# Initial holdings of bonds with delta hedging (Black Scholes model)

Consider the Black Scholes model so $$dS_t = \mu S_t dt + \sigma S_t dW_t, \;\;\; dB_t = rB_t dt$$ I want to delta hedge an European call option with strike price $K$ and strike time $T$. It is known that the price of the option at $t=0$ is $C_0 = S_0\Phi(d_1) - \exp(-rT)K\Phi(d_2)$ where $d_1, d_2$ are well known but not relevant for my question.

Now the delta of the stock is $\Phi(d_1)$, so my initial portfolio consists of $\Phi(d_1)$ shares in $S$ the portfolio value is then $S_0\Phi(d_1)$. But as I understand we want the initial portfolio value ($V_0$) equal to $C_0$. But that would mean we need to put another $- \exp(-rT)K\Phi(d_2)/B_0$ bonds in the portfolio so that $V_0 = C_0$. This confuses me. How can we buy a negative amound of bonds? What am I doing wrong?

## 2 Answers

It just simply means you have to borrow money

Holding a negative amount is also known as short sale.

https://en.wikipedia.org/wiki/Short_(finance)