Options and Black-Scholes Implied Volatility Financial Markets, Day 2, Class 3 Jun Pan Shanghai Advanced Institute of Finance (SAIF) Shanghai Jiao Tong University April 19, 2019 Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 1 / 38 Outline An overview of the options market: I Why options? I History, trading volume and market size. I Options exchanges and market participants. The Black-Scholes option pricing model: I The Black-Scholes model. I Risk-neutral pricing. I The Black-Scholes formula. Using the Black-Scholes formula: I The pricing of at-the-money options. I Black-Scholes option implied volatility. Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 2 / 38 Modern Finance Mortgage 4 200 9 1960 1982 1First Stock 00 5 5 1 1 9 2 2 8 0 9 9 Backed 8 3 1 Investments and 9 Index3 Futures 0 6 0 1 6 8 1 Capital Structure Securities 1 0 5 0 9 2 2 19 9 (Fannie 8 8 0 (Modigliani and Miller) 2 1 4 0 Two-Fund 6 WorldCom 0 Mae) 19 7 7 2 0 Separation 1 Financial 5 Scandal 2 9 2 19 9 79 85 (Tobin) 00 Crisis 1 9 Enron 6 OTC Derivatives 01 8 3 1 56 Scandal 20 98 Interest 2 19 78 1 00 19 6 Rate 64 1 00 CAPM Rise of 19 Dot-Com 9 1 Swaps 55 20 987 (Sharpe) Junk Bonds Peak 20 77 96 19 (Michael 9 Stock 10 Dodd-Frank 1 4 Efficient Markets 5 Milken) 1 Market 1999 988 20 19 95 Hypothesis Index Mutual 76 Crash European 9 LTCM (Samuelson, Fama) 11 66 1 31 Funds (Bogle) 1 Sovereign 5 Crisis 9 5 1998 98 2 1 7 S&L Bailout 01 Crisis 19 9 7 01 9 9 Asian 6 Collapse of 2 1 22 7 19 52 Crisis 8 4 199 9 Junk Bonds Trade 1 1 7 90 0 1 9 9 7 First Credit 1 War 6 Mutual Funds 1 20 3 99 Portfolio 1 51 8 TIPS Derivatives 3 7 1 Study (Jensen) 9 Theory 1 7 2 1 9 First US Options9 (CDS) 0 9 6 1 0 1 1 (Markowitz)019 6 9 3 2 9 7 Exchange, CBOE Chinese4 9 1 9 95 1 1 6 9 2 9 1 Stock 2 1 7 7 1 0 19 Option Pricing Theory9 0 1971 5 1 0 Trump Birth of Index (Black, Scholes, Merton)2 9 Market 5 2 1 9 9 1 Crash Funds (McQuown) 9 4 3 199Large Derivatives Losses Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 3 / 38 Sampling the Right Tail Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 4 / 38 Sampling the Left Tail Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 5 / 38 A Brief History 1973: CBOE founded as the first US options exchange, and 911 contracts were traded on 16 underlying stocks on first day of trading. 1975: The Black-Scholes model was adopted for pricing options. 1977: Trading in put options begins. 1983: On March 11, index option (OEX) trading begins; On July 1, options trading on the S&P 500 index (SPX) was launched. 1987: Stock market crash. 1993: Introduces CBOE Volatility Index (VIX). 2003: ISE (an options exchange founded in 2000) overtook CBOE to become the largest US equity options exchange. 2004: CBOE Launches futures on VIX. Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 6 / 38 Trading Volumes OCC Monthly Report for September 2015 Equity ETF Index SPX VIX avg daily contract (million) 7.57 7.33 2.04 1.14 0.76 avg daily premium ($ billion) 1.58 1.50 3.21 2.78 0.14 avg premium per contract ($) 211 205 1,575 2,474 195 put/call ratio of contract 0.80 1.48 1.21 1.87 0.60 put/call ratio of dollar volume 1.14 2.05 1.43 1.48 0.35 For Sept 2015, the average daily trading in options is 16.94 million contracts and $6.30 billion; the average daily trading in stocks is 7.92 billion shares and $321 billion. End of Sept 2015, the open interest for equity and ETF options is 292 million contracts, and 23.7 million contracts for index options. The overall market size: about $95.7 billion. Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 7 / 38 Trading Volumes, Stocks vs Options Source: NYSE and OCC (The Options Clearing Corporation) Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 8 / 38 Leverage Embedded in Call Options Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 9 / 38 Leverage Embedded in Put Options Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 10 / 38 Options Exchanges After 6 years of litigation, CBOE in 2012 was able to retain its exclusive licenses on options on the S&P 500 index. As a result, CBOE remains its dominance in index options with over 98% of the market share. The trading of equity and ETF options, however, is spread over 12 options exchanges: OCC Monthly Report for September 2015 CBOE PHLX ISE BATS ARCA AMEX Equity (%) 16.32 17.50 10.53 14.37 11.81 8.89 ETF (%) 15.93 15.02 15.55 10.48 10.14 10.14 This level of decentralized trading is not an option-only phenomenon. By now, US stocks are regularly traded on 11 exchanges (lit market) and many alternative platforms (dark pools). Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 11 / 38 Market Participants By types, I Designated market makers: facilitate trades and provide liquidity. I Customers from full service brokerage firms: e.g., hedge funds. I Customers from discount brokerage firms: e.g., retail investors. I Firm proprietary traders: prop trading desks in investment banks. By their activities against the market makers: I open buy: buy options to open a new position. I open sell: sell/write options to open a new position. I close buy: buy options to close an existing position. I close sell: sell/write options to close an existing position. Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 12 / 38 Option Trading Volume by Investor Types open buy open sell close buy close sell put call put call put call put call Small Stocks avg volume 16 53 18 49 8 18 9 26 % Firm Proprietary 7.48 4.46 5.42 4.09 4.42 4.84 3.83 3.75 % Discount Broker 7.35 12.92 9.96 11.97 7.81 11.14 6.74 11.89 % Full-Service Broker 72.61 71.73 75.84 73.66 77.90 72.09 75.96 71.60 Medium Stocks avg volume 38 96 36 89 17 39 21 57 % Firm Proprietary 10.87 8.81 9.89 7.62 8.19 8.17 6.76 6.85 % Discount Broker 8.49 12.48 9.38 9.97 8.67 9.34 9.73 12.27 % Full-Service Broker 69.22 67.90 71.38 72.37 71.42 69.89 69.36 68.14 Large Stocks avg volume 165 359 135 314 66 159 90 236 % Firm Proprietary 14.45 11.36 13.61 10.14 11.18 9.86 9.19 8.25 % Discount Broker 9.77 13.18 7.83 8.02 7.73 7.55 11.31 13.64 % Full-Service Broker 63.60 64.70 69.68 71.98 68.72 69.95 65.27 65.84 S&P 500 (SPX) avg volume 17398 10254 12345 11138 7324 7174 10471 6317 % Firm Proprietary 23.51 34.29 35.71 25.51 32.51 20.05 20.10 28.24 % Discount Broker 4.22 4.19 1.38 1.59 1.48 1.72 4.45 4.78 % Full-Service Broker 58.24 48.16 48.81 59.45 49.75 63.79 59.58 51.72 Source: Pan and Poteshman (2006), CBOE data from 1990 through 2001. Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 13 / 38 A Nobel-Prize Winning Formula Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 14 / 38 The Black-Scholes Model The Model: Let St be the time-t stock price, ex dividend. Prof. Black, Merton, and Scholes use a geometric Brownian motion to model St : dSt = (µ − q) St dt + σ St dBt : Drift: (µ − q) St dt is the deterministic component of the stock price. The stock price, ex dividend, grows at the rate of µ − q per year: I µ: expected stock return (continuously compounded), around 12% per year for the S&P 500 index. I q: dividend yield, round 2% per year for the S&P 500 index. Diffusion: σ St dBt is the random component, with Bt as a Brownian motion. σ is the stock return volatility, around 20% per year for the S&P 500 index. Financial Markets, Day 2, Class 3 Options and Black-Scholes Implied Volatility Jun Pan 15 / 38 Brownian Motion Independence of increments: For all 0 = t0 < t1 < : : : < tm, the increments are independent: B(t1) − B(t0); B(t2) − B(t1);:::; B(tm) − B(tm−1) Translating to Finance: stock returns are independently distributed.