Numerical Sensitivity of Discrete Sums with Continuous Limits

# How to compute derivatives of discrete sums with respect to continuous upper limit for sensitivity analysis?


# How to compute derivatives of discrete sums with respect to continuous upper limit for sensitivity analysis?












In several quantitative finance contexts (discrete Asian options, credit derivatives, variance swaps with discrete monitoring), we encounter sums of the form:

$$S(n) = \sum_{k=1}^{n} f(k)$$

where $f(k)$ has no closed-form antidifference.

For sensitivity analysis (Greeks), I need to compute derivatives treating the upper limit as continuous:

$$\frac{\partial S}{\partial x}\bigg|_{x=n}, \quad \frac{\partial^2 S}{\partial x^2}\bigg|_{x=n}$$

Concrete example (from stochastic volatility context):

$$S(x) = \sum_{k=1}^{x} e^{-rk - b k^{3/2}}$$

with $r = 0.05$, $b = 0.1$, evaluated at $x = 10^6$.

What I have tried:

- Finite differences:

$$\frac{dS}{dx} \approx \frac{S(n+h) - S(n-h)}{2h}$$

$$\frac{d^2S}{dx^2} \approx \frac{S(n+h) - 2S(n) + S(n-h)}{h^2}$$

Problem: This requires extending to non-integer $n$. I tried interpolating $S(n)$ but second derivatives accumulate significant numerical noise, especially for large $n$.

- Euler-Maclaurin formula:

$$\sum_{k=1}^{n} f(k) \approx \int_1^n f(x)dx + \frac{f(1)+f(n)}{2} + \sum_{j=1}^{p} \frac{B_{2j}}{(2j)!}\left(f^{(2j-1)}(n) - f^{(2j-1)}(1)\right)$$

Problem: For $f(k) = e^{-rk - bk^{3/2}}$, the higher derivatives become increasingly complex, and the asymptotic expansion does not converge well due to the $k^{3/2}$ term.

- Analytic continuation via Lerch/Hurwitz zeta:

$$\Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(k+a)^s}$$

Problem: Not applicable when $f(k)$ mixes exponential decay with fractional powers $k^{3/2}$.

- Direct numerical integration of the "summand density":

Tried replacing the sum with:

$$S(x) \approx \int_1^x f(t) dt + \text{correction terms}$$

Problem: The correction terms for non-integer limits are not straightforward to compute with high precision.

My questions:

- What numerical methods do practitioners use for computing such "fractional Greeks" when no closed form exists?

- Is there established literature on extending discrete sums to continuous limits for sensitivity calculations?

- For large $n$ (e.g., $10^6$ terms in HFT or risk systems), what precision is typically achievable in production?

Any references to papers or production implementations would be appreciated.