This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Volatility Trading Concept Volatility Trading Concept and Application Option Trading Strategies What Is Volatility? Index Index Volatility and Option Pricing DeltaNeutral Strategy Case Study 1 Case Study 2 Risks in Volatility Trading Application of Option Volatility Trading
–Option Risk Management –Equity Derivative Structured Product Implied Standard Deviations Implied Standard Deviations Of the six input factors for the BlackScholesMerton stock option pricing model, only the stock price volatility is not directly observable. A stock price volatility estimated from an option price is called an implied standard deviation (ISD) or implied volatility (IVOL). Calculating an implied volatility requires: All other input factors, and Either a call or put option price
3 Implied Standard Deviations, Implied Standard Deviations, Cont. Sigma can be found by trial and error, or by using the following formula. This simple formula yields accurate implied volatility values as long as the stock price is not too far from the strike price of the option contract.
2 2π T ( Y − X) 2 Y−X Y−X C − σ≈ + C − − Y+X 2 2 π Y = Se − yT X = Ke rT
4 Example, Calculating an ISD Example, Calculating an ISD
Stock Price: Strike Price: Time (in years): Riskless Rate (%): Dividend Yield (%): Volatility (%): 50.00 45.00 0.2500 6.00 2.00 39.25 Discounted Stock, Y: Discounted Strike, X: 49.75 44.33 Estimated Volatility (Sigma, in %): 38.89 Percent Error: 0.92% d(1): N(d1): d(2): N(d2): 0.68595 0.75363 0.48970 0.68783
T ( 2Π/T)/Y+X C (YX)/2 (C(YX)/2)^2 ((YX)^2)/ π < 1%, not bad! 0.05 4.29 18.40 9.35 BSM Call Price: $7.00 Observed Call Price: $7.00 5 CBOE Implied Volatilities for CBOE Implied Volatilities for Stock Indexes The CBOE publishes data for two implied volatility indexes: S&P 100 Index Option Volatility, ticker symbol VIX Nasdaq 100 Index Option Volatility, ticker symbol VXN Each of these volatility indexes are calculating using ISDs from eight options: 4 calls with two maturity dates: 2 slightly out of the money 2 slightly in the money 2 slightly out of the money 2 slightly in the money 4 puts with two maturity dates: The purpose of these indexes is to give investors information about market 6 volatility in the coming months. VIX vs. S&P 100 Index Realized VIX vs. S&P 100 Index Realized Volatility 7 VXN vs. Nasdaq 100 Index Realized VXN vs. Nasdaq 100 Index Realized Volatility 8 Hedging a Portfolio with Index Hedging a Portfolio with Index Options Many institutional money managers use stock index options to hedge the equity portfolios they manage. To form an effective hedge, the number of option contracts needed can be calculated with this formula:
Number of Option Contracts = Portfolio Beta ×Portfolio Value Option Delta ×Underlying Value ×100 Note that regular rebalancing is needed to maintain an effective hedge over time. Why? Well, over time: Underlying Value Changes Option Delta Changes Portfolio Value Changes Portfolio Beta Changes
9 Example: Calculating the Number of Example: Calculating the Number of Option Contracts Needed to Hedge an Equity Portfolio Your $45,000,000 portfolio has a beta of 1.10. You decide to hedge the value of this portfolio with the purchase of put options. The put options have a delta of 0.31 The value of the index is 1050. Number of Option Contracts = Portfolio Beta Portfolio Value Option Delta × Underlying Value × 100 = 1.10 × 45,000,000 = 1,520.74 0.31× 1050 × 100 So, you buy 1,521 put options.
10 Implied Volatility Skews Implied Volatility Skews Volatility skews (or volatility smiles) describe the relationship between implied volatilities and strike prices for options. Recall that implied volatility is often used to estimate a stock’s price volatility over the period remaining until option expiration. 11 Implied Volatility Skews, Cont. Implied Volatility Skews, Cont. 12 Graph of Volatility Skews Graph of Volatility Skews 13 Logically, there can be only one stock price volatility, because price volatility is a property of the underlying stock. However, volatility skews do exist. There is widespread agreement that the major cause is stochastic volatility. Stochastic volatility is the phenomenon of stock price volatility changing randomly over time. Recall that the BlackScholesMerton option pricing model assumes that stock price volatility is constant over the life of the option. Nevertheless, the BlackScholesMerton option pricing model yields accurate option prices for options with strike prices close to the current stock price. Implied Volatility Skews, Implied Volatility Skews, Summary 14 Useful Websites Useful Websites www.jeresearch.com (for more information on option formulas) www.cboe.com (for a free option price calculator) www.numa.com (for options trading strategies and a lot more on options) www.ivolatility.com (for applications of implied 15 Option Trading Strategy Option Trading Strategy Leverage Trading / Directional Trading Strategy
Take buy call, sell put, call spread, etc buy put, sell call, put spread, etc aview on the market direction Volatility Trading Strategy
Take a view on the market volatility A measure of the degree of the fluctuations What Is Volatility? What Is Volatility?
