Lecture 5 - Dividends and Options on Other Instruments

Dividends and Options on Other Instruments

FNCE30007 Derivative Securities / Lecture 5

Schedule

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Introduction to Options

Properties of Stock Options

The Binomial Model

The Black‐Scholes‐Merton Model

Dividends and Options on Other

InstrumentsThe GreeksFutures Markets

Hedging with Futures and Forwards

Forward and Futures Prices Futures Options Swaps

Outline and Readings

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Outline Known discrete dividend American options and dividends Known continuous dividend Options on other instruments Binomial trees Portfolio insurance Range forward contracts

Readings Hull, 7th/8th ed., section 13.10, chapter 15

Known Discrete Dividend

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We assume that the stock price falls by the amount of the dividend on the ex‐dividend date.

The BSM model assumes a smooth path for stock prices. Since in the case of a dividend, there is a jump in the stock price, we can not use the BSM model.

We need a new model.


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When there is a dividend, we can decompose the stock price into two as a risky component and a riskless component.

The risky component has a smooth path and can be used to model the stock price.

If we set the risky component of the stock price to S (in the BSM model), then the BSM model is correct.

We still assume that the markets are frictionless, the risk‐free rate is constant, there are no arbitrage opportunities, and security trading is continuous.

Valuing a European Option

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In addition to the standard assumptions of the BSM model (the previous slide), we assume that the stock pays a dividend over the life of the option and the timing and the amount of the dividend are known with certainty.

The stock price net off the present value of the dividend follows the model assumed in the BSM .

We can easily extend the one dividend assumption to accommodate multiple dividends. How?


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We want to find the price of an option on a stock that pays a dividend, but we are going to use a stock price reduced by the present value of the dividend in the BSM. Is this correct? Are we going to get the price of the option on a dividend paying stock?

To value a European option on a dividend paying stock: Find the present value of all the dividends the stock is expected to pay during the life of the option.

Reduce the stock price by the amount of the present value. Call this S*.

Wherever you see S in the Black‐Scholes model use S*.


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Note that the volatility of the risky component is assumed to be the same as the volatility of the stock price. In theory these are different, but we will assume that the two are same.

Example

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The stock price nine months from the expiration of an option is $100, the exercise price of the option is $110, the risk‐free rate is 5% per annum, and the volatility is 40% per annum. The ex‐dividend dates are in three and six months. The dividend on each ex‐dividend date is expected to be $0.75. Find the prices of the call and put on the stock.

Example

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S = 100, K = 110, r = 0.05, σ = 0.40, T = 9/12 = 0.75 Present value of the dividends:

Find d1 and d2. Then compute N(x).

4721.1$75.075.0 )12/6)(05.0()12/3)(05.0( ee

2

1

2 1

1 2

1 2

ln((100 1.4721) /110) (0.05 0.4 / 2)(0.75)0.0365 0.04

0.4 0.750.0365 0.4 0.75 0.3829 0.38

( ) 0.4840, ( ) 0.3520( ) 0.5160, ( ) 0.6480

d

d d TN d N dN d N d

Example

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Finally plug everything into the formulas.

( 0.05)(0.75)

( 0.05)(0.75)

98.5279(0.4840) 110 (0.3520) $10.39

110 (0.6480) 98.5279(0.5160) $17.82

c e

p e

American Options and Dividends

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The Black‐Scholes formula can also be used to price an American call option on a non‐dividend paying stock.

Put option prices obtained from the Black‐Scholesformula do not reflect early exercise for American putsand, thus, are extremely biased. A binomial model wouldbe necessary to get an accurate price.


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An American call on a non‐dividend‐paying stock should never be exercised early.

An American call on a dividend‐paying stock should only be exercised immediately prior to an ex‐dividend date. High dividends and low time value lead to early exercise.


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It may be optimal to exercise an American call option before its maturity right before an ex‐dividend date.

If there are multiple dividends, in most cases we consider the last ex‐dividend date. Proof is in the appendix to chapter 13.

But, we still need to examine each dividend date for certainty.

We can value an American call option on a dividend‐paying stock using the Black’s approximation.

Black’s Approximation

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Set the American price equal to the maximum of two European prices:– The 1st European price is for an option maturing at the same time as the American option.

– The 2nd European price is for an option maturing just before the final ex‐dividend date.

