Series Expansion and Taylor Approximation

Series expansions allow us to approximate complex nonlinear functions with polynomial terms around a known point.
In quantitative finance, Taylor expansions are used for risk approximation, Greeks computation, and convexity adjustments — expressing prices and sensitivities in powers of small perturbations.

1. Power Series and Analytic Functions

A power series centered at $a$ is a sum of the form:

\[\sum_{n=0}^\infty c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots\]

If the series converges for $

x-a

<R$, where $R$ is the radius of convergence, the function defined by this series is said to be analytic at $a$.

Example:
The exponential function has the representation: $e^x = \sum_{n=0}^\infty \frac{x^n}{n!}.$

The sine and cosine functions similarly: $\sin x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}, \quad \cos x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}.$

2. Taylor Series

If $f(x)$ is infinitely differentiable near $a$, its Taylor series expansion around $a$ is:

\[f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f^{(3)}(a)}{3!}(x-a)^3 + \cdots\]

Or compactly, $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n.$

The remainder term after truncating at order $n$ (Taylor’s theorem) is: $R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}, \quad \text{for some } \xi \text{ between } a \text{ and } x.$

Example:
Expanding $e^x$ around $a=0$: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

3. Maclaurin Series

A special case with $a=0$: $f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots$

Examples:

$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots$, $ x <1$
$(1+x)^r = 1 + rx + \frac{r(r-1)}{2!}x^2 + \frac{r(r-1)(r-2)}{3!}x^3 + \cdots$

The binomial series extends to non-integer $r$, crucial in continuous compounding and yield approximations.

4. First and Second Order Approximations

Truncating the series gives linear and quadratic approximations:

First order (linearization): $f(x) \approx f(a) + f'(a)(x-a).$

Second order (quadratic): $f(x) \approx f(a) + f'(a)(x-a) + \frac{1}{2}f''(a)(x-a)^2.$

These approximations capture the slope and curvature near $a$.
They form the foundation for Greeks expansion and hedging analysis.

Example (Option Price Sensitivity): $dV \approx \Delta\,dS + \frac{1}{2}\Gamma\,(dS)^2.$ Here, $\Delta = \frac{\partial V}{\partial S}$ and $\Gamma = \frac{\partial^2 V}{\partial S^2}$ are the first and second derivatives with respect to $S$.

5. Taylor Expansion in Several Variables

For $f(x_1,\ldots,x_n)$, the first- and second-order Taylor expansion around $\mathbf{a}$ is:

\[f(\mathbf{x}) \approx f(\mathbf{a}) + \nabla f(\mathbf{a})^\top (\mathbf{x}-\mathbf{a}) + \frac{1}{2} (\mathbf{x}-\mathbf{a})^\top H(f)(\mathbf{a}) (\mathbf{x}-\mathbf{a}),\]

where $\nabla f$ is the gradient and $H(f)$ is the Hessian matrix.

In Finance:

$\nabla f$: vector of sensitivities (Greeks)
$H(f)$: curvature matrix of cross-Greeks
The expansion provides the second-order P&L approximation used in risk aggregation.

6. Error and Convergence

The accuracy of a truncated Taylor expansion depends on the remainder $R_n(x)$ and the behavior of higher derivatives.

If $\lim_{n\to\infty} R_n(x)=0$ for $

x-a

<R$, then the Taylor series converges to $f(x)$ in that interval.

However, convergence does not always guarantee equality — some functions are infinitely differentiable but not equal to their Taylor series outside the convergence domain.

Example:
$f(x) = e^{-1/x^2}$ for $x\neq 0$ and $f(0)=0$ has all derivatives zero at 0, but $f(x)\neq 0$ for $x\neq 0$.

7. Applications in Quantitative Finance

7.1 Option Pricing Approximations

Small perturbations in inputs allow for local approximations: $V(S+\Delta S) \approx V(S) + \Delta\,\Delta S + \tfrac{1}{2}\Gamma (\Delta S)^2.$ This expansion isolates linear (delta) and quadratic (gamma) risk components, forming the backbone of delta-gamma hedging.

7.2 Convexity Adjustment

In interest rate products, the expectation of a nonlinear function of rates can be expanded: $E[f(r)] \approx f(E[r]) + \tfrac{1}{2}f''(E[r])Var(r).$ The correction term is the convexity adjustment, arising from Jensen’s inequality for convex functions.

7.3 Volatility Expansion

For implied volatility $\sigma = \sigma_0 + \epsilon$, one can expand price $V(\sigma)$: $V(\sigma) \approx V(\sigma_0) + \nu (\sigma - \sigma_0) + \tfrac{1}{2}\text{Vomma}(\sigma - \sigma_0)^2,$ capturing vega and higher-order vol sensitivities.

7.4 Duration and Convexity of Bonds

Bond price as function of yield $y$: $P(y) \approx P(y_0) - D\,P(y_0)(y - y_0) + \tfrac{1}{2}C\,P(y_0)(y - y_0)^2.$ This gives the duration-convexity decomposition for yield curve risk management.

8. Asymptotic Expansions

Taylor series often provide local approximations; for extreme values or limits, asymptotic expansions are used:

\[f(x) \sim a_0 + \frac{a_1}{x} + \frac{a_2}{x^2} + \cdots, \quad x \to \infty.\]

Used in:

Approximation of option tails (e.g., far OTM Black–Scholes prices)
Large deviation theory
Laplace and saddlepoint methods in risk analytics

9. Logarithmic and Exponential Approximations

Frequently used first-order expansions:

Function	Approximation near 0
$\ln(1+x)$	$x - \tfrac{x^2}{2} + \tfrac{x^3}{3}$
$e^x$	$1 + x + \tfrac{x^2}{2}$
$(1+x)^r$	$1 + rx + \tfrac{r(r-1)}{2}x^2$