FE5101 Derivatives - lecture notes (ver 140130)

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Information about FE5101 Derivatives - lecture notes (ver 140130)
Finance

Published on February 15, 2014

Author: CEGB

Source: slideshare.net

Description

Second half of the 'Fixed Income & Derivatives' module, as a general introduction to convexity, volatility forecasting and implied volatility. Construction and interpretation of the implied volatility surface (term structure, smile) is covered in depth.

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 FE5101 FIXED INCOME & DERIVATIVES Lectures 6-10 Charles Brown October 2013 Singapore Page 1 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 TABLE OF CONTENTS 1 Course Summary ..................................................................................................................................................... 4 1.1 Mode of Teaching and Learning ..................................................................................................................... 4 1.2 Reading List........................................................................................................................................................ 4 1.3 Contact Details ................................................................................................................................................... 4 1.4 Some reminders ................................................................................................................................................. 4 2 Estimating Volatility................................................................................................................................................ 5 2.1 Objectives for this Section ................................................................................................................................ 5 2.2 Actual versus Implied Volatility ..................................................................................................................... 5 2.3 Central Moments of a Random Variable ........................................................................................................ 5 2.4 Standard Approach to Estimating Volatility ................................................................................................. 6 2.5 Exponential Weighted Moving Average Model ........................................................................................... 7 2.5.1 Weighting scheme.................................................................................................................................... 7 2.5.2 Attractions of EWMA .............................................................................................................................. 8 2.6 GARCH(1,1) model ........................................................................................................................................... 8 2.6.1 Key terms and definitions....................................................................................................................... 8 2.6.2 GARCH(p,q) ............................................................................................................................................. 9 2.6.3 Mean Reversion........................................................................................................................................ 9 2.6.4 Stochastic Variance .................................................................................................................................. 9 2.6.5 Model Calibration using the Maximum Likelihood Method........................................................... 10 2.6.6 Variance Targeting................................................................................................................................. 11 2.6.7 How Good is the Model? ...................................................................................................................... 12 2.6.8 Estimating Future Daily Variance ....................................................................................................... 13 2.6.9 Volatility Term Structure ...................................................................................................................... 13 2.6.10 Impact of Volatility Changes................................................................................................................ 14 2.7 Further Reading ............................................................................................................................................... 15 3 The Implied Volatility Surface : Term Structure................................................................................................ 16 3.1 Objectives for this Section .............................................................................................................................. 16 3.2 Option Markets ................................................................................................................................................ 16 3.2.1 Observable Option Prices ..................................................................................................................... 16 3.2.2 Option Markets by Asset Class ............................................................................................................ 17 3.3 Black-Scholes.................................................................................................................................................... 17 3.3.1 The Black-Scholes Equation & Formula.............................................................................................. 17 3.3.2 Moneyness .............................................................................................................................................. 19 3.3.3 Put Call Parity ........................................................................................................................................ 20 3.3.4 Volatility as a Parameter ....................................................................................................................... 21 3.3.5 Implied Volatilities by Strike................................................................................................................ 21 3.3.6 Characteristics of a Long Call Option ................................................................................................. 22 3.4 Term Structure of Implied Volatility ............................................................................................................ 26 3.4.1 Volatility Cones ...................................................................................................................................... 