Derivatives - UniBg

Degree in Management Engineering

Finance I – Financial Markets

Derivatives

Michele Meoli - A.Y. 2019/20

20/01/2020 2

Definitions

Forward contract

OTC market

Forward vs futures

Hedging

Options pricing

Black & Scholes

Black & Scholes (2)

Derivative securities

https://www.investopedia.com/terms/f/forwardcontract.asp

http://www.investopedia.com/terms/o/over-the-countermarket.asp

http://www.investopedia.com/ask/answers/06/forwardsandfutures.asp

http://www.investopedia.com/terms/h/hedge.asp

http://www.investopedia.com/university/options-pricing/

http://www.investopedia.com/terms/b/blackscholes.asp

https://www.youtube.com/watch?v=Xy_txjKPNyg

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Agenda

1. Definition and classification

2. Markets for derivatives

3. Pricing of forward contracts

4. Pricing options: the binomial tree

5. Pricing options: the Black & Scholes formula


1. Definition and classification

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Definitions

Three definitions for Derivative Security:

1. A security with a rate of return linked to a price index;

2. A security that, opportunely combined with other securities (or goods), behaves as a

risk-free security;

3. In a contract, an agreement to exchange the underlying asset (or equivalent cash

flows) at a future date at pre-determined conditions.


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Classification

Derivatives securities can be classified under the two general categories:

1. Forward;

2. Options.

Notwithstanding the variety of existing derivative contracts, all of them can be classified under these categories, or decomposed in a summation of securities of these categories;

In this lecture we therefore analyse these categories, and consider possible combinations;


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Classification

Analysing derivative securities, we aim to understand:

1. the elements of each contract;

2. the variables affecting value;

3. the economic role and the application in practice.

We anticipate an important aspect with regard to the economic role:

All derivatives can be employed either to manage (or transfer) financial risk, or to speculate on financial positions;

Both roles can be assumed with the use of any financial derivative, provided the recur to correct quantities and techniques.


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Forward

A forward contract is an agreement between two parties to buy or sell an asset (which can be

of any kind) at a pre-agreed future point in time;

Main features of a forward contract are the following:

The object of trade (the underlying) and its quantity;

The date of expiration of the contract, or alternatively, the time to expiration

(maturity);

The pre-agreed price (strike, or exercise price);

Forward contracts usually specify other parameters, with limited economic value (delivery

clauses, extraordinary events conditions, etc), especially when the underlying is a kind of

goods;


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Forward

An interesting clause is that of settlement condition: the contract may not involve the actual

delivery of the underlying (in exchange of the payment of strike price), but the payment of

the difference between market and strike prices;

When a forward contract is issued, two securities are generated on the market, corresponding

to the position of the buyer (long position) and that of the seller (short position).


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Options

An option contract is an agreement in which the buyer (holder) has the right (but not the obligation) to exercise by buying or selling an asset at a set price (strike price) on (European style option) or before (American style option) a future date (the exercise date or

expiration); and the seller (writer) has the obligation to honour the terms of the contract;

When the right (without obligation) is given to the long position, the security is called call option.

When the right (without obligation) is given to the short position, the security is called put option;

In both cases, when an option is issued, two securities are actually circulating: that of the

holder, and that of the writer.


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Graphical representation of Derivatives

In order to univocally identify a derivative, a graphic representation is employed in

practice, plotting the outcome on expiration date as a function of the underlying price (ST) on expiration (T), given a certain strike price (X);

In the following slides, we report valuation for forward, call options and put options,

long (on the left) and short (on the right) positions;

Naturally, all these contracts have a (positive or negative) acquisition price, and

therefore the outcome on expiration should be shifted (upwards or downwards)

accordingly.


