Models Used in Variance Swap Pricing


 Daniel Andrews
 2 years ago
 Views:
Transcription
1 Models Used in Variance Swap Pricing Final Analysis Report Jason Vinar, Xu Li, Bowen Sun, Jingnan Zhang Qi Zhang, Tianyi Luo, Wensheng Sun, Yiming Wang Financial Modelling Workshop 2011 Presentation Jan 15, 2011 Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
2 Report Summary Variance Swap Introduction (Tianyi Luo) Definition and Payoff Pricing Methods Intro: Rule of Thumb, Replication and Simulation Models of Volatility Skew: Stochastic Volatility Inspired and 7Parameter Skew Model Parametrization (Tianyi Luo) Constructing The Volatility Surface (Wensheng Sun) Analysis of Rule of Thumb Pricing (Jingnan Zhang) Analysis of Continuous and Discrete Replication (Bowen Sun) Analysis of Pricing Using Simulation (Qi Zhang, Yiming Wang) Discussion of Future Work (Xu Li) Variance Swap with Stochastic Interest Rate The Bound of Value of Variance Swap Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
3 What is variance swap? A variance swap is an instrument which allows investors to trade future realized volatility against current implied volatility. It is a popular way to add volatility exposure to a portfolio since It comes without the directional risk of the underlying security. The variance swap quotes are based on the implied volatility while the final payoff is based on the realized volatility. The additivity of variance allows the investor to easily take a forward volatility position. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
4 The Payoff The payoff of the variance swap can be expressed as: P = N(σ 2 R K var ) where N is notional amount, K var is the strike quoted in annualized variance, and σr 2 is the realized variance over the life of the contract defined as σr 2 = 252 D (ln S i ) 2 D S i 1 with D being the number of trading days during the contract. i=1 Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
5 The Payoff Continued Note the payoff of the variance swap is a convex function of the realized volatility. Here is an example of the payoff of a variance swap with notional amount $ and strikes at Payoff of Variance Swap Strikes at Payoff Realized Volatility Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
6 Pricing Method: Rule of Thumb The rule of thumb pricing includes the following two methods: 90% put: take 90% of the forward price as the strike and plug that into the volatility function. 25% delta put: search for the strike that has a 25% delta (on the put side) and plug that strike into the volatility function. These two methods are empirically based and were justified by practitioners for nonsteep volatility skew. They are used as quick ways to estimate the fair strike of variance swap. Under different scenarios, these rule of thumb might not be accurate enough thus other pricing methods would be suggested. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
7 Pricing Method: Discrete Replication Theoretically, the payoff of variance swap can be replicated statically using infinite many puts and calls (some of them are not traded in the market). So we also used two replication methods to compute the fair strike. Using weights that are inversely proportional to the square of the strike produces a constant dollar gamma. In a discrete setting the fair strike has the form: K vs = 2er N put T T [ N put(k) call call(k) w i + w i ] F F i=1 where w i = c with c = 1 k 2 N 1, k = K F, put(k) is the put price and call(k) is the call price. This approach uses a range of percent strikes (of the forward) to find the dollar strike, e.g. 50% to 150% by 5% increments. i=1 Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
8 Pricing Method: Continuous Replication In a continuous setting, the fair strike is calculated as the following: 2e K vs = rt F T [ 1 put(k) 1 call(k) 0 k 2 dk + F F k 2 dk] F This approach requires a parametrized volatility surface, gaps and nonexist strikes should be filled and extrapolated. Intuitively, the accuracy of this approach depends highly on the reliability of volatility surface. The derivation of the theoretical strike will be included in our final report. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
9 Pricing Method: Local Volatility & Monte Carlo Simulation With a parametrized implied volatility surface, we can construct the local volatility surface used Monte Carlo simulation. Conceptually, this is defined as: σ local (s, t) = f (σ imp (K, T )) After the translation, we simulate the underlying index with daily time steps. Using the fact that payoff = N i=1 (ln S i+1 S i ) 2 K vs and at time zero, the contract has no value, we can solve for a fair strike. We will be giving more on this method in the later part of our presentation. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
10 Modelling Vol Skew: SVI We will use Gatheral s Stochastic Volatility Inspired (SVI) and 7Parameter Skew Model to model the volatility skew. Define k = ln(k/f ), where K is the strike and F is the forward price. Gatheral s SVI Model reads a + b[ρ(k m) + (k m) σ imp (F, T, K) = 2 + σ 2 ] T 0.65 Illustration of the SVI Model Implied Vol Moneyness k=log(k/f) This is an illustration with a = 0.026, m = 0, b = 0.1, ρ = 0.6, σ = 0.6 Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
