Why Black–Scholes Call Delta Is Known at a Given Time

# Why not delta of Call option is stochastic or random variable?


# Why not delta of Call option is stochastic or random variable?












Delta of an option is defined as ratio of change in price of call option to change in price of underlying securities. If, $c_t$ is call option price at time $t$ and $S_t$ is the price of underlying securities, then the delta of call option is:

$$\Delta(t)=\frac{\partial c_t}{\partial S_t}$$

If change in $dS_t$ ie $(S_{t+dt} -S_t)$ is random, which by definition is true, then delta ($\Delta (t)$, computed at time t) must be random variable (instead of known constant) as it involve $dS_t$ in denominator. But under the Black-Scholes model, the delta of European of Call option (which is written as $N(d_1)$ or $\Phi (d_2)$) is deterministic variable (ie known with certainty) at time $t$.

I want to know why delta of a call option is deterministic quantity, why not it is random variable? If possible, please provide both logical reasoning and formal derivation.

## Answer by Gordon (score 3)

https://quant.stackexchange.com/a/24935

Note that, at time $t$, \begin{align*} d_1(t) = \frac{\ln \frac{S_t}{K} + (r+\frac{\sigma^2}{2}) (T-t)}{\sigma \sqrt{T-t}}, \end{align*} which is a function of $S_t$, and then, it is a random quantity. Consequently, the delta $N(d_1(t))$ is also a random quantity.

Note that, at time $t$, $N(d_1(t))$ is known. However, $\{N(d_1(t)) \mid t \ge 0\}$ is a stochastic process adapted to the filtration of the equity process $\{S_t \mid t \ge 0\}$, that is, $\mathcal{F}_t$.

## Answer by RandyF (score 3)

https://quant.stackexchange.com/a/24936

Since we are dealing with the change in the value of the call option in relation to the change in the value of the stock price, we are looking at the simple slope of the function. We do not need to know how much the stock price will change in order to estimate the slope.

In the absence of gamma, the stock price could change 2% or 1% and this wouldn't change the delta. Let's say we have a simple equation, y = mx + b. m is the impact that a change in x has on the change in y, which is the same idea behind delta. If we know how y responds to changes in x, we can deterministically define m and it is not a random variable.