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Graphic representation of Forwards


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Graphic representation of Options


Right to sellRight to sell for other party

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Notes on graphic representations (1/2)

We notice that:

1. The outcome of derivative securities obviously depends on the behaviour of the underlying, and

in particular on its future price/value;

2. Long positions are similar to bets on price-increase expectations, while short positions are similar

to bets on price-decrease expectations;

3. On expiration date, the pair of positions sums up to a zero-sum game, i.e. what a position wins

equals what the other loses;

4. A forward contract may lead to positive or negative outcomes for both parties, while an optiondetermines only positive (or at least null) results for a party (long position in a call option, and

short position in a put option), while negative (or at best null) for the other, if we do not

consider acquisition costs;


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Notes on graphic representations (2/2)

Therefore, in general, because of the behaviour of the underlying price, all derivatives may be consider

more or less attractive (for both the writer and the holder);

Participation in a derivative contract is, of course, an act of will; therefore, at time of issue, either

the value is null for both parties, or an acquisition cost is paid by one party to the other;

At any following point in time, up to expiration date, a certain value can be attributed to a

derivative security, reflecting expectations on the underlying price;

A DCF technique cannot be employed to price derivatives, because the risk of holding a derivative

is not trivially quantifiable; by contrast, a parity condition logic is employed.


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American and European style options American style options can be converted at any time up to their expiration;

Nevertheless, it is quite easy to show how an American style call on an underlying that does not produce any cash flows is not to be converted before expiration, and in this sense, its value is equal to that of a European style analogous option;

In fact, by exercising before expiration, the time value of an option is lost, and it is therefore more convenient to close the position (i.e. sell the underlying) rather than exercise;

Test this by observing that the value of a call, before expiration, is always greater than the value obtained by exercising (St – X);

This principle does not hold when:

1. The underlying pays cash flows:

In fact, converting before expiration may be convenient if dividends (or other cash flows) are high enough;

2. When considering a put option:

In fact, the option can be deep in the money (far from the threshold that makes the exercise inconvenient), as like in the case when the underlying price is close to zero.


2. Markets for derivatives

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Organised markets for derivatives (1/4)

The organisation of a market for derivative securities, where standard contracts (such as futures) are traded, is more difficult than that of another financial market:

This is due to the nature of exchanges that involve interactions of parties that are not concluded on spot;

In fact, a second transition is to be carried out on expiration, involving a wealth transfer at the expenses of the one party (i.e. the need to buy underlying at a higher price with respect to the market, or symmetrically, sell the underlying asset at a lower price than the market);


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Organised markets for derivatives (2/4) As a consequence, a trust problem arise, because of the risk of default:

Consequently, parties are interested in the acquisition of information on their counterparty;

But this fact would involve a loss of anonymity that is, by contrast, guaranteed in spot contracts, and would ultimately decrease participation;

Obviously, in order to boost participation, default risk needs to be minimised, and at the same time anonymity has to be guaranteed;

Two solutions are employed to solve the issue:

1. Clearing houses;

2. Marking-to-market mechanism.


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Clearing houses:

A clearing house is a financial services company that provides clearing and settlement services for financial transactions and often acts as central counterparty;

The economic effect is that the derivative is split in two contracts, and the clearing house acts as counterparty in both deals, assuming default risk;


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Marking-to-market mechanisms:

In order not to charge default risk on transition costs, the clearing house assumes a minimum risk by employing a margin mechanism;

In practise, the clearing house requires a cash (or underlying-asset) deposit compensating eventual default losses;

Evidently, a deposit covering the whole value of the traded underlying would fully cover default risk, but would require an excessive immobilisation of financial capital, ultimately reducing participation;

Therefore, a marking-to-market system has been introduced, reducing the financial commitment to the daily-measured loss of each party;

A good mechanism of marking-to-market requires expensive information systems and money transfers: this is the main reason why the development of financial derivative markets has been slower than for other markets.


3. Pricing of forward contracts

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Pricing forward contracts (1/4)

In order to describe the parity condition logic, we employ an example, that allows us to pricea forward contract:

Consider the two following portfolios:

1. Portfolio A: a Forward (+1, T, X) and a Zero-coupon bond (T, X):

fw + X e-rT

2. Portfolio B: a unit of the underlying:

S0

(Note that +1 indicates a long position)

If the two portfolios are equivalent, they should have the same value:

fw (+1, T, X) + X e -rT = S0 fw (+1, T, X) = S0 – X e -rT

This equivalence is named no arbitrage condition, because when it does not hold,

arbitrage opportunities arise (see following slide).