11 Modelling Vol Skew: 7Parameter Skew The 7Parameter Skew Model reads A L e λlk + β 4 σ imp (F, T, K) = β 1 + kβ 2 T + k2 R β 3 2T A R e λrk + β 6 if k < k L if k L k K R if k R < k Illustration of the Skew Model Implied Vol Moneyness k=log(k/f) This is an illustration with β 1 = 20% implied vol at the at the money forward, β 2 = 0.1, β 3 = 0.5, β 4 = 50%, β 5 = 0.75, β 6 = 15%, β 7 = 1. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
12 Data Set Structure & Model Parametrization The data set structure: Market data on 20 market dates For each market date, we have 15 maturities For each maturity, we have options with different strikes 300 realized market fair strike as benchmark Which means: we have 300 volatility skews to calibrate, 20 volatility surface to construct. We followed the these steps to calibrate both the SVI and skew models: Loss function is defined as Σ N i=1 (σmodel i σ imp i ) 2. Constrains are added to both the set of parameters. Minimize the loss function using Matlab function lsqnonlin.m under prescribed constrains. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
13 Illustration of Parametrization Here, we show a typical fit of these two models. This is a fit for a relatively short maturity A Parameterization of SVI & Skew Model on Short Maturity SVI Fit Market Skew Fit 0.6 Implied Vol Moneyness k=log(k/f) Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
14 Illustration of Parametrization This is a fit for a relatively long maturity A Parameterization of SVI & Skew Model on Long Maturity SVI Fit Market Skew Fit 0.4 Implied Vol Moneyness k=log(k/f) Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
15 Conclusion of Parametrization After hundreds of tests of fit, we come to the following conclusion about the parametrization: The accuracy of both models is higher for shorter maturity and lower for longer maturity. Generally, 7Parameter Skew model performs better (with accuracy of the order 10 6 ) than SVI does (with accuracy of the order 10 4 ). Please note: our data exhibit volatility skew other than smile which means conclusion can reverse when data shows a smile! We will use these parametrization to construct our volatility surface and later local volatility surface. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
16 Constructing The Volatility Surface How to interpolate the volatility surface? Since the variance is additive, we used the forward volatility formula to interpolate the desired implied volatility to construct the surface: σ 2 2T 2 = σ 2 1T 1 + σ 2 F (T 2 T 1 ) for T 2 > T 1 Then given T star, T 1 T star T 2, define dt = T star T 1, we have the interpolated volatility: σ1 2 σ star = T 1 + σf 2 dt T 1 + dt Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
17 Constructing The Volatility Surface A Visual Illustration: Step 1 σ k T Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
18 Constructing The Volatility Surface A Visual Illustration: Step 2 σ T_star k T Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
19 Constructing The Volatility Surface A Visual Illustration: Step 3 σ T_star T_lower k T Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
20 Constructing The Volatility Surface A Visual Illustration: Step 4 σ T_star T_lower T_upper k T Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
21 Constructing The Volatility Surface A Visual Illustration: Step 5 σ T_star k T Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
22 Constructing The Volatility Surface A Visual Illustration: Step 6 σ T_star k T Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
23 Constructing The Volatility Surface A Visual Illustration: Step 7 Surface σ k T Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
24 Constructing The Volatility Surface Extrapolate Out of The Maturity Range In constructing local volatility surface, we need to use daily time step which might be less than the smallest maturity and our desired variance swap maturity might be longer than the longest the data provided, so we need to extrapolate out of the maturity range the data specified. We did the following: Assign σ(k star, T star ) = σ(k star, T (1)) if T star < T (1) Assign σ(k star, T star ) = σ(k star, T (end)) if T star > T (end) Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
25 Constructing The Volatility Surface We built a MATLAB function to calculate the volatility surface: surface = vol surf (para, T, Model, T star, K star) para: parameter set from the calibration (a matrix) T: data maturity range (a column vector) Model: SVI or skew (string) T star: specified maturity (an arbitrary m by 1 vector) k star: specified moneyness (an arbitrary n by 1 vector) surface: interpolated volatility surface (an m by n matrix) In conclusion, this function searches the neighbouring maturity slices T 1 and T 2, evaluates σ 1 and σ 2 and then interpolates in between to give σ(k star, T star ). Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
26 Constructing The Volatility Surface This is how the volatility surface look like: Volatility Surface Implied Volatility Moneyness = log(k/f) Maturity Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