Intuitively, which one do you think is more volatile? For example, compare the two listed companies in Taiwan TSMC (ticker: 2330) Chung Hwa (ticker: 2412) What Is Volatility?
Volatility in statistical language
1 1N σY = ∑ (ri − r ) 2 ∆T N − 1 i =1 σ D = σ Y / annual trading days
2412
Daily Return 0.73% 1.45% 4.41% 0.00% 2.23% 1.48% 1.50% 3.86% 1.56% 3.80% 5.80% 0.70% 2.12% 0.27% Last Daily Last Date Price Return Date Price 10May 51.5 0.98% 31May 53.5 13May 51 0.98% 3Jun 53 14May 51.5 0.00% 4Jun 52 15May 51.5 0.98% 5Jun 52.5 16May 51 0.99% 6Jun 52.5 17May 50.5 1.00% 7Jun 53 20May 50 0.00% 10Jun 52.5 21May 50 1.00% 11Jun 52.5 22May 50.5 0.00% 12Jun 54.5 23May 50.5 0.99% 13Jun 55 24May 51 1.94% 14Jun 53.5 27May 52 1.94% 17Jun 53.5 28May 51 4.79% 18Jun 53.5 29May 53.5 1.85% 19Jun 53 30May 54.5 1.85% 20Jun 52.5 Daily Last Return Date Price 0.94% 21Jun 53 1.90% 24Jun 53 0.96% 25Jun 54.5 0.00% 26Jun 54.5 0.95% 27Jun 55 0.95% 28Jun 54 0.00% 1Jul 53 3.74% 2Jul 54 0.91% 3Jul 54.5 2.77% 4Jul 54 0.00% 5Jul 53.5 0.00% 8Jul 53.5 0.94% 9Jul 55 0.95% 10Jul 54.5 0.95% Average Return Daily Return 0.00% 2.79% 0.00% 0.91% 1.83% 1.87% 1.87% 0.92% 0.92% 0.93% 0.00% 2.77% 0.91% 0.13% Daily volatility
2330 Last Daily Last Daily Last Date Price Return Date Price Return Date Price 10May 78.6 1.76% 31May 77.73 0.59% 21Jun 69 13May 77.3 2.33% 3Jun 77.27 4.20% 24Jun 68.5 14May 79.1 3.39% 4Jun 74.09 0.62% 25Jun 69.5 15May 81.8 1.69% 5Jun 74.55 1.85% 26Jun 66.5 16May 80.5 1.12% 6Jun 73.18 7.08% 27Jun 66.5 17May 81.4 2.83% 7Jun 68.18 0.00% 28Jun 68 20May 79.1 0.57% 10Jun 68.18 1.34% 1Jul 67 21May 78.6 1.71% 11Jun 67.27 2.74% 2Jul 66 22May 80 1.14% 12Jun 65.45 4.76% 3Jul 63.5 23May 79.1 0.58% 13Jun 68.64 0.00% 4Jul 64.5 24May 79.6 1.15% 14Jun 68.64 2.02% 5Jul 67 27May 78.6 0.59% 17Jun 67.27 2.02% 8Jul 71 28May 78.2 0.58% 18Jun 68.64 0.20% 9Jul 71.5 29May 77.7 1.16% 19Jun 68.5 0.00% 10Jul 70 30May 78.6 1.16% 20Jun 68.5 0.73% Average Return σ Y = 38% σ D = 2.4% σ Y = 25% σ D = 1.6% Volatility and Option Pricing Volatility and Option Pricing Underlying Stock Price Strike price Maturity Dividend Option price is influenced by Interest rate Volatility Assume that all the other factors are equal, will you pay the same price for the option written on TSMC and Chung Hwa? Volatility and Option Pricing Volatility and Option Pricing From buyers’ point of view, higher volatility means The price of a call option increases when the underlying stock becomes more volatile. More chances to expire in the money; Higher hedging cost From issuers’ point of view , higher volatility means implied volatility Two types of volatility
