Derivative Securities Pricing - Riassunto

EXTENSION OF THE BASIC B-S FORMULA TO STOCK PAYING A DIVIDEND YIELD

MKTEx dividend price = Market value of the stock - Present value of the dividends

If we use the dividend yield: -q TS * = S e0 0discount factor, but instead of subtracting the interest, we subtract the dividend yield

The cost of carry of the underlying asset reduces because it produces dividends.

European Index Options

To price European index options we can use the formula for an option on a stock paying a known dividend yield

Set S = current index level0

Set q = average dividend yield expected during the life of the option

Use q to decrease the risk free rate in the standard B-S formula

ALTERNATIVE FORMULAS: USING FORWARD/FUTURES PRICES

Given that

So that -rT -qTF e = S e0 0(the underlying asset net of the dividend payment, the ex dividend, is equal to the forward times the discount factor)

We may also rewrite the previous formula as follows

CURRENCY OPTIONS

Currency options are used by corporations to buy

THE FOREIGN INTEREST RATE

We denote the foreign interest rate py r .f

When a trader buys one unit of euros against USD on the spot market it has an investment of S USD (ex.01.2 $/€)

The return from investing at the ‘foreign’ rate is r S USDf 0

This shows that the foreign currency provides a “dividend yield” at rate r f

The future formula is

Proof:

QUOTATION METHODS

➢ Indirect quotation:

The exchange rate of a foreign currency is expressed as equivalent to a certain number of units of the domestic currency →(ex. 1.25 USD per one Euro 1.25 $/€)

➢ Direct quotation:

The exchange rate of the domestic currency is expressed as equivalent to a certain number of units of a foreign currency →(ex. 0.8 Euros per one USD 10.8 €/$)

The domestic rate (R) refers to the numerator, the foreign rate (Rf) refers to the denominator

In the FX (foreign exchange) conventions we use the inderect quotation, for

exampleEURUSD→(EUR is the base and USD the quote the local value is the dollar)

Note that: USDYPJ

The first two letters represent the country, the last one represents the currency

When we are dealing with correncies we look at the pips, because the changes in the currencies are verylow pips

EURUSD =PRICING EUROPEAN CURRENCY OPTIONS

Remember that:

If we have a call on the euro:

• If the price of the euro goes up we have a profit (we can buy euro at a lower price)
• That also means that the price of the dollars goes down and, as the currency option is a put on thedollar too, we have a loss

A foreign currency is an asset that provides a “dividend yield” equal to rf

We can use the formula for an option on a stock paying a dividend yield:

• Set S = current exchange rate0
• Set q = r f

The formulas are:

Example σ

The euro/USD spot price is 1.25 $/€ and the $/€ exchange rate has a volatility ( ) of 4% per year

The risk-free rates of interest for the US and the

Euroarea are 5% (R) and 4% (Rf) respectively

Questions:

1. Calculate the value of a European call option to buy one Euro using USD at 1.25 $/€ (1.25 USD against 1 Euro) in nine months
2. Use the put-call parity to calculate the price of a European put option to sell one Euro at 1.25 $/€ in nine months
3. What is the price of a call option to buy one USD at 0.8 (1/1.25) €/$ in nine months?

We have to adjust the put-call parity: Put + discounted value of the underlying asset (using the foreign interest rate)

Alternative formulas

RANGE FORWARD CONTRACTS

It's a product we can use to hedge just a part of the portfolio and not all of it, because it is less expensive and a full coverage it is not needed.

OPTIONS ON FUTURES

With an option on futures we have a double derivative: we have a derivative written on a derivative contract.

In time T we will exercise the option and we will acquire a future contract with expiration date in time T.

We will discuss:

1. Mechanics of call future
1. Options - Description of the Instrument
   • In discrete time model: binomial model
   • In continuous time model: Black model
2. Pricing
   • In discrete time model: binomial model
   • In continuous time model: Black model

1) MECHANICS OF CALL FUTURE OPTIONS - DESCRIPTION OF THE INSTRUMENT

When a future call option is exercised the holder acquires:

1. A long position in the futures, just like a call option (notice that there is no price to pay, as in any future)
2. A cash amount equal to the difference between the current future price and the strike price (S - X)

When a put futures option is exercised the holder acquires:

1. A short position in the futures, just like a put option (notice that no price will be received, as in any future)
2. A cash amount equal to the difference between the strike price and the current future price (X - S)

Example

An investor has a September futures call option contract on copper with a strike price of 240 cents per pound; one futures contract is on 25.000 pounds of copper

Suppose the futures price of copper for delivery in...