27 Constructing The Volatility Surface To verify this is a nonarbitrage volatility surface, we look at the total variance surface. Total Variance = σ 2 impt 2 Total Variance Surface 1.5 Implied Volatility Maturity Moneyness = log(k/f) Unlike the implied vol, the total variance is an increasing function of T for any k, so this volatility surface implies the nonarbitrage assumption. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
28 Test of Rule of Thumbs To get a quick view of the fair strike for variance swaps, one can use either of the following rule of thumb calculations. We can price variance swaps using rule of thumb: 90% put: take 90% of the forward strike as the strike and plug that into the volatility function 25% delta put: search for the strike that has a 25% delta (on the put side) and plug that strike into the volatility function Review that we already have two models: Gatherals SVI Model Skew Model We are going to show comparisons of these roles of thumb calculations to the market prices in hopes of creating a guide line for the use of these two methods. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
29 Test of Rule of Thumbs Next, we apply these two rules into SVI Model and Skew Model Respectively and compare model data with market data over both maturity and calendar time (market date), then do statistics analysis. The guideline will cover the following dimensions: Time to maturity or Calendar Time Type of volatility surface Rule of Thumb Method Review that we already have two models: Gatherals SVI Model Skew Model After determining the guidelines, we also wanted to check the appropriateness of the 90% and 25% values used in the respective methods and make a recommendation for better levels given the time series data available. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
30 Test of Rule of Thumbs Here we calculate volatility derived from market data of 10/27/2010 as example: Figure: Rule of thumb in SVI model (left) and skew model (right) Two Figures above have applied rule of thumb into SVI Model and Skew Model with same market date, therefore the same strike is taken. 1 1 Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
31 Test of Rule of Thumbs From previous graphs, we find: The shape of two Figures is similar which means choice of volatility surface does not seem to show a big difference, and Market Fair Price is almost higher than other two ways, we notice when the Maturity goes increasing, both rules approximate to same volatility. It shows how each method performs relative to the time to maturity for fixed market date. 25% Delta put performs very well for maturities less than 2 years and turns out not so well when maturity is increasing to large period of time. Moreover, the 90% Strike does not work well at all due to the poor match of the market data. At 10 years maturity, the difference is 4 vol points of 25% Delta Put, however at 2 years, the difference is very tiny, how does this spread? We need to do different version for calendar time of market date instead of Maturity. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
32 Test of Rule of Thumbs Expiration date of 12/21/2012 = 2Y Expiration date of 12/21/2013 = 3Y 1 Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
33 Test of Rule of Thumbs We did similar plot on longer maturity. The conclusion is: When maturity goes further, 90% Forward rules seems plausible than 25% rule. While, for short maturities, the 25% Delta Put matches market better. 90% forward rule is good for long maturity, but lower than market. 25% Delta Put is good for short maturity, but lower than market. There are no significant differences between Skew Model and SVI Model using these two rules of thumb. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
34 Test of Rule of Thumbs We also sampled from the market fair strike and our model strike price randomly and then Ftest is performed on the random samples. And we find that the statistical analysis is in line with our assumption from graphs before. The 90% strike does not seems to work for any maturity in our data sample though it does perform better than 25% for the long maturity, but not good enough to use. The 25% delta put works very well for short term maturity variance swaps, but becomes ineffective when the maturity goes longer than 2 years. Given the reservation stated in 1 and 2, each pricing method works equally well with either volatility model. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
35 Conclusion of Discrete & Continuous Replication Discrete and continuous replication is theoretically true to price the variance swap. Commonly, discrete replication uses a range of percent strikes to find the dollar strike. Usually we use 50%150%. We also did trials on 10%190%, 30%170%, 70%130% and 90%110%. Statistical analysis shows that: Discrete Replication + SVI is better than Discrete Replication + Skew Continuous Replication + SVI and Continuous Replication + skew are equivalent. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