Market Value of Option Option Pricing Model Implied Volatility actual volatility Actual Volatility Option Pricing Model Fair Value of Option DeltaNeutral Strategy DeltaNeutral Strategy Delta Delta = change of option price change of stock price short position in stock = Delta Deltaneutral is the position where long position in call In a Deltaneutral position, small changes in stock price will not change the value of the stockoption portfolio. An example
0.50%
c hg. o f po rt f o lio v a lue DATE 20/05/2002 Stock price Option Price Delta Long call Short stock 79.09 5.42 0.6011 1000 601
Stock Price chg. of stock price Option Price chg. of portfolio 79.485 0.50% 5.66 0.01% 79.881 1.00% 5.91 0.03% 80.276 1.50% 6.16 0.06% 79.09 0.00% 5.42 0.00% 78.695 0.50% 5.19 0.01% 78.299 1.00% 4.96 0.03% 77.904 1.50% 4.73 0.06%
2.00% 1.00% c hg. o f s ha re pric e 0.00% 0.00% 0.50% 1.00% 2.00% 1.00% Buy 1,000 call option on TSMC. Assume that – – – – – – Case Study 1 Case Study 1 European style, expire in two months; Sold at the money; One option exchanged for one share; Interest rate r=2%p.a.; No dividend will be paid; actual annual volatility σY= 38%. Sell underlying stock to keep the portfolio Deltaneutral by rehedging it from time to time. Case Study 1 Case Study 1 The benchmark for rehedging decision is σD= 2.4%, which means Enjoy a leisure day Daily change of stock price < 2.4% Adjust the stock position to achieve Daily change of stock price ≥ 2.4% Deltaneutral In our example, altogether there are 10 rehedges during the two months’ life of the call option 10.00% 5.00% 0.00% 5.00% 10.00% Case Study 1 When σY=38%, 48% and 28%, the outcomes of this strategy Fair are: Price
Implied Volatility 38% 48% 28% $ 4.99 6.27 3.72 Price of % of stock price on Call Option t he day of issuance 6.35% 7.97% 4.73% Call Option Premium Paid 4,992.98 6,268.19 3,715.91 P/L on Stock Hedging Position 5,013.88 5,734.95 4,108.06 Expiring Value of Call Options 0.00 0.00 0.00 Total P/L 20.90 533.24 392.15 When an investor buy an option whose implied volatility is lower than its actual one, he makes money no matter to which direction the market moves ! Case Study 2 Case Study 2 Now consider buying 1,000 options on Chunghwa and short sell the underlying stock to hedge. Assume that all factors are the same as in the example of TSMC, except that actual annualised volatility is 25%. when σY=25%, 35% and 15%, the results are as followed: Fair Price Implied Volatility 25% 35% 15% $ 2.18 3.02 1.35 Price of % of stock price on Call Option the day of issuance 4.24% 5.86% 2.61% Call Option Premium Paid 2,182.42 3,019.03 1,345.37 P/L on Stock Hedging Position 785.60 1,646.68 1,027.83 Expiring Value of Call Options 3,000.00 3,000.00 3,000.00 Total P/L 31.98 1,665.71 626.80 Case Study 2 Please note that the price for options written on Chung Hwa is relatively cheaper than that on TSMC (i.e. the former has a lower percentage price).