26 3.4.2 Term Structure - Drivers ....................................................................................................................... 26 3.4.3 Strategy Definition : Delta Neutral Straddle ...................................................................................... 27 3.4.4 Strategy Definition : Calendar Spread ................................................................................................ 31 3.4.5 Forward volatility and no-arbitrage constraints................................................................................ 32 3.4.6 Term Structure Interpolation................................................................................................................ 33 3.4.7 Accounting for holidays and events.................................................................................................... 34 3.5 Further Reading ............................................................................................................................................... 35 4 The Implied Volatility Surface : Smile ................................................................................................................ 36 4.1 Objectives for this Section .............................................................................................................................. 36 4.2 The Observed Smile ........................................................................................................................................ 36 4.3 The Smile – Constraints .................................................................................................................................. 38 4.3.1 Put Spreads and Call Spreads .............................................................................................................. 38 4.3.2 Butterflies ................................................................................................................................................ 39 4.3.3 Option Price Asymptotes...................................................................................................................... 40 4.3.4 Examples (from Hull) ............................................................................................................................ 41 Page 2 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 4.4 The Smile - Drivers.......................................................................................................................................... 42 4.4.1 Principal Component Analysis ............................................................................................................ 42 4.4.2 Quoting Conventions ............................................................................................................................ 43 4.4.3 Smile Dynamics and Exotics................................................................................................................. 43 4.4.4 Interpretations of the Smile .................................................................................................................. 43 4.5 The Smile - Skew.............................................................................................................................................. 44 4.5.1 Strategy Definition : Risk Reversal ...................................................................................................... 44 4.5.2 Vanna....................................................................................................................................................... 47 4.6 The Smile - Wings............................................................................................................................................ 48 4.6.1 Strategy Definition : Market or 1Vol Strangle.................................................................................... 48 4.6.2 Volga ........................................................................................................................................................ 48 4.6.3 Strategy Definition : Vega Neutral Butterfly...................................................................................... 48 4.6.4 Butterfly variants ................................................................................................................................... 52 4.7 Further Reading ............................................................................................................................................... 52 5 Smile Interpolation, Extrapolation and Fitting .................................................................................................. 54 5.1 Objectives for this Section .............................................................................................................................. 54 5.2 The Market Smile............................................................................................................................................. 54 5.3 The Equivalent Smile ...................................................................................................................................... 54 5.3.1 Definitions............................................................................................................................................... 56 5.3.2 Exercise : Reconstruct the Smile........................................................................................................... 56 5.3.3 Exercise : Deconstruct the Smile .......................................................................................................... 57 5.4 Solving for the Equivalent Smile ................................................................................................................... 57 5.5 The VannaVolga Model .................................................................................................................................. 60 5.5.1 Pricing with the Equivalent Smile ....................................................................................................... 60 5.5.2 Pricing with market structures............................................................................................................. 60 5.5.3 Pricing directly by risk .......................................................................................................................... 60 5.6 Extrapolation of the Smile .............................................................................................................................. 61 5.6.1 Overview................................................................................................................................................. 