Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am currently reading Volatility Trading, I have only just started, but I am trying to understand a "derivation from first principles" of the BSM pricing model.

I understand how the value of a long call ($C$) and delta-hedged short position ($\Delta S$) in the underlying is given by:

$$ C - \Delta S_t $$


  • $C$ is the value of the long call option
  • $S_t$ is the spot price of the underlying at time $t$
  • $\Delta$ is the hedge ratio.

On page 9, I also understand that the change in the value of said portfolio, as the underlying moves from $S_{t}$ to $S_{t+1}$ is given by:


$$ C(S_{t+1}) - C(S_t) -\Delta(S_{t+1} - S_t) + r(C-\Delta S_t) $$

where the last term is money earned from reinvesting net received funds obtained in establishing the position at a rate $r$.

The change in the option value is then obtained via a second-order Taylor series approximation:


$$ \Delta(S_{t+1} - S_t) + \frac{1}{2}(S_{t+1} - S_t)^2\frac{\partial^2C}{\partial{S}^2} + \theta - \Delta(S_{t+1}-S_t) + r(C- \Delta S_t) $$

where $\theta$ is time decay.

I don't see how the author moves from equation 1.2 to equation 1.3, as it is not clear (at least to me) what function $f(x)$ he is approximating in 1.3

I would be grateful if someone could explain how the author makes the leap from equation 1.2 to the Taylor approximation (1.3) given on page 9.

share|improve this question
The author of the book "Volatility trading" (there will be a new edition in 2013) is quite active on nuclear phynance, there is even a long thread on the book: nuclearphynance.com/Show%20Post.aspx?PostIDKey=110391 – philippe Feb 23 '13 at 9:05
up vote 5 down vote accepted

He is approximating $C(S_{t+1})$ around $t$:

$$C(S_{t+1})=C(S_{t}) + \frac{\partial C(S_{t})}{\partial S_{t}}(S_{t+1}-S_{t})+\frac{(S_{t+1}-S_t)²}{2}\frac{\partial^{2}C(S_{t})}{\partial S_{t}^{2}} + ...$$

In addition, he takes the time value of $C(S_t)$ into account (and I look only at the time contribution here):

$$C(S_{t+1})-C(S_t)=\Delta t\frac{\partial C}{\partial t}+...=\Delta t\cdot\theta+..$$

There, the first equation is just the derivative of the option with regard to t. Usually, $\theta$ is the loss of the option value in a day, so it is just a question of normalization here. If you put everything together, you get the step you are looking at.

share|improve this answer
Thanks. +1 for the effort. I now get the first part of the derivation. I am trying to understand the theta derivation (i.e. derivation wrt time), and how that leads to the final equation presented in 1.3 – Homunculus Reticulli Feb 23 '13 at 20:01
Ok, I finally got it phew. Thanks – Homunculus Reticulli Feb 24 '13 at 3:15

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.