EXAM REVISION
🔥 MASTER NUMERICAL REVISION LIST (Units 1–4)
This is your closed list of everything you need for the exam numerically.
Nothing extra. Nothing missing.
We’ll group by topic exactly like your lecturer builds it.
🧠 SECTION A — ARBITRAGE & ONE-PERIOD MODEL (Unit 2)
✅ Q1 — Arbitrage profit calculation
📍 From: Unit 2, Problem 2.1
You must be able to:
Identify arbitrage opportunity
Calculate:
borrowing cost
interest earned
final profit
Pattern:
Borrow → invest → forward contract → compare outcomes
✅ Q2 — Risk-neutral probability (1-period)
📍 From: Unit 2, Example 2.1 + formula
Core formula (MUST memorise):
qU=1+r−DU−D,qD=U−(1+r)U−DqU=U−D1+r−D,qD=U−DU−(1+r)
You must:
Solve for qq
Check: qU+qD=1qU+qD=1
✅ Q3 — Pricing European call (1-period)
📍 From: Unit 2, Example 2.1
Steps (EXAM EXACT):
Compute payoffs:
max(SU−K,0),max(SD−K,0)max(SU−K,0),max(SD−K,0)
Take expectation under QQ
Discount:
C=11+rEQ[payoff]C=1+r1EQ[payoff]
✅ Q4 — Replicating portfolio (hedging)
📍 From: Unit 2, section with θ1,θ2θ1,θ2
Solve system:
θ1SU+θ2=D1θ1SU+θ2=D1θ1SD+θ2=D2θ1SD+θ2=D2
Then:
V=θ1S+θ2V=θ1S+θ2
✅ Q5 — State price / pricing via cashflows
📍 From: Unit 2 equation (2.5)
V=11+r(qUD1+qDD2)V=1+r1(qUD1+qDD2)
✅ Q6 — Trinomial model probabilities
📍 From: Unit 2, Problem 2.2
You must:
Solve:
EQ(S1)=S0EQ(S1)=S0
With:
q1+q2+q3=1q1+q2+q3=1
👉 This gives infinite solutions
✅ Q7 — Hedging in trinomial model
📍 From: Unit 2, Problem 2.2 (iii)
Solve:
θ1Si+θ2=Diθ1Si+θ2=Di
👉 Leads to constraint:
3D1−5D2+2D3=03D1−5D2+2D3=0
✅ Q8 — Two-period tree probabilities
📍 From: Unit 2, Problem 2.3
You must:
Work forward through tree
Solve probabilities from expectations
✅ Q9 — Two-period option pricing
📍 From: Unit 2, Problem 2.3
Steps:
Compute payoffs at final nodes
Work backwards or use expectation:
C=EQ[max(S2−K,0)]C=EQ[max(S2−K,0)]
✅ Q10 — Hedging strategy (multi-step)
📍 From: Unit 2, Problem 2.3 (d)
Solve system for actions A1,A2,A3,A4A1,A2,A3,A4
🧠 SECTION B — MARTINGALES (Unit 3)
✅ Q11 — Conditional expectation
📍 From: Unit 3 §3.1
Discrete:
E[X∣Y=y]=∑xP(X=x∣Y=y)E[X∣Y=y]=∑xP(X=x∣Y=y)
✅ Q12 — Check martingale property
📍 From: Unit 3 definition
You must verify:
E(Mn+1∣past)=MnE(Mn+1∣past)=Mn
✅ Q13 — Random walk martingale
📍 Example 1
Show:
Mn=M0+∑ξiMn=M0+∑ξi
is a martingale if E[ξi]=0E[ξi]=0
✅ Q14 — Multiplicative process
📍 Example 2
Xn=X0∏ζiXn=X0∏ζi
Check:
martingale if E[ζi]=1E[ζi]=1
super/sub otherwise
✅ Q15 — Variance martingale
📍 Example 3
Mn=Sn2−vnMn=Sn2−vn
👉 Show expectation of increment = 0
✅ Q16 — Exponential martingale
📍 Example 4
Mn=eθSnϕ(θ)nMn=ϕ(θ)neθSn
🧠 SECTION C — MULTI-PERIOD BINOMIAL MODEL (Unit 4)
✅ Q17 — Build binomial tree
📍 Unit 4 §4.1
Sn=S0UkDn−kSn=S0UkDn−k
✅ Q18 — Risk-neutral probabilities (multi-period)
📍 Theorem 4.1
qU=1+r−DU−DqU=U−D1+r−D
(same as before — but MUST connect to martingale idea)
✅ Q19 — Binomial distribution of stock price
📍 Unit 4
P(Sn=S0UkDn−k)=(nk)qUkqDn−kP(Sn=S0UkDn−k)=(kn)qUkqDn−k
✅ Q20 — European call pricing (CRR formula)
📍 Section 4.3
C=1(1+r)N∑(Nk)qUkqDN−kmax(SN−K,0)C=(1+r)N1∑(kN)qUkqDN−kmax(SN−K,0)
✅ Q21 — Find cutoff k0k0
📍 Same section
k0=smallest integer s.t. S0UkDN−k≥Kk0=smallest integer s.t. S0UkDN−k≥K
✅ Q22 — European put pricing
📍 Section 4.4
Same structure as call but:
max(K−SN,0)max(K−SN,0)
✅ Q23 — American put (BACKWARD INDUCTION)
📍 Problem 4.3
THIS IS IMPORTANT
At each node:
value=max(exercise,continuation)value=max(exercise,continuation)
✅ Q24 — Compare American vs European
📍 Unit 4
American ≥ European
Must justify using computed values
✅ Q25 — Parameter construction (U, D, r)
📍 Section 4.8
U=eσdt,D=e−σdtU=eσdt,D=e−σdt
🚨 THIS IS YOUR COMPLETE NUMERICAL SYLLABUS
If you can do ALL 25 of these → you are fully covered.
No gaps. No wasted time.
🔥 NEXT STEP (IMPORTANT)
We now start mastering them in the correct order:
Order I recommend (matches learning flow):
Q2 → Q3 → Q4 (core foundation)
Q8 → Q9 (multi-step logic)
Q17 → Q20 (binomial model full)
Q23 (hardest, exam favourite)
Then fill gaps