MTH 105
FINANCIAL MARKETS
 What is a Financial market:
- A financial market is a mechanism that allows
people to trade financial security.
 Transactions occur either:
- either in an Exchange; a building where
securities are traded
- Over the counter; electronically or telephone
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FINANCIAL MARKETS
 Types of Financial Markets:
- Capital Markets: Long term securities (> 1yr)
Stock Markets
Bond Markets
- Commodity Markets
- Money Markets: Short term (< 1yr)
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FINANCIAL MARKETS
- Derivative markets
- Foreign Exchange Markets
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FINANCIAL MARKETS
 Capital Markets:
are for trading securities with original maturity that
is greater than 1yr (Stocks and Bonds)
 Stocks: -represent ownership
-shareholders receive dividends
-they can also decide to sell their shares
NB: Stocks are risky assets(uncertain future
return)
High risk, High return
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FINANCIAL MARKETS
 Bonds: represents a debt owned by the issuer to
the investor
-Bonds obligate the issuer to pay a specific
amount at a given date, generally with periodic
interest payments
NB: Bonds issued by the Government are risk free
: Low risk, Low return
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FINANCIAL MARKETS
STOCKS
BONDS
 Risky
 Risk-free
(uncertain future returns)
 No maturity date
 Have control over the firm
(vote, firm’s activities, etc)
 Receives dividends
(not guaranteed)
 Receiving capital invested is
not guaranteed
(future return is certain)
 Maturity date
 Have no control over the firm
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 Receives interest payments
(guaranteed)
 Receives capital invested
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FINANCIAL MARKETS
 Capital Market Participants:
1.Government: Issues bonds to finance projects
such as schools etc. or pay off its debt.
NB: Government never issues stock
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FINANCIAL MARKETS
2.Companies: Issue both stocks and bonds to fund
investment projects
3.Investors/Households: Purchasers of capital
market securities
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A SIMPLE MARKET MODEL
 Assume that only two assets are traded in the
financial market:
- Stocks: Risky assets
[uncertain future returns]
- Bonds: Risk-free
[future returns are certain]
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A SIMPLE MARKET MODEL
 The time scale to be used:
t=0; represents today
t=1; represents the future (tomorrow, 1yr time etc)
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A SIMPLE MARKET MODEL
 For Stocks let:
S(t): the price of a share at time t
S(0): the price of a share today (certain)
S(1): the price of a share in future (uncertain)
Ks: Return on Stocks
Ks = S(1) – S(0)
S(0)
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(uncertain)
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A SIMPLE MARKET MODEL
 For Bonds let:
A(t): the price of a bond at time t
A(0): the price of a bond today
A(1): the price of a bond in future
KA = A(1) – A(0)
A(0)
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(certain)
(certain)
(certain)
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A SIMPLE MARKET MODEL
 For a Portfolio let:
x: number of shares held by an investor
y : number of bonds held by an investor
Portfolio (x shares and y bonds)
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A SIMPLE MARKET MODEL
 The total wealth of an investor holding x shares
and y bonds at time t is given by:
V(t) = x S(t) + y A(t)
NB: S(t), A(t) ; price per share and bond
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A SIMPLE MARKET MODEL
 At t=0
V(0) = x S(0) + y A(0)
 At t=1
V(1) = x S(1) + y A(1)
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A SIMPLE MARKET MODEL
 Return on a Portfolio (x, y)
Kv = V(1) – V(0)
V(0)
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uncertain
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A SIMPLE MARKET MODEL
 Examples:
 If S(0) = GH 50 and,
S(1) =
GH 52 with probability p,
GH 48 with probability 1 − p,
for a certain 0 < p < 1.
Find the return on this stock or find Ks.
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A SIMPLE MARKET MODEL
 If A(0) = 100 and A(1) = 110 Ghana cedis. What
will be the return on an investment in this bond?
Or find KA.
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A SIMPLE MARKET MODEL
 If S(0) = GH 50, A(0) = GH100, A(1) = GH110
S(1) =
GH 52 with probability p,
GH 48 with probability 1 − p,
for a certain 0 < p < 1.
For a portfolio of x = 20 shares and y = 10 bonds
find; V(0): value of the portfolio at t=0
V(1): value of the portfolio at t=1
Kv: return on this portfolio
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A SIMPLE MARKET MODEL
 If S(0) = GH 40, A(0) = GH 50, A(1) = GH 70
S(1) =
GH 50 with probability p,
GH 35 with probability 1 − p,
For a portfolio of x = 10 shares and y = 200 bonds
find; Ks, KA, and Kv
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A SIMPLE MARKET MODEL
 Let A(0) = 90, A(1) = 100, S(0) = 25 dollars
S(1) = 30 with probability p
20 with probability 1−p
where 0 <p<1.
For a portfolio with x = 10 shares and y = 15 bonds
calculate Ks, KA and Kv
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A SIMPLE MARKET MODEL
 Assumption 1: Randomness
The future stock price S(1) is a random variable
with at least two different values. The future price
A(1) of the risk-free security is a known number.
