Options on stocks paying dividends
Options on stocks paying dividends If the asset is a stock, the current stock price is the PV of expected future dividends • • •
Corporate finance: In theory, share price will drop by the amount of the dividend after the dividend has been paid Thus, the BS formula will still work in this case, as long as we reduce the current stock price by the PV of the dividend Only dividends that are expected within the maturity of the option are considered
%$&' ()*(,) .
2 3
0 10 4 3 × 6
•
𝑑" =
• • • •
𝑑7 = 𝑑" − 𝜎 𝑇 𝐶 = (𝑆= − 𝑃𝑉(𝐷))𝑁(𝑑" ) − 𝑋𝑒 D16 𝑁(𝑑7 ) 𝑃 = 𝑋𝑒 D16 𝑁(−𝑑7 ) − (𝑆= − 𝑃𝑉 𝐷 )𝑁(−𝑑" ) 𝑃 = 𝐶 − 𝑆= − 𝑃𝑉 𝐷 + 𝑋𝑒 D16
4 6
Lecture example 3: • • • • • •
WPL share price= $10.50 Standard deviation= 30% p.a. Risk free rate= 5% p.a. Call strike= $10 Time to expiry= 6 months A dividend of $0.80 is expected in 2 months
𝑃𝑉 𝐷 = 0.8𝑒 D=.=I × 7/"7 = 0.793 •
Thus, 𝑆= − 𝑃𝑉 𝐷 = $10.50 − $0.793 = $9.707
Pricing European stock index options Some analysts prefer to use dividend yield instead of known dividend. Also, if the underlying asset is a stock index, then we definitely have to calculate dividend yield rather than known dividend •
𝑑" =
• • • •
𝑑7 𝐶R 𝑃R 𝑃R
%$&' ) .
2 3
0 1DQ 0 4 3 × 6 4 6
= 𝑑" − 𝜎 𝑇 = 𝑆= 𝑒 DQ6 𝑁(𝑑" ) − 𝑋𝑒 D16 𝑁(𝑑7 ) = 𝑋𝑒 D16 𝑁(−𝑑7 ) − 𝑆= 𝑒 DQ6 𝑁(−𝑑" ) = 𝐶R − 𝑆= 𝑒 DQ6 + 𝑋𝑒 D16
Lecture example 4: A European call is written on a market index with T=6 months to maturity • • • • •
Current market value of the index= 900 Strike price= 900 Risk free rate= 5% p.a. Volatility of the index= 20% p.a. Dividend yield= 0.2% in 1 month and 0.3% in 2 months
Thus, average dividend yield: •
=.==70=.==S
• •
𝐴𝑛𝑛𝑢𝑎𝑙 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑞 = 0.25 × 12 = 3% 𝑝. 𝑎. Call price= $54.27; Put price= $45.44
7
= 0.25% 𝑝𝑒𝑟 𝑚𝑜𝑛𝑡ℎ
Pricing European currency options
1
•
𝑑" =
• • • •
𝑑7 𝐶R 𝑃R 𝑃R
%$&' ) .
2
0 1D1d 0 4 3 × 6 3 4 6
= 𝑑" − 𝜎 𝑇 = 𝑆= 𝑒 D1d 6 𝑁(𝑑" ) − 𝑋𝑒 D16 𝑁(𝑑7 ) = 𝑋𝑒 D16 𝑁(−𝑑7 ) − 𝑆= 𝑒 D1d 6 𝑁(−𝑑" ) = 𝐶R − 𝑆= 𝑒 D1d 6 + 𝑋𝑒 D16
Options on stock: Stock options
What this means
Effect
Long a call on stock
You have the right to buy the stock
You receive stock and pay in $
Long a put on stock
You have the right to sell the stock
You sell the stock and thus receive $
Options on currency A vs currency B: Currency options
What this means
Effect
Long a call on currency A
You have the right to buy currency A
You receive currency A and pay in currency B
Long a put on currency B
You have the right to sell currency B
You sell currency B and thus receive currency A
•
Long a call on currency A= Long a put on currency B
Lecture example 5: A European call on AUD is created with T=6 months to expiry • • • • •
Current spot exchange rate= AUD$1.2/USD Strike price= AUD$1.3/USD Australian risk free rate= 5% p.a. American risk free rate= 7% p.a. Volatility= 20% p.a.
