Reviews for Mathematical finance

Phuong Le – Faculty of Economic Mathematics, UEL

1 Introduction to financial markets

1.1 Currencies
ˆ National currencies

ˆ Foreign currencies

1.2 Financial markets

ˆ Financial institutions: banking system, insurance companies, corpora-
tions providing electronic payment services, investment funds, securi-
ties corporations.

ˆ Traders: arbitrageurs, speculators, hedgers.

ˆ Trade: long and short positions, risk and return.

ˆ Efficient market hypothesis / theory.

1.3 Financial products

ˆ Debt securities: bonds.

ˆ Equity securities: common stock, preferred stocks, investment certifi-

cates.

ˆ Derivatives: forwards, futures, swaps, options.

2 Interest and cash flows

2.1 Indexes
If a value changes from P at t0 to F at t1 , then

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 ˆ the percentage change is F −P
P
× 100%,
ˆ the scale factor is F
P
= 1+percentage change,
ˆ the index at t1 , taking t0 as the base, is F
P
× 100,

2.2 Interest and discounting

2.2.1 Compound interest
If the interest is
ˆ compounded n times per year, then the future value in t years is
 r nt
F =P 1+ ,
n
ˆ compounded continuously, then the future value in t years is
F = P ert .

2.2.2 Discounting
If the interest is
ˆ compounded n times per year, then the present value is
 r −nt
P =F 1+ ,
n
ˆ compounded continuously, then the present value is
P = F e−rt .

Note 1:
1. To compute annuity, regular saving or payment, we may use the geo-
metric series formula
q n+1 − 1
a + aq + aq 2 + · · · + aq n = a for q ̸= 1.
q−1
When n goes to infinity:
∞
X a
aq k = for |q| < 1.
k=0
1−q

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 2. To compare interest rates of different periods, we may use APR. Annual
percentage rate (APR) for the interest
ˆ compounded n times per year is
 r n
AP R = 1 + − 1,
n
ˆ compounded continuously is
AP R = er − 1.
3. To find the real data (adjusted data) from the raw data (nominal data),
we may use the compound interest and discounting formulae, taking
inflation rate as the interest rate (compounded annually).

2.3 Investment appraisal

2.3.1 Net present value (NPV)
The net present value (NPV) of an investment is equal to the present value
of the revenue flow minus the present sum of all related costs.

2.3.2 Internal rate of return (IRR)

The internal rate of return (IRR) is the annual interest rate for which the
NPV equals zero.

2.3.3 Investment appraisal

ˆ For the same cost, the project with higher NPV is better.
ˆ For any cost, the project with higher IRR is better.

3 Bonds and related matters

3.1 Bond pricing
ˆ The market price of a bond which gives a total cash flow ci at time ti ,
(i = 1, 2, . . . , n) is
Xn
B= ci e−ri ti ,
i=1

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 where ci is the zero rate / spot rate for maturity ti with continuous
compounding.

ˆ The bond yield is the discount rate that makes the present value of the
cash flows on the bond equal to the market price of the bond. Hence
the bond yield (continuously compounded) is the rate y which solves
n
X
ci e−yti = B.
i=1

ˆ The par yield for a certain maturity is the coupon rate that causes
the bond price to equal its face value. If m is the number of coupon
payments per year for a bond with face value of $100, then the par
yield c is given by
n−1
X c −ri ti  c  −rn tn
e + 100 + e = 100.
i=1
m m

Hence
(100 − 100d)m
c= ,
A
where d is the present value of $1 received at maturity and A is the
present value of an annuity of $1 on each coupon date.

3.2 Spot interest rate and zero curve

Spot interest rate and zero curve can be determined from available bonds in
the market by the bootstrap method.

3.3 Forward interest rate

ˆ The forward rate for the period between times T1 and T2 is

R2 T2 − R1 T1 R2 − R1
RF (T1 , T2 ) = = R2 + T1 ,
T2 − T1 T2 − T1
where R1 and R2 are the zero rates for time periods T1 and T2 with
both rates continuously compounded.

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 ˆ The instantaneous forward rate for a maturity T is the forward rate
that applies for a very short time period starting at T
∂R
RF (T ) = R + T ,
∂T
where R is the T -year rate (R could be a function of time).

3.4 Macaulay duration

The Macaulay duration of the bond when the yield y is expressed with con-
tinuous compounding is defined as
n
X ci e−yti
D= ti × ,
i=1
B

Applications:
ˆ Approximation of the change of bond value:
∆B
≈ −D∆y or ∆B ≈ −DB∆y.
B
ˆ When the yield y is expressed with compounding m times per year,
then
∆B
≈ −Dm ∆y,
B
where
D
Dm :=
1 + y/m
is referred to as the “modified duration”.

3.5 Convexity
The convexity C of a bond is defined as
Pn 2 −yti
1 ∂ 2B i=1 ci ti e
C= = .
B ∂y 2 B
Applications: The convexity provides a more accurate approximation
∆B C
≈ −D∆y + (∆y)2 .
B 2
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4 Forward, futures and swaps
4.1 Forward and futures
Forward and futures are used for hedging purpose.
Notations:

ˆ S0 : Spot price today,

ˆ F0 : Futures / forward price today (the delivery price for a contract

with zero value),

ˆ T : Time until delivery date,

ˆ r: Risk-free interest rate for maturity T .

