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Itô's lemma applied to F(t, S) where dS = μS dt + σS dW gives dF equal to which expression?

(∂F/∂t + μS·∂F/∂S) dt + σS·∂F/∂S dW

(∂F/∂t + μS·∂F/∂S + ½σ²S²·∂²F/∂S²) dt + σS·∂F/∂S dW

μS·∂F/∂S dt + σS·∂F/∂S dW

(∂F/∂t + ½σ²S²·∂²F/∂S²) dt + σS·∂F/∂S dW

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2026 Statistics

Key Facts: INV 201 Exam

~5 hrs

Exam Time

SOA INV 201 syllabus

~$1,275

Standard Fee

SOA fee table

3 sittings

Mar / Jul / Nov

SOA exam schedule

20%

Stochastic Calc Weight

INV 201 syllabus

300-500 hrs

Recommended Study

QFI track guidance

ASA

Prerequisite Designation

SOA FSA pathway

Under SOA's 2025 FSA restructuring, the legacy QFI Quantitative Finance (QFI-QF) exam was renamed INV 201 and now sits as the second of the two QFI Quantitative Finance courses on the FSA Investment track. INV 201 is a roughly 5-hour written-response exam built around a case study, sat in the March, July, and November windows, and concentrates weight on stochastic calculus and risk-neutral pricing (about 20%), with substantial coverage of interest-rate models, credit risk, risk measurement, equity and volatility models, hedging, fixed-income derivatives, and insurance quantitative applications. Candidates need ASA designation as a prerequisite, and SOA expects approximately 300 to 500 study hours for this advanced-level fellowship exam.

Sample INV 201 Practice Questions

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1Itô's lemma applied to F(t, S) where dS = μS dt + σS dW gives dF equal to which expression?

A.(∂F/∂t + μS·∂F/∂S) dt + σS·∂F/∂S dW

B.(∂F/∂t + μS·∂F/∂S + ½σ²S²·∂²F/∂S²) dt + σS·∂F/∂S dW

C.μS·∂F/∂S dt + σS·∂F/∂S dW

D.(∂F/∂t + ½σ²S²·∂²F/∂S²) dt + σS·∂F/∂S dW

Explanation: Itô's lemma includes a second-order correction term ½σ²S²·∂²F/∂S² because (dW)² = dt is non-negligible. The drift adds three pieces: time decay, first-order S move, and the convexity correction; the diffusion remains σS·∂F/∂S.

2Under the risk-neutral measure Q, the discounted price of a non-dividend-paying tradable asset is best described as which process?

A.A submartingale with positive drift

B.A martingale

C.A mean-reverting Ornstein-Uhlenbeck process

D.A deterministic exponential growth process

Explanation: The fundamental theorem of asset pricing states that absence of arbitrage is equivalent to the existence of an equivalent measure under which discounted tradable prices are martingales. This martingale property is the mathematical core of risk-neutral pricing.

3Girsanov's theorem is used in derivatives pricing to accomplish which task?

A.Convert a Brownian motion under P into a drifted Brownian motion under an equivalent measure Q

B.Solve the heat equation underlying Black-Scholes

C.Decompose a semimartingale into finite variation and martingale parts

D.Diagonalize the variance-covariance matrix of asset returns

Explanation: Girsanov's theorem provides the Radon-Nikodym derivative that lets you change measure from physical P to risk-neutral Q. Under Q the drift of the Brownian motion shifts so that discounted asset prices become martingales.

4In the Black-Scholes derivation, the self-financing replicating portfolio holds Δ = ∂C/∂S shares of the underlying. What is the key consequence of this delta-hedge for pricing?

A.It eliminates the dW term in the portfolio dynamics, leaving a riskless return that must equal r

B.It guarantees positive expected profit equal to the equity risk premium

C.It eliminates both the drift and diffusion of the option value

D.It makes the portfolio insensitive to interest-rate changes

Explanation: Holding Δ shares cancels the σS·∂C/∂S dW term shared by the option and stock. The remaining portfolio is locally riskless, and no-arbitrage forces its return to equal the risk-free rate r, which yields the Black-Scholes PDE.

