Understanding and Solving Societal Problems with Modeling and Simulation
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1 Understanding and Solving Societal Problems with Modeling and Simulation Lecture 12: Financial Markets I: Risk Dr. Heinrich Nax & Matthias Leiss Dr. Heinrich Nax & Matthias Leiss / 39
2 Outline Risk in financial markets Definition Market risk Insurance against risk: options Understanding options Valuing options Using options to extract risk information Risk neutrality Empirical densities Tails Results Summary Dr. Heinrich Nax & Matthias Leiss / 39
3 What is risk? General: A situation involving exposure to danger. (Oxford Dictionary) Finance: Any event or action that may adversely affect an organization s ability to achieve its objectives and executes its strategies. (ETH RiskLab) Market risk: the risk of losses in positions arising from movements in market prices. Credit risk: the risk of a counterparty failing to meet its obligations in accordance with agreed terms. Operational risk: the risk of a loss resulting from inadequate or failed internal processes, people and systems or from external events. Important: the outcomes (e.g. prices) are observable risk is not! Dr. Heinrich Nax & Matthias Leiss / 39
4 Market risk Jan 2006: When should you buy / sell? How much? S&P 500 Index S&P 500 (USD) Dr. Heinrich Nax & Matthias Leiss / 39
5 Market risk Sept 2007: When should you buy / sell? How much? S&P 500 Index S&P 500 (USD) Dr. Heinrich Nax & Matthias Leiss / 39
6 Market risk Mar 2008: When should you buy / sell? How much? S&P 500 Index S&P 500 (USD) Dr. Heinrich Nax & Matthias Leiss / 39
7 Market risk Sept 2008: When should you buy / sell? How much? S&P 500 Index S&P 500 (USD) Dr. Heinrich Nax & Matthias Leiss / 39
8 Market risk Jan 2010: When should you buy / sell? How much? S&P 500 Index S&P 500 (USD) Dr. Heinrich Nax & Matthias Leiss / 39
9 Market risk Feb 2014: When should you buy / sell? How much? S&P 500 Index S&P 500 (USD) Dr. Heinrich Nax & Matthias Leiss / 39
10 Outline Risk in financial markets Definition Market risk Insurance against risk: options Understanding options Valuing options Using options to extract risk information Risk neutrality Empirical densities Tails Results Summary Dr. Heinrich Nax & Matthias Leiss / 39
11 Options: an introduction A European option gives the buyer the right but not the obligation to buy / sell an underlying asset at a specified price on a specified date. Buyer of the option Seller of the option Call option Right to buy asset Obligation to sell asset Put option Right to sell asset Obligation to buy asset Exercise price / strike price: the price at which you buy / sell the asset. Exercise date / maturity: the date on which the option can be exercised. Dr. Heinrich Nax & Matthias Leiss / 39
12 Example: options traded at the Chicago Board Options Exchange Quotes for S&P 500 options on October 10, Index value of S&P 500: (USD) Type Maturity Strike price (USD) Price (USD) Open Interest Call Call Call Put Put Let us assume that the market participants, on aggregate, assess the market risk correctly (individual, unbiased priors). Dr. Heinrich Nax & Matthias Leiss / 39
13 Value of a call option at maturity Call option value (payoff diagram) given a $60 exercise price. AT MATURITY Call option value $15 Out of the money At the money In the money Share Price Dr. Heinrich Nax & Matthias Leiss / 39
14 Value of a put option at maturity Put option value (payoff diagram) given a $60 exercise price. AT MATURITY Put option value $10 In the money At the money Out of the money Share Price Dr. Heinrich Nax & Matthias Leiss / 39
15 Value of a call option before maturity Option Price Increasing volatility and time to maturity Stock Price Dr. Heinrich Nax & Matthias Leiss / 39
16 Valuing options: a milestone in finance (and a Nobel prize) Robert C. Merton and Myron S. Scholes have, in collaboration with the late Fischer Black, developed a pioneering formula for the valuation of stock options. Their methodology has paved the way for economic valuations in many areas. It has also generated new types of financial instruments and facilitated more efficient risk management in society. Black, Merton and Scholes made a vital contribution by showing that it is in fact not necessary to use any risk premium when valuing an option. This does not mean that the risk premium disappears; instead it is already included in the stock price. Dr. Heinrich Nax & Matthias Leiss / 39
17 The Black-Scholes model Strategy: eliminate risk with a replicating portfolio ( hedging ), i.e. a combination of cash and asset that dynamically matches the option. Then the price of a risk-free security is S 2 = Time decay + Volatility = long cash short asset V: price of the option, S: asset price, r: risk-free rate, σ: volatility Dr. Heinrich Nax & Matthias Leiss / 39
18 Black-Scholes, inverted: Black-Scholes options pricing model inputs Output Black-Scholes options pricing model reversed Dr. Heinrich Nax & Matthias Leiss / 39
19 Outline Risk in financial markets Definition Market risk Insurance against risk: options Understanding options Valuing options Using options to extract risk information Risk neutrality Empirical densities Tails Results Summary Dr. Heinrich Nax & Matthias Leiss / 39
