Negative drift when calibrating GBM parameters

Setup for question: Consider a basket of $$N$$ stocks $$\{S^1, S^2, \dots, S^N\}$$. For fixed strike $$K$$, each stock in the basket, $$S^i$$, follows the SDE

$$dS_t^i = \mu^i(t) S_t^i dt + \sigma^i(K, t) S_t^i dW_t^i$$

where $$W_t^i$$ is a standard brownian motion, $$\mu^i(t) = \dfrac{1}{t}\log\left(\dfrac{F^i(t)}{S^i_t}\right)$$ and $$\sigma^i(k, t)$$ are the implied volatilities from standard call options in the market.

At time $$T$$, the SDE has the analytic solution

$$\dfrac{S_T^i}{S_0^i} = e^{\left(\mu^i(T) -\frac{{\sigma^i(K, T)}^{2}}{2}\right)T+ \sigma^i(K, T) W_T^i }.$$

For ease of notation, put

$$\overline{S^i_t} := \dfrac{S_t^i}{S_0^i}$$ $$\overline{\mu^i(t)} :=\left(\mu^i(t) -\frac{{\sigma^i(K, t)}^{2}}{2}\right)t = \log\left(\dfrac{F^i(t)}{S^i_t}\right) -\frac{{\sigma^i(K, t)}^{2}}{2}t$$

Then $$\overline{S^i_T} = e^{\overline{\mu^i(T)} )T+ \sigma^i(K, T) W_T^i }$$

My question: When I use real forward prices in the market and real volatility surfaces, I'm finding that $$\overline{\mu^i(t)}$$ is negative! Obviously, stock prices cannot be negative. So I'm not sure what I'm doing wrong! So basically, I find that $$\log\left(\dfrac{F^i(t)}{S^i_t}\right) -\frac{{\sigma^i(K, t)}^{2}}{2}t < 0$$ when I use real forward, stock and vol surfaces. What am I doing wrong?

• Hmm. Shouldn’t you use risk neutral measure? And in a GBM the drift and volatility is constant. This is not the case for you. Why do you call it GBM? – Sanjay Aug 16 '19 at 22:55