What is the price to long a USD call? •
𝑑" =
%$2.3) 2.e
2 3
0 =.=ID=.=f 0 =.73 × =.7
g 23
g 23
i
= −0.56599N d1= 0.2843
•
𝑑7 = 0.56599 − 0.2
•
𝐶R = 1.2𝑒 D=.=f×i/"7 ×0.2843 − 1.3𝑒 D=.=I×23 ×0.2389 = $0.027
"7
= −70740.2389 g
What is the price to long a USD put? •
𝑃R = 0.027 − 1.2𝑒 D=.=f×=.I + 1.3𝑒 D=.=I×=.I = $0.136
What is the price to long an AUD call? (= long USD put) "
•
𝑆= =
•
𝑋=
•
𝑑" =
".7 "
= 𝑈𝑆𝐷0.8333 = 𝑈𝑆𝐷0.7692
".S '.leee 2 g $% 0 =.=fD=.=I 0 =.73 × '.mgn3
3
=.7
g 23
23
= 0.7074 0.7611
2
i
•
𝑑7 = 0.7074 − 0.2
• •
𝐶R = 0.8333𝑒 D=.=I×i/"7 ×0.7611 − 0.7692𝑒 D=.=f×23 ×0.7157 = $0.08698 Garman-Kohlhagen’s model: 𝐶R × 𝑆= × 𝑋 = 0.08698 × 1.2 × 1.3 = 𝐴𝑈𝐷0.136
"7
= 0.56599 0.7157 g
4. Hedging with option contracts Hedging stock market risk An alternative way of hedging the risk that prices will move adversely is with options. In this case, there is an upfront cost of hedging; an option premium is paid at the time of entering the hedge in order to insure against bad outcomes. However, the upside potential (should prices happen to move favourably) is retained. • •
SPI200 futures uses $25 multiplier ASX200 index options uses $10 multiplier
Lecture example 6: Assume that today is November and the ASX200 is 5430. You hold a portfolio that tracks the ASX200; current market value of your portfolio is $100 million. Concerned about a market correction sometime during December, the manager wishes to insure the portfolio against a crash. This can be done by using SPI200 futures. The manager, however, is unwilling to give up potential increases in the market over December à by using options, we are not exposed to a price fall and we can also gain if price rises. Hedging strategy: • •
Identify the exposure à what movement in the underlying asset will hurt us? Establish an option strategy which makes money if this adverse movement occurs
To hedge, we will want to protect our portfolio value against a market decline. Both ASX200 index call and out option (expiring in Dec) have strike price= 5450 Do we use calls or puts? •
Long put option à will make money if stock price falls
How many put options will we buy? •
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑜𝑛𝑒 𝑝𝑢𝑡 𝑜𝑝𝑡𝑖𝑜𝑛 = 5450 × $10 = 54 500
•
𝑁𝑜. 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑠 =
"== === === Ir I==
= 1835 𝑝𝑢𝑡 𝑜𝑝𝑡𝑖𝑜𝑛𝑠
Assume ASX200 falls to 4600 in Dec 2010: ri==
•
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑣𝑎𝑙𝑢𝑒 =
•
Puts are in the money (ST < X), so exercise puts and receive payoff of: 𝑃𝑎𝑦𝑜𝑓𝑓 = 1835 × 10 × 5450 − 4600 = $15.598 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 𝑇𝑜𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 = 84.715 + 15.598 = $100.313 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 ~ 100 𝑚𝑖𝑙𝑙𝑖𝑜𝑛
•
IrS=
× 100 000 000 = $84.715 𝑚𝑖𝑙𝑙𝑖𝑜𝑛
Assume ASX200 rises to 5900 in Dec 2010: Iu==
•
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑣𝑎𝑙𝑢𝑒 =
• • •
Puts are out of the money (ST > X), so don’t exercise puts 𝑇𝑜𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 = $108.66 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 > $100 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 Compare this to Lecture 1 where we gain nothing on SPI200 futures when ASX200 rises to 5900 (net value was around 100 million)
IrS=
× 100 000 000 = $108.66 𝑚𝑖𝑙𝑙𝑖𝑜𝑛
Hedging foreign exchange risk 3
If a company was prepared to pay an option premium up front, it could rule out any adverse movements in the exchange rate while retaining the ability to capture any favourable movements. Lecture example 7: Your company makes a sale of machines to a UK customer. The sale price is GBP10 million payable at the end of the year and you are worried that the GBP will move unfavourably between now and the end of the year. How can you insure against any unfavourable moves but capture the benefits of favourable moves in the exchange rate? • •