5The Black-Scholes PDE for a European derivative V on a non-dividend stock is:

A.∂V/∂t + rS·∂V/∂S + ½σ²S²·∂²V/∂S² − rV = 0

B.∂V/∂t + μS·∂V/∂S + ½σ²S²·∂²V/∂S² − rV = 0

C.∂V/∂t + rS·∂V/∂S − ½σ²S²·∂²V/∂S² + rV = 0

D.∂V/∂t − rS·∂V/∂S + ½σ²·∂²V/∂S² − rV = 0

Explanation: The Black-Scholes PDE uses the risk-free rate r in both the drift and discounting terms because of risk-neutral pricing; the real-world drift μ does not appear. The convexity term is +½σ²S²·∂²V/∂S² and the discount term is −rV.

6The closed-form Black-Scholes price of a European call on a non-dividend stock is:

A.C = S·N(d2) − K·e^(−rT)·N(d1)

B.C = S·N(d1) − K·e^(−rT)·N(d2)

C.C = K·e^(−rT)·N(d1) − S·N(d2)

D.C = S·e^(−rT)·N(d1) − K·N(d2)

Explanation: Black-Scholes prices a call as the spot weighted by N(d1) less the discounted strike weighted by N(d2). N(d1) is the delta and N(d2) is the risk-neutral probability the option finishes in the money.

7Put-call parity for European options on a non-dividend stock states:

A.C − P = S − K·e^(−rT)

B.C + P = S + K·e^(−rT)

C.C − P = K·e^(−rT) − S

D.P − C = S − K·e^(−rT)

Explanation: Put-call parity is C − P = S − K·e^(−rT) for European options on a non-dividend stock, derived from the static portfolio long call short put paying S − K at expiry. Violations imply arbitrage.

8Which statement about the d1 and d2 quantities in Black-Scholes is correct?

A.d1 = d2 + σ√T

B.d1 = d2 − σ√T

C.d1 and d2 are unrelated

D.d1 = d2 + r·T

Explanation: By definition d1 = [ln(S/K) + (r + ½σ²)T] / (σ√T) and d2 = d1 − σ√T. Equivalently d1 = d2 + σ√T, which is why N(d1) − N(d2) drives the call's vega.

9A geometric Brownian motion satisfies dS = μS dt + σS dW. What is the distribution of ln(S_T / S_0)?

A.Normal with mean (μ − ½σ²)T and variance σ²T

B.Normal with mean μT and variance σ²T

C.Lognormal with mean μT and variance σ²T

D.Chi-squared with T degrees of freedom

Explanation: Applying Itô to ln S yields d(ln S) = (μ − ½σ²) dt + σ dW. So ln(S_T/S_0) is normal with mean (μ − ½σ²)T and variance σ²T, which makes S_T lognormal.

10Under the risk-neutral measure for a non-dividend stock, the drift of S becomes:

A.Zero

B.The real-world drift μ

C.The risk-free rate r

D.The Sharpe ratio times σ

Explanation: Girsanov's theorem replaces the real-world drift μ with the risk-free rate r under Q, since discounted prices must be martingales. The diffusion coefficient σ is unchanged by the measure change.

About the INV 201 Exam

SOA INV 201 Quantitative Finance is the second exam in the two-course Quantitative Finance and Investment (QFI) sequence on the FSA Investment (INV) track, testing risk-neutral pricing, interest-rate and credit models, equity volatility modeling, risk measurement, hedging, fixed-income derivatives, and insurance applications.

Assessment

Written-response case-study exam covering quantitative finance topics across the QFI track, with an integrated case study and a written-answer section.

Time Limit

~5 hours

Passing Score

Pass mark set by SOA

Exam Fee

~$1,275 (Society of Actuaries (SOA))

INV 201 Exam Content Outline

20%

Stochastic Calculus and Risk-Neutral Pricing

Itô's lemma, Girsanov's theorem and change of measure, Black-Scholes derivation and closed-form pricing, put-call parity, and the martingale approach to derivative valuation.

15%

Interest Rate Models

Vasicek and CIR short-rate models, Hull-White extensions, the Heath-Jarrow-Morton forward-rate framework, and the LIBOR Market Model (BGM).

15%

Credit Risk Models

Merton structural model, Black-Cox first-passage, KMV distance-to-default, reduced-form intensity models (Jarrow-Turnbull, Duffie-Singleton), and CDS pricing including the credit triangle and CDS-bond basis.

