Quant Academy Math · Probability · Quant Finance
Phase 9 Exam

Stochastic Calculus - Phase Exam

18 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 teaches
Questions 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 Medium
Why can't the integral against a Brownian path be defined pathwise as a Riemann–Stieltjes integral?
Brownian paths are discontinuous
Brownian paths have infinite first variation on every interval
Brownian motion is not adapted
The integrand is not measurable
Q2 Easy
In the Itô multiplication table, \((dW)^2\) equals:
\(0\)
\(dt\)
\(dW\)
\((dt)^2\)
Q3 Medium
The solution of \(dS=\mu S\,dt+\sigma S\,dW\) is:
\(S_0 e^{\mu t+\sigma W_t}\)
\(S_0 e^{(\mu-\sigma^2/2)t+\sigma W_t}\)
\(S_0(1+\mu t+\sigma W_t)\)
\(S_0 e^{(\mu+\sigma^2/2)t+\sigma W_t}\)
Q4 Medium
Under an equivalent change of measure, which of a Brownian-driven SDE changes?
The volatility \(\sigma\)
The drift
Both drift and volatility
Neither
Q5 Medium
Feynman–Kac says the solution of \(v_t+av_x+\tfrac12 b^2 v_{xx}-rv=0,\ v(T,\cdot)=h\) equals:
\(\E[h(X_T)\mid X_t=x]\)
\(\E[e^{-\int_t^T r\,du}h(X_T)\mid X_t=x]\)
\(h(x)\)
\(e^{-r(T-t)}h(x)\)
Q6 Medium
The risk-neutral price of a claim paying \(V_T\) at \(T\) is:
\(\E_\Prob[e^{-rT}V_T]\)
\(\E_{\mathbb{Q}}[e^{-rT}V_T]\)
\(e^{-\mu T}\E_\Prob[V_T]\)
\(\E_{\mathbb{Q}}[V_T]\)
Q7 Medium
\(\int_0^T W_s\,dW_s\) equals:
\(\tfrac12 W_T^2\)
\(\tfrac12 W_T^2-\tfrac12 T\)
\(W_T^2-T\)
\(\tfrac12 W_T^2+\tfrac12 T\)
Q8 Medium
For \(dS=\mu S\,dt+\sigma S\,dW\), the SDE for \(Y=\log S\) is:
\(dY=\mu\,dt+\sigma\,dW\)
\(dY=(\mu-\tfrac12\sigma^2)dt+\sigma\,dW\)
\(dY=(\mu+\tfrac12\sigma^2)dt+\sigma\,dW\)
\(dY=\mu S\,dt+\sigma S\,dW\)
Q9 Medium
\(\E[S_t]\) for GBM equals:
\(S_0 e^{(\mu-\sigma^2/2)t}\)
\(S_0 e^{\mu t}\)
\(S_0 e^{\sigma^2 t/2}\)
\(S_0\)
Q10 Medium
To turn \(dS=\mu S\,dt+\sigma S\,dW\) into \(dS=rS\,dt+\sigma S\,d\tilde W\), the market price of risk is:
\(\theta=\mu-r\)
\(\theta=(\mu-r)/\sigma\)
\(\theta=(r-\mu)/\sigma\)
\(\theta=\sigma/(\mu-r)\)
Q11 Easy
The PDE arises from Itô because a martingale must have:
Zero volatility
Zero drift
Constant value
Positive drift
Q12 Easy
Under the risk-neutral measure \(\mathbb{Q}\), a non-dividend stock has drift:
\(\mu\)
\(r\)
\(\mu-r\)
\(0\)
Q13 Medium
The Itô isometry states that \(\E[(\int_0^t\Delta\,dW)^2]\) equals:
\((\E\int_0^t\Delta\,ds)^2\)
\(\E\int_0^t\Delta_s^2\,ds\)
\(\int_0^t\Delta_s^2\,ds\)
\(t\,\E[\Delta_t^2]\)
Q14 Medium
Why is \(W_t^2-t\) a martingale but \(W_t^2\) is not?
Both are martingales
Subtracting \(t\) removes the \(dt\) drift, leaving a pure Itô integral
Because \(W_t^2\) is negative
Because \(t\) is random
Q15 Easy
In Euler–Maruyama, the noise increment is:
\(\sigma S\,\Delta t\,Z\)
\(\sigma S\,\sqrt{\Delta t}\,Z\)
\(\sigma S\,Z\)
\(\sigma S\,(\Delta t)^2 Z\)
Q16 Easy
The Radon–Nikodym derivative \(Z=d\mathbb{Q}/d\Prob\) must satisfy:
\(\E_\Prob[Z]=0\)
\(\E_\Prob[Z]=1\) and \(Z\gt 0\)
\(Z\lt 0\)
\(Z=1\) always
Q17 Medium
When Feynman–Kac is used for option pricing, the drift in the PDE is:
The physical drift \(\mu\)
The risk-free rate \(r\)
Zero
The dividend yield
Q18 Medium
What guarantees the existence of a risk-neutral measure \(\mathbb{Q}\)?
Market completeness
Absence of arbitrage
Constant volatility
Zero interest rates
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