Why Black-Scholes is still so famous among practitioners?

One of the first things that everybody says talking about the Black-Scholes model is that it is not a good model. This is mainly true for many reason, such as the unrealistic assumptions of no transaction costs, continuous trading and constant volatility of the underlying.

As a consequence, there is a question that should arise spontaneously in your mind: why this model is still so famous and used by financial practitioners? One of the possible answers to this question simply relies on a very famous mathematical result, known as the “robustness of the Black-Scholes formula”.

Let’s recall the diffusion model of the underlying under the standard BS model:

dS_t = St (\mu dt + \hat{\sigma} dW_t),

where \mu stands for the drift parameter and \hat{\sigma} for the constant volatility. However, everybody knows that the hypothesis of constant volatility is not good at all, as observed from markets’s implied volatility surfaces exhibiting different smile behaviours.

Let’s then suppose that the volatility is not constant but it behaves as a stochastic process \sigma_t. The diffusion of our stock S_t is then:

dS_t = St (\mu dt + \sigma_t dW_t).

Let’s also suppose that we want to sell a Call option C_t. We then create our self-financing hedging portfolio V_t just
– buying \delta_t of the underlying;
– investing the remaining amount of money \theta_t in the risk-free asset.
Then, our portfolio V_t evolves under the following dynamics:

dV_t = \delta_t (dS_t + q_t S_t dt) - dC_t + \theta_t r_t dt = \delta_t (dS_t + q_t S_t dt) - dC_t + (V_t - \delta_t S_t + C_t) r_t dt,

where the parameters r_t, q_t and \sigma_t refers to the risk-free interest rate, the dividend yield and the volatility, which are supposed to be unknown stochastic processes.

If we consider the Call option price C(S_t,t), as a function of the stock price and of time, we can then apply the Itô formula and we clearly obtain:

dC_t = \frac{\partial C}{\partial t} + \frac{\partial C}{\partial S} dS_t + \frac{1}{2}S_t^2 \sigma_t^2 \frac{\partial^2 C}{\partial S^2}.

This implies, after equating \delta_t = \frac{\partial C}{\partial S}, that

dV_t = \left[ - \frac{\partial C}{\partial t} + \frac{\partial C}{\partial S}S_t q_t dt - \frac{1}{2}S_t^2 \sigma_t^2 \frac{\partial^2 C}{\partial S^2} + (V_t - \frac{\partial C}{\partial S} S_t + C_t) r_t \right] dt.

Let’s now consider a trader that, not caring about the stochastic dynamics of volatility, rates and dividends decides to price and hedge using the standard BS approach with constant volatility \hat{\sigma}, interest rate \hat{r} and dividend rate \hat{q}. In this case we know that the Call option price must satisfy the Black-Scholes PDE with dividends, which gives us:

- \frac{\partial C}{\partial t} = -C \hat{r} + S(\hat{r} - \hat{q}) \frac{\partial C}{\partial S} + \frac{1}{2}S_t^2 \hat{\sigma}^2 \frac{\partial^2 C}{\partial S^2}.

Then, we can substitute this expression into the portfolio dynamics dV_t. Looking at the PnL of our portfolio, defined as the return exceeding the risk-free rate, we finally obtain:

PnL = dV_t - r_t V_t dt = \left[ \frac{\partial C}{\partial S}S_t (q_t - \hat{q}) + \frac{1}{2}S_t^2 (\hat{\sigma}^2 - \sigma_t^2) \frac{\partial^2 C}{\partial S^2} + \left(C_t - \frac{\partial C}{\partial S} S_t\right) (r_t - \hat{r}) \right] dt.

Let’s now consider a null interest rate framework or the case in which r_t = \hat{r}, neglecting in this way the last component of the previous formula. We then have:

PnL = \left[ \frac{\partial C}{\partial S}S_t (q_t - \hat{q}) + \frac{1}{2}S_t^2 (\hat{\sigma}^2 - \sigma_t^2) \frac{\partial^2 C}{\partial S^2} \right] dt.

This final result is really important because it implies that for a payoff which is delta and gamma positive, the trader is sure to have a positive PnL if:
– he has overestimated the volatility, that is \hat{\sigma} \geq \sigma_t;
– he has underestimated the dividends, that is \hat{q} \leq q_t.

To simplify, let’s also neglect the dividend component. We can then conclude that if \hat{\sigma} \geq \sigma_t, the Black-Scholes delta hedging is a super hedging strategy which gives us a superheding price \overline{C_t} for our Call option. On the other side, if \hat{\sigma} \leq \sigma_t, the BS delta hedging is a subheding strategy which gives us a subheding price \underline{C_t} for our Call option. Formally we have:

\overline{C_t} \geq C_t \geq \underline{C_t}.

In this way we have demonstrated that the true Call option price lies between two boundaries, determined by different levels of volatility with respect to the true one. In practice it means that traders will always sell delta positive options pricing them with a higher volatility with respect to the expected one, in order to have more chances to obtain a positive profit from the hedging strategy.

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