Current call option value. S o. = Current stock price. N(d) = probability that a random draw from a normal distribution
Options dan Futures DWITYAPOETRA S. BESAR MATA KULIAH : ANALISIS INVESTASI DAN MANAJEMEN RISIKO
2013 PROG STUDI MAGISTER MANAJEMEN UNIVERSITAS TRISAKTI
Bahan Bodie, Kane, and Marcus (BKM), 2009, Investments, 8th (global) edition, McGraw-Hill / Irwin. Kuliah sesi ini: BKM, Bab 20, 21 dan 22 Soal:
Outline Bagian 1: Pasar Opsi
Bagian 2: Valuasi Opsi Bagian 3: Pasar Futures
Bagian 1: Pasar Opsi
Terminologi Opsi
• • • • •
Buy - Long Sell - Short Call Put Key Elements – Exercise atau Strike Price – Premium atau Price – Maturity atau Expiration
Pasar Opsi
Hubungan harga pasar dan eksekusi In the Money - exercise of the option would be profitable Call: market price>exercise price Put: exercise price>market price Out of the Money - exercise of the option would not be profitable Call: market priceX 0 if ST < X Profit to Call Writer Payoff + Premium
Opsi
Grafik Payoff dan Profit to Call Option saat Expiration
Opsi
Grafik Payoff dan Profit to Call Writers saat Expiration
Opsi - Puts
Payoffs and Profits at Expiration - Puts Payoffs to Put Holder 0 if ST > X (X - ST) if ST < X
Profit to Put Holder Payoff - Premium
Opsi
Payoffs and Profits at Expiration – Puts Continued
Payoffs to Put Writer 0 if ST > X -(X - ST) if ST < X
Profits to Put Writer Payoff + Premium
Opsi
Strategi Opsi Straddle (Same Exercise Price) Long Call and Long Put Spreads - A combination of two or more call options or put options on the same asset with differing exercise prices or times to expiration. Vertical or money spread: Same maturity Different exercise price Horizontal or time spread: Different maturity dates
Opsi Table 20.3 Value of a Straddle Position at Option
Expiration
Opsi
Put Call Parity If the prices are not equal arbitrage will be possible X C S0 P T (1 rf )
Opsi
Contoh Put Call Parity - Disequilibrium Stock Price = 110 Call Price = 17 Put Price = 5 Risk Free = 5% Maturity = 1 yr X =X 105 C S0 P T (1 rf ) 117 > 115 Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternative
Opsi
Tabel Strategi Arbitrasi
Opsi
Sekuritas yang mirip Opsi • • • •
Callable Bonds Convertible Securities Warrants Collateralized Loans
Opsi
Exotic Options • • • • •
Asian Options Barrier Options Lookback Options Currency Translated Options Digital Options
Outline Bagian 2: Valuasi Opsi
Valuasi Opsi
• Nilai intrinsic (Intrinsic value) - profit that could be made if the option was immediately exercised – Call: stock price - exercise price – Put: exercise price - stock price • Nilai waktu - the difference between the option price and the intrinsic value
Valuasi Opsi
Grafik Call Option Value sebelum Expiration
Valuasi Opsi
Tabel Faktor Determinan Nilai Call Option
Valuasi Opsi
Restriksi pada nilai opsi : Call • Value cannot be negative • Value cannot exceed the stock value • Value of the call must be greater than the value of levered equity C > S0 - ( X + D ) / ( 1 + rf )T
C > S0 - PV ( X ) - PV ( D )
Valuasi Opsi
Grafik Range of Possible Call Option Values
Valuasi Opsi
Grafik Call Option Value as a Function of the Current Stock Price
Valuasi Opsi
Contoh: Binomial Option Pricing 120
100
10
C 90
Stock Price
0 Call Option Value X = 110
Valuasi Opsi
Binomial Option Pricing: Alternative Portfolio Buy 1 share of stock at $100 Borrow $81.82 (10% Rate) 18.18 Net outlay $18.18 Payoff Value of Stock 90 120 Repay loan - 90 - 90 Net Payoff 0 30
30
0 Payoff Structure is exactly 3 times the Call
Valuasi Opsi
Binomial Option Pricing: 30
30 18.18
C 0
3C = $18.18 C = $6.06
0
Valuasi Opsi
Expanding to Consider Three Intervals: • Assume that we can break the year into three intervals • For each interval the stock could increase by 5% or decrease by 3% • Assume the stock is initially selling at 100
Valuasi Opsi
Expanding to Consider Three Intervals Continued S+++ S++ S++-
S+
S+-
S
S+-SS-S---
Valuasi Opsi
Possible Outcomes with Three Intervals Event
