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CHAPTER 18 Derivatives and Risk Management

Derivatives: Forward, futures, options

Put call parity, Black Scholes Formula

Other derivatives: swaps, rights, warrants

Hedging with derivatives

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What is a derivative?

A derivative is a financial contract between two parties to transact an asset at a fixed price at a future date.

It derives value from other assets or events.

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Definitions

Buyer: one who buys the derivative.

Writer: one who sells the derivative.

Long position: the position of the buyer.

Short position: the position of the writer.

Expiry date: the date when cash flows would be exchanged.

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Definitions

Underlying asset: the asset to be transacted.

Strike price (or exercise price): the transaction price of the underlying asset at the expiry date.

Counter parties: the opposite party in the derivative contract

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The Forward Contract

A financial contract which allows the buyer to buy a specific asset at a specific price on a specific future date.

The seller has to sell to the buyer that asset at that price and at that future date.

Delivery date: expiry date.

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The Forward Contract Payoff

Payoff: the profit brought about by the contract.

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The Forward Contract Payoff

Payoff: the profit brought about by the contract.

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The Futures Contract

Similar to forward contracts

Specifications standardized: underlying asset, contract size, expiry date.

Traded in exchanges

Many types: e.g. commodity, interest rates, equity, FX etc.

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Features of Futures Contract

Margin account: Initial margin

Maintenance margin

Margin call

Mark to market: Delivery price is updated at the end of

every trading day

Gains and losses are updated into margin account.

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What is an option?

A contract that gives its holder the right, but not the obligation, to buy (or sell) an asset at some predetermined price within a specified period of time.

It’s important to remember:

It does not obligate its owner to take action.

It merely gives the owner the right to buy or sell an asset.

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Option terminology

Call option – an option to buy a specified number of shares of a security within some future period.

Put option – an option to sell a specified number of shares of a security within some future period.

Exercise (or strike) price – the price stated in the option contract at which the security can be bought or sold.

Option price – option contract’s market price.

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Option terminology (con’t)

Expiration date – the date the option matures.

Exercise value – the value of an option if it were

exercised today (Current stock price - Strike price).

In-the-money call – a call option whose exercise price is less than the current price of the underlying stock.

Out-of-the-money call – a call option whose exercise price exceeds the current stock price.

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The Call Option Payoff (long position)

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The Call Option Payoff (short position)

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Determining option exercise value and option premium

Stock price

Strike price

Exercise value

Option price

Option premium

$25.00 $25.00 $0.00 3.00 3.00

30.00 25.00 5.00 7.50 2.50

35.00 25.00 10.00 12.00 2.00

40.00 25.00 15.00 16.50 1.50

45.00 25.00 20.00 21.00 1.00

50.00 25.00 25.00 25.50 0.50

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Call Option Intrinsic Value and Time Value

Intrinsic Value: the value of the call option if exercised now

Time value (or premium): the difference between the value of the call option and the intrinsic value

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Call Option Intrinsic Value and Time Value

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Relationship of Call Value with other Factors

Factor Change: An increase in… Call Value

Change

Relation-

ship

spot price of the underlying asset Increase Positive

time to expiry date Increase Positive

strike price Decrease Negative

risk-free interest rate Increase Positive

the return volatility of the

underlying asset

Increase Positive

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The Put Option Payoff (long position)

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The Put Option Payoff (short position)

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Relationship of Put Value with other Factors

Factor Change: An increase in… Call Value

Change

Relation-

ship

spot price of the underlying asset Decrease Negative

time to expiry date Increase Positive

strike price Increase Positive

risk-free interest rate Decrease Negative

the return volatility of the

underlying asset

Increase Positive

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Put Call Parity

Relates the call price and the put price with the strike price and the spot price

P = K exp(-rT ) - S + C

Arbitrage opportunities exist if put and call prices violate the relationship

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The Black-Scholes option pricing model

)]S[N(-d -)][N(-dKe P

)][N(dKe - )]S[N(d C

Tσ - d d

Tσ

T 2

r ln(S/K)

d

12

Tr-

2

Tr-

1

12

2

RF

1

RF

RF

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Use the B-S OPM to find the option value of a call option with S = $27, K = $25, rRF = 6%, T = 0.5 years, and σ2 = 0.11.

0.6327 0.1327 0.5000 N(0.3391) )N(d

0.7168 0.2168 0.5000 N(0.5736) )N(d

textbook the in C Appendix From

0.3391 .7071)(0.3317)(0 - 0.5736 d

0.5736 .7071)(0.3317)(0

(0.5) )]2

0.11 [(0.06 )ln($27/$25 d

2

1

2

1

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Solving for option value

$4.0036 C

[0.6327]$25e - ]$27[0.7168 C

)][N(dKe - )]S[N(d C

)(0.06)(0.5-

2

T-r

1RF

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Swaps

The exchange of cash payment obligations between two parties, usually because each party prefers the terms of the other’s debt contract.

An interest rate swap is a financial contract based on a notional amount, whereby the buyer of the contract pays a fixed interest based on the notional amount periodically to the seller, and the seller of the contract pays a floating rate interest based on the same notional amount periodically to the buyer.

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Other Types of Derivatives

Rights and Warrants: like call options allowing the holder to buy stocks at a strike price.

The Shares as a Call Option: shares have a limited liability, hence it is like a call option.

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The Need to Hedge

Better debt capacity and cost. Smoother budget funding. Reduced cases of extreme

financially-poor performance. Better comparative advantage in

hedging. Beneficial tax effects.

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An Approach to Risk Management

Identify the situations when the firm would make a loss—quantify the loss.

Find a hedging instrument that rewards when the loss-making situations occur—quantify the rewards.

Compute the satisfactory quantity of hedging instrument to purchase.

Purchase the satisfactory quantity of the hedging instrument.

Monitor the cash flows necessary to maintain the hedge.

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Why Derivatives are Good Hedging and Speculating Instruments

Good speculating instrument: built in leverage magnifies investment risk and return.

Good hedging instrument: built in leverage allows little overhead cost to get into hedge position.