Actuarium

SOA QFI QFQuantitative Finance & Investment β€” Quantitative Finance

Fellowship (FSA)
5 hoursΒ·β‰ˆ10–12 written-answer questions with derivationsΒ·β‰ˆ450 study hours

Syllabus learning objectives

Official syllabus

Paraphrased from the SOA syllabus so the study-plan builder and practice sets track the topics you will actually be examined on. Weights are the official topic ranges.

Stochastic calculus & derivative pricing

35%
  • Apply ItΓ΄'s lemma and the risk-neutral measure to price European and path-dependent derivatives.
  • Derive the Black–Scholes PDE and interpret the Greeks.
  • Price options via binomial trees, Monte Carlo and finite-difference methods.
  • Explain the volatility smile and its implications for model choice.

Interest-rate models

30%
  • Describe and compare short-rate models (Vasicek, CIR, Hull–White) and their term-structure implications.
  • Price bonds and bond options in an affine model.
  • Explain the HJM and LIBOR market model frameworks.
  • Calibrate a term-structure model to caps/swaptions.

Credit risk & structured products

20%
  • Compare structural (Merton) and reduced-form credit models.
  • Price a credit default swap from hazard rates and recovery assumptions.
  • Explain the mechanics and risks of securitisations and tranching.
  • Assess counterparty credit risk (CVA) concepts.

Numerical methods & model risk

15%
  • Apply variance-reduction techniques (antithetic, control variates, quasi-random).
  • Assess discretisation error and convergence in simulation.
  • Identify model risk in derivative pricing and hedging.
  • Explain the limits of calibration and parameter stability.

Lecture videos for this exam

Open the full video library β†’

How to Derive the Black-Scholes Equation

Roman Paolucci Β· Derivatives & Quantitative Finance

The Easiest Way to Derive the Black-Scholes Model

Perfiliev Financial Training Β· Derivatives & Quantitative Finance

Overview

QF is the mathematical core of the QFI track: stochastic calculus, derivative pricing, interest-rate and credit models, and the numerical methods that make them work.

Duration
5 hours
Questions
β‰ˆ10–12 written-answer questions with derivations
Style
Computer-based written answer
Credit
QFI track FSA requirement
Passing
β‰ˆ 60–65% of points

Syllabus map

Stochastic calculus & derivative pricing
35%
Interest Theory and Time Value of Money
Interest-rate models
30%
Interest Theory and Time Value of Money
Credit risk & structured products
20%
Risk Measures and Capital
Numerical methods & model risk
15%

Key formulas

ItΓ΄'s lemma β€…β€Šdf=(ft+ΞΌfx+12Οƒ2fxx)dt+Οƒfx dW\;df=\left(f_t+\mu f_x+\tfrac12\sigma^2 f_{xx}\right)dt+\sigma f_x\,dW

Black–Scholes β€…β€ŠC=SΞ¦(d1)βˆ’Keβˆ’rTΞ¦(d2),d1,2=ln⁑(S/K)+(rΒ±Οƒ2/2)TΟƒT\;C=S\Phi(d_1)-Ke^{-rT}\Phi(d_2),\quad d_{1,2}=\dfrac{\ln(S/K)+(r\pm\sigma^2/2)T}{\sigma\sqrt T}

Vasicek β€…β€Šdr=a(bβˆ’r) dt+σ dW;P(t,T)=A(t,T)eβˆ’B(t,T)rt,β€…β€ŠB=1βˆ’eβˆ’a(Tβˆ’t)a\;dr=a(b-r)\,dt+\sigma\,dW;\quad P(t,T)=A(t,T)e^{-B(t,T)r_t},\;B=\dfrac{1-e^{-a(T-t)}}{a}

CIR β€…β€Šdr=a(bβˆ’r) dt+Οƒr dW\;dr=a(b-r)\,dt+\sigma\sqrt r\,dW (Feller: 2ab>Οƒ22ab>\sigma^2)

CDS spread (flat hazard Ξ»\lambda, recovery RR) β€…β€Šsβ‰ˆΞ»(1βˆ’R)\;s\approx\lambda(1-R)

Merton distance to default β€…β€ŠDD=ln⁑(V/D)+(ΞΌβˆ’ΟƒV2/2)TΟƒVT\;DD=\dfrac{\ln(V/D)+(\mu-\sigma_V^2/2)T}{\sigma_V\sqrt T}

Study strategy

  1. Derivations are graded step by step β€” write every step, even the obvious ones.

  2. Memorise the Vasicek/CIR/Hull–White comparison table (mean reversion, negativity, analytic bonds, fit to initial curve).

  3. Practise short qualitative answers: why the smile exists, why calibration is unstable.

Common traps

  • Forgetting the 12Οƒ2fxx\tfrac12\sigma^2 f_{xx} term.

  • Applying risk-neutral drift rr in a real-world projection.

  • Using the CIR formula when Feller's condition fails and rates hit zero.

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