Lecture 1: An Overview of Financial Markets
Intermediate Asset Pricing
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Contacts Information
Role
Office
E-mail
Ambrogio Dalò Lectures/Tutorials/Assignment/Coordinator DUI 861 a.dalo@rug.nl
Gosse Alserda
Lectures/Tutorials/Assignment
DUI 851 g.a.g.alserda@rug.nl
Boris Ginzburg Lectures/Tutorials/Assignment
DUI 707 b.ginzburg@rug.nl
Jennifer Amens Secretary
DUI 836 j.j.amens@rug.nl
Fama, E.F., 2014, Two pillars of asset pricing, American Economic Review, 104(6), pp.
1467-85.
Shiller, R.J., 2014, Speculative asset prices, American Economic Review, 104(6), pp.
1486-1517.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Aims of the Course
Most of you, if not all, are already familiar with concepts coming from the course Asset Pricing
& Capital Budgeting (EBP032A05) and Capital Structure and Financial Planning (EBB060A05).
Assuming that you acquired all the relevant topics and information provided by these two courses,
your knowledge in terms of financial literacy during this course you should follow this path:
Why you need this? Well, you need this for (1) your thesis, (2) your master, and more importantly,
(3) your career after.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Structure of the Course and of Each Lecture/Tutorial
We plan to help you in achieving the previous by trough the following course structure:
 Lecture: overview of important issues;
 Tutorial:
To stimulate you to come prepared; at a more basic level than the exam;
Work on exercises (often from old exams);
Additionally, the final result for this course will be based on Grade on Assignment (15% – also
valid of the next academic year) Grade on the written examination (85%).
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Content of Today’s Lecture
This lecture is meant to be mainly a refresh of topics that you should have covered during the
course of Asset Pricing & Capital Budgeting (EBP032A05) and Capital Structure and Financial
Planning (EBB060A05).
 what is a corporation and its connection to the financial markets;
 the existing difference between real and financial assets;
 what is the risk/return trade-off;
 the concept of efficiency on financial markets;
 which are the major categories of financial assets.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Types of Firms
In the real world, there exist different kinds of firms (i.e., solo proprietorship, partnerships, etc.).
Each of them has its advantages/disadvantages (i.e., for a sole proprietorship is owned and run
by one person, the lenders can claim the owners’ assets; this is due to the unlimited personal
liability of the owner to the firm’ debt).
During this course, we will mostly treat a specific type of firm called: Corporations. The main
characteristics of a corporation with respect to all the existing models of firms are:
 i) it is a legal entity separate from its owners;
 ii) the best feature of a corporation is that it is solely responsible for its obligations, meaning that its
owners are not liable for any liability the corporation enters into;
 iii) the ownership is divided among the fixed amount of stocks issued, and due to that, the owner of a
share is known as a: shareholder, stockholder, or equity holder;
 iv) the sum of all the share outstanding is called Equity (E);
 v) there is no limit to the number of shareholders, and thus the amount of funds a company can raise by
selling shares;
 vi) the shareholders are entitled to dividend payments and have voting rights;
 vii) the corporations can be private and public,
 and viii) the shareholders are subject to the so call double taxation issue.
From now on, we are going to focus our attention only on Public Corporations because the stocks
of these firms are the only ones that trade on the Financial Market.
Introduction
Corporation on Financial Markets
Graphical Overview of a Corporation
Risk And Return of Stocks and Indexes
Other Financial Assets
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Capital Structure of a Corporation
Firms’ capital can be decomposed in Debt and Equity. Thus firms can raise money issuing debt
(e.g., bonds) and new shares. The proportions of debt and equity constitute the firm’s Capital
Structure.
Most of the firms decide to finance them by equity alone, or by a combination of equity and debt.
Concerning the equity, previously, we have assumed that there exists only one type of share:
ordinary shares.In practice, firms can issue at least two kinds of shares:
 Ordinary Shares: represent the basic voting shares of a corporation. Holders of ordinary
shares are typically entitled to one vote per share and do not have any predetermined dividend
amounts;
 Preferred Shares: is a class of ownership in a corporation that has a higher claim on its
assets and earnings than common stock. Preferred shares generally have (1) dividend that
must be paid out before dividends to common shareholders, and (2) the shares usually do
not carry voting rights.