36 Conclusion of Discrete & Continuous Replication Discrete and continuous replication is theoretically true to price the variance swap. Commonly, discrete replication uses a range of percent strikes to find the dollar strike. Usually we use 50%150%. We also did trials on 10%190%, 30%170%, 70%130% and 90%110%. Statistical analysis shows that: Discrete Replication + SVI is better than Discrete Replication + Skew Continuous Replication + SVI and Continuous Replication + skew are equivalent. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
37 Transfer Implied Volatility to Local Volatility The formula that transfer implied volatility to local volatility reads: σ(s, t) 2 loc = 1 k w w T w k ( w + k2 w 2 )( w k ) w k 2 where k = ln( S F ) and w = σ2 impt. Derivatives are found numerically: we use forward approximation to calculate the first order derivative and central approximation to calculate the second derivative. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
38 Monte Carlo Simulation We assume underlying asset follows the stochastic equation: ds t S t = µ(t)dt + σ(s, t)dw t where µ(t) = r(t) q(t) and σ(s, t) is the local volatility. Every step of simulation we generate a new normal distributed random number and plug in local volatility and local drift term to produce a new underlying price. And the final price can be obtained using: payoff = N i=1 (ln S i+1 S i ) 2 K vs Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
39 Monte Carlo Simulation & Error Analysis To reach a more accurate payoff, we should simulate large number of paths of price evolution. Then, take the expectation of the total number of payoff, which will be the fair volatility strike that let the initial value of the variance swap equal to zero. We finally compare the simulated value to market price: Percentage Error = abs(simulated valuemarket data) / market data Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
40 Monte Carlo Simulation & Error Analysis Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
41 Discussion of Future Work: Variance Swap with Stochastic Rate So far, the work we have been done is under the assumption that he underlying price process is an Itô process with a deterministic short rate. we have shown that rule of thumbs and the discrete model and continuous model work well in this setting for a relative short maturity, that is, less than 2 years for rule of thumbs and 3 to 5 years for the discrete and continuous models. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
42 Variance Swap with Stochastic Interest Rate The market has a stochastic interest rate; The value of an equity variance swap depends on the interest rate volatility; Longdated variance swaps will be sensitive to the interest rate volatility; Longdated variance swaps do happen; Price becomes more complicated. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
43 Variance Swap with Stochastic Interest Rate The rest is an upper bound of the fair strike developed by Per Hörfelt and Olaf Torn. P(t, T ) bond price at time t that matures at T. S(t) stock price at time t. dp(t, T ) P(t, T ) = r(t)dt + σ P(t, T ) dw 1 (t), (1) ds(t) S(t) = r(t) dt + σ S(t) dw 2 (t). (2) r(t) is the continuous compounded short rate at time t, Q is a riskneutral measure, (W 1, W 2 ) is a 2dim QWiener process with joint correlation ρ. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
44 Discussion of Future Work: The Bound of Value of Variance Swap Introduce The solution of (1) is t β(t) = exp( r(θ) dθ), 0 P(t, T ) t = β(t) exp( σ P (θ, T )dw 1 (θ) 1 t σp 2 (θ, T ) dθ). (3) P(0, T ) It follows β(t) = 1 t P(0, t) exp( σ P (θ, T )dw 1 (θ) + 1 t σp 2 (θ, T ) dθ). (4) Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
45 The Bound of Value of Variance Swap Define Tforward measure Q T by dq T dq = P(t, T ) Ft P(0, T )β(t) Then Girsanov theorem implies that W T defined by dw T 1 (t) = dw 1 (t) σ P (t, T ) dt, dw T 2 (t) = dw 2 (t) ρσ P (t, T ) dt, is a Wiener process with respect to Q T. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
46 The Bound of Value of Variance Swap Let V vs be the fair variance, V 0 vs be the fair variance in a market with zero bond volatility. V bvs the total bond variance. Some simple calculation yields V vs = E Q T [ 1 T ] [ σs 2 T dt = Vvs 0 + E Q T 1 T ] 2ρσ P σ S σp 2 0 T dt 0 V 0 vs + 2ρV 1 2 vs V 1 2 bvs V bvs. χ ρ Vvs χ ρ ( V 0 vs (1 ρ 2 )V bvs + ρ V bvs ) where χ ρ is the sign function of ρ. Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
47 References More Than You Ever Wanted To Know About Volatility Swaps Just What You Need To Know About Variance Swaps The Volatility Surface: A practitioner s guide The Fair Value of a Variance Swap  a question of interest Team Three (U of Minnesota) Models Used in Variance Swap Pricing Jan 15, / 47
European Options Pricing Using Monte Carlo Simulation
European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical
More informationMarket Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk
Market Risk: FROM VALUE AT RISK TO STRESS TESTING Agenda The Notional Amount Approach Price Sensitivity Measure for Derivatives Weakness of the Greek Measure Define Value at Risk 1 Day to VaR to 10 Day
More information金融隨機計算 : 第一章. BlackScholesMerton Theory of Derivative Pricing and Hedging. CH Han Dept of Quantitative Finance, Natl. TsingHua Univ.
