Formules

Question 1

Garch volatility estimator

Answer 1

σ_n^2 = γ * VL + αu_{n-1}^2 + βσ_{n-1}^2

With 7 being gamma, VL being long-term volatility, the second term being alpha * return^2 and the third term being beta * var

Question 2

EWMA volatility estimator

Answer 2

σ_n^2 = λσ_{n-1}^2 + (1 - λ)u_{n-1}^2

Lambda * var+ (1-lamda) * return^2

Question 3

T-day X% value at risk with

Answer 3

VaR = Pt√TσpN^(-1)(X)

With PT = portfolio value
N^(-1)(X) equalling Z-value of the Xth percentage

Question 4

Stock price path in the risk-neutral world

Answer 4

S_t+δt = S_t * e^((r - (σ^2)/2)δt + σ√δtZ)

So stock price * exp (return-var/2)*delta t + stdev * sqrt (delta T) * Z

Z being random number of deviations from the mean

Question 5

Sharpe ratio

Answer 5

Sp = (rp - rf) / σp

Question 6

Information ratio

Answer 6

αp / σ(εp)

So alpha portfolio / idiosyncratic stdev

Question 7

Treynor ratio

Answer 7

Tp = (rp - rf) / βp

Question 8

Gamma and vega hedging

Answer 8

[ option1* ] = [ r1 g1 ]^-1 [ -Γp ]
[ option2* ] [ v0 g2 ] [ -νp ]

So Mmult matrix inverse of option greeks * portfolio values

Then recalculate delta

Hedge delta with -delta stocks

Question 9

State prices and risk-neutral state prices

Answer 9

State-prices:

qu = (R - D) / (R(U - D))
qd = (U - R) / (R(U - D))

Risk-neutral state prices

πU = (R - D) / (U - D)
πD = (U - R) / (U - D)

With pay-off for risk-neutral needing to be discounted

Question 10

Opinion-adjusted weights (so for black litterman/envelope portfolio/market portfolio with RF rate)

Answer 10

z = S^(-1)(E(r)^adj - rf)

With S being the var/covar matrix and ^-1 being its inverse

Question 11

Black-Litterman steps with formulas

Answer 11

1. E(R) market= E(r)_market = Sm* + rf -> so not weights * av. return!
2. Normalization factor: (m^opinion - rf) / (m^T Sm)

(M^opinon / E(r)) also seems to work!

3. Normalized expected returns

E(r)_normalized = Sm^opinion - rf + rf / (m^T Sm* + rf)

Although old return * normalization factor also seems to work

4. Adjust returns for N individual stock opinions

μ^adj_i = μ_i + σ_i,1^2 * τ_1 + … + σ_i,j^2 * τ_j + … + σ_i,N^2 * τ_N

Which is basically the old stock price + (varstock/varotherstock) * delta for otherstock

Of course a delta for the stock itself will have varstock/varstock = 1 and thus be fully incorporated

5. calculate opinion adjusted weights

z = S^(-1)(E(r)^adj - rf)

With S being var/covar matrix
M* being weights
M^opinion opinion on market returns

Question 12

Portfolio var

Answer 12

σp^2 = w^T Σw

σp^2 is the variance of the portfolio.
w is the vector of weights of the assets in the portfolio.
w^T is the transpose of the vector of weights.
Σ (Sigma) is the covariance matrix of the asset returns.

Question 13

Delta

Answer 13

N(D1) for call options

N(-D1) for put option

Which is norm.s.dist function (or perhaps normsdist also works)

Question 14

Calculation of expected shortfall

Answer 14

Average loss within VaR

So if 5% VaR would be a loss of 6% which equals 60k

The average loss if the loss of 6% is met or exceeded could be 7.5%
Then expected shortfall would be 75k

Question 15

Stock movement in up state

Answer 15

U = e^(stdevsqrt(delta t)
or exp(stdevsqrt(delta t)

D = exp(stdev*-sqrt(delta t)

Question 16

Optimization through shrinkage

Answer 16

Create var-covar matrix, and another version where only the variance is included

Then weigh them, so for example 0.3 * S + 0.7 * S-shrunk

Makes extreme covariances smaller and thus outcomes more realistic

Question 17

Put-call parity

Answer 17

C + Ke^(-rT) = P + S

where:
C = call premium
Ke^(-rT) = present value of the strike
P = put premium
S = the current price of the underline

Question 18

Option-price change formula

Answer 18

First order approximation df = Δ x δS
Second order approximation df = Δ x δS + 1/2 Γ x (δS)^2

So you can approximate the delta in option price with delta * delta option price

However to be accurate, take this, and 0.5 * gamma * delta stock^2

So for example a delta is 0.6, and gamma is 0.03, stock price increases with 1.

Estimated new option price is 0.6 higher (0.6 * 1)
Accurate estimate includes gamma: 0.6 * 1 + 0.03 * 0.05 * 1^2 = 0.0615

New delta after price increase is 0.6 + 0.03 = 0.63

Question 19

An option has a delta of 0.5. Is it a put or a call?

Answer 19

Call, puts always have 0 or lower delta.

Question 20

What is a volatility smile?

Answer 20

-Distribution of rates tends to exhibit kurtosis (fat tails)
-Lognormal distribution assumed by BS model underestimates the probability of extreme price movements, especially for FX rates
-OTM calls and puts have higher probability of ending ITM than under lognormal distribution -> obeserved market prices reflect this probability and are thus higher than implied by BS
-So higher prices due to higher implied volatility
-Called a smile because the implied volatility for both extremely low and high strike prices is higher, thus creating a smiley without eyes.

Question 21

Describe volatility skew

Answer 21

-Typically seen for equity options
-Stock prices have negative skewness (heavy left tail) -> equity options with lower prices usually have higher volatility
-Extreme negative movements are underestimated by BS
-OTM call has lower probability of ending up ITM than under lognormal distribution -> lower expected payoff -> lower price and IV
-OTM put has higher probability of ending up ITM than underl ognormal distribution -> higher expected payoff -> higher price and IV
-Called a skew because volatility decreases as strike price increases