As we will see in the next slides, this distinction has some significant consequences of firms’
capital structure.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Capital Structure of a Corporation, Example – Apple
Below you can take a look of Apple’ capital structure:
Source: own calculation based on Thomson Reuther – Eikon data.
Other Financial Assets
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Capital Structure of a Corporation, Example – FaceBook
Below you can take a look of FaceBook’ capital structure:
Source: own calculation based on Thomson Reuther – Eikon data.
Other Financial Assets
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Financial Markets
For both private and public company the equity is divided in a given amount of share. The
difference is that for the private company the shares are traded privately, while for public company
the shares are traded on stock markets (alternatively, stock exchanges).
The main aim of a financial markets is to:
 allow for the separation of Ownership and Management (leads to the so called agency
problems);
 transfer money across individuals and time;
 provides liquidity to shareholders;
 allocation of risk (by creating a portfolio of assets, we will be more in this respect next
week);
 they are divided in Primary and Secondary market.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Corporate Management Team
In a corporation, ownership and direct control are typically separated. Specifically, shareholders,
who own the company, can indirectly control the firm by electing the Board of Directors. The
Board of Directors is elected by a voting procedure in which each share counts for one vote.
 Board of Directors, is the ultimate decision-making authority and set-up the main strategy
followed by the firm. Additionally, the member of the board, hire a Chief Executive Officer
(CEO);
 CEO, is the one entitled to run day-to-day the firm;
 Chief Financial Officer, is the one entitled to take investment decisions, take financing
decisions, and managing the cash for the treasury.
The main aim of shareholders is to maximize the value of their shares, but they DO NOT run
directly firm. Sometimes this can lead to a specific issue called: agency problem. We have
agency problems when the managers work for their self-interests (e.g. by rising their salary).
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Tools Against the Agency Problem
The shareholders have three possible tools to enforce that the managers work to maximize the
value of the shares rather then their own wealth. Specifically, shareholders can:
 limit the managers’ compensations;
 tie the management’s compensation to firm performance. In this way, managers and shareholders have the same aim but there is a drawback. Since managers are paid according
to the performance of the firm, under a weak designed compensation policy they have all
the incentive to take even the most risk strategies to achieve such a target. Taking too
risky projects can lead to the default and in such case, the shareholders lost completely their
capital;
 replacing the CEO by trough a so-called hostile takeover. Shareholders with deep pockets
can go to the stock market and buy the (additional) equity necessary so that they obtain
the lower bound of 50% + 1 stock. Once they own slightly more the half of the company,
they can vote to replace the board of directors and then, replace indirectly the CEO.
The hostile takeover allows us to formally introduce the concept of the stock market.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Primary and Secondary Markets, an Example
Other Financial Assets
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Real Vs Financial Investments/Assets
From the investor perspective, buying the stocks of a corporation is equivalent to investing. The
investments can be of two types: Real and Financial. An investment in stocks can then be seen
as a Financial Investment. The following differences between the two investment type exist:
 Real Assets:
i) have productive capacity;
ii) examples: land, buildings, machines, intellectual property.
 Financial Assets:
i) they are claims on real assets;
ii) they do not contribute directly to productive capacity;
iii) fixed-income security (promises a fixed stream of income determined by a specified formula; e.g.,
bonds);
iv) equity (represents ownership share in a corporation; e.g stocks);
v) derivative: provide payoffs that are determined by the prices of other assets (e.g., stock market
indexes, stocks, etc);
vi) investment in currency: the payoffs are determined by the fluctuation of the related currency (e.g.
US dollars, Euros, etc);
vii) commodity futures: the payoffs are determined by the fluctuation of the commodity’s prices (e.g.
oil, gold, beef, etc);
This course will cover the financial instruments from the sub-point from iii) to v).