This is only an approximation as the true probability of early exercise is not considered.

The calculation is more complex if there are multiple dividends over the life of the option.

Black’s Approximation

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Today Nov Dec Jan Feb Mar

Latest Ex‐Dividend Date

Second European Option First European Option

American Option

Example

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Consider a one‐year American call option on a stock that is expected to pay a dividend of $2 in the next six months. The current stock price is $50, the strike price is $45, the risk‐free rate is 5% per annum and the volatility is 30% per annum. Find the value of the option.

Example

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First find the value of the short option. S = 50, K = 45, r = 0.05, σ = 0.30, D = 0, T = 6/12 = 0.50 Using the BSM, we find the value as $7.71.

Next, find the value of the long option. S = 50, K = 45, r = 0.05, σ = 0.30, D = 2, T = 12/12 = 1 Find the PV of D = $1.951 and reduce the stock price by PV. Using the BSM, we find the value as $8.43.

Finally, compare the two values and choose the highest one as the option’s value. Option’s value is $8.43.

Known Continuous Dividend

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Dividend yield is defined as per annum rate at which the dividend is paid, expressed as a % of the stock price rather than as a dollar amount.

When a discrete dividend is paid the stock price falls on the ex‐dividend date.

When a continuous dividend is paid, the stock falls continuously over time.

This fall, however, is not noticeable.


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The expected growth rate in the price of a non‐dividend paying stock is μ per annum in the BSM.

But if the stock now pays a constant continuous dividend of q per annum, then the expected growth rate in the stock price is now μ – q.

We still use the standard assumptions as in the BSM. This time the stock pays a known continuous dividend yield of q% per annum.


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Since there is a dividend and there is also a jump, we can’t use the standard BSM.

Following two cases result in the same probability distribution. The stock starts at price S and provides a dividend yield = q. The stock starts at price Se–q T and provides no income.

The second case has no jump. We can value European options by reducing the stock price to Se–q T and then behaving as though there is no dividend.

Example

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The stock price nine months from the expiration of a call option is $100, the exercise price of the call option on the stock is $110, the risk‐free rate is 5% per annum, and the volatility is 40% per annum. Assume that the stock pays a continuous dividend yield of 4% per annum. Find the price of the option.

Example

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S = 100, K = 110, r = 0.05, σ = 0.40, T = 0.75, q = 0.04

1

2 1

1 2

(-0.04)(0.75) (-0.05)(0.75)

2ln(100/110)+(0.05-0.04+0.40 /2)(0.75)d = 0.08030.40 0.75

d =d - 0.40 0.75 0.4267

( ) 0.4681, ( ) 0.3336

c=100e (0.4681)-110e (0.3336)=10.08

N d N d

Options on Other Instruments

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Options on Stock Indices

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If we assume that the dividend paid by each stock in the index is relatively small compared to the total number of stocks in the index, and the ex‐dividend dates are spread out reasonably across the year, then it can be assumed that a stock index is similar to a stock which pays a known continuous dividend yield.

Options on Stock Indices

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Index options are settled in cash. Upon exercise a call holder receives: (ST – K)(index multiplier) Upon exercise a put holder receives: (K – ST)(index multiplier) Writer pays the same amount in cash.

Consider a call option on an index with a strike price of 5,460. Suppose one contract is exercised when the index level is 5,580. Assume an index multiplier of $10.

What is the payoff? (5580 ‐ 5460)*($10) = $1,200

Example: Index

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Consider a four‐month European call option on a stock index. Assume that the current value of the index is 5,500. Further assume that the exercise price is 5,300, risk‐free rate is 5% per annum and the volatility on the index is 30% per annum. The dividend yield on the index is expected to be 4%. Assume an index multiplier of $1. Find the value of the call.

Options on Currencies

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We denote the foreign interest rate by rf . When an Australian company buys one unit of the foreign currency it has an investment of S dollars.

The return from investing at the foreign rate is rf S dollars. This shows that the foreign currency provides a “dividend yield” at rate rf .

We can use the formula for an option on a stock paying a continuous dividend yield: Set S = current exchange rate (One unit foreign currency costs x in domestic currency).

Set q = rƒ

Example: Currency

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Consider a four‐month European call option on euros. Suppose that the current €/AUD exchange rate is 1.4633, the exercise price is 1.45, the risk‐free rate is 6% in Australia and 3% in the Euro Zone. Assume a 20% volatility. Find the value of the call.