4.1.1 Forward / futures price

◦ For an investment asset which provides

ˆ no income:
F0 = S0 erT ,

ˆ a known income:
F0 = (S0 − I)erT ,
where I is the present value of the income during life of forward /
futures contract,

ˆ a known yield:
F0 = S0 e(r−q)T ,
where q is the average yield during the life of the contract (expressed
with continuous compounding). If the storage cost per unit time as a
percent of the asset value q is considered, then

F0 = S0 e(r+u−q)T ,

where c = r + u − q is called the cost of carry.

◦ The third formula can be applied to calculate the futures price of

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 ˆ stock index:
F0 = S0 e(r−q)T ,
where q is the average dividend yield on the portfolio represented by
the index during life of contract,

ˆ foreign currency:
F0 = S0 e(r−rf )T ,
where rf is the foreign risk-free interest rate.

◦ For consumption assets with

ˆ the storage costs:

F0 ≤ (S0 + U )erT ,

ˆ the storage cost per unit time:

F0 ≤ S0 e(r+u)T ,

ˆ the storage cost per unit time and income yield:

F0 ≤ S0 e(r+u−q)T .

4.1.2 Valuing a forward / futures contract

ˆ the value of a long forward / futures contract:

(F0 − K)e−rT ,

ˆ the value of a short forward / futures contract:

(K − F0 )e−rT ,

where K is the delivery price and F0 is the forward / futures price for a
contract that would be negotiated today.

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4.2 Interest rate swap
Interest rate swaps are used to convert investments and liabilities from fixed
rate to floating rate or from floating rate to fixed rate.
Consider a swap which involves paying the fixed rate r per annum and
receiving floating rate on an amount M . The exchanges occurs m times per
year, in t1 , t2 ,..., tn years, where the corresponding zero rates are r1 , r2 ,...,
rn (continuous compounding). To valuate the swap:

ˆ Calculate floating forward rates (with continuous compounding)

ri ti − ri−1 ti−1
RF (ti−1 , ti ) = ,
ti − ti−1

then convert them to rates RF′ (ti−1 , ti ) which compound m times per
year m
RF′ (ti−1 , ti )

1+ = eRF (ti−1 ,ti ) .
m
ˆ Calculate the swap cash flows that will occur if floating forward rates
are realized
R′ (ti−1 , ti ) r
CFi = M × F −M × .
m m
ˆ Discount these swap cash flows at zero rates
n
X
Swap value = CFi × e−ri ti .
i=1

Time Fixed Floating Net PV of net

(yrs) cash flow cash flow cash flow cash flow
R′ (t ,t )
t1 −M × mr M × F m0 1 CF1 CF1 × e−r1 t1
R′ (t ,t )
t2 −M × mr M × F m1 2 CF2 CF2 × e−r2 t2
··· ··· ··· ··· ···
Total Swap value

4.3 Currency swap

Currency swaps are used to transform investments and liabilities in one cur-
rency to another currency.

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 Consider a fixed for fixed currency swap which exchanges the domestic in-
terest rate R on an amount M for the foreign interest rate Rf on an amount
Mf . Payments are made annually. The exchanges occurs in t1 , t2 ,..., tn
years, where the corresponding domestic zero rates are r1 , r2 ,..., rn and for-
eign zero rates are rf1 , rf2 ,..., rfn . The spot exchange rate is ES per unit in
domestic currency.

4.3.1 Valuation currency swaps in terms of forward exchange rates

ˆ Calculate forward exchange rate

EFi = ES e(ri −rfi )ti .

ˆ Calculate the swap cash flows

(
−M × R + Mf × Rf × EFi for i < n,
CFi =
−M × (R + 1) + Mf × (Rf + 1) × EFi for i = n.

ˆ Discount these swap cash flows at zero rates

n
X
Swap value = CFi × e−ri ti .
i=1

Time Domestic Foreign Forward Net Present

cash cash exchange cash value
flow flow rate flow
t1 −M × R Mf × Rf EF1 CF1 CF1 × e−r1 t1
t2 −M × R Mf × Rf EF2 CF2 CF2 × e−r2 t2
··· ··· ··· ··· ··· ···
tn−1 −M × R Mf × Rf EFn−1 CFn−1 CFn−1 × e−rn−1 tn−1
tn −M × (R + 1) Mf × (Rf + 1) EFn CFn CFn × e−rn tn
Total Swap value

4.3.2 Valuation currency swaps in terms of bonds

ˆ Discount the domestic swap cash flows at zero rates
(
M × R × e−ri ti for i < n,
CFi =
M × (R + 1) × e−ri ti for i = n.

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 ˆ Discount the domestic swap cash flows at zero rates
(
Mf × Rf × e−rfi ti for i < n,
CFfi = −rfi ti
Mf × (Rf + 1) × e for i = n.

ˆ The swap value is the difference between two these cash flows
n n
1 X X
Swap value = CFfi − CFi .
ES i=1 i=1

Time Domestic PV of Domestic Foreign PV of Foreign

Cash flows Cash flows Cash flows Cash flows
1 M ×R CF1 Mf × Rf CFf1
2 M ×R CF2 Mf × Rf CFf2
··· ··· ··· ··· ···
n M × (R + 1) CF
Pn n Mf × (Rf + 1) CF
Pn fn
Total i=1 CFi i=1 CFfi

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