金融隨機計算 : 第一章 BlackScholesMerton Theory of Derivative Pricing and Hedging CH Han Dept of Quantitative Finance, Natl. TsingHua Univ. Derivative Contracts Derivatives, also called contingent claims, are
More informationPricing Exotics under the Smile 1
Introduction Pricing Exotics under the Smile 1 The volatility implied from the market prices of vanilla options, using the Black Scholes formula, is seen to vary with both maturity and strike price. This
More information2013 CBOE Risk Management Conference Variance and Convexity: A Practitioner s Approach
013 CBOE Risk Management Conference Variance and Convexity: A Practitioner s Approach Vishnu Kurella, Portfolio Manager Table of Contents I. Variance Risk Exposures and Relationship to Options 3 II. Forward
More informationImplied Vol Constraints
Implied Vol Constraints by Peter Carr Bloomberg Initial version: Sept. 22, 2000 Current version: November 2, 2004 File reference: impvolconstrs3.tex I am solely responsible for any errors. I Introduction
More informationCaput Derivatives: October 30, 2003
Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor
More informationLecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing
Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Key concept: Ito s lemma Stock Options: A contract giving its holder the right, but not obligation, to trade shares of a common
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationHedging Barriers. Liuren Wu. Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/)
Hedging Barriers Liuren Wu Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/) Based on joint work with Peter Carr (Bloomberg) Modeling and Hedging Using FX Options, March
More informationOption Valuation. Chapter 21
Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of inthemoney options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price
More informationConsider a European call option maturing at time T
Lecture 10: Multiperiod Model Options BlackScholesMerton model Prof. Markus K. Brunnermeier 1 Binomial Option Pricing Consider a European call option maturing at time T with ihstrike K: C T =max(s T
More informationHow to Manage the Maximum Relative Drawdown
How to Manage the Maximum Relative Drawdown Jan Vecer, Petr Novotny, Libor Pospisil, Columbia University, Department of Statistics, New York, NY 27, USA April 9, 26 Abstract Maximum Relative Drawdown measures
More informationImplied Volatility Surface
Implied Volatility Surface Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 16) Liuren Wu Implied Volatility Surface Options Markets 1 / 18 Implied volatility Recall
More informationMonte Carlo Methods in Finance
Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction
More informationVanna. Sensitivity of vega (also known as kappa) to a change in the underlying price,
Vanna Sensitivity of vega (also known as kappa) to a change in the underlying price, the vanna is a second order cross Greeks. Like any other cross Greeks, the vanna can be defined in many ways: (a) (b)
More informationChapter 1: Financial Markets and Financial Derivatives
Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange
More informationCalculating VaR. Capital Market Risk Advisors CMRA
Calculating VaR Capital Market Risk Advisors How is VAR Calculated? Sensitivity Estimate Models  use sensitivity factors such as duration to estimate the change in value of the portfolio to changes in
More informationIL GOES OCAL A TWOFACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES 15 // MAY // 2014
IL GOES OCAL A TWOFACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES 15 MAY 2014 2 MarieLan Nguyen / Wikimedia Commons Introduction 3 Most commodities trade as futures/forwards Cash+carry arbitrage
More informationFinancial Modeling. An introduction to financial modelling and financial options. Conall O Sullivan
Financial Modeling An introduction to financial modelling and financial options Conall O Sullivan Banking and Finance UCD Smurfit School of Business 31 May / UCD Maths Summer School Outline Introduction
More informationHedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies
Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative
More informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationStudy on the Volatility Smile of EUR/USD Currency Options and Trading Strategies
Prof. Joseph Fung, FDS Study on the Volatility Smile of EUR/USD Currency Options and Trading Strategies BY CHEN Duyi 11050098 Finance Concentration LI Ronggang 11050527 Finance Concentration An Honors
More informationK 1 < K 2 = P (K 1 ) P (K 2 ) (6) This holds for both American and European Options.
Slope and Convexity Restrictions and How to implement Arbitrage Opportunities 1 These notes will show how to implement arbitrage opportunities when either the slope or the convexity restriction is violated.
More informationMATH3075/3975 Financial Mathematics
MATH3075/3975 Financial Mathematics Week 11: Solutions Exercise 1 We consider the BlackScholes model M = B, S with the initial stock price S 0 = 9, the continuously compounded interest rate r = 0.01 per
More informationGuaranteed Annuity Options
Guaranteed Annuity Options Hansjörg Furrer Marketconsistent Actuarial Valuation ETH Zürich, Frühjahrssemester 2008 Guaranteed Annuity Options Contents A. Guaranteed Annuity Options B. Valuation and Risk
More informationLecture 6: Option Pricing Using a Onestep Binomial Tree. Friday, September 14, 12
Lecture 6: Option Pricing Using a Onestep Binomial Tree An oversimplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationBinomial lattice model for stock prices
Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationBlackScholes and the Volatility Surface
IEOR E4707: Financial Engineering: ContinuousTime Models Fall 2009 c 2009 by Martin Haugh BlackScholes and the Volatility Surface When we studied discretetime models we used martingale pricing to derive
More informationQuanto Adjustments in the Presence of Stochastic Volatility
Quanto Adjustments in the Presence of tochastic Volatility Alexander Giese March 14, 01 Abstract This paper considers the pricing of quanto options in the presence of stochastic volatility. While it is
More informationQuantitative Strategies Research Notes
Quantitative Strategies Research Notes March 999 More Than You Ever Wanted To Know * About Volatility Swaps Kresimir Demeterfi Emanuel Derman Michael Kamal Joseph Zou * But Less Than Can Be Said Copyright
More informationGN47: Stochastic Modelling of Economic Risks in Life Insurance
GN47: Stochastic Modelling of Economic Risks in Life Insurance Classification Recommended Practice MEMBERS ARE REMINDED THAT THEY MUST ALWAYS COMPLY WITH THE PROFESSIONAL CONDUCT STANDARDS (PCS) AND THAT
More informationThe BlackScholes Model
The BlackScholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The BlackScholes Model Options Markets 1 / 19 The BlackScholesMerton
More informationCS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options
CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common
More informationIntroduction to ArbitrageFree Pricing: Fundamental Theorems
Introduction to ArbitrageFree Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 810, 2015 1 / 24 Outline Financial market
More informationOPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options
OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis binomial model replicating portfolio single period artificial (riskneutral)
More informationLecture 4: The BlackScholes model
OPTIONS and FUTURES Lecture 4: The BlackScholes model Philip H. Dybvig Washington University in Saint Louis BlackScholes option pricing model Lognormal price process Call price Put price Using BlackScholes
More informationArbitrageFree Pricing Models
ArbitrageFree Pricing Models Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models 15.450, Fall 2010 1 / 48 Outline 1 Introduction 2 Arbitrage and SPD 3
More informationUsing the SABR Model
Definitions Ameriprise Workshop 2012 Overview Definitions The Black76 model has been the standard model for European options on currency, interest rates, and stock indices with it s main drawback being
More information104 6 Validation and Applications. In the Vasicek model the movement of the shortterm interest rate is given by
104 6 Validation and Applications 6.1 Interest Rates Derivatives In this section, we consider the pricing of collateralized mortgage obligations and the valuation of zero coupon bonds. Both applications
More informationA variance swap is a forward contract on annualized variance, the square of the realized volatility. n 1. 1 n 2 i=1.