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A SIMPLE MARKET MODEL
 Assumption 2: Positivity of Prices
All stock and bond prices are strictly positive,
A(t) > 0 andS(t) > 0 for t =0 ,1.
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A SIMPLE MARKET MODEL
 Assumption 3: Divisibility, Liquidity and
Short Selling
An investor may hold any number of x shares and
y bonds, whether integer or fractional, negative,
positive
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A SIMPLE MARKET MODEL
- Divisibility: The fact that one can hold a fraction
of a share or bond is referred to as divisibility.
- Liquidity: It means that any asset can be
bought or sold on demand at the market price in
arbitrary quantities.
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A SIMPLE MARKET MODEL
+ x : long position in shares (buying shares)
+ y : long position in bonds (buying bonds)
Long Position;
If the number of securities of a particular kind held
in a portfolio is positive, we say that the investor
has a long position.
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A SIMPLE MARKET MODEL
- x : short position in shares (short selling shares)
- y : short position in bonds (issuing bonds)
Short Position;
If the number of securities of a particular kind held
in a portfolio is negative, we say that the investor
has a short position.
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A SIMPLE MARKET MODEL
- Short Position in Stocks:
Short selling:
- Borrow stocks (-x)
- Sell stocks
- Use the proceeds to make some other
investments
NB: Owner of stocks keeps all rights to it
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A SIMPLE MARKET MODEL
 Closing the short position:
Bonds: Paying interest and Face value
Shares: Repurchase the stock and return to the
owner
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A SIMPLE MARKET MODEL
 Assumption 4: Solvency
The wealth of an investor must be non-negative at
all times, V (t) ≥ 0 for t =0 ,1.
- Admissible Portfolio: A portfolio satisfying this
condition is called admissible
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A SIMPLE MARKET MODEL
 Assumption 5: Discrete Unit Prices
The future price S(1) of a share is a random
variable taking only finitely many values.
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A SIMPLE MARKET MODEL
 Assumption 6: No-Arbitrage Principle
- We shall assume that the market does not allow
for risk-free profits with no initial investment
- If the initial value of an admissible portfolio is
zero, V (0) = 0, then V (1) =0
- There is no admissible portfolio with initial value
V (0) = 0 such that V (1) > 0
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A SIMPLE MARKET MODEL
- If a portfolio violating this principle did exist, we
would say that an arbitrage opportunity was
available.
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A SIMPLE MARKET MODEL
 Suppose that investor A in Accra offers to buy 100 shares
at GH 2 per share while investor B in Kumasi sells 100
shares at GH1per share.
If this were the case, the investors would, in effect, be
handing out free money.
An investor with no initial capital could realise a profit of
200 − 100 = 100 Ghana cedis by taking simultaneously a
long position (thus buying from investor B) and a short
position (selling to investor A).
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BINOMIAL MODEL
 BINOMIAL MODEL:
- The choice of stock and bond prices in a
binomial model is constrained by the No-
Arbitrage Principle.
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BINOMIAL MODEL
- Suppose that:
S(0)=A(0)
S(1)= S u
Sd
when stocks go up
when stocks go down
- Then: Sd <A(1) <S u, or else an arbitrage
opportunity would arise.
(Insert Diagram).
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BINOMIAL MODEL
 Show that an arbitrage opportunity would arise
when A(1) ≤ Sd
A(1) ≥ Su.
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BINOMIAL MODEL
 Example:
If S(0) = A(0) = GH 100,
S(1) = 125 with probability p
105 with probability 1-p
Proof that an arbitrage opportunity will arise if
A(1)=GH 90
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BINOMIAL MODEL
 Assignment
If S(0) = A(0) = GH 100,
S(1) = 125 with probability p
105 with probability 1-p
Proof that an arbitrage opportunity will arise if
A(1)=GH 130
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RISK AND EXPECTED RETURN
 Risk
- The uncertainty associated with receiving future
returns
 Expected Return
The weighted average of all possible returns from
a portfolio
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RISK AND EXPECTED RETURN
 Example:
Given that; S(0)=25, A(0)=90, A(1)=100
S(1) = 30 with probability 0.6
20
with probability 0.4
if x=5 shares and y=10 bonds, find the expected
return and risk on:
- Stocks
- Bonds
- Portfolio
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RISK AND EXPECTED RETURN
 Given the choice between two assets or
portfolios with the same expected return, any
investor would obviously prefer that involving
lower risk.
 Similarly, if the risk levels were the same, any
investor would opt for higher return.
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OPTIONS
 An option is a financial derivative which gives the
holder the right (but not the obligation) to buy
(call option) or sell (put option) an asset on a
specific pre-determined future date and price.
 Call Option: Gives the holder the right to buy an
asset on a fixed future date and price.
 Put Option: Gives the holder the right to sell an
asset on a fixed future date and price.
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OPTIONS
 TERMS:
- Strike price: the agreed upon price to buy or sell
an asset in future
- Delivery date: the agreed upon future date to buy
or sell an asset
- Exercising the option: buying or selling an asset
at a pre-determined price and date
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OPTIONS
 2 TYPES OF OPTIONS
- Call Option: gives the holder the right to buy an
asset at a pre-determined date and price.