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Computing the Prices
If you buy a stock of a generic firm and keep it for one year. Your investment will have the
following timeline:
0
1
P0
D1 + P1
Then the price of such stock according will be equal to:
P0
=
D1 + P1
(1 + r)
Where P0 and P1 are the prices of the stock when you buy it and sell it respectively, D1 is the
dividend paid by the firm after the first year and r, from the firm perspective, is the equity cost of
capital. Form the investor perspective instead; it represents the “Expected Return” on its shares.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Computing the Price, Example 1
Suppose you buy a stock which you wish to sell in one year and expect will have a price at the
end of the first year equal to 47.00$ ( P1 = 47.00$), a dividend equal to 0.56$ ( D1 = 0.56) per
share and an expected return equal to 0.068 ( r = 0.068). The timeline of your investment is the
following:
0
1
P0
0.56 + 47.00
How much is the price of the stock today? ( P0 =?)
P0
=
0.56 + 47.00
(1 + 0.068)
=
44.53$
Which in turn implies that to buy such stock today you have to pay on financial markets 44.53$.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Assets’ Price Series
Suppose that there exist three assets on financial markets: Bund (a bond issued by the German
government), the OATs (a bond issued by the French government), and Renault’s stocks. Their
price series will look as follows:
Source: Thomson Reuther – Eikon. Sample period: 01/2001 − 12/2004.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Prices Vs Returns
There are at least two good reasons to use the return series of financial securities instead of price
series:
1
investors are concerned about the return of his assets;
2
given their properties, return series can be handled easily concerning price series (e.g.,
average returns vary around a constant).
The differences between price and return series of the S&P500:
It is possible to verify how risky an asset is by using the return series of assets.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Computing the Return
The return can be decomposed into two parts. Starting from the central equation of the previous
slide and solving for r , we can write down:
P0
=
1+r
=
r
=
D1 + P1
(1 + r)
D1 + P1
P0
D1 + P1 − P0
P0
→
(1 + r)P0
=
→
r
=
→
r
=
D1 + P1
D1 + P1
−1
P0
P − P0
D1
+ 1
P0
P
|{z}
| {z0 }
Dividend Yield
→
→
Capital Gain
Where written like this, r is called Total Net Return.
The total return decomposes profits of the shareholders in two parts: the first one indicates how
much of these profits come from divides, while the second part suggests how much comes from
selling the stock on financial markets.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Computing a Return Timeseries
Starting from stock’s price series, we can retrieve their return series.
0
1
2
3
N
Pi,0
Di,1 + Pi,1
Di,2 + Pi,2
Di,3 + Pi,3
Di,N + Pi,N
Precisely, we can compute the return at the end of each year:
0
1
2
3
N
ri,1
ri,2
ri,3
ri,N
where ri,1 , ri,2 , ri,3 , . . . , ri,4 are defined as:
ri,1 =
ri,N =
Di,1 + Pi,1 − Pi,0
Pi,0
,
r2 =
Di,2 + Pi,2 − Pi,1
Pi,1
, ri,3 =
Di,3 + Pi,3 − Pi,2
Pi,2
, ...
Di,N + Pi,N − Pi,N −1
Pi,N −1
Why do we need to compute the time-series of returns? Because we need it to understand which
relationships exist between performance and risk.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Risk and Return of Financial Assets
The figure below synthesizes the existing relation between Risk and Return among these three
assets previously showed (2Y Bund, 10Y OAT, and Renault’s stocks) within the Risk-Return
space:
Note that to higher risk (σ) is associated with a higher average net return ( r̄ ).
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Short Selling
Given what we have seen up to now, the investor will buy a stock on the financial market and
wait that the price rises before sell it. In this way, investors make profits from the increase in
prices. But the other way around is also possible.