September is currently 251 cents and on the last settlement was 250250 (yesterday) 251 last settlement

If the option is exercised, the investor receives a payoff of $2,500 = 25,000(F-K) = 25,000(250-240) cents + along position in the futures contract

EFFECTIVE PAYOFFS

If the futures position is closed out immediately, we have to consider the change in the futures price since the last settlement.

In the example: 25,000(Ft-Ft-1) = 25,000 (251-250) cents = 25,000 cents = 250$

POTENTIAL ADVANTAGES OF FUTURES OPTIONS OVER SPOT OPTIONS

These advantages can be:

• Futures contracts may be easier to trade than underlying asset, they are more liquid
• Exercise of option does not lead to delivery of underlying asset
• Futures options may entail lower transactions costs

2.A) PRICING – BINOMIAL MODEL

A 1-month call option on futures has a strike price of 29.

T = 1/12

K = 29

F = 300

We have two possible scenarios:

- Up state of the world: the price goes up by 10%. We are going to pay 29 for an asset

worth 33 and we can easily calculate the final value of this option as equal to 4•

Down state of the world: the price of the asset is equal to 28 we are not going to exercise the option and its value is 0

RISKLESS PORTFOLIO

Then, we can create a riskless portfolio: we have

• A long position in Δ future
• A short position in 1 call option

We can offset the money that we lose on the option with the money that we get in the underlying asset.

We have the two states of the world:

• In the up state of the world, we have a gain (for which future that we are holding) for 3Δ – 4
• The price goes up, we have a loss as we have a short position
• In the down state, we have a gain in the underlying asset of -2Δ and the call option we sold won’t be exercised

The portfolio is riskless when 3Δ – 4 = – 2Δ Δ = 0.8

The riskless portfolio is:

• Long 0.8 futures
• Short 1 call option

In the 2 states of the world the gain will always be

– 1.6 (seen from the bank perspective, if we have a negative value we have a liability).

The value of the portfolio today, if the risk free rate is 6%, is –0.06/12– 1.6e = – 1.592

The value of the option, given that the value of the future at inception is zero – 1 (short 1 option) = – 1.592

The value is 1.592

GENERALIZATION

Stock price (in the up state of the world)

Option price (in the up state of the world)

Consider the portfolio that is long Δ futures and short 1 derivative

The riskless portfolio is change in the option price change in the stock price

The value of the portfolio at time T is

The value of the portfolio today

Hence

Substituting for Δ we obtain

Value of the option is the present value of:

❶ option value in the up state of the world

❷ option value in the down state of the world

Weighted by the probability p and 1 – p

The probability is

2.B) PRICING – BLACK MODEL (CONTINUOUS TIME MODEL)

The idea behind the pricing of the stock

When we have dividends, the formula is S* = S - PV(Div)MKT

The ex-dividend price is equal to the market price - the present value of the dividends

When we have a dividend yield the formula is -qTS * = S e0 0

The ex-dividend price is equal to the current stock price S times the discount factor (the dividend yield). It's a way to subtract from the stock price the amount of dividends that will be paid

Furthermore, in a risk neutral world the expected value of the future stock price is rTE(S ) = S * eT 0

This relationship is telling us that, in a risk neutral world, the growth rate for a risky asset is equal to the risk-free rate

We can use the formula for an option on a stock paying a dividend yield and we:

Set S = F (current futures price), since the underlying is the futures contract0 0

Set q = r (domestic risk-free rate ), so that the expected growth of F in a risk-neutral world is zero (the discount factor is equal to the growth rate)

BLACK

MODEL

Starting from the formula (r – q)TF = S e0,T 0

We can rewrite it (– q)T (– r)TS e = F e0 0,T

The stock price net of the dividend is equal to the future net of the interest rate

Using the last formula, the Black model can be rewritten in:

Right hand side of the formula above

BLACK MODEL – OPTION ON THE SPOT

European futures options and spot options are equivalent when futures contract matures at the same time as the option (convergence spot-future).

This enables Black’s model to be used to value a European option on the spot price of an asset using a futures contract with maturity equal to the maturity of the option.

For example, a 6-month European call option on spot gold is the same as a 6-month European option of the 6-month futures price.

FUTURE STYLE OPTIONS

Future style options are options without upfront payment of the premium (just like a future, in which everything is paid at the end)

Example

Call expiring out of the money

The strike price is 100 and the

The underlying price is 101.

The call price is 1.39.

The price of the future at the end of the day is 100 (perfectly at the money).

The price of the call is 0.74 and, as we entered in this position at 1.39, we are losing 0.65.

At day 2, the value is 0.52 as we are approaching to the end, and we are losing even more money.

At day 3 we lose all the money (loss equal to 1.39).

Upfront (at day 1) we didn't pay anything, but we paid at the end.

Call expiring in the money.