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CM2 formula cheat sheet

Financial engineering's greatest hits, on one page: what each formula says, why it matters, and the plain Memori syntax that types it. The grey text under each name doubles as a reference for writing your own cards.

Portfolio theory

Mean-variance: how risks add, and the straight line that beats the curve.

Two-asset portfolio variance

The correlation term is where diversification lives

sigma_p^2 = w^2 sigma_1^2 + (1-w)^2 sigma_2^2 + 2w(1-w) rho sigma_1 sigma_2
σ2p = w2 σ21 + (1-w)2 σ22 + 2w(1-w) ρ σ1 σ2

Capital market line

Slope is the market Sharpe ratio; efficient portfolios only

E[R_p] = r_f + ((E[R_M] - r_f)/(sigma_M)) sigma_p
E[Rp] = rf + (E[RM] - rfσM) σp

CAPM

Security market line

Prices any asset, efficient or not, by its beta

E[R_i] = r_f + beta_i (E[R_M] - r_f)
E[Ri] = rf + βi (E[RM] - rf)

Beta

Only systematic risk is rewarded

beta_i = (Cov(R_i, R_M))/(sigma_M^2)
βi = Cov(Ri, RM)σ2M

Stochastic calculus and lognormal returns

Geometric Brownian motion

The lognormal share-price model

dS_t = mu S_t dt + sigma S_t dW_t
dSt = μ St dt + σ St dWt

GBM solution

The −σ²/2 is the volatility drag

S_t = S_0 e^{(mu - sigma^2/2)t + sigma W_t}
St = S0 e(μ - σ2/2)t + σ Wt

Ito's lemma for f(t, S)

The extra second-order term is what makes Ito different

df = f_t dt + f_S dS + (1)/(2) sigma^2 S^2 f_{SS} dt
df = ft dt + fS dS + 12 σ2 S2 fSS dt

Black-Scholes

The call price and the two d's; everything else follows.

Call price

Put via parity or symmetry

c = S Phi(d_1) - K e^{-rT} Phi(d_2)
c = S Φ(d1) - K e-rT Φ(d2)

d1

d_1 = (ln(S/K) + (r + sigma^2/2)T)/(sigma sqrt(T))
d1 = ln(S/K) + (r + σ2/2)Tσ √(T)

d2

d_2 = d_1 - sigma sqrt(T)
d2 = d1 - σ √(T)

Put-call parity

Model-free: portfolio replication only

c + K e^{-rT} = p + S
c + K e-rT = p + S

Greeks

Call delta

Put delta is Φ(d₁) − 1

Delta = Phi(d_1)
Δ = Φ(d1)

Vega

φ is the standard normal density; same for calls and puts

nu = S sqrt(T) phi(d_1)
ν = S √(T) φ(d1)

Binomial model

Risk-neutral probability

Price = discounted expectation under q, not the real-world p

q = (e^{r delta t} - d)/(u - d)
q = er δ t - du - d

Utility and risk measures

Absolute risk aversion

Relative risk aversion multiplies by w

A(w) = -(U''(w))/(U'(w))
A(w) = -U''(w)U'(w)

Value at risk

A quantile of the loss distribution; TailVaR averages beyond it

VaR_p = F^{-1}(p)
VaRp = F-1(p)

Every one of these has an interactive tool. Try the efficient frontier, Brownian motion simulator, option payoffs or the binomial tree. Memori is a flashcard app built by actuarial students, with a ready-made CM2 set in the shop. Join the beta.