Phase 15 Exam
Derivatives & Volatility - 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 MediumIn the Black–Scholes derivation, delta-hedging removes the random term so that the portfolio must earn the risk-free rate. The economic content is that:
The option is riskless
The stock has zero drift
An option can be replicated, so no-arbitrage pins its price regardless of the stock’s real drift
Volatility does not affect the price
Q2 EasyImplied volatility is best described as:
The market’s forecast of realized volatility
The \(\sigma\) that makes the Black–Scholes price equal the market price
The historical standard deviation of returns
A number that is the same for all strikes
Q3 MediumThe Dupire equation is significant because it shows that:
Local vol equals implied vol at every strike
There is a unique one-factor diffusion that reprices all of today’s European options, extractable from the surface
Volatility must be stochastic
Option prices are independent of the density
Q4 MediumIn the Heston model the parameter that primarily controls the direction (sign) of the volatility skew is:
The mean-reversion speed \(\kappa\)
The long-run variance \(\theta\)
The spot–variance correlation \(\rho\)
The initial variance \(v_0\)
Q5 MediumIn total-variance coordinates, the calendar no-arbitrage condition is:
\(\partial_k w\ge0\)
\(\partial_T w(k,T)\ge0\) for every fixed moneyness
\(w\le1\)
\(\partial_{kk}w=0\)
Q6 MediumA trader is long a delta-hedged straddle. Over the next month realized volatility comes in far below the implied vol paid. The position:
Profits, because it is delta-hedged
Loses, because gamma P&L cannot offset the theta paid
Is unaffected by realized vol
Profits only if the stock rises
Q7 MediumThe existence of a non-flat volatility smile directly implies that:
Black–Scholes has an arithmetic error
A single constant \(\sigma\) cannot reproduce all option prices simultaneously
Options are mispriced and arbitrageable
Vega is negative for some strikes
Q8 MediumFor the Dupire local variance to be positive and well defined, the call surface must be free of:
Vega risk
Calendar arbitrage (numerator>0) and butterfly arbitrage (denominator>0, non-negative density)
Any skew
Interest-rate risk
Q9 MediumThe Feller condition \(2\kappa\theta\ge\xi^2\) ensures that:
The smile is symmetric
The variance process stays strictly positive and never hits zero
The model has a closed-form call price
Volatility risk is hedgeable
Q10 MediumA butterfly-spread value \(C(K-\Delta)-2C(K)+C(K+\Delta)\) that is negative signals:
A steep but valid skew
A negative implied risk-neutral density - butterfly arbitrage
High interest rates
Correct convexity
Q11 EasyWhich statement about the Greeks is correct?
Gamma and vega both peak deep in-the-money
Theta is always positive for a long call
Call delta lies in \([0,1]\) and equals \(N(d_1)\)
Rho is typically the largest first-order Greek
Q12 MediumAn equity index shows implied vol rising sharply for low strikes (OTM puts). Via Breeden–Litzenberger this indicates:
A thinner left tail than lognormal
A fatter left tail: extra risk-neutral probability of large downward moves
That vega is zero ATM
That the risk-free rate is negative
Q13 HardThe central practical weakness of the pure local-volatility model is that:
It cannot fit today’s vanilla prices
It has too many free parameters
It fits vanillas exactly but predicts unrealistic forward-smile dynamics, mispricing forward-starting exotics
It ignores the risk-free rate
Q14 HardThe main advantage of Heston over pure local volatility is that it:
Fits today’s vanilla surface more exactly
Produces realistic forward-smile dynamics via a genuine second (correlated) volatility factor
Requires fewer parameters
Makes the market complete
Q15 HardWhy must arbitrage-free smoothing (e.g. SVI) precede calibrating a local-vol or Heston model?
To reduce the number of strikes
Because Dupire’s derivatives amplify noise and any surface arbitrage yields negative local variance or nonsensical fits
Because SVI is faster to evaluate than Black–Scholes
It is not necessary