Alexander Gairat In collaboration with IVolatility.com Variance swaps Introduction The goal of this paper is to make a reader more familiar with pricing and hedging variance swaps and to propose some practical
More informationThe Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models
780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond marketmaker would deltahedge, we first need to specify how bonds behave. Suppose we try to model a zerocoupon
More informationHedging Exotic Options
Kai Detlefsen Wolfgang Härdle Center for Applied Statistics and Economics HumboldtUniversität zu Berlin Germany introduction 11 Models The Black Scholes model has some shortcomings:  volatility is not
More informationVannaVolga methods applied to FX derivatives: a practitioner s approach
VannaVolga methods applied to FX derivatives: a practitioner s approach F. Bossens ARABKVBA Summer School 78 September 2009 1 Presentation layout Problem statement: o Vanilla prices : the implied volatility
More information1 Introduction Outline The Foreign Exchange (FX) Market Project Objective Thesis Overview... 1
Contents 1 Introduction 1 1.1 Outline.......................................... 1 1.2 The Foreign Exchange (FX) Market.......................... 1 1.3 Project Objective.....................................
More informationSome Theory on Price and Volatility Modeling
Some Theory on Price and Volatility Modeling by Frank Graves and Julia Litvinova The Brattle Group 44 Brattle Street Cambridge, Massachusetts 02138 Voice 617.864.7900 Fax 617.864.1576 January, 2009 Copyright
More informationIntroduction to Options. Derivatives
Introduction to Options Econ 422: Investment, Capital & Finance University of Washington Summer 2010 August 18, 2010 Derivatives A derivative is a security whose payoff or value depends on (is derived
More informationLecture Quantitative Finance
Lecture Quantitative Finance Spring 2011 Prof. Dr. Erich Walter Farkas Lecture 12: May 19, 2011 Chapter 8: Estimating volatility and correlations Prof. Dr. Erich Walter Farkas Quantitative Finance 11:
More informationINSTITUTE AND FACULTY OF ACTUARIES EXAMINATION
INSTITUTE AND FACULTY OF ACTUARIES EXAMINATION 14 April 2016 (pm) Subject ST6 Finance and Investment Specialist Technical B Time allowed: Three hours 1. Enter all the candidate and examination details
More informationPrivate Equity Fund Valuation and Systematic Risk
An Equilibrium Approach and Empirical Evidence Axel Buchner 1, Christoph Kaserer 2, Niklas Wagner 3 Santa Clara University, March 3th 29 1 Munich University of Technology 2 Munich University of Technology
More informationVolatility Index: VIX vs. GVIX
I. II. III. IV. Volatility Index: VIX vs. GVIX "Does VIX Truly Measure Return Volatility?" by Victor Chow, Wanjun Jiang, and Jingrui Li (214) An Exante (forwardlooking) approach based on Market Price
More informationThe BlackScholes Formula
ECO30004 OPTIONS AND FUTURES SPRING 2008 The BlackScholes Formula The BlackScholes Formula We next examine the BlackScholes formula for European options. The BlackScholes formula are complex as they
More informationFINANCIAL ECONOMICS OPTION PRICING
OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.
