Using a Taylor Expansion to Derive Delta-Hedged Option P&L
Summary
The explanation shows how to approximate the change in an option’s value as the underlying price and time change. Expand the option price around the current underlying price: the first-order term is delta times the price move, and the second-order term captures curvature through gamma. The first-order term cancels the change in the short underlying hedge when the hedge ratio equals delta.
Time decay enters through the option’s derivative with respect to time, represented by theta, while reinvestment of the portfolio’s net funds contributes the interest-rate term. Together, these terms connect the discrete portfolio change to the familiar delta-hedged approximation involving gamma, theta, and financing. This is a local approximation that omits higher-order terms; its accuracy depends on small moves and the chosen time step. The answer clarifies the expansion conceptually but does not supply a full derivation of the Black–Scholes–Merton model.
Key ideas
- Expand the option value around the current underlying price to approximate its change.
- The first-order price effect is delta times the underlying move.
- Delta hedging cancels that first-order price effect in the portfolio.
- Gamma captures the second-order effect of the underlying move.
- Theta accounts for time decay, while financing contributes an interest-rate term.
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# Taylor series expansion (Volatility Trading book) explanation sought
# Taylor series expansion (Volatility Trading book) explanation sought
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:
$$ C - \Delta S_t $$
where
- $C$ 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:
(1.2)
$$ C(S_{t+1}) - C(S_t) -\Delta(S_{t+1} - S_t) + r(C-\Delta S_t) $$
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:
(1.3)
$$ \Delta(S_{t+1} - S_t) + \frac{1}{2}(S_{t+1} - S_t)^2\frac{\partial^2C}{\partial{S}^2} + \theta - \Delta(S_{t+1}-S_t) + r(C- \Delta S_t) $$
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.
## Answer by philippe (score 6, accepted)
https://quant.stackexchange.com/a/7349
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.