•
Use options Let’s suppose that you want to guarantee that the exchange rate will be AUD1.00= GBP0.4050, or better. You could do this by purchasing a put option on the GBP with exercise price of GBP0.4050/AUD This means we will get at least 1/GBP0.4050= AUD2.4691/GBP for each of the GBP10 million we receive
Assume exchange rate is GBP0.4400/AUD in Dec (i.e. GBP weakens against the AUD): • • • •
Each GBP is worth only 1/0.440= AUD2.2727 If we receive GBP 10 million and convert it at the spot rate à we receive AUD22 727 000 Under put option we have the right to sell GBP 10 million at the exchange rate of GBP0.4050/AUD We would exercise the option and receive AUD24 691 000
Assume exchange rate is GBP0.3850/AUD in Dec (i.e. GBP strengthens against AUD) • • • •
Each GBP is worth 1/0.3850= AUD 2.5974 In this case, the company will not exercise the option to sell The GBP 10 million received can be converted at the spot rate, yielding AUD25 974 000 If we had used a forward contract, we would not have benefited from the strengthening GBP
Hedging interest rate risk It is possible for a firm to protect itself against adverse movements in interest rates and to benefit from favourable movements via interest-rate options. Note that for a December call option on a December BAB futures contract, both the call and the futures mature at the same time. Also recall that the futures price converges to the price of the underlying asset at maturity (or else it is trivial to execute an arbitrage). At maturity, therefore, the futures option is identical to an option directly on the underlying bank bill (because the futures price and the underlying bank bill price are the same). Lecture example 8: Your accountants have done a cash flow analysis and have determined that you will have a cash shortage of (roughly) $1 million in the first quarter of next year. To finance this cash shortage, the firm will issue ten 90-day bank bills each with a face value of $100 000; this will raise just under $1 million. 90-days later you will have to repay the face value of the bill ($1 million) The difference between monies raised and the $1 million repaid is interest. You are concerned that interest rates will rise between now and the end of the year, hence increasing the interest cost of this money. Interest rates are currently around 8%. How can the firm protect itself from increases in the interest rate but at the same time capture the benefits of any decrease in interest rates? •
Enter into a put futures option contract, under which you sell a 90-day bank bill at a prearranged price only when it is in your interest to do so
Consider buying a put option struck at $92. If interest rates have risen to 10% p.a. by Dec: 4
• • •
When you sell the 90-day bank bills you receive: 𝐹𝑢𝑛𝑑𝑠 𝑟𝑎𝑖𝑠𝑒𝑑 =
"== === "0
n' egx
×=."=
× 10 𝑏𝑖𝑙𝑙𝑠 =
$975 935.83 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑐𝑜𝑠𝑡 = 1 000 000 − 975 935.83 = $24 064.17 Your firm would elect to exercise the put option to sell a bank bill at a price implied by a 100 – 92= 8% interest rate. The gain on this put option is the difference between the price you can get by exercising the option and the price you could get on the open market " === ===
" === ===
•
𝑃𝑢𝑡 𝑜𝑝𝑡𝑖𝑜𝑛 𝑝𝑎𝑦𝑜𝑓𝑓 =
•
The net cost is the interest payable minus the gain on the option: 𝑁𝑒𝑡 𝑐𝑜𝑠𝑡 = 24 064.17 − 4719.73 = $19 344.44. This is equivalent to interest payable at 8% p.a.