- A call option is exercised only when the market
price S(1) is above the strike price.
- If the stock price falls below the strike price, the
option will be worthless
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OPTIONS
 Example:
Let A(0) = 100, A(1) = 110, S(0) = 100 GH and
S(1) = 120 with probability p,
80 with probability 1 − p,
where 0 < p < 1.
Strike Price: GH 100
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OPTIONS
- Hence at t=1, the value of a call option is:
C(1) = S(1) – Strike price
= 120 - 100
80 - 100
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= 20
0
with prob. p
with prob. 1-p
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OPTIONS
 At t=0, the value of a call option is found in 2
steps:
Step 1 (replicating the option)
- Construct a portfolio of x shares and y bonds
such that the value of the investment at time 1 is
the same as that of the option,
xS(1) + yA(1) = C(1)
120x + 110y = 20
80x + 110y = 0
solve for x and y
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OPTIONS
x=1
2,
y= −4
11
To replicate the option we need to buy 1̸ 2 a share
of stock and take a short position of − 4̸ 11 in
bonds.
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OPTIONS
 Step 2: (Valuing the Option)
Compute the time 0 value of the investment in
stock and bonds. It will be shown that it must be
equal to the option price,
xS(0) + yA(0) = C(0)
xS(0) + yA(0) =1̸2 × 100 −4̸11 × 100 = 13.6364GH
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OPTIONS
 Put Option: gives the holder the right to sell an
asset at a pre-determined date and time
- A put option is exercised only when the market
price S(1) is below the strike price.
- If the stock price rises above the strike price, the
option will be worthless
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OPTIONS
 Put Option:
Let A(0) = 100, A(1) = 110, S(0) = 100 GH and
S(1) = 120 with probability p,
80 with probability 1 − p,
where 0 < p < 1.
Strike Price: GH 100
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OPTIONS
- Hence at t=1, the value of put option is:
P(1) = Strike price - S(1)
= 100-120
100 -80
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= 0 with prob. p
20 with prob. 1-p
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OPTIONS
 EXAMPLE:
Let A(0) = 100, A(1) = 110, S(0) = 100 GH and
S(1) = 120 with probability p,
80 with probability 1 − p
- For a call option with strike price GH 100, find
C(0)
- For a put option with strike price GH 100, find
P(0)
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OPTIONS
 Solution:
C(0) is found in 2 steps
Step 1 (Replicating the Option):
Construct a portfolio of x shares and y bonds such
that the value of the portfolio at t = 1 is the same
as that of the option.
At t=1;
V(1) = C(1)
xS(1) + yA(1) = C(1)
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OPTIONS
Step 2: (Valuing or Pricing the Option)
At t=0, the value of the portfolio in stock and bonds
must be equal to the value of the option.
At t=0;
V(0) = C(0)
xS(0) + yA(0) = C(0)
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OPTIONS
 Assignment
Let A(0) = 100, A(1) = 110, S(0) = 100 GH and
S(1) = 120 with probability p,
80 with probability 1 − p
Compute the value of a call option C(0) if:
- Strike price = GH 90
- Strike price = GH 110
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OPTIONS
 Portfolio (x,y,z)
In a market in which options are available, it is
possible to invest in a portfolio (x, y, z) consisting
of x shares of stock, y bonds and z options.
At t=0 and t=1, the value of such a portfolio is:
V (0) = xS(0) + yA(0) + zC(0)
V (1) = xS(1) + yA(1) + zC(1)
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FORWARD CONTRACT
 Forward contract is a financial derivative which
obligates the holder to buy (long forward
contract) or sell (short forward contract) an asset
on specific pre-determined future date and at a
fixed price.
 Long forward: Obligates the holder to buy an
asset on a fixed future date and price.
 Short forward: Obligates the holder to sell an
asset on a fixed future date and price
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At t=1, the value of a Long Forward Contract:
LF(1) = S(1) – Forward price
At t=1, the value of a Short Forward Contract:
SF(1) = Forward price-S(1)
At t=0.
F(0)=0
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no value
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SIMILARITIES
OPTION
FORWARD CONTRACT
 Right to buy
 Obligation to buy
(call option)
(long forward contract)
 Right to sell
 Obligation to sell
(put option)
(short forward contract)
 Fixed price
 Fixed price
(strike price)
(forward price)
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SIMILARITIES
 In both option and forward contracts, agreement
is made today, but transaction takes place in
future.
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DIFFERENCES
OPTIONS
FORWARD CONTRACT
 Not compulsory to buy
or sell assets
 Compulsory to buy or
sell assets
 Down payment
required
 No down payment is
required
(Premium)
(no premium)
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FORWARD CONTRACT
 Portfolio (x,y,z)
It is possible to invest in a portfolio (x, y, z) consisting of x
shares of stock, y bonds and z future contracts.
At t=0
V (0) = xS(0) + yA(0)
no value for F(0)
At t=1
V (1) = xS(1) + yA(1) + zF(1)
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