Binomial Trees

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Binomial Trees

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The binomial model can be easily used to value options on dividend paying stocks.

Binomial trees can also be used to value American options on indices and currencies. It may be optimal to exercise American options on indices and currencies before maturity.

For an option on a continuous dividend paying stock (q% per annum), we can apply the binomial model in the usual way with the following adjusted value of p:

For an option on a discrete dividend paying stock, see section 18.3 in Hull (7th and 8th editions). Not Examinable.

(r-q)Te -dp=u-d

Portfolio Insurance

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Index Options and Portfolio Insurance

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Suppose the value of the index is S and the strike price is K.

If a portfolio has a β of 1.0, the portfolio insurance is obtained by buying 1 put option contract on the index for each 10S dollars held.

If the β is not 1.0, the portfolio manager buys β put options for each 10S dollars held.

In both cases, K is chosen to give the appropriate insurance level.

Index Options and Portfolio Insurance

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Payoff

Protective Put Portfolio Net Position

Example

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A portfolio has a beta of 1.0. The portfolio is currently worth $110,000. Suppose that the ASX200 index currently stands at 5,500. What trade is necessary to provide insurance against the portfolio value falling below $90,000?

Example

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The strategy is to buy two put option contracts with a strike of 4,500.

Why? (current index)*($10) = (5,500)*($10) = $55,000

The portfolio is worth two times the index. 90,000/2 = $45,000

Exercise price = $45,000/$10 = 4,500

Example

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If the index drops to 4,200, then the portfolio will be: Percentage change in the index: (4200 – 5500)/5500 = ‐23.6% Portfolio value: (1 ‐ 0.236)*( $110,000) = $84,000 The payoff from the options: (2)*(4,500 – 4,200)*($10) = $6,000

Total portfolio value: $84,000 + $6,000 = $90,000

Beta ≠ 1.0

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Portfolio has a beta of 2.0? Portfolio is twice as sensitive to index changes Slope of underlying exposure is twice as steep Double up the put option position to offset this

Portfolio has a beta of 0.5? Portfolio is half as sensitive to index changes Slope of underlying exposure is half as steep Halve the put option position to offset this

Strike Price When Beta ≠ 1.0

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We can use the capital asset pricing model (CAPM) to find the strike price when the portfolio’s beta is not 1.0.

Remember the CAPM equation is:

ri rf βi(rm‐rf)

Reconsider previous example with a beta of 2.0. The risk‐free rate is 12% per annum, the dividend yield both on the index and the portfolio is 4% per annum.

Number of put options required = 2 110,0005,500 10 =4

Example: Finding the Strike Price

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If index rises to 5,775 in three months, it provides a 275/5500 or 5% return.

Total return from the index (incl. dividends)= 5 + (4/4) = 6%.

Excess return over risk‐free rate= 6 – (12/4) = 3%.

ri = 0.03 + 2(0.06‐0.03) = 0.09

Increase in Portfolio Value= 9 – 1 = 8%. Expected Portfolio value=$110,000*(1.08) = $118,800.


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Value of Index in 3 months Expected Portfolio Value in 3 months ($)

6,000 127,800

5,775 118,800

5,250 97,800

5,055 90,000

4,750 77,800


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An option with a strike price of 5,055 will provide protection against a 18.18% decline in the portfolio value.

If the index drops to 4,750, then the portfolio will be: Portfolio value is $77,800 The payoff from the options: (4)*(5,055 – 4,750)*($10) = $12,200

Total portfolio value: $ 77,800 + $ 12,200 = $90,000

Range Forward Contracts

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These contracts have the effect of guaranteeing that the exchange rate paid or received will lie within a certain range.

Long range forward: when currency is to be paid it involves selling a put with strike K1 and buying a call with strike K2.

Short range forward: when currency is to be received it involves buying a put with strike K1 and selling a call with strike K2.

Normally the price of the put equals the price of the call.


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K1 K2Payoff

Asset Price‐15

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K1 K2Payoff

Asset Price

Short Range ForwardLong Range Forward

Lecture 5 - Dividends and Options on Other Instruments

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Transcript of Lecture 5 - Dividends and Options on Other Instruments