Specifically, investors can short sale an asset and make profits from the decline of prices. To
compare the two look to the table below, reporting the Cash Flows coming from the two strategies:
Purchase of Stock
Time
Action
Cash Flow
0
Buy Share
- Initial Price
1
Receive the Divided (if any),
Sell Share
+ Dividend Yield
+ Capital Gain
Short Sale of Stock
Time
Action
Cash Flow
0
Borrow and Sell Share
+ Initial Price
1
Repay the Divided (if any),
Buy Back Share
- Dividend Yield
- Capital Gain
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Market Efficiency
For Efficient Market Hypotheses (EMH) assumes that the price of assets fully reflects all the
available information. The term “efficient market” was introduced by Fama (1965). There exist
three forms of market efficiency:
Weak-form: tests whether all the information coming from the historical prices and volumes
is fully reflected in current stock prices. If this hypothesis is verified, an investor can’t
gain excess profit from the time series analysis (this kind of analysis is also called technical
analysis);
Semistrong-form: tests whether publicly available information is fully reflected in current
stock prices. If this hypothesis is verified, an investor can’t gain excess profit from the
balance sheet analysis (this kind of analysis is also called fundamental analysis);
Strong-form: tests whether all the information, public or private, is fully reflected in current
stock prices. If this hypothesis is verified, an investor can’t gain excess profit from any public
and private information.
More generally, to test the Weak-form of market efficiency, we examine the predictability of
stocks returns, whether to check the Semistrong-form and Strong-form of market efficiency, we
use the Event Study approach.
You will study the market efficiency hypothesis more in detail the remaining of this course.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
EMH Example 1: Dot-Com Bubble (1990s)
What Happened?
Technology stocks surged to excessive valuations.
Companies with little or no revenue saw massive price increases.
The bubble burst in 2000, leading to significant market losses.
Why EMH Failed: Prices did not reflect intrinsic value and were driven by speculative investor
behavior and irrational exuberance.
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Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
EMH Example 3: GameStop Short Squeeze (2021)
What Happened?
Retail investors on Reddit’s r/WallStreetBets coordinated a short squeeze.
GameStop stock (GME) rose from $17 to over $400 despite no fundamental changes.
Institutional short sellers faced massive losses.
Why EMH Failed: Price movements were driven by a social movement rather than fundamental
value, highlighting the impact of sentiment and speculative trading.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Mutual Funds
A mutual fund is a pooled investment vehicle managed by professionals. Investors purchase shares
in the fund, which represent a portion of the total holdings. Funds are diversified across asset
classes such as stocks, bonds, and other securities.
A type of mutual fund that primarily invests in stocks.
Designed for long-term capital growth.
Can be categorized into large-cap, mid-cap, and small-cap funds based on market
capitalization.
Generally higher risk but offers the potential for greater returns.
They provide liquidity, diversification, and professional management.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Mutual funds: Stylized facts
Delegated portfolio management industry plays a crucial role in financial markets. According to
the Investment Company Institute, in fact, at year-end 2024:
▷ Total worldwide assets invested in regulated open-end funds reached 68.9 trillion US dollars.
US open-end funds (8,582 mutual funds and 3,304 ETFs);
▷ In particular, the US mutual fund industry remained the largest in the world with 33.6 trillion
in total net assets (vs. 150 billion at the end of ’80s);
▷ Almost 54.4% of US households invested in mutual funds (only 5% in 1980);
▷ Equity mutual funds represented 59% of the overall US mutual funds industry;
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Bonds
A bond is generally defined as a loan issued by corporations or governments. The bonds issued
by a corporation are considered riskier most of the time, while government bond is considered
risk-free. There exist crucial terminology related to such instruments that are important to know:
 Bond Certificate: states the terms of the bond as well as the amounts and dates of all
payments to be made;
 Coupons: the promised bond payments. It usually occurs every six months and is
determined by the coupon rate stated in the certificate;
 Maturity Date: the final repayment date of a bond. Payments (coupons) continue till to
this date;
 Principal/Face Value/Par Value: the amount used to compute the interest payment. It
corresponds to the initial investment and is paid back at the end of the contract;
 Term: the time remaining until the repayment date.
Moreover, it is essential to know that there exist two kinds of bonds: Zero-Coupon Bonds, which
do not pay any coupon during their duration, and Coupon Bonds that do pay a coupon during
their duration.