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Pricing forward contracts: Arbitrage mechanism (2/4)


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Pricing forward contracts (3/4)

Assuming no arbitrage conditions, the price of a forward in long position is:

fw (+1, T, X) = S0 – X e -rT

Analogously, we derive the price of a forward in short position:

fw (–1, T, X) = – S0 + X e -rT

This confirms that a forward is a zero-sum game:

fw (–1, T, X) = – fw (+1, T, X)


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Pricing forward contracts: Arbitrage example (4/4)

Consider the following example:

I am committed to buy within one year at the price of 1 something valued 0.9 today on the market;

Given a r = 3%, I am holding a contract with a value of [S0 – X e-rT], i.e. [0,90 – 0,97] = - 0.07 (a

positive value for the opposite position);

If on the market an analogous contract is priced differently, an arbitrage opportunity arises;

Suppose a short position is quoted 0.08, I have the chance to short-sell the underlying, lend 0.98 and

take long position in forward; while my current outcome is null, on expiration date I can buy the

underlying at strike price to cover my short sale, then cash money lent earning interest for an amount

of 1.01 (this is the money machine mechanism!);

Because the hypothesis of market efficiency supposes arbitrage opportunities to be immediately

absorbed, - 0.07 is likely to be the market price of this contract.


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Forward price (1/2)

Going back to the former example, an individual willing to activate a forward with the same conditions, taking the short position, would not find a rational third party taking the long position without an equal compensation of 0.07;

Conversely, 0.07 is going to be the maximum price the individual is willing to pay in order to take the short position;

Therefore, a forward contract as such is to be born only together with a cash payment;


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Forward price (2/2)

Of course, under different conditions, a certain set of parameters may determine a null value for both securities (long and short forwards), and no cash payment would be required;

Usually, parameters for forward contracts issued over-the-counter are set in order to determine a null value for both parties;

This price, setting an equilibrium between long and short forward positions in named forward price, and can be easily determined as follows:

XFW = S0 e +rT

The forward price set on the market under the conditions in the former example would be 0.9274.


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Introducing other costs and revenues

The value of a forward, as determined so far, is valid only supposing that the

underlying does not originate any positive or negative cash flows during its life (e.g.:

dividends, interests, maintaining costs, etc.);

Otherwise, the holder of the underlying has to consider the cash flows F1, F2, … at

time t1, t2, … up to time T, and the correct formula to determine the value of a

forward contract is:

fw (+1, T, X) = S0* – X e -rT

where: S0* = S0 – Fi e –rti


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Example: forward on foreign currency (1/2)

Forward contracts on foreign currency are examples of derivative securities producing cash

flows during their time to maturity;

In fact, an amount in foreign currency is an underlying asset yielding interests according to a

foreign rate of return (e.g. the interest rate on deposits) for the holder;

In general, considering derivatives in foreign currencies, annual compounded rate of returns

are employed, so that:

fw (+1, T, X) = S0* – X /(1+r)T

XFW = S0* (1+r)T

where: S0* = S0 – Fi /(1+r)Ti


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Example: forward on foreign currency (2/2)

For example, consider a forward on a US dollar. A US dollar yields interests equal to

[(1+r$)T –1] when deposited in a deposit for the time T, so that:

S0* = S0 – S0 [(1+r$)T –1] /(1+r$)

T = S0 /(1+r$)T

where: S0 is the spot exchange rate for dollars.

Consequently we derive the so-called formula for exchange-rate dealers:

XFW = S0 (1+r)T / (1+r$)T


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Hedging with forward contracts (1/2) It is straightforward to notice that the price for a forward (say in long position) changes in

absolute value at any change in the value of the underlying;

The forward value changes of the same absolute value, but in percentage it changes at a higher

rate;

Of course, a portfolio composed by a unit of underlying, and a forward in short position, is a

free-risk portfolio. In other words, this portfolio is equivalent to a free-risk asset, because the

value of the underlying does not affect the investor’s wealth:

S0 – fw (+1, T, X) = X e -rT


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Hedging with forward contracts (2/2) The high recur to risk-free interest rates in derivatives trading is due to the no arbitrage

condition: in fact, a free-risk portfolio (composed by a risky underlying and the relative

short forward), is to be linked to a free-risk rate of return;

From these observation, we understand that a forward can be held for:

Hedging: when it covers an opposite position in an underlying;

Speculation: when it does not cover any position.