•
𝐼𝑚𝑝𝑙𝑖𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 1 000 000 −
"0
n' egx
×=.={
−
"0
n' egx
×=."=
" === === "0
n' egx
×=.={
= $4719.73
= $19 344.44
If interest rates have fallen to 6% p.a. by Dec: • • • • • •
When you sell the 90-day bank bills you receive: 𝐹𝑢𝑛𝑑𝑠 𝑟𝑎𝑖𝑠𝑒𝑑 =
"== === "0
n' egx
×=.=i
× 10 𝑏𝑖𝑙𝑙𝑠 =
$985 421.17 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑐𝑜𝑠𝑡 = 1 000 000 − 985 421.17 = $14 578.83 Your firm would elect not to exercise the put option to sell a bank bill for $980 655.56 because the market price is $985 421.17. Therefore, the put option expires with zero value 𝑁𝑒𝑡 𝑐𝑜𝑠𝑡 = $14 578.83 The option hedge has placed a cap of $19 344.44 (interest at 8% p.a.) without sacrificing the potential benefits if the interest rate should happen to fall This insurance is not free and the firm will have to pay an option premium up front
Drawing payoff and profit diagrams for combined options Lecture example 9: Consider the following strategy: • •
Long one call with X=$10, call price is $0.50 Short one call with X=$12, call price is $0.40
Draw the payoff and profit diagrams: Solution: Start by completing the payoff table and use its outcome to guide T=0
ST < 10
10 < ST < 12
ST > 12
Long a call (X=10)
(0.5)
0
ST – 10
ST -10
Short a call (X=12)
0.4
0
0
-(ST – 12)
(0.1)
0
ST – 10
2
5
Lecture example 10: Consider the following strategy: • •
Own one share Short one call with X=10
Draw the payoff diagram for the above strategy ST < 10
ST > 10
Own one share
ST
ST
Short a call (X=10)
0
-(ST – 10)
ST
10
Lecture 5 Part 1 (Week 6): Pricing Options Using the Binomial Approach Binomial method: 1. 2. 3.
Replicating portfolio method Delta hedging method Risk-neutral method
1. Expectation pricing In modern financial theory, an asset’s value is equal to its expected future cash flows discounted to present value at a rate appropriate with the risk of an asset. For example, the value of a project – its NPV – is the expected after tax cash flows discounted at the relevant rate: •
𝑁𝑃𝑉 =
Š R|}~•€~• •‚ƒ„ …†‡ˆ ‰Š ‹~‚1 € €•" "01 Œ
− 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑜𝑢𝑡𝑙𝑎𝑦
Similarly, the current value/price of a bond is the coupon interest payments discounted at the appropriate yield as well as the discounted face value on maturity: 6
•
𝐵𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒 =
Š •‡•}‡Š ‰Š€~1~ƒ€ ‚€ €‰‘~ € ‰•" "0‹‰~†• Œ
+
’‚•~ “‚†•~ "0‹‰~†• ”
Similarly, under the dividend theory of value, the current share price is the expected future dividends of the company discounted at the appropriate return on equity: •
𝑃= =
• R|}~•€~• •‰“‰•~Š• ‚€ €‰‘~ € €•" "01 Œ
The above methodology is known as expectation pricing. There are three key elements to expectation pricing: 1. The timing of future payoffs 2. A magnitude of the expected payoffs, and 3. A discount rate appropriate with the risk of the asset A derivative security is just another type of asset, so we should be able to value derivatives using expectation pricing.
2. Replicating a derivative Derivative assets are redundant in the sense that we can recreate the payoffs to some derivatives using clever strategies involving the underlying asset and the bank account. This ability to replicate gives us a way of valuing that derivative. E.g. put-call parity.
3. The binomial model for stock-price movements The value of a derivative security, by definition, derives from the value of the underlying asset. We may know the share price today, but if the option has (say) one year to expiry, we need to estimate the likely share price one year forward. Thus, to value an option, we need a model which describes the evolution of stock price through time. Two features of stock prices we need to incorporate are: 1. Stock prices can never go negative 2. The distribution of possible future prices is lognormal
7
Corporate finance: In theory, share price will drop by the amount of the dividend after the dividend has been paid Thus, the BS formula will still work in this case, as long as we reduce the current stock price by the PV of the dividend Only dividends that are expected within the maturity of the option are considered
%$&' ()*(,) .