10%

Equity and Volatility Models

Local volatility (Dupire), stochastic volatility (Heston), and jump-diffusion (Merton), and how each addresses the implied volatility smile and forward-smile dynamics.

15%

Risk Measurement

Value at Risk (parametric, historical, Monte Carlo), Expected Shortfall as a coherent risk measure, EWMA and GARCH(1,1) volatility models, and copulas including Gaussian, t, Clayton, and Gumbel for tail dependence.

10%

Hedging Strategies

Delta, gamma, vega, theta, and rho Greeks; dynamic hedging error and discrete-rebalancing P&L; static replication including variance-swap replication; multi-Greek hedging of exotic positions.

10%

Fixed Income Derivatives

Black's model for caplets, floorlets, and European swaptions; cap-floor parity and payer/receiver swaptions; MBS pricing with prepayment models, OAS, and Monte Carlo over rate paths.

Insurance Quantitative Applications

Variable annuity GMxB riders (GMDB, GMAB, GMIB, GMWB) and dynamic hedging across equity, rates, and vega; equity-indexed annuity replication; AG43 stochastic reserves and CTE-70 standards.

How to Pass the INV 201 Exam

What You Need to Know

Passing score: Pass mark set by SOA
Assessment: Written-response case-study exam covering quantitative finance topics across the QFI track, with an integrated case study and a written-answer section.
Time limit: ~5 hours
Exam fee: ~$1,275

Keys to Passing

Complete 500+ practice questions
Score 80%+ consistently before scheduling
Focus on highest-weighted sections
Use our AI tutor for tough concepts

INV 201 Study Tips from Top Performers

1Master the Itô-to-Black-Scholes derivation end-to-end. Be able to write Itô's lemma for F(t, S), construct the self-financing replicating portfolio, derive the Black-Scholes PDE, and quote the closed-form call/put plus put-call parity from memory.

2Build a one-page reference for short-rate and forward-rate models. Include Vasicek, CIR (with Feller condition), Hull-White, HJM no-arbitrage drift, and the LMM, and learn which numeraire each model naturally uses.

3Drill the credit triangle s ≈ λ(1 − R) and Merton's d2 by hand. Switching fluently between hazard rates, survival probabilities, CDS spreads, and distance-to-default makes case-study questions much faster.

4Practice all five Greeks together with their hedging implications. Be ready to quote Δ, Γ, vega, θ, ρ for European calls/puts and explain long-gamma versus short-gamma P&L under realized versus implied volatility.

5Use exam-style written-response practice for variable annuity hedging. Walk through a GMxB rider end-to-end: identify equity/rates/vol risks, propose a dynamic hedge, and connect to AG43 stochastic reserves and CTE-70 reporting.

Frequently Asked Questions

What is INV 201 and how does it relate to the old QFI exams?

INV 201 is the second course in the two-exam Quantitative Finance sequence under SOA's 2025 FSA restructuring of the Investment (INV) track. It replaces the legacy QFI-QF (QFI Quantitative Finance) exam and continues to test stochastic calculus, derivative pricing, interest-rate and credit models, risk measurement, hedging, and insurance applications at fellowship depth.

When is INV 201 administered?

SOA offers INV 201 in the March, July, and November sittings each year. It is delivered as a written-response exam over approximately 5 hours, with an integrated case study that ties multiple topics together.

How much does INV 201 cost?

The standard exam fee is approximately $1,275 per sitting. Always confirm the current price on the SOA fee table before registering, since SOA periodically updates exam fees.

What are the prerequisites for INV 201?

Candidates need to have attained the ASA designation before sitting INV 201. The syllabus also assumes prior exposure to INV 101 (the first Quantitative Finance course) and to FAM/ALTAM-level material on derivatives, time series, and corporate finance.

How many hours should I plan to study for INV 201?

Plan for roughly 300 to 500 study hours over four to six months. The math (Itô calculus, Girsanov, HJM/LMM) is dense, and the written-response format rewards practicing full case-study answers under exam time conditions, not just multiple-choice drills.

What weights should I expect across topics?

Stochastic calculus and risk-neutral pricing is the largest topic at about 20%, followed by interest-rate models (15%), credit risk (15%), and risk measurement (15%). Equity/volatility models (10%), hedging (10%), and fixed-income derivatives (10%) carry equal mid-weight, with insurance quantitative applications at about 5%.