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
Other Financial Assets
Zero-Coupon Bonds
The Zero-Coupon Bonds timeline is as follows:
Zero-Cupon Bonds
0
N
−P
FV
To evaluate bonds (Zero-Coupon or Coupon Bonds), we use the so-called: Yield to Maturity
(YTM). The YTM is the interest rate requested by the market for holding a bond with a particular maturity. Suppose that we want to compute the price ( P ) that we should pay for a bond
reimbursing in one year a Face Value ( F V ) equal to 100, 000$ with a YTM equal to 4%. We can
use the so-called present value formula:
P
=
FV
(1 + Y T M )
=
100, 000$
(1 + 0.04)
=
96, 154$
It might happen that instead, we know the price paid to buy a Zero-Coupon Bond and the face
value, but not the YTM. In this case, suppose that the face value remains the same wrt to the
previous example, but the price paid on financial markets is 96, 618$, we need to rearrange the
last equation to retrieve the YTM:
96, 618$
=
100, 000$
(1 + Y T M )
−→
YTM
=
100, 000$
−1
96, 618$
=
0.035
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Risk And Return of Stocks and Indexes
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Coupon Bonds
The Coupon Bonds timeline instead is as follows:
0
1
2
Cupon Bonds
3
4
N
−P
C
C
C
C
C + FV
There are two possible ways to evaluate a coupon bond. The first one assumes that the YTM is
constant across time. The price of such a bond can be then computed as:
C
C
C
FV
+
P
=
+ ··· +
+
(1 + Y T M ) (1 + Y T M )2
(1 + Y T M )N
(1 + Y T M )N
C
C
C
FV
+ ··· +
P
=
+
+
(1 + Y T M ) (1 + Y T M )2
(1 + Y T M )N
(1 + Y T M )N
|
{z
}


‚
ŒN
C
1
Proof
−−This is an Annuity!
1−
=
YTM
1+YTM

‹N C
1
FV
P
=
1−
+
YTM
1+YTM
(1 + Y T M )N
Firms generally issue new bonds at parity, meaning that P = F V .. If instead, the bond is issued
at at discount P < F V (the price paid is lower than reimbursement received by the investor at the
expiration date of the bond). If the bond is issued at premium P > F V (the price paid is higher
than reimbursement received by the investor at the expiration date of the bond).
Introduction
Corporation on Financial Markets
Risk And Return of Stocks and Indexes
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Coupon Bonds, Example
Suppose that you want to buy a coupon bond that pays 5.5% yearly, and a face value of 1000$
and an expiration date of five years (use an YTM of 5.5%). The timeline of such investment will
appear as follows:
0
1
2
3
4
5
−P
55
55
55
55
55 + 1000
Using the formula of the previous slide, we can compute the price as follow:
P
=

‹5 1
55
1, 000
1−
+
0.055
1.055
(1.055)5
=
1, 000
Is this bond sold at a parity, discount, or premium?
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Risk And Return of Stocks and Indexes
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Coupon Bonds, Example Second Method
If instead, you assume that the Y T M is not constant and you have the following yield curve:
The price of the bond with payments as in the previous slide will be equal to:
C
C
C
C
C + FV
P=
+
+
+
+
(1 + Y T M1 ) (1 + Y T M2 )2 (1 + Y T M3 )3 (1 + Y T M4 )4 (1 + Y T M5 )5
P=
55
55
55
55
55 + 1000
+
+
+
+
(1 + 0.046) (1 + 0.048)2 (1 + 0.05)3 (1 + 0.052)4 (1 + 0.055)5
= 1023
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The Yield Curve
So far, we assumed that there exists one interest rate for all maturities. In reality, the EARs differ
over maturities, and the tool used to check, which is the interest rate for the different maturities
is the Yield (YLD) Curve. Below, an example of a “standard” YLD curve:
According to the YLD curve, a cash flow received in two years is discounted at a two-year interest
rate, while a cash flow received in three years is discounted at a three-year interest rate. Thus,
our r depends on the timing of the cash flow. Furthermore, notice that a longer maturity implies
a higher interest rate.
