Contents
Exam Overview
The Fundamentals of Actuarial Mathematics – Financial (FAM-F) exam tests candidates on financial economics, derivatives, and investment concepts. The exam is three hours long with approximately 30 multiple-choice questions. Let’s explore each major topic with its essential formulas and concepts.
Interest Rate Theory
Short Rate Models
The instantaneous short rate r(t) forms the foundation of interest rate modeling. Under the Ho-Lee model, we have:
r(t) = r(0) + θ(t) + σW(t)
Where:
- r(0) is the initial short rate
- θ(t) is the drift function
- σ is the volatility
- W(t) is a Wiener process
For the Vasicek model, the short rate follows:
dr(t) = a(b – r(t))dt + σdW(t)
Where:
- a is the mean reversion speed
- b is the long-term mean rate
- σ is the volatility
Forward Rates
The relationship between spot rates and forward rates:
f(t,T) = -[∂/∂T ln P(t,T)]
Where P(t,T) is the price of a zero-coupon bond.
Derivatives Pricing
Option Pricing Fundamentals
The Black-Scholes formula for European call options:
C = S₀N(d₁) – Ke^(-rT)N(d₂)
Where:
- d₁ = [ln(S₀/K) + (r + σ²/2)T]/(σ√T)
- d₂ = d₁ – σ√T
- S₀ is the current stock price
- K is the strike price
- r is the risk-free rate
- T is time to expiration
- σ is volatility
- N() is the standard normal CDF
For European put options:
P = Ke^(-rT)N(-d₂) – S₀N(-d₁)
Put-Call Parity
For European options:
C – P = S₀ – Ke^(-rT)
Option Greeks
Delta (Δ):
- Call: N(d₁)
- Put: N(d₁) – 1
Gamma (Γ):
Γ = N'(d₁)/(S₀σ√T)
Theta (Θ):
- Call: -[S₀N'(d₁)σ/(2√T)] – rKe^(-rT)N(d₂)
- Put: -[S₀N'(d₁)σ/(2√T)] + rKe^(-rT)N(-d₂)
Vega (ν):
ν = S₀√T × N'(d₁)
Rho (ρ):
- Call: KTe^(-rT)N(d₂)
- Put: -KTe^(-rT)N(-d₂)
Binomial Option Pricing
Parameters
Up factor: u = e^(σ√(T/n))
Down factor: d = 1/u
Risk-neutral probability: p = (1 + r – d)/(u – d)
Option Value
At each node (i,j):
V(i,j) = e^(-r×dt)[pV(i+1,j+1) + (1-p)V(i+1,j)]
Financial Risk Management
Value at Risk (VaR)
For normal distribution:
VaR = μ + σ × z_α
Where:
- μ is the mean return
- σ is the standard deviation
- z_α is the standard normal quantile
Portfolio Risk Measures
Portfolio variance:
σ_p² = Σ[i=1 to n]Σ[j=1 to n] w_i × w_j × σ_i × σ_j × ρ_ij
Where:
- w_i, w_j are portfolio weights
- σ_i, σ_j are asset standard deviations
- ρ_ij is correlation between assets i and j
Investment Strategies
Duration and Convexity
Modified Duration:
D_mod = -(1/P) × (dP/dr)
Convexity:
C = (1/P) × (d²P/dr²)
Price approximation:
ΔP/P ≈ -D_mod × Δr + (C/2) × (Δr)²
Derivatives Markets
Forward Contracts
Forward price:
F₀ = S₀(1 + r – q)ᵗ
Where:
- S₀ is spot price
- r is risk-free rate
- q is dividend yield
- t is time to maturity
Futures
Daily settlement adjustment:
Mt = Ft – Ft₋₁
Where:
- Mt is the margin adjustment
- Ft is today’s futures price
- Ft₋₁ is yesterday’s futures price
Important Relationships and Applications
Option Strategies
Straddle value:
V = max(S – K, 0) + max(K – S, 0)
Strangle value:
V = max(S – K₁, 0) + max(K₂ – S, 0), where K₂ > K₁
Butterfly spread value:
V = max(S – K₁, 0) – 2max(S – K₂, 0) + max(S – K₃, 0)
Risk-Neutral Pricing
The fundamental theorem of asset pricing states that the price of any derivative is:
V₀ = E_Q[e^(-rT)V_T]
Where:
- E_Q denotes expectation under risk-neutral measure
- V_T is the payoff at maturity
Study Strategies
Understanding these concepts requires both theoretical knowledge and practical application. Here are some effective study approaches:
- Start with the fundamentals of financial mathematics and build up to more complex derivatives concepts.
- Practice implementing these formulas with different numerical values to develop intuition.
- Focus on understanding the relationships between different pricing models and their assumptions.
- Work through example problems that combine multiple concepts, as the exam often tests integrated knowledge.
- Develop a systematic approach to option pricing problems, considering:
- Market conditions
- Contract specifications
- Mathematical assumptions
- Practical limitations
Exam Tips
When applying these formulas during the exam:
- Watch for special cases where simplified formulas can be used.
- Pay attention to units and time conventions (annual, continuous, etc.).
- Consider whether the assumptions of each model are satisfied.
- Double-check your inputs, especially for option pricing formulas.
Remember that this exam tests not just the ability to apply formulas, but also understanding of when and why to use particular approaches.