More informationLecture 12: The BlackScholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 12: The BlackScholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The BlackScholesMerton Model
More informationFinancial Options: Pricing and Hedging
Financial Options: Pricing and Hedging Diagrams Debt Equity Value of Firm s Assets T Value of Firm s Assets T Valuation of distressed debt and equitylinked securities requires an understanding of financial
More informationMaster of Mathematical Finance: Course Descriptions
Master of Mathematical Finance: Course Descriptions CS 522 Data Mining Computer Science This course provides continued exploration of data mining algorithms. More sophisticated algorithms such as support
More informationApplications of Stochastic Processes in Asset Price Modeling
Applications of Stochastic Processes in Asset Price Modeling TJHSST Computer Systems Lab Senior Research Project 20082009 Preetam D Souza November 11, 2008 Abstract Stock market forecasting and asset
More informationArbitrage Theory in Continuous Time
Arbitrage Theory in Continuous Time THIRD EDITION TOMAS BJORK Stockholm School of Economics OXTORD UNIVERSITY PRESS 1 Introduction 1 1.1 Problem Formulation i 1 v. 2 The Binomial Model 5 2.1 The One Period
More informationFinancial Modeling. Class #06B. Financial Modeling MSS 2012 1
Financial Modeling Class #06B Financial Modeling MSS 2012 1 Class Overview Equity options We will cover three methods of determining an option s price 1. BlackScholesMerton formula 2. Binomial trees
More informationOptions Trading Strategies
Options Trading Strategies Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: ) Liuren Wu (Baruch) Options Trading Strategies Options Markets 1 / 18 Objectives A strategy
More informationInformation Content of Right Option Tails: Evidence from S&P 500 Index Options
Information Content of Right Option Tails: Evidence from S&P 500 Index Options Greg Orosi* October 17, 2015 Abstract In this study, we investigate how useful the information content of outofthemoney
More informationOption Portfolio Modeling
Value of Option (Total=Intrinsic+Time Euro) Option Portfolio Modeling Harry van Breen www.besttheindex.com Email: h.j.vanbreen@besttheindex.com Introduction The goal of this white paper is to provide
More informationHypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam
Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests
More informationMoreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing
More informationExample 1: Calculate and compare RiskMetrics TM and Historical Standard Deviation Compare the weights of the volatility parameter using,, and.
3.6 Compare and contrast different parametric and nonparametric approaches for estimating conditional volatility. 3.7 Calculate conditional volatility using parametric and nonparametric approaches. Parametric
More informationBlackScholes Equation for Option Pricing
BlackScholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there
More informationLecture 11: The Greeks and Risk Management
Lecture 11: The Greeks and Risk Management This lecture studies market risk management from the perspective of an options trader. First, we show how to describe the risk characteristics of derivatives.
More informationValuing Stock Options: The BlackScholesMerton Model. Chapter 13
Valuing Stock Options: The BlackScholesMerton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The BlackScholesMerton Random Walk Assumption
More informationEquityBased Insurance Guarantees Conference November 12, 2010. New York, NY. Operational Risks
EquityBased Insurance Guarantees Conference November , 00 New York, NY Operational Risks Peter Phillips Operational Risk Associated with Running a VA Hedging Program Annuity Solutions Group Aon Benfield
More informationThe Heston Model. Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014
Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014 Generalized SV models Vanilla Call Option via Heston Itô s lemma for variance process EulerMaruyama scheme Implement in Excel&VBA 1.
More informationINTEREST RATES AND FX MODELS
INTEREST RATES AND FX MODELS 8. Portfolio greeks Andrew Lesniewski Courant Institute of Mathematical Sciences New York University New York March 27, 2013 2 Interest Rates & FX Models Contents 1 Introduction
More informationExam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D
Exam MFE Spring 2007 FINAL ANSWER KEY Question # Answer 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D **BEGINNING OF EXAMINATION** ACTUARIAL MODELS FINANCIAL ECONOMICS
More informationInvesco Great Wall Fund Management Co. Shenzhen: June 14, 2008
: A Stern School of Business New York University Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008 Outline 1 2 3 4 5 6 se notes review the principles underlying option pricing and some of
More informationHedging Variable Annuity Guarantees
p. 1/4 Hedging Variable Annuity Guarantees Actuarial Society of Hong Kong Hong Kong, July 30 Phelim P Boyle Wilfrid Laurier University Thanks to Yan Liu and Adam Kolkiewicz for useful discussions. p. 2/4
More informationLisa Borland. A multitimescale statistical feedback model of volatility: Stylized facts and implications for option pricing
EvnineVaughan Associates, Inc. A multitimescale statistical feedback model of volatility: Stylized facts and implications for option pricing Lisa Borland October, 2005 Acknowledgements: Jeremy Evnine
More informationPart V: Option Pricing Basics
erivatives & Risk Management First Week: Part A: Option Fundamentals payoffs market microstructure Next 2 Weeks: Part B: Option Pricing fundamentals: intrinsic vs. time value, putcall parity introduction
More informationPricing variance swaps by using two methods: replication strategy and a stochastic volatility model
