Phase 14 Exam
Mathematical Finance - Phase Exam
15 interleaved questions drawn from every lesson in this phase. Interleaving mixes topics on purpose - that difficulty is what builds durable, transferable understanding. Aim for 70%+ before advancing; below that, revisit the flagged lessons.
How this exam teachesQuestions are shuffled across lessons (not blocked by topic) so you practice choosing the right idea. Missed questions are added to your review queue automatically.
Q1 MediumAn arbitrage opportunity in the one-period market is a portfolio that:
Has positive expected return
Costs \(\le0\) today and pays \(\ge0\) in every state, \(\gt 0\) in some state
Always beats the stock
Has zero variance
Q2 MediumUnder the risk-neutral measure \(\mathbb{Q}\), the discounted stock price \(\tilde S_t=S_t/R^t\) is:
A submartingale with positive drift
A martingale: \(\E_{\mathbb{Q}}[\tilde S_{t+1}\mid\mathcal F_t]=\tilde S_t\)
Deterministic
A martingale only under \(\Prob\)
Q3 MediumThe First Fundamental Theorem of Asset Pricing states that no arbitrage is equivalent to:
Market completeness
Existence of at least one equivalent martingale measure
Uniqueness of the martingale measure
The stock being a martingale under \(\Prob\)
Q4 MediumThe Black–Scholes PDE is derived by:
Assuming investors are risk-neutral
Constructing a delta-hedged portfolio that is instantaneously risk-free and must earn \(r\)
Maximizing expected utility
Averaging historical returns
Q5 MediumWith a risk-free asset, two-fund separation says every mean–variance investor holds:
A different risky portfolio each
The risk-free asset plus the same tangency portfolio, scaled by risk appetite
Only the risk-free asset
Only the highest-return asset
Q6 EasyThe one-period market is arbitrage-free if and only if:
\(R\lt d\)
\(d\lt R\lt u\)
\(R\gt u\)
\(u=d\)
Q7 MediumWhy does the replication cost equal the discounted risk-neutral expectation?
By assumption
Because the algebra of the hedge collapses exactly to \(\tfrac1R(qC_u+(1-q)C_d)\)
Only approximately, for small moves
Because \(p=q\)
Q8 MediumAn arbitrage-free market is complete if and only if:
It has two states
The equivalent martingale measure is unique
The stock has zero drift
There are more states than assets
Q9 MediumIn the call formula \(C=S_0N(d_1)-Ke^{-rT}N(d_2)\), the term \(N(d_2)\) is:
The real-world probability of exercise
The risk-neutral probability \(\mathbb{Q}(S_T\gt K)\)
The option’s delta
Always 0.5
Q10 MediumThe Merton optimal risky fraction for CRRA utility is:
\(w^*=(\mu-r)/\sigma\)
\(w^*=(\mu-r)/(\gamma\sigma^2)\)
\(w^*=\gamma(\mu-r)\)
\(w^*=r/\sigma^2\)
Q11 MediumIn the replication price, the real-world probability \(p\) of an up move:
Equals \(q\)
Determines the price
Does not enter the price at all
Must exceed \(q\)
Q12 EasyTo price an \(n\)-period claim by backward induction you:
Average terminal payoffs under \(\Prob\)
Roll one-period risk-neutral values back node by node, discounting by \(R\) each step
Discount the maximum payoff
Use \(q^n\) only
Q13 HardIn an incomplete market, a non-replicable claim has:
No arbitrage-free price
Exactly one arbitrage-free price
A range of arbitrage-free prices, one per EMM
A negative price
Q14 MediumWhich statement about the model’s assumptions is correct?
Volatility is observed to be constant across strikes
A volatility smile shows the constant-\(\sigma\) assumption fails
The formula accounts for jumps
Trading is assumed costly
Q15 HardWhich measure and drift govern the optimal-investment problem?
Risk-neutral \(\mathbb{Q}\) with drift \(r\)
Real-world \(\Prob\) with the physical drift \(\mu\)
Neither; it is preference-free
Both give the same answer