What is meant by $\frac {\Phi (d_2)}{\Phi (d_1)}$ in the Black Scholes call option price?

I found it in a solution as $\frac{\text{short position in cash}}{(\text{number of shares})(\text{strike price discounted to time zero})}$

Reference can be found here

Q. Number 4 on page number 19, and Its solution on page number 26

  • 1
    $\begingroup$ We need more background to understand your question. $\endgroup$
    – Gordon
    Commented Jan 27, 2016 at 19:40
  • $\begingroup$ Hi @Gordon, please see in the Q, I edited it. $\endgroup$ Commented Jan 28, 2016 at 5:01
  • $\begingroup$ This term is not part of the BS formula for a call price as you can see here: en.wikipedia.org/wiki/Black%E2%80%93Scholes_model in the section "Black-Scholes formula". $\endgroup$
    – Richi Wa
    Commented Jan 28, 2016 at 6:57
  • $\begingroup$ @Rchard Yes but N (d_1) and N (d_2) are parts of the BS formula... and in solution given in link used N (d_2) divide by N (d_1) which I am confised about. $\endgroup$ Commented Jan 28, 2016 at 7:57
  • $\begingroup$ Your question is really hard to understand, it does not become clear to me, what you are asking for. Consider editing it to improve the quality (and quantity) of answers. $\endgroup$ Commented Jan 28, 2016 at 8:08

1 Answer 1


For a call option with price given by \begin{align*} c = S_0 \Phi(d_1) - K e^{-rT}\Phi(d_2), \end{align*} the delta hedge ratio $\Phi(d_1)$ is the number of shares to hold. That is, $S_0 \Phi(d_1)$ is the total holding share value for hedging, while $K e^{-rT}\Phi(d_2)$ is the total cash amount in short.

In the question, it says that, for $N$ options, 250,000 shares of the stock are hold, and the amount of $ £413,057$ is in short. The strike price is $K=2.0$. Therefore, \begin{align*} N \Phi(d_1) = 250000, \mbox{ and } N K e^{-rT}\Phi(d_2) = £413057. \end{align*} Consequently, \begin{align*} \frac{\Phi(d_2)}{\Phi(d_1)} &= \frac{N K e^{-rT}\Phi(d_2)}{N\Phi(d_1)}\frac{N}{N K e^{-rT}}\\ &=\frac{N K e^{-rT}\Phi(d_2)}{N\Phi(d_1) K e^{-rT}} \\ &=\frac{413057}{250000 \times 2.0 \times e^{-0.03 \times 0.5}}\\ &= 0.8386. \end{align*}

  • $\begingroup$ $\Phi (d_1)$ is the numbers of shares hold PER OPTION, no? $\endgroup$ Commented Jan 28, 2016 at 16:55
  • $\begingroup$ @HemantRupani: yes. $\endgroup$
    – Gordon
    Commented Jan 28, 2016 at 16:56

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