2 3
0 10 4 3 × 6
•
𝑑" =
• • • •
𝑑7 = 𝑑" − 𝜎 𝑇 𝐶 = (𝑆= − 𝑃𝑉(𝐷))𝑁(𝑑" ) − 𝑋𝑒 D16 𝑁(𝑑7 ) 𝑃 = 𝑋𝑒 D16 𝑁(−𝑑7 ) − (𝑆= − 𝑃𝑉 𝐷 )𝑁(−𝑑" ) 𝑃 = 𝐶 − 𝑆= − 𝑃𝑉 𝐷 + 𝑋𝑒 D16
4 6
Lecture example 3: • • • • • •
WPL share price= $10.50 Standard deviation= 30% p.a. Risk free rate= 5% p.a. Call strike= $10 Time to expiry= 6 months A dividend of $0.80 is expected in 2 months
𝑃𝑉 𝐷 = 0.8𝑒 D=.=I × 7/"7 = 0.793 •
Thus, 𝑆= − 𝑃𝑉 𝐷 = $10.50 − $0.793 = $9.707
Pricing European stock index options Some analysts prefer to use dividend yield instead of known dividend. Also, if the underlying asset is a stock index, then we definitely have to calculate dividend yield rather than known dividend •
𝑑" =
• • • •
𝑑7 𝐶R 𝑃R 𝑃R
%$&' ) .
2 3
0 1DQ 0 4 3 × 6 4 6
= 𝑑" − 𝜎 𝑇 = 𝑆= 𝑒 DQ6 𝑁(𝑑" ) − 𝑋𝑒 D16 𝑁(𝑑7 ) = 𝑋𝑒 D16 𝑁(−𝑑7 ) − 𝑆= 𝑒 DQ6 𝑁(−𝑑" ) = 𝐶R − 𝑆= 𝑒 DQ6 + 𝑋𝑒 D16
Lecture example 4: A European call is written on a market index with T=6 months to maturity • • • • •
Current market value of the index= 900 Strike price= 900 Risk free rate= 5% p.a. Volatility of the index= 20% p.a. Dividend yield= 0.2% in 1 month and 0.3% in 2 months
Thus, average dividend yield: •
=.==70=.==S
• •
𝐴𝑛𝑛𝑢𝑎𝑙 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑞 = 0.25 × 12 = 3% 𝑝. 𝑎. Call price= $54.27; Put price= $45.44
7
= 0.25% 𝑝𝑒𝑟 𝑚𝑜𝑛𝑡ℎ
Pricing European currency options
1
•
𝑑" =
• • • •
𝑑7 𝐶R 𝑃R 𝑃R
%$&' ) .
2
0 1D1d 0 4 3 × 6 3 4 6
= 𝑑" − 𝜎 𝑇 = 𝑆= 𝑒 D1d 6 𝑁(𝑑" ) − 𝑋𝑒 D16 𝑁(𝑑7 ) = 𝑋𝑒 D16 𝑁(−𝑑7 ) − 𝑆= 𝑒 D1d 6 𝑁(−𝑑" ) = 𝐶R − 𝑆= 𝑒 D1d 6 + 𝑋𝑒 D16
Options on stock: Stock options
What this means
Effect
Long a call on stock
You have the right to buy the stock
You receive stock and pay in $
Long a put on stock
You have the right to sell the stock
You sell the stock and thus receive $
Options on currency A vs currency B: Currency options
What this means
Effect
Long a call on currency A
You have the right to buy currency A
You receive currency A and pay in currency B
Long a put on currency B
You have the right to sell currency B
You sell currency B and thus receive currency A
•
Long a call on currency A= Long a put on currency B
Lecture example 5: A European call on AUD is created with T=6 months to expiry • • • • •
Current spot exchange rate= AUD$1.2/USD Strike price= AUD$1.3/USD Australian risk free rate= 5% p.a. American risk free rate= 7% p.a. Volatility= 20% p.a.