61 5.6.2 Extreme Strikes....................................................................................................................................... 62 5.7 Fitting ................................................................................................................................................................ 65 5.7.1 Fitting an Exogenous Smile Surface .................................................................................................... 66 5.8 Further Reading ............................................................................................................................................... 68 Page 3 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering 1 FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 Course Summary 1.1 Mode of Teaching and Learning    Five 3-hour lectures and one 3-hour Tutorial An assignment to be submitted before Tutorial; this assignment will contribute 25% towards the total score for the module One half of the exam will focus on the content of this second part, contributing 25% towards the total score for the module 1.2 Reading List Title and Author Edition/Year Publisher Options, Futures, and Other Derivatives Author: John C Hull 8e / 2011 Pearson Education Compulsory Principles of Financial Engineering Author: Salih N. Neftci 2e / 2008 Academic Press Compulsory - / 2006 John Wiley & Sons Supplementary The Volatility Surface : a practitioner's guide Author: Jim Gatheral The FX Smile Author: Antonio Castagna Optional Volatility and Correlation Author: Riccardo Rebonato Optional 1.3 Contact Details Lecturer : Charles Brown <-> - Teaching Assistant : Lin Yupeng <-> 1.4 Some reminders – – – – Switch your hand phone to SILENT mode Questions in class and on IVLE forum are encouraged Chatting is discouraged 10 minute break midway Page 4 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering 2 FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 Estimating Volatility 2.1 Objectives for this Section  Appreciation of historical volatility and importance to option valuation  Ability to implement and use EWMA model to estimate daily variance  Appreciation of differences between EWMA and GARCH(1,1)  Ability to implement and use GARCH(1,1) model to o estimate daily variance by using :   Variance Targeting to enhance stability  o Maximum Likelihood Method to solve for parameters And test these results using Ljung-Box statistics build term structure of estimates of daily variance and annualized volatility o derive changes to the entire term structure that are internally consistent with GARCH(1,1) 2.2 Actual versus Implied Volatility 2.3 Central Moments of a Random Variable The kth moment about the mean of a random variable X with probability density function f(x) is    x     k  E  X  E X   k  k f  x dx  All probability distributions have an infinite number of ‘ moments’. Typically, only the first four are of interest: First central moment 1  0 (where the first moment = the mean) Page 5 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014       Second central moment (the variance)  2  E  X  E X 2 Third central moment (the skewness) 3  E  X  E X 3 Fourth central moment (the kurtosis)  4  E  X  E X 4 For the valuation of non-linear derivative contracts, the third and fourth moments are of special interest. Skewness : relative length of tails  Third standardized moment 3 3 Skewness > 0 means right tail is longer, left tail is heavier Skewness < 0 means left tail is longer, right tail is heavier Kurtosis : the 'peakedness' of the p.d.  Fourth standardized moment 4 4 Kurtosis > +3 means higher, sharper peak and fatter tails ('leptokurtotic') Kurtosis = 3 means the distribution is 'Normal' ('mesokurtotic') Kurtosis < +3 means lower, flatter peak and thinner tails ('platykurtic') Excess kurtosis = Kurtosis - 3 2.4 Standard Approach to Estimating Volatility  Define Si as the value of market variable at end of day i  Define ui= ln(Si / Si-1) as the daily return, with mean  Define σn as the volatility per day between day n-1 and day n, as estimated at u 1 m  un  i m i 1 end of day n-1 2 n  Page 6 of 68 1 m  (un i  u )2 m  1 i 1 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 Simplifications usually made:  Define ui as (Si −S i-1)/Si-1  Assume that the mean value of ui is zero  Replace m−1 by m (see ‘Maximum Likelihood Method’ later) This gives 2 n  1 m 2  un  i m i 1 2.5 Exponential Weighted Moving Average Model 2.5.1 Weighting scheme Instead of assigning equal weights to the observations we can set 2 2  n  i 1 iun  i m where m  i 1 i  1. In an exponentially weighted moving average model, the weights assigned to the u2 decline exponentially as we move back through time. Since 1  x1  x 2  x 3  ....  1 1 x then 1  x 1   x 2  x 3  ....  1  x   1 and this leads to 2 2 2  n   n 1  (1   )un 1 The higher λ is, the more weight is given to the previous variance estimate and the less weight is given to the previous squared return, i.e. a high λ implies more Page 7 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 persistence in the variance estimate. A low λ implies a variance estimate that will be more reactive to recent (squared) returns.1 2.5.2 Attractions of EWMA  Relatively little data needs to be stored : we need only remember the current estimate of the variance rate and the most recent observation on the market variable  Tracks volatility changes more responsively than with equal weights 2.6 GARCH(1,1) model GARCH stands for "Generalized Autoregressive Conditional Heteroskedasticity". 2.6.1 Key terms and definitions Autoregression Autoregression is a linear prediction model of the form p X t  c    i X t i   t i 1 The modeling is done via the φ terms, or weights. In GARCH, the variance estimate is a linear function of the previous variance estimate, the long term variance, and the latest squared return. Skedasticity : homogeneity of variance If all random variables in a sequence have the same variance, the sequence is homoskedastic; otherwise, heteroskedastic. Since for financial time series, the variance (of returns) does itself vary over time, the nature of the variability is something we would like to be able to model for forecasting purposes. GARCH looks at the correlations between variances over different lags. 1 RiskMetrics uses λ = 0.94 for daily volatility forecasting. Page 8 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 Conditional variance (or skedastic function) is the variance of a conditional probability function (e.g. distribution of a random variable Y given that the value of a random variable X takes the value x). GARCH models this conditionality by using an auto regression function. 