1 today options option pricing applications: currency risk and convertible bonds reading brealey,...
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1
Today
Options
•Option pricing
•Applications: Currency risk and convertible bonds
Reading
• Brealey, Myers, and Allen: Chapter 20, 21
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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
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