What is the price to long a USD call? •
𝑑" =
%$2.3) 2.e
2 3
0 =.=ID=.=f 0 =.73 × =.7
g 23
g 23
i
= −0.56599N d1= 0.2843
•
𝑑7 = 0.56599 − 0.2
•
𝐶R = 1.2𝑒 D=.=f×i/"7 ×0.2843 − 1.3𝑒 D=.=I×23 ×0.2389 = $0.027
"7
= −70740.2389 g
What is the price to long a USD put? •
𝑃R = 0.027 − 1.2𝑒 D=.=f×=.I + 1.3𝑒 D=.=I×=.I = $0.136
What is the price to long an AUD call? (= long USD put) "
•
𝑆= =
•
𝑋=
•
𝑑" =
".7 "
= 𝑈𝑆𝐷0.8333 = 𝑈𝑆𝐷0.7692
".S '.leee 2 g $% 0 =.=fD=.=I 0 =.73 × '.mgn3
3
=.7
g 23
23
= 0.7074 0.7611
2
i
•
𝑑7 = 0.7074 − 0.2
• •
𝐶R = 0.8333𝑒 D=.=I×i/"7 ×0.7611 − 0.7692𝑒 D=.=f×23 ×0.7157 = $0.08698 Garman-Kohlhagen’s model: 𝐶R × 𝑆= × 𝑋 = 0.08698 × 1.2 × 1.3 = 𝐴𝑈𝐷0.136
"7
= 0.56599 0.7157 g
4. Hedging with option contracts Hedging stock market risk An alternative way of hedging the risk that prices will move adversely is with options. In this case, there is an upfront cost of hedging; an option premium is paid at the time of entering the hedge in order to insure against bad outcomes. However, the upside potential (should prices happen to move favourably) is retained. • •
SPI200 futures uses $25 multiplier ASX200 index options uses $10 multiplier
Lecture example 6: Assume that today is November and the ASX200 is 5430. You hold a portfolio that tracks the ASX200; current market value of your portfolio is $100 million. Concerned about a market correction sometime during December, the manager wishes to insure the portfolio against a crash. This can be done by using SPI200 futures. The manager, however, is unwilling to give up potential increases in the market over December à by using options, we are not exposed to a price fall and we can also gain if price rises. Hedging strategy: • •
Identify the exposure à what movement in the underlying asset will hurt us? Establish an option strategy which makes money if this adverse movement occurs
To hedge, we will want to protect our portfolio value against a market decline. Both ASX200 index call and out option (expiring in Dec) have strike price= 5450 Do we use calls or puts? •
Long put option à will make money if stock price falls
How many put options will we buy? •
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑜𝑛𝑒 𝑝𝑢𝑡 𝑜𝑝𝑡𝑖𝑜𝑛 = 5450 × $10 = 54 500
•
𝑁𝑜. 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑠 =
"== === === Ir I==
= 1835 𝑝𝑢𝑡 𝑜𝑝𝑡𝑖𝑜𝑛𝑠
Assume ASX200 falls to 4600 in Dec 2010: ri==
•
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑣𝑎𝑙𝑢𝑒 =
•
Puts are in the money (ST < X), so exercise puts and receive payoff of: 𝑃𝑎𝑦𝑜𝑓𝑓 = 1835 × 10 × 5450 − 4600 = $15.598 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 𝑇𝑜𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 = 84.715 + 15.598 = $100.313 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 ~ 100 𝑚𝑖𝑙𝑙𝑖𝑜𝑛
•
IrS=
× 100 000 000 = $84.715 𝑚𝑖𝑙𝑙𝑖𝑜𝑛
Assume ASX200 rises to 5900 in Dec 2010: Iu==
•
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑣𝑎𝑙𝑢𝑒 =
• • •
Puts are out of the money (ST > X), so don’t exercise puts 𝑇𝑜𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 = $108.66 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 > $100 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 Compare this to Lecture 1 where we gain nothing on SPI200 futures when ASX200 rises to 5900 (net value was around 100 million)
IrS=
× 100 000 000 = $108.66 𝑚𝑖𝑙𝑙𝑖𝑜𝑛
Hedging foreign exchange risk 3
If a company was prepared to pay an option premium up front, it could rule out any adverse movements in the exchange rate while retaining the ability to capture any favourable movements. Lecture example 7: Your company makes a sale of machines to a UK customer. The sale price is GBP10 million payable at the end of the year and you are worried that the GBP will move unfavourably between now and the end of the year. How can you insure against any unfavourable moves but capture the benefits of favourable moves in the exchange rate? • •