Probability
Final Stock Price
3 up
1/8
100 (1.05)3
=115.76
2 up 1 down
3/8
100 (1.05)2 (.97)
=106.94
1 up 2 down
3/8
100 (1.05) (.97)2
= 98.79
3 down
1/8
100 (.97)3
= 91.27
Valuasi Opsi
Valuasi dengan model Black-Scholes Co = SoN(d1) - Xe-rTN(d2) d1 = [ln(So/X) + (r + 2/2)T] / (T1/2) d2 = d1 + (T1/2) where Co = Current call option value So = Current stock price N(d) = probability that a random draw from a normal distribution will be less than d
Valuasi Opsi
Black-Scholes Option Valuation X = Exercise price e = 2.71828, the base of the natural log r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option) T = time to maturity of the option in years ln = Natural log function Standard deviation of annualized cont. compounded rate of return on the stock
Valuasi Opsi
Grafik Kurva Distribusi Normal
Valuasi Opsi
Contoh Call Option: So = 100 X = 95 r = .10 T = .25 (quarter) = .50 d1 = [ln(100/95) + (.10+(5 2/2))] / (5.251/2) = .43 d2 = .43 + ((5.251/2) = .18
Valuasi Opsi
Probabilities dari Distribusi Normal N (.43) = .6664 Table 21.2 d N(d) .42 .6628 .43 .6664 Interpolation .44 .6700
Valuasi Opsi
Probabilities from Normal Distribution Continued
N (.18) = .5714 Table 21.2 d N(d) .16 .5636 .18 .5714 .20 .5793
Valuasi Opsi
Tabel Cumulative Normal Distribution
Valuasi Opsi
Nilai Opsi Call Co = SoN(d1) - Xe-rTN(d2) Co = 100 X .6664 - 95 e- .10 X .25 X .5714 Co = 13.70 Implied Volatility Using Black-Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock?
Valuasi Opsi
Contoh Spreadsheet utk menghitung nilai opsi Black-Scholes
Valuasi Opsi
Menggunakan Goal Seek utk mendapatkan Implied Volatility
Valuasi Opsi
Grafik Implied Volatility of the S&P 500 (VIX Index)
Valuasi Opsi
Black-Scholes Model dengan Dividend • The call option formula applies to stocks that pay dividends • One approach is to replace the stock price with a dividend adjusted stock price Replace S0 with S0 - PV (Dividends)
Valuasi Opsi
Put Value Using Black-Scholes P = Xe-rT [1-N(d2)] - S0 [1-N(d1)] Using the sample call data S = 100 r = .10 X = 95 g = .5 T = .25 95e-10x.25(1-.5714)-100(1-.6664) = 6.35
Valuasi Opsi
Put Option Valuation: Using Put-Call Parity P = C + PV (X) - So = C + Xe-rT - So Using the example data C = 13.70 X = 95 S = 100 r = .10 T = .25 P = 13.70 + 95 e -.10 X .25 - 100 P = 6.35
Valuasi Opsi
Using the Black-Scholes Formula Hedging: Hedge ratio or delta The number of stocks required to hedge against the price risk of holding one option Call = N (d1) Put = N (d1) - 1 Option Elasticity Percentage change in the option’s value given a 1% change in the value of the underlying stock
Valuasi Opsi
Portfolio Insurance • Buying Puts - results in downside protection with unlimited upside potential • Limitations – Tracking errors if indexes are used for the puts – Maturity of puts may be too short – Hedge ratios or deltas change as stock values change
Valuasi Opsi
Hedging On Mispriced Options Option value is positively related to volatility: • If an investor believes that the volatility that is implied in an option’s price is too low, a profitable trade is possible • Profit must be hedged against a decline in the value of the stock • Performance depends on option price relative to the implied volatility
Valuasi Opsi
Hedging dan Delta The appropriate hedge will depend on the delta Recall the delta is the change in the value of the option relative to the change in the value of the stock Delta =
Change in the value of the option Change of the value of the stock
Outline Bagian 3: Pasar Futures
3. Pasar Futures
Futures dan Forwards • Forward - an agreement calling for a future delivery of an asset at an agreed-upon price • Futures - similar to forward but feature formalized and standardized characteristics • Key difference in futures – Secondary trading - liquidity – Marked to market – Standardized contract units – Clearinghouse warrants performance
3. Pasar Futures
Key Terms untuk Futures Contracts • • • •