Fair Price Fair Price Implied Volatility 38% 48% 28% $ 4.99 6.27 3.72 Price of A % of stock price on Call Option t he day of issuance 6.35% 7.97% 4.73% Cost of Call Options 4,992.98 6,268.19 3,715.91 P/L on Stock Position 5,013.88 5,734.95 4,108.06 Expiring Value of Call Options 0.00 0.00 0.00 Total P/L 20.90 533.24 392.15 Implied Volatility 25% 35% 15% $ 2.18 3.02 1.35 Price of A % of stock price on Call Option t he day of issuance 4.24% 5.86% 2.61% Cost of Call Options 2,182.42 3,019.03 1,345.37 P/L on Stock Position 786.50 1,646.68 1,027.83 Expiring Value of Call Options 3,000.00 3,000.00 3,000.00 Total P/L 31.08 1,665.71 626.80 This is because the volatility of TSMC’s stock is higher than that of Chung Hwa’s. Risks in Volatility Trading Risks in Volatility Trading Volatility trading strategy may be subjected to potential loss if the writer/buyer of option estimates the market volatility incorrectly. – A single shock to stock price (e.g. 911 event, corporate action etc, whether positive or negative, may lead to great increase/decrease of the actual volatility of the underlying stock The daily up and down limit on underlying stock may obstruct timely rehedging and other friction in the underlying market (transaction cost, bid/offer spread, liquidity) Option model assumptions Regulatory risks such as foreign ownership limit, short selling restriction – – – Application of Option Volatility Trading Application of Option Volatility Trading (1) Option Risk Management Market makers usually reduce optionality risks by buying/selling options of same/different strike, maturity and hedge the net delta position between different options. For example
– Option portfolio may consist of three parts: Short call with higher implied volatility(CH) Long call with lower implied volatility(CL) Long underlying stock – – The premium of CL eats up part of their profit When market volatility moves up unexpectedly, the profit in CL partially offset the loss in CH Application of Option Volatility Trading (1) Option Risk Management — Examples Covered warrant risk management buy short term single stock options to cover gamma risks in the the warrant book Index option volatility vs. single stock volatility hedging or arbitrage between single stock volatility and index volatility CB volatility vs. single stock option volatility take advantage on volatility differential between implied volatility from CB and single stock options Application of Option Volatility Trading (2) Equity Derivative Structured Products One common example of equity derivative structured product is Equity Linked Note (ELN). Most popular examples are: –
– Principal Guaranteed Notes High Yield Notes (HYN)
Bond + Equity Derivatives
Structure: Bull/Bear/Range Underlying: Stock/Basket/Index = Equity Linked Note
Return: Coupon/Redemption (fixed or dependent on underlying performance) Equity Derivative Structured Products
s s s s Principle Guaranteed Notes Considerations: Structure: Pricing: U/L, participation, protected portion Investor + note + options participation = (unprotected portion + interest) / option value Types: range / bull / bear Equity Derivative Structured Products Equity Derivative Structured Products
Principle Guaranteed Notes — Example 1 U/Ls: TSMC Tenor: 1/2 year on notes Options: + 100~110 call spread Notes: + zero coupon note Protection: 97% Issue price: 100% Participation: 100% of the appreciation of U/L on maturity Redemption: on maturity, if * appreciation of U/L < 100%, redemption will be 97% * 100% <= appreciation of U/L < 110%, redemption will be 97% + appreciation of U/L * appreciation of U/L >= 110%, redemption will be 97% + 10% Equity Derivative Structured Products Equity Derivative Structured Products Principle Guaranteed Notes — Example 2 U/Ls: TSMC Tenor: 1/2 year on notes Options: + 100~110 call spread Notes: + zero coupon note Protection: 94% Issue price: 100% Participation: 100% of the appreciation of U/L on maturity Redemption: on maturity, if * appreciation of U/L < 100%, redemption will be 94% * 100% <= appreciation of U/L < 110%, redemption will be 94% + appreciation of U/L * appreciation of U/L >= 110%, redemption will be 94% + 10% The difference in protection rate above indicates a different implied volatility in the embedded call options Equity Derivative Structured Equity Derivative Structured Products High Yield Notes
Considerations: Structure: Pricing: Types: U/L, issue price, annual yield Investor + note options issue price = PV(par) option value bull / bear / range Equity Derivative Structured Products Equity Derivative Structured Products
High Yield Notes — Example U/Ls: TSMC at $78.64 Tenor: 60 days on notes Options: 90% put, strike at $ 70.78 Notes: + zero coupon note Issue price: 98% of par Ann. Yield: 12.2% Redemption: on maturity, if * U/L close >= 90%, redemption will be at 100% of par * U/L close < 90%, redemption will be the stock price on maturity / $70.78 Summary Summary Volatility trading concept and application
Making profit without taking directional view but view on market volatility through deltaneutral strategy. (Provided that short selling facility on the underlying is Hedging option portfolio possible) volatility risk (Gamma and Vega risk) liquidity risk (the discontinuous movement of stock price) Equity derivatives structured product combining equity options and fixed income securities whose feature depends on options premium paid/ sold ...
View
Full
Document
This note was uploaded on 02/21/2010 for the course FINA 221 taught by Professor Na during the Spring '09 term at HKUST.
 Spring '09
 na
 Derivatives, Volatility

Click to edit the document details