•
Use options Let’s suppose that you want to guarantee that the exchange rate will be AUD1.00= GBP0.4050, or better. You could do this by purchasing a put option on the GBP with exercise price of GBP0.4050/AUD This means we will get at least 1/GBP0.4050= AUD2.4691/GBP for each of the GBP10 million we receive
Assume exchange rate is GBP0.4400/AUD in Dec (i.e. GBP weakens against the AUD): • • • •
Each GBP is worth only 1/0.440= AUD2.2727 If we receive GBP 10 million and convert it at the spot rate à we receive AUD22 727 000 Under put option we have the right to sell GBP 10 million at the exchange rate of GBP0.4050/AUD We would exercise the option and receive AUD24 691 000
Assume exchange rate is GBP0.3850/AUD in Dec (i.e. GBP strengthens against AUD) • • • •
Each GBP is worth 1/0.3850= AUD 2.5974 In this case, the company will not exercise the option to sell The GBP 10 million received can be converted at the spot rate, yielding AUD25 974 000 If we had used a forward contract, we would not have benefited from the strengthening GBP
Hedging interest rate risk It is possible for a firm to protect itself against adverse movements in interest rates and to benefit from favourable movements via interest-rate options. Note that for a December call option on a December BAB futures contract, both the call and the futures mature at the same time. Also recall that the futures price converges to the price of the underlying asset at maturity (or else it is trivial to execute an arbitrage). At maturity, therefore, the futures option is identical to an option directly on the underlying bank bill (because the futures price and the underlying bank bill price are the same). Lecture example 8: Your accountants have done a cash flow analysis and have determined that you will have a cash shortage of (roughly) $1 million in the first quarter of next year. To finance this cash shortage, the firm will issue ten 90-day bank bills each with a face value of $100 000; this will raise just under $1 million. 90-days later you will have to repay the face value of the bill ($1 million) The difference between monies raised and the $1 million repaid is interest. You are concerned that interest rates will rise between now and the end of the year, hence increasing the interest cost of this money. Interest rates are currently around 8%. How can the firm protect itself from increases in the interest rate but at the same time capture the benefits of any decrease in interest rates? •
Enter into a put futures option contract, under which you sell a 90-day bank bill at a prearranged price only when it is in your interest to do so
Consider buying a put option struck at $92. If interest rates have risen to 10% p.a. by Dec: 4
• • •
When you sell the 90-day bank bills you receive: 𝐹𝑢𝑛𝑑𝑠 𝑟𝑎𝑖𝑠𝑒𝑑 =
"== === "0
n' egx
×=."=
× 10 𝑏𝑖𝑙𝑙𝑠 =
$975 935.83 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑐𝑜𝑠𝑡 = 1 000 000 − 975 935.83 = $24 064.17 Your firm would elect to exercise the put option to sell a bank bill at a price implied by a 100 – 92= 8% interest rate. The gain on this put option is the difference between the price you can get by exercising the option and the price you could get on the open market " === ===
" === ===
•
𝑃𝑢𝑡 𝑜𝑝𝑡𝑖𝑜𝑛 𝑝𝑎𝑦𝑜𝑓𝑓 =
•
The net cost is the interest payable minus the gain on the option: 𝑁𝑒𝑡 𝑐𝑜𝑠𝑡 = 24 064.17 − 4719.73 = $19 344.44. This is equivalent to interest payable at 8% p.a.
