackexchange.com/a/49038
"Whose price function $V$ fluctuates according to the actual market price of that derivative"—this is not true. The reason being that we are 'modeling' the derivative price (where a model is a simplified version of reality).
So $V$ tells us what the derivative price would be under our model—and since this model doesn't use the actual derivative price as an input, $V$ doesn't depend upon the actual derivative price in any explicit way.
The fact that the value of $V$ given by our model ends up being close to the actual derivative price is a consequence of our model assumptions being reasonably close to reality.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.