Futures price - agreed-upon price at maturity Long position - agree to purchase Short position - agree to sell Profits on positions at maturity Long = spot minus original futures price Short = original futures price minus spot
3. Pasar Futures
Tabel Futures Listing
3. Pasar Futures
Grafik Profits to Buyers and Sellers of Futures and Option Contracts
3. Pasar Futures
Tabel Contoh Kontrak Future
3. Pasar Futures
Trading Mechanics • Clearinghouse - acts as a party to all buyers and sellers – Obligated to deliver or supply delivery
• Closing out positions – Reversing the trade – Take or make delivery – Most trades are reversed and do not involve actual delivery
• Open Interest
3. Pasar Futures
Skema Panel A, Trading without a Clearinghouse. Panel B, Trading with a Clearinghouse
3. Pasar Futures
Margin and Trading Arrangements Initial Margin - funds deposited to provide capital to absorb losses Marking to Market - each day the profits or losses from the new futures price are reflected in the account Maintenance or variation margin - an established value below which a trader’s margin may not fall
3. Pasar Futures
Margin and Trading Arrangements Continued Margin call - when the maintenance margin is reached, broker will ask for additional margin funds Convergence of Price - as maturity approaches the spot and futures price converge Delivery - Actual commodity of a certain grade with a delivery location or for some contracts cash settlement Cash Settlement – some contracts are settled in cash rather than delivery of the underlying assets
3. Pasar Futures
Strategi Perdagangan • Speculation – short - believe price will fall – long - believe price will rise
• Hedging – long hedge - protecting against a rise in price – short hedge - protecting against a fall in price
3. Pasar Futures
Basis dan Basis Risk • Basis - the difference between the futures price and the spot price – over time the basis will likely change and will eventually converge • Basis Risk - the variability in the basis that will affect profits and/or hedging performance
3.3.Pasar PasarFutures Futures Grafik Hedging Revenues Using Futures (Futures Price = $97.15)
3. Pasar Futures
Harga Futures Spot-futures parity theorem - two ways to acquire an asset for some date in the future • Purchase it now and store it • Take a long position in futures • These two strategies must have the same market determined costs
3. Pasar Futures
Spot-Futures Parity Theorem • With a perfect hedge the futures payoff is certain -- there is no risk • A perfect hedge should return the riskless rate of return • This relationship can be used to develop futures pricing relationship
3. Pasar Futures
Contoh Hedge : • Investor owns an S&P 500 fund that has a current value equal to the index of $1,500 • Assume dividends of $25 will be paid on the index at the end of the year • Assume futures contract that calls for delivery in one year is available for $1,550 • Assume the investor hedges by selling or shorting one contract
3. Pasar Futures
Contoh Hedge Value of ST
1,510
1,550
1,610
(1,550 - ST)
40
0
-60
Dividend Income
25
25
25
1,575
1,575
Payoff on Short
Total
1,575
3. Pasar Futures
Rate of Return for the Hedge ( F0 D) S 0 S0 (1,550 25) 1,500 5% 1,500
3. Pasar Futures
General Spot-Futures Parity ( F0 D ) S 0 rf S0 Rearranging terms
F0 S 0 (1 rf ) D S 0 (1 rf d ) dD
S0
3. Pasar Futures
Kemungkinan melakukan Arbitrasi • If spot-futures parity is not observed, then arbitrage is possible • If the futures price is too high, short the futures and acquire the stock by borrowing the money at the risk free rate • If the futures price is too low, go long futures, short the stock and invest the proceeds at the risk free rate
3. Pasar Futures
Spread Pricing: Parity for Spreads T1 (1 r d ) F (T1 ) S0 f T2 (1 r d ) F (T2 ) S0 f
F (T2 ) F (T1 )(1 rf d )
(T 2 T 1)
3. Pasar Futures
Teori Harga Futures • • • •
Expectations Normal Backwardation Contango Modern Portfolio Theory
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