•
𝐼𝑚𝑝𝑙𝑖𝑒𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 1 000 000 −
"0
n' egx
×=.={
−
"0
n' egx
×=."=
" === === "0
n' egx
×=.={
= $4719.73
= $19 344.44
If interest rates have fallen to 6% p.a. by Dec: • • • • • •
When you sell the 90-day bank bills you receive: 𝐹𝑢𝑛𝑑𝑠 𝑟𝑎𝑖𝑠𝑒𝑑 =
"== === "0
n' egx
×=.=i
× 10 𝑏𝑖𝑙𝑙𝑠 =
$985 421.17 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑐𝑜𝑠𝑡 = 1 000 000 − 985 421.17 = $14 578.83 Your firm would elect not to exercise the put option to sell a bank bill for $980 655.56 because the market price is $985 421.17. Therefore, the put option expires with zero value 𝑁𝑒𝑡 𝑐𝑜𝑠𝑡 = $14 578.83 The option hedge has placed a cap of $19 344.44 (interest at 8% p.a.) without sacrificing the potential benefits if the interest rate should happen to fall This insurance is not free and the firm will have to pay an option premium up front
Drawing payoff and profit diagrams for combined options Lecture example 9: Consider the following strategy: • •
Long one call with X=$10, call price is $0.50 Short one call with X=$12, call price is $0.40
Draw the payoff and profit diagrams: Solution: Start by completing the payoff table and use its outcome to guide T=0
ST < 10
10 < ST < 12
ST > 12
Long a call (X=10)
(0.5)
0
ST – 10
ST -10
Short a call (X=12)
0.4
0
0
-(ST – 12)
(0.1)
0
ST – 10
2
5
Lecture example 10: Consider the following strategy: • •
Own one share Short one call with X=10
Draw the payoff diagram for the above strategy ST < 10
ST > 10
Own one share
ST
ST
Short a call (X=10)
0
-(ST – 10)
ST
10
Lecture 5 Part 1 (Week 6): Pricing Options Using the Binomial Approach Binomial method: 1. 2. 3.
Replicating portfolio method Delta hedging method Risk-neutral method
1. Expectation pricing In modern financial theory, an asset’s value is equal to its expected future cash flows discounted to present value at a rate appropriate with the risk of an asset. For example, the value of a project – its NPV – is the expected after tax cash flows discounted at the relevant rate: •
𝑁𝑃𝑉 =
Š R|}~•€~• •‚ƒ„ …†‡ˆ ‰Š ‹~‚1 € €•" "01 Œ
− 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑜𝑢𝑡𝑙𝑎𝑦
Similarly, the current value/price of a bond is the coupon interest payments discounted at the appropriate yield as well as the discounted face value on maturity: 6
•
𝐵𝑜𝑛𝑑 𝑝𝑟𝑖𝑐𝑒 =
Š •‡•}‡Š ‰Š€~1~ƒ€ ‚€ €‰‘~ € ‰•" "0‹‰~†• Œ
+
’‚•~ “‚†•~ "0‹‰~†• ”
Similarly, under the dividend theory of value, the current share price is the expected future dividends of the company discounted at the appropriate return on equity: •
𝑃= =
• R|}~•€~• •‰“‰•~Š• ‚€ €‰‘~ € €•" "01 Œ
The above methodology is known as expectation pricing. There are three key elements to expectation pricing: 1. The timing of future payoffs 2. A magnitude of the expected payoffs, and 3. A discount rate appropriate with the risk of the asset A derivative security is just another type of asset, so we should be able to value derivatives using expectation pricing.
2. Replicating a derivative Derivative assets are redundant in the sense that we can recreate the payoffs to some derivatives using clever strategies involving the underlying asset and the bank account. This ability to replicate gives us a way of valuing that derivative. E.g. put-call parity.
3. The binomial model for stock-price movements The value of a derivative security, by definition, derives from the value of the underlying asset. We may know the share price today, but if the option has (say) one year to expiry, we need to estimate the likely share price one year forward. Thus, to value an option, we need a model which describes the evolution of stock price through time. Two features of stock prices we need to incorporate are: 1. Stock prices can never go negative 2. The distribution of possible future prices is lognormal
7