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The Yield Curve and its Variations, Flat Yield Curve
Sometimes, the Yield Curve departs from what we have just seen. Specifically, we can have a Flat
Yield (YLD) Curve, and more importantly, an Inverted Yield (YLD) Curve. Both, but especially
the latter, give us some indication of the general economic conditions. Specifically, a Flat Yield
Curve appears as follows:
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The Yield Curve and its Variations, Inverted Yield Curve
While an inverted Inverted Yield Curve appears as follows:
Usually, an Inverted Yield Curve indicates the approaching of an economic crisis. The investors are
forecasting that there is much uncertainty shortly so that they prefer to buy short-term bonds instead of long term bonds. To check if the yield curve is inverted, financial economists take the difference between the long and short-term yields as showed here: 10-Years Yield minus 3-Months Yield.
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Yield Curve, Focus on the Euro Area, 1 of 2
The YLD curve below as reported by the European Central Bank two years ago:
We have all the characteristic that a YLD Curve should have, but for maturities lower than
5 years the interest rate is negative!!!
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Yield Curve, Focus on the Euro Area, 2 of 2
The YLD curve below as reported by the European Central Bank:
We now have a curve almost entirely sifted down and below zero, and in the first part (between
0 and 10 years, instead of 5), we also have an inversion indicating that most likely, there could
be a new crisis in the future.
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Derivatives
A derivative is a security that gets its value from the value of another asset, such as commodity
prices, bond, and stock prices, or market index values.
There are two kinds of options:
 the Call Option gives you the right to buy the underlying asset at the strike price (K ). The
value of calls increases as the price of the underlying asset increases;
 the Put Option gives you the right to sell the underlying asset at the strike price (K ). The
value of puts increases as the price of the underlying asset decreases;
Such instruments are an agreement made today regarding the delivery of an asset (or in some
cases, its cash value) at a specified maturity date for a given price called the futures price, to be
paid at contract maturity.
The main difference between the two contracts is that while the future holder is obliged to make or
take delivery, while the option holder has the right, but not the obligation, to buy or sell the
underlying assets. Thus the option holder will execute the contract if and only if it is profitable.
So when the value (Vt ) is greater then zero (Vt > 0).
You will be more in this respect in the last weeks of this course.
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Call (long) Payoff, Example
The figure below report the graph of payoff for a generic Call option with a strike price K
With a price of the underling asset between 0 and K (0 < P < K ) the value of the Call is equal to
zero (in Red). From K onwards ( P > K ), the value of the Call starts to increase (in Green).
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Put (long) Payoff, Example
The figure below report the graph of payoff for a generic Put option with a strike price K
With a price of the underling asset between 0 and K (0 < P < K ) the value of the Put start to
decrease as we move to K (in Green). From K onwards ( P > K ), the value of the Call it is zero
(in Red).
Geometric Series, General Case
Given the start term and the common ratio, the geometric series (S ) is equal to:
S
A + A × R + A × R2 + · · · + A × RN −1
=
=
N
−1
X
A × Ri
(1)
i=0
we then multiply both sides of the equation times R:
R×S
=
A × R + A × R2 + A × R3 + · · · + A × RN
=
N
X
A × Ri
(2)
i=1
subtract equation (2) from equation (1) and rearrange the terms as follow:
S−R×S
S−R×S
S−R×S
S(1 − R)
=
=
=
=
S
=
A + A × R + A × R2 + · · · + A × RN −1 − (A × R + A × R2 + A × R3 + · · · + A × RN )
A ×
R −
A ×
R2 − A ×
R3 − · · · − A × RN
A ×
R +
A ×
R2 + · · · + A ×
RN −1 − A +
N
A− A× R
A(1 − RN )
A(1 − RN )
(3)
(1 − R)
Moreover, for N −→ ∞ and |R| < 1, RN −→ 0 and the (3) becomes:
S
=
A
(1 − R)
(4)
we are going to use equation (3) and (4) as shortcuts to compute the present values of a different
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stream of cash flows.