MATH 2510
Rolf Poulsen
December 1, 2010
Financial Mathematics 2: Facts and — Mostly — Formulas
The annuity symbol. The present value of a £1 n-year annuity when the interest rate is i is
an i =
1 − vn
,
i
where v = 1/(1 + i).
Puts and calls. The pay-off of a strike-K, expiry-T call-option (right but not obligation to buy) is
Call(T ) = (S(T ) − K)+ := max(S(T ) − K, 0). The pay-off a put-option (right but not obligation to
sell) is P ut(T ) = (K − S(T ))+ = max(K − S(T ), 0).
The put-call parity links put and call prices
Call(0) − P ut(0) = S(0) − PT K,
where PT is the maturity-T zero-coupon bond price.
Forwards and futures. Arbitrage-free time-0, delivery/maturity/expiry-T forward prices
P
• Deterministic cash dividends: F wd(0, T ) = (S(0) − ti dti Pti )/PT
• Dividend yield (∼ foreign interest rate): F wd(0, T ) = e(r−q)T S(0)
With constant interest rates, forward and futures prices are equal.
The holder of a long position in a futures contract receives (gets/pays) F ut(t, T ) − F ut(t − dt, T ) at
time t
P
Bonds and interest rates. Zero-coupon bonds prices Pt satisfy P ricej = t cj,t Pt for any bond j
with time-t cash-flow cj,t .
Zero-coupon spot rates satisfy Pt = (1 + yt )−t ⇒ yt = (Pt )−1/t − 1.
The 1-year-ahead forward rates are ft = Pt /Pt+1 − 1 with P0 = 1.
The yearly payment for a £100, n-year annuity bond with coupon rate c is 100/an c .
The yield to maturity, ytmj , of bond j is the solution of the non-linear equation
P ricej =
X
t
cj,t
.
(1 + ytmj )t
The n-year par yield is
1 − Pn
(py)n = Pn
.
t=1 Pt
Duration and immunization. The — or: our — use of Macauley duration requires — or: assumes
— a flat yield curve; Pt = (1 + y)−t .
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Macauley duration is
τ=
1 X t × ct
,
P V t (1 + y)t
P
where ct ’s are cash-flows and P V = t ct (1 + y)−t is the present value
The Macauley duration of an n-year bond with coupon payments D and redemption R is
τbond =
D × (IA)n y + R × nv n
,
D × an y + R × v n
where v = 1/(1 + y) and the ”increasing annuity” symbol is
(IA)n y =
(1 + y)an y − nv n
.
y
Putting R = 0 gives annuity results.
The duration of a portfolio (x1 , . . . , xn ) is the value-weighted average of the durations of its constituents
n
X
P Vj × τj
τpf =
.
xj
P Vpf
j=1
To achieve Redingtion immunization, present values and durations of assets and liabilities must match.
Furthermore the convexity of assets must be bigger than the convexity of liabilities — meaning that
the asset payments are more dispersed than the liability payments.
Random variables. For a discrete random variable X and a function f we have
X
E(f (X)) =
f (x)Prob(X = x).
x
For a continuous random variable, the right hand side in the equation above must be substituted by
the integral of f wrt. the density function.
For any random variable X we have var(X) = E(X 2 ) − (E(X))2 . Variance is non-negative, and
standard deviation is the square root of variance. Q
Q
For independent random variables Xi , we have E( i Xi ) = i E(Xi )
2
X ∼ N (µ, σ 2 ) ⇒ E(eX ) = eµ+σ /2
X ∼ N (µ, σ 2 ) ⇒ aX + b ∼ N (aµ + b, a2 σ 2 )
A sum of independent normal variables is normal with mean and variance equal to the sum of the
means and variances of the constituents.
Y is log-normal if ln Y is normal.
Investments with random rates of return. For the time-n value of a single, £1 investment made
at time 0 — the accumulated wealth, Sn — we have
Snk =
n
Y
(1 + it )k ,
t=1
where the it ’s are the yearly (effective) rates of return and k is some integer.
For an annuity investment of £1 per year the accumulated wealth, At , satisfies
Akt = (1 + it )k (1 + At−1 )k ,
with the convention that At is the value immediately before the time-t investment is made.
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