4. Pricing options: the binomial tree

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Pricing Options: the binomial case (1/2)

Consider the following example:

Suppose that the ordinary share ENI, today priced 5 euro, may be priced only 5.2 or 4.7 within a month;

Let us consider now an option on a single share ENI, that expires in a month at the strike price 5.1 euro;

Let us combine a portfolio as follows:

P 1 share ENI – 5 call (ENI, 1 month, X = 5,1)


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Pricing Options: the binomial case (2/2) Within a month, in both scenarios (share priced either 5.2 or 4.7) this portfolio takes a price

of 4.7 euro, and right now it can’t be but the price of a zero-coupon bond with one month to maturity and a face value of 4.7.

Consequently:

S0 – 5 call = 4,7 e–rT

call = 1/5 S0 – 1/5 4,7 e–rT

Given r = 3%, the call price is 0.062 euro.

If the price is different, I would compose a zero-coupon bond at a price inferior or superior to the market, creating an arbitrage opportunity;

When considering non-binomial distribution of probabilities for the share value, the former formula changes, but the principle is the same;

Moving to a continuous distribution of share values, and supposing a Normal distribution for returns, we get the Black & Scholes formula.


5. Pricing options: the Black & Scholes

formula

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Agenda

1. The Black and Scholes formula

2. The put-call parity

3. Option price and parameters

4. Special cases: warrants and exotic options


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The Black and Scholes formula

A recommended video on Black and Scholes Formula


https://www.youtube.com/watch?v=pr-u4LCFYEY

5.1. The Black & Scholes formula

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Pricing options: Black & Scholes formula (1/2)

The idea that a combination of underlying and an option can compose a free-risk portfoliohelps us in determining a price for an option allowing returns to be normally continually

distributed;

Remember the example:

S0 – 5 call = 4,7 e–rT

Or

S0 – 4,7 e–rT = 5 call

This pricing technique is based on the direct relationship between underlying and option

values (of course not 1:1 as in the case of forward contracts);


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Pricing options: Black & Scholes formula (2/2)

The Black & Scholes formula derives the price for a call option as follows:

call (1, T, X) = N(d1) S0 – N(d2) X e -rT

where: N(.) is the function of cumulated probability for a standard

normal distribution;

T

T2

rX

Sln

d

20

1

T

T2

rX

Sln

d

20

2


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Example: Pricing an option on Telecom shares


5.2. The put-call parity

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Put-Call Parity Black & Scholes formula refers to a call option;

The correspondent value of a put option can be derived from the same formula, by considering the putas a combination of other securities;

Consider the following portfolios:

Portfolio A: 1 call (1, T, X) + zero coupon bond (T, X) call + X e-rT

Portfolio B: 1 put (1, T, X) + 1 unit of underlying put + S

On expiration:

As the two portfolios are equivalent on expiration, they need to be equivalent at any time, therefore:

put = call + X e-rT – S0

When modeling returns as normal and continually distributed, the Black & Scholes formula applies:

put (1, T, X) = [N(d1) – 1] S0 – [N(d2) – 1] X e -rT

ST > X ST < X

Portfolio A:

Exercise the call with cash from zero-coupon bond: get a

unit of underlying as outcome

Portfolio A:

Do not exercise the call, and cash the zero-coupon bond,

getting a sum of X as outcome

Portfolio B:

Do not exercise the put: the outcome is a unit of

underlying

Portfolio B:

Exercise the put, delivering the underlying: the outcome is

a sum X


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Put-Call Parity: an example

The former slide explains how a portfolio composed by a call option, a credit of X and a short-sale of

underlying is equivalent to a put option;

Going back to the example of options on Telecom Italia stock shares, we can show how the no-arbitrage condition pushes market prices to their theoretical values;

The theoretical value of a put can be determined as (see page 29):

put = 1,090 + 15,418 – 14,874 = 1,634

while it is traded at 1.568 on the market;

Check market prices for money machines… good luck!


5.3. Option price and parameters

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Option Price and Parameters (1/2)

Five parameters determine the price of a call option:

S0 : current price of underlying asset;

X : strike price;

r : risk-free interest rate (continuously compounded);

T : time to expiration;

: volatility of underlying asset.

