MPF_RRFI - Lecture 03
How Traders Manage Their Risks (Chapter 8)
Delta
is a small increase in the value of the variable
is the resulting change in the value of the portfolio
Using calculus terminology, delta is the partial derivative of the portfolio value with respect to the value of the
variable
Eliminate delta exposure delta hedging delta neutral portfolio
Linear products (forward, futures, swap) constant delta
Non-linear products (options, exotics) time-varying delta
Gamma
Gamma is the rate of change of the portfolio’s delta with respect to the price of the underlying asset (the second
partial derivative)
Vega
Vega, is the rate of change of the value of the portfolio with respect to the volatility, , of the underlying asset
price
Theta
Theta is the rate of change of the value of the portfolio with respect to the passage of time, with all else
remaining the same
Rho
Rho is the rate of change of a portfolio with respect to the level of interest rates
Delta = =
ΔP
ΔS
∂P
∂S
ΔS
ΔP
→ →
⟹
⟹
Gamma =
P∂
2
∂S
2
Vega =
∂P
∂σ
σ
Interest Rate Risk (Chapter 9)
net interest income liquidity preference theory
Types of rates:
Treasury rate
LIBOR
The OIS rate
Repo rate
Duration
Duration measures the sensitivity of percentage changes in the bond’s price to changes in its yield.
is a bond's yield, is the bond market price
Convexity
Convexity measures curvature of the bond-yield relationship
→
D = − − = ⟹ ΔB = −DBΔy
1
B
ΔB
Δy
1
B
∂B
∂y
∑
n
i=1
ci
ti
e
−yti
B
y B
C = = ⟹ = −DΔy + C(Δy
1
B
B∂
2
∂y
2
∑
n
i=1
ci
t
2
i
e
−yti
B
ΔB
B
1
2
)
2
Volatility (Chapter 10)
volatility clustering, long memory
realized volatility = standard deviation of continuously compounded returns per unit of time
implied volatility = implied from option prices, Black-Scholes-Merton model
The Exponentially Weighted Moving Average (EWMA) Model
is a constant between 0 and 1 (weight)
is volatility
is daily percentage return
The GARCH(1,1) Model
is a long-run average variance rate
, , and are coefficients
= λ + (1 − λ)σ
2
n
σ
2
n−1
u
2
n−1
λ
σ
u
= γ + α + βσ
2
n
VL
u
2
n−1
σ
2
n−1
VL
γ α β
Correlations and Copulas (Chapter 11)
correlation diversification risk management decisions
correlation measures only linear dependence
copulas are often used to the calculation of the distribution of default rates for loan portfolios
⟹ ⟹