# 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?

• The Black Scholes model isn't perfect and won't typically output the values found in the market. The derivation of the option price assumes a specific dynamics for the stock price, which doesn't exactly match what an actual stock price will do, and so the 'perfect' delta hedging is only perfect in reality if the model assumptions are true, which they aren't. – Slade Sep 30 '19 at 18:30
• This is how all scientific theories work: they assume that (under some assumptions) the real world phenomenon will match exactly what the theory predicts. It is left to experimental work to check wether this is true or not. – Alex C Oct 2 '19 at 18:04

"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.