It is straightforward to obtain, by means of a spreadsheet software, a tool such as that in the previous

slide, to derive option price as a function of these parameters;

Notice that if the underlying produces cash flows before maturity, they have to be discounted as in the

case of forward contracts;


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Option Price and Parameters (2/2)

Observe that:

A call option in long position has always a greater value than an homologous forward, because of

the time value of an option, i.e. the right to not accomplish the exchange;

The price of a call option in long position:

Is positively linked to S0 , r, T and ;

Is negatively linked to X;

The only novelties with respect to forward pricing is the role of : this is due to the fact that volatility measure the financial risk, which is censored by options.


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Implicit Volatility

Prices of option derivatives traded in official markets univocally define a level of

volatility (once given the contract clauses, underlying current price and free-risk rate

of return);

This measure of volatility is known as implicit volatility, and it is usually considered

the best available measure of expected volatility, because it contains all information

about market expectations;

Financial newspaper usually report implicit volatility when listing option pricing on

stock shares and stock indices.


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Hedging ratio with options

An option price always reacts to underlying-price variations;

In particular, we can show that, when the underlying price changes by a currency unit, the

price of a call option reacts by N(d1), and the price of a put option reacts by [N(d1)-1];

Empirically, contrast results in the example formerly provided, and consider the analogies

with the binomial case;

We observe that:

A portfolio composed by a quantity N(d1) units of underlying and a call option in short position

A portfolio composed by a quantity [N(d1)-1] units of underlying and a put option in short position

are both risk-free portfolios;

A similar result can be obtained by getting a short position in a forward contract (for the

corresponding units of underlying), and keeping it until the end of the period we need to

hedge.


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-hedging

The unit of underlying asset covered by a certain derivative is called ;

-hedging is the practice of covering risk by composing a portfolio with N units of

underlying and (-N) units of derivative;

This procedure is imperfect, overall for small variations of the underlying price: this

is due to the fact that considers variations in S0 , but not in other parameters, that in fact change contemporaneously;

A perfect hedging could be carried out, under theoretical conditions, only

considering all parameter variations. This leads to consider all the so-called greeks, a

set of hedging parameters with reference to all sources of price volatility.

Nevertheless, is the most important of greeks.


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-hedging : example

This example shows how a portfolio composed by a single Telecom share can be hedged by

means of a quantity of call options (notice how -hedging is imperfect):


5.4. Special cases: warrants and exotic

options

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Warrants Warrants are call options on stock shares, with the peculiarity that the writer is the underlying

issuer;

This special feature affects the theoretical value:

By exercising, the underlying price is diluted;

Therefore, there is a further effect on expiration for the holder, inferior to what it looks just before conversion;

As a consequence, a warrant is worth a little less than an analogous call option;

The correct value of a warrant can be approximated by recurring to the Black & Scholes formula, and employing a measure of the underlying as follows:

where: N being the total number of shares issued, and NW being the number of circulating warrants.

Bonds with warrants are securities composed of a bond and a warrant, that before expiration can be detached and traded separately;

Covered warrants are simple options. The name is due to the fact that they are usually issued with a long time-to-expiration, as in the case of warrants. Because of their nature, they do not suffer the dilution effect above described.

W

wrT*

0*DIL,0

NN

NXeNSS


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Exotic derivatives: composition of simple derivatives Besides the types of derivatives consider so far, in general a security is called a derivative when its return varies as a

function of some underlying;

The problem is now that of recognise what derivatives compose a security, and in particular how to split them:

Consider the following contract:

The intermediary pays its customer, within a month, 5 euro if the price of ENI share is lower than 5.5 euro; the

difference between market price and 5.5 euro if the price is between 5.5 and 6 euro; 5.5 euro if the market price

is higher than 6 euro.

Plotting the income-price curve:

Notice that this derivative can be easily split in a portfolio containing: a zero-coupon bond with a face value of 5

euro, a call option in long position with strike price 5.5, and a call option in short position with strike price at 6

euro;

Supposing a yearly volatility of 37%, r=3%, a current price for the ENI share of 5.25, the call at 5.5 is worth

0.1296 and that at 6 is worth 0.0318, the zero coupon bond 4.9875: the derivative has an equilibrium price of

5.0853.


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