20 Extracting the density: Theory Under risk neutrality: P 0 =e rt Z 1 1 Z T f( )d Price at time 0 = discounted expected payoff Z 1 C 0 (K) =e rt (S T K)f(S T )ds T K C0(K) 0 =e rt [F (K) 1], C 00 1 st der. = PV [CDF - 1], 2 nd 0 (K) =e rt f(k) der. = PV [PDF] Dr. Heinrich Nax & Matthias Leiss / 39
21 Naïve result: distribution Figure 1: Risk Neutral Distribution from Raw Options Prices Probability S&P 500 Index Distribution from put prices Distribution from call prices Dr. Heinrich Nax & Matthias Leiss / 39
22 Naïve result: density Figure 2: Risk Neutral Density from Raw Options Prices Probability S&P 500 Index Density from put prices Density from call prices Dr. Heinrich Nax & Matthias Leiss / 39
23 Fitting in price space: splines Figure 3: Market Option Prices with Cubic Spline Interpolation Option price S&P 500 Index Spline interpolated call price Spline interpolated put price Market call prices Market put prices Dr. Heinrich Nax & Matthias Leiss / 39
24 Fitting in price space: densities II Figure 4: Densities from Option Prices with Cubic Spline Interpolation Density S&P 500 Index Density from interpolated put prices Density from interpolated call prices Dr. Heinrich Nax & Matthias Leiss / 39
25 Figlewski method Figlewski (2009) proposes the following method: Transform data to implied volatility space using inverted Black-Scholes Fit in implied volatility space Transform back using Black-Scholes and calculate empirical risk neutral density Add tails using GEV functions Dr. Heinrich Nax & Matthias Leiss / 39
26 Fitting in implied volatility space ( volatility smile ) Figure 5: Implied Volatilities from All Calls and Puts Minimum Bid Price th degree Spline Interpolation (1-knot) Implied volatility S&P 500 Index 4th degree polynomial on combined IVs Traded Call IVs Traded Put IVs Dr. Heinrich Nax & Matthias Leiss / 39
27 Fitting in implied volatility space ( volatility smirk ) Polynomial fit and empirical bid ask spread in IV space on for Implied volatility IV fit Empirical IVs Strikes Dr. Heinrich Nax & Matthias Leiss / 39
28 The empirical density (missing tails!) Risk Neutral Density on for Density S&P 500 Index Dr. Heinrich Nax & Matthias Leiss / 39
29 Adding the tails Risk Neutral Density and Fitted GEV Tail Functions on for Density Empirical RND Left tail GEV function Right tail GEV function Connection points S&P 500 Index Dr. Heinrich Nax & Matthias Leiss / 39
30 Figlewski s method, step-by-step: 1. Begin with bid and ask quotes for calls and puts for a given maturity. 2. Discard quotes for very deep OTM options (<$.50). 3. Use the Black-Scholes model to transform option prices to implied volatility space. 4. Combine put and call options by blending implied volatilities to use only OTM and ATM option values. 5. Fit a 4 th order polynomial in IV space through midprice points weighted by open interest.* 6. Transform back to price space and determine empirical density. 7. Connect GEV tails. * Different weights to Figlewski s original approach. Dr. Heinrich Nax & Matthias Leiss / 39
31 Full density (voila!) Risk Neutral Density on for Density S&P 500 Index Dr. Heinrich Nax & Matthias Leiss / 39
32 A first impression Risk Neutral Densities for selected dates 71 days before expiration Density S&P 500 Index Dr. Heinrich Nax & Matthias Leiss / 39
33 A short film Dr. Heinrich Nax & Matthias Leiss / 39
34 Option implied expected returns Annualized Expected Return incl. Dividend Yield implied by Quarterly Options Annualized Expected Log Return + Dividend Yield Dr. Heinrich Nax & Matthias Leiss / 39
35 Expected vs. realized prices (a dynamic wisdom of crowds ) Realized vs expected index value implied by quarterly options S&P 500 (USD) Realized Expected Dr. Heinrich Nax & Matthias Leiss / 39
36 Outline Risk in financial markets Definition Market risk Insurance against risk: options Understanding options Valuing options Using options to extract risk information Risk neutrality Empirical densities Tails Results Summary Dr. Heinrich Nax & Matthias Leiss / 39
37 From real-world financial gambles to PDFs ( and back) You have learned today: One of the various risks related to financial markets is market risk. Options give the right but not the obligation to buy (call) or sell (put) a specified asset at a specified price on a specified date. The Black-Scholes model is used for valuing options. Option exchange markets collect the wisdom of crowds of option traders with respect to future developments of the underlying asset. This information is used in practice by banks and hedge funds for estimating risk. Dr. Heinrich Nax & Matthias Leiss / 39
38 Additional information Quantitative Risk Management Paul Embrechts Financial Market Risks Didier Sornette ETH Risk Center Seminar Series Nobel prize for Merton and Scholes press.html Figlewski method : S. Figlewski, Estimating the Implied Risk Neutral Density for the U.S. Market Portfolio; Volatility and Time Series Econometrics: Essays in Honor of Robert F. Engle (Bollerslev, Russell and Watson, eds.) Dr. Heinrich Nax & Matthias Leiss / 39
39 Thank you for listening!
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