Technical report, IDE835, October 19, 28 Pricing variance swaps by using two methods: replication strategy and a stochastic volatility model Master s Thesis in Financial Mathematics Danijela Petkovic School
More informationIntroduction to Equity Derivatives
Introduction to Equity Derivatives Aaron Brask + 44 (0)20 7773 5487 Internal use only Equity derivatives overview Products Clients Client strategies Barclays Capital 2 Equity derivatives products Equity
More informationSensex Realized Volatility Index
Sensex Realized Volatility Index Introduction: Volatility modelling has traditionally relied on complex econometric procedures in order to accommodate the inherent latent character of volatility. Realized
More informationChapter 5 Financial Forwards and Futures
Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Question 5.2. Description Get Paid at Lose Ownership of Receive Payment
More informationVolatility Surfaces: Theory, Rules of Thumb, and Empirical Evidence
Volatility Surfaces: Theory, Rules of Thumb, and Empirical Evidence Toby Daglish School of Economics and Finance Victoria University of Wellington Email: toby.daglish@vuw.ac.nz John Hull Rotman School
More informationModeling the Implied Volatility Surface. Jim Gatheral Stanford Financial Mathematics Seminar February 28, 2003
Modeling the Implied Volatility Surface Jim Gatheral Stanford Financial Mathematics Seminar February 28, 2003 This presentation represents only the personal opinions of the author and not those of Merrill
More informationChapter 10. Chapter 10 Topics. Recent Rates
Chapter 10 Introduction to Risk, Return, and the Opportunity Cost of Capital Chapter 10 Topics Rates of Return Risk Premiums Expected Return Portfolio Return and Risk Risk Diversification Unique & Market
More informationInstitutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)
Copyright 2003 Pearson Education, Inc. Slide 081 Institutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared
More informationOption Values. Option Valuation. Call Option Value before Expiration. Determinants of Call Option Values
Option Values Option Valuation Intrinsic value profit that could be made if the option was immediately exercised Call: stock price exercise price : S T X i i k i X S Put: exercise price stock price : X
More informationLecture 11: RiskNeutral Valuation Steven Skiena. skiena
Lecture 11: RiskNeutral Valuation Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena RiskNeutral Probabilities We can
More informationUCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall MBA Final Exam. December Date:
UCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall 2013 MBA Final Exam December 2013 Date: Your Name: Your Equiz.me email address: Your Signature: 1 This exam is open book, open
More information2. How is a fund manager motivated to behave with this type of renumeration package?
MØA 155 PROBLEM SET: Options Exercise 1. Arbitrage [2] In the discussions of some of the models in this course, we relied on the following type of argument: If two investment strategies have the same payoff
More informationAn introduction to ValueatRisk Learning Curve September 2003
An introduction to ValueatRisk Learning Curve September 2003 ValueatRisk The introduction of ValueatRisk (VaR) as an accepted methodology for quantifying market risk is part of the evolution of risk
More informationSome Practical Issues in FX and Equity Derivatives
Some Practical Issues in FX and Equity Derivatives Phenomenology of the Volatility Surface The volatility matrix is the map of the implied volatilities quoted by the market for options of different strikes
More informationOption Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013
Option Pricing Chapter 4 Including dividends in the BS model Stefan Ankirchner University of Bonn last update: 6th November 2013 Stefan Ankirchner Option Pricing 1 Dividend payments So far: we assumed
More informationNumerical Methods for Option Pricing
Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly
More informationOPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION
OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION NITESH AIDASANI KHYAMI Abstract. Option contracts are used by all major financial institutions and investors, either to speculate on stock market
More informationForward Price. The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow.
Forward Price The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow. The forward price is the delivery price which makes the forward contract zero
More informationBrownian Motion and Ito s Lemma
Brownian Motion and Ito s Lemma 1 Introduction 2 Geometric Brownian Motion 3 Ito s Product Rule 4 Some Properties of the Stochastic Integral 5 Correlated Stock Prices 6 The OrnsteinUhlenbeck Process Brownian
More informationPricing and calibration in local volatility models via fast quantization
Pricing and calibration in local volatility models via fast quantization Parma, 29 th January 2015. Joint work with Giorgia Callegaro and Martino Grasselli Quantization: a brief history Birth: back to
More informationBeyond Black Scholes: Smile & Exo6c op6ons. Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles
Beyond Black Scholes: Smile & Exo6c op6ons Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles 1 What is a Vola6lity Smile? Rela6onship between implied
More informationCaps and Floors. John Crosby
Caps and Floors John Crosby Glasgow University My website is: http://www.johncrosby.co.uk If you spot any typos or errors, please email me. My email address is on my website Lecture given 19th February
More informationReturn to Risk Limited website: www.risklimited.com. Overview of Options An Introduction
Return to Risk Limited website: www.risklimited.com Overview of Options An Introduction Options Definition The right, but not the obligation, to enter into a transaction [buy or sell] at a preagreed price,
More information