1 Today Options Option pricing Applications: Currency risk and convertible bonds Reading Brealey, Myers, and Allen: Chapter 20, 21.

1

Today

Options

•Option pricing

•Applications: Currency risk and convertible bonds

Reading

• Brealey, Myers, and Allen: Chapter 20, 21

2

Options

Gives the holder the right to either buy (call option) or sell (put option) at a specified price.

•Exercise, or strike, price

•Expiration or maturity date

•American vs. European option

•In-the-money, at-the-money, or out-of-the-money

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Option payoffs (strike = $50)

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Valuation

Option pricing

How risky is an option? How can we estimate the expected cashflows, and what is the appropriate discount rate?

Two formulas

•Put-call parity

•Black-Scholes formula *

* Fischer Black and Myron Scholes

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Option Values

• Intrinsic value - profit that could be made if the option was immediately exercised (內在價值、真實價值 )– Call: stock price - exercise price– Put: exercise price - stock price

• Time value - the difference between the option price and the intrinsic value

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Time Value of Options: Call

Option value

XStock Price

Value of Call Intrinsic Value

Time value

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Factors Influencing Option ValuesFactor

Stock price

Exercise price

Volatility of stock price

Time to expiration

Interest rate

Dividend Rate

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Put-call parity

Relation between put and call prices

P + S = C + PV(X)

S = stock price P = put price C = call price X = strike price PV(X) = present value of $X = X / (1+r)t r = riskfree rate

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Option strategies: Stock + put

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Put-Call Parity Relationship

Long a call and write a put simultaneously.

Call and put are with the same exercise price and maturity date.

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Put-Call Parity Relationship

ST < X ST > X

Payoff for

Call Owned 0 ST - X

Payoff for

Put Written-( X -ST) 0

Total Payoff ST - X ST – X

PV of (ST-X)= S0 - X / (1 + rf)T

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Payoff of Long Call & Short Put

Long Call

Short Put

Payoff

Stock Price

Combined =Leveraged Equity

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Arbitrage & Put Call Parity

Since the payoff on a combination of a long call and a short put are equivalent to leveraged equity, the prices must be equal.

C - P = S0 - X / (1 + rf)T

If the prices are not equal, arbitrage will be possible

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Put Call Parity - Disequilibrium Example

Stock Price = 110, Call Price = 17, Put Price = 5

Risk Free = 5% per 6 month (10.25% effective annual yield)

Maturity = .5 yr X = 105

C - P > S0 - X / (1 + rf)T

17- 5 > 110 - (105/1.05)

12 > 10

Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternative

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Put-Call Parity Arbitrage

ImmediateImmediate Cashflow in Six MonthsCashflow in Six Months

PositionPosition CashflowCashflow SSTT<105<105 SSTT>> 105 105

Buy StockBuy Stock -110-110 S STT S STT

Borrow 100Borrow 100X/(1+r)X/(1+r)TT = 100 = 100 +100+100 -105-105 -105-105

Sell CallSell Call +17+17 0 0 -(S-(STT-105)-105)

Buy PutBuy Put -5 -5 105-S105-STT 0 0

TotalTotal 22 0 0 0 0

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Example

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Black-Scholes

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Cumulative Normal Distribution

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Example

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Option pricing

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Using Black-Scholes

Applications

•Hedging currency risk

•Pricing convertible debt

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Currency risk

Your company, headquartered in the U.S., supplies auto parts to Jaguar PLC in Britain. You have just signed a contract worth ₤18.2 million to deliver parts next year. Payment is certain and occurs at the end of the year.

The $ / ₤ exchange rate is currently S$/₤ = 1.4794.

How do fluctuations in exchange rates affect $ revenues? How can you hedge this risk?

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S$/₤, Jan 1990 – Sept 2001

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$ revenues as a function of S$/₤

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Currency risk

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$ revenues as a function of S$/₤

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Convertible bonds

Your firm is thinking about issuing 10-year convertible bonds. In the past, the firm has issued straight (non-convertible) debt, which currently has a yield of 8.2%.

The new bonds have a face value of $1,000 and will be convertible into 20 shares of stocks. How much are the bonds worth if they pay the same interest rate as straight debt?

Today’s stock price is $32. The firm does not pay dividends, and you estimate that the standard deviation of returns is 35% annually. Long-term interest rates are 6%.

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Payoff of convertible bonds

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Convertible bonds

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Convertible bonds

Call option

X = $50, S = $32, σ = 35%, r = 6%, T = 10 Black-Scholes value = $10.31

Convertible bond