2.6.2 GARCH(p,q) GARCH regresses on “lagged” or historical terms. The lagged terms are either variance or squared returns. The generic GARCH (p, q) model regresses on (p) squared returns and (q) variances. p q i 1 j 1 2 2 2  n      i u n  i    j n  j Therefore, GARCH (1, 1) “lags” or regresses on last period’s squared return (i.e., just 1 return) and last period’s variance (i.e., just 1 variance). The GARCH(1,1) variance estimate is calculated as follows: 2 2 2  n  VL  un 1   n 1 (The EWMA model is a particular case of GARCH(1,1) where γ = 0, α = 1-λ, β = λ.) 2.6.3 Mean Reversion GARCH(1,1) adapts the EWMA model by giving some weight (γ) to a long run average variance rate (VL). Whereas the EWMA has no mean reversion, GARCH(1,1) is consistent with a mean-reverting variance rate model, recognizing that over time the variance tends to get pulled back to a long-run average level of VL. 2.6.4 Stochastic Variance GARCH(1,1) is equivalent to a model where the variance V follows the stochastic process: dV  a  VL  V   dt    V  dz where Page 9 of 68 a  1     and     2 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 2.6.5 Model Calibration using the Maximum Likelihood Method In maximum likelihood methods we calibrate the model by choosing parameters that maximize the likelihood of the observations occurring Example 1 We observe that a certain event happens one time in ten trials. What is our estimate of the proportion of the time, p, that it happens? The probability of the event happening on one particular trial and not on the others is p  (1  p )9 We maximize this to obtain a maximum likelihood estimate. Result: p  0 .1 Example 2 Estimate the variance v of a variable X (normally distributed with zero mean) from m observations on X (u1, u2, ... , um). For a normal distribution, the likelihood of X is gX   X2  exp  2 2   2 2   1 The likelihood of ui being observed is the probability distribution function for X when X=ui : g ui     u2  1 exp i   2v  2v   The likelihood of m observations occurring in the order of actual observation is  1   u 2  exp i    2v  2v  i 1    m Maximizing this is equivalent to maximizing m   2   2 v     ui   m lnv      ui    ln  v  i 1  i 1  v    m Differentiate wrt v Page 10 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 m  u2   d    m ln v     i     m  12  dv  v v i 1  v    m u i 1 2 i For this to be zero (a maximum) then the maximum likelihood estimator of v is 1 m 2 v   ui m i 1 which was our equation for variance earlier (replacing m-1 for an unbiased estimate with m for a maximum likelihood estimate). Application in Excel This section refers to the spreadsheet SS1 posted to IVLE. We choose parameters that maximize m  i 1  u2  1 exp  i   2v  2vi i   or equally,  u2   ln(vi )  i   vi  i 1  m  Start with trial values of ω, α, and β  Update variance estimates for the whole time series  Calculate  i 1  m  u i2   i     ln( v )  v i Use solver to search for values of ω, α, and β that maximize this objective function  Important note: set up the spreadsheet so that you are searching for three numbers that are the same order of magnitude (this is how the solver in Excel works best). 2.6.6 Variance Targeting There are four unknowns in the GARCH(1,1) function above : α, β, γ, VL. Since weights must sum to 1, then α+β+γ=1 Page 11 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 which removes one degree of freedom. We define ω = γVL so that the GARCH(1,1) function becomes : 2 2 2  n    un 1   n 1 and then solve for α, β and ω. Solving for α and β gives us γ, and solving for ω gives us VL since VL   1   Since solving for a maximum with three parameters can result in unstable solutions, one way of implementing GARCH(1,1) to resolve this problem is by using variance targeting: we set the long-run average volatility VL equal to the sample variance (the actual long term average variance), leaving only the two other parameters to be determined. 2.6.7 How Good is the Model? The objective of the model is to produce an estimate of the future daily variance, driven by the autocorrelation, or clustering, of the variance. Clustering reduces the randomness of the variance. If the model is successful, it will fully account for the source of non-randomness in the variance. We can test for this by measuring the randomness of the squared returns, before and after standardizing them with the GARCH(1,1) variance estimates.  The Ljung-Box statistic tests for autocorrelation of a time series η K m k k2 k 1 for all correlations with lag k, where ωk = (m+2)/(m-k) and m = number of observations  We compare the autocorrelation of the squared returns ui2 with the autocorrelation of the standardized squared returns ui2/σi2 Page 12 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 For the model to be successful, the standardized squared returns should be truly random (i.e. without autocorrelation) compared to the unstandardized returns. In this case, the model has explained the source of non-randomness (autocorrelation) in the variance. 2.6.8 Estimating Future Daily Variance We can use GARCH(1,1) to calculate estimates of daily variance at points in the future as well. The GARCH(1,1) function is 2 2 2  n  1       VL    un 1     n 1 Simply re-arranging this gives 2 2 2  n  VL    un 1  VL      n 1  VL  and for day n+t 2 2 2  nt  VL    unt 1  VL      nt 1  VL  Then we can take the expectation of this 2        V  2 2 2 E  n t  VL  E   un t 1  VL  E    n t 1  VL        E  2 n  t 1  L And recurse backwards to day n       2 2 E  n t  VL        n  VL   t 2 2 E  n t  VL        n  VL t This formula enables us to derive the daily variance estimates for any t. 2.6.9 Volatility Term Structure Define a  ln 2 1   * The expectation of the squared return on day n+t-1 is the variance estimate itself Page 13 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 And rewrite the term structure formula above as   2 V t   E  n t  VL  e  at  V 0   VL  Then the average daily variance from t=0 to t=T is an integral of the form T 1 1  e  aT V t dt  VL   V 0  VL  T aT 0 The average daily variance over some period T can be converted into an annualized volatility for option tenor T as  T   252 avgdailyvar and  T  2  aT  VL  1  e  252   aT    0 2    VL    252    where 252 is the number of observations per annum. 2.6.10 Impact of Volatility Changes For risk-management purposes, we would like to ‘bump’ the volatility and revalue the portfolio. But how to reflect this ‘bump’ consistently in the rest of the term structure? We bump the unmodified volatility volatility  0 to  0   0 and re-write the modified  * T  as follows     * T 2  252  VL  1  e  aT aT   0    0 2    VL     252   Then take the difference between the squared unmodified and modified volatilities  1  e  aT  aT  * T 2   T 2       2   0   0   Rearrange the left hand side    * T 2   T 2   * T    T 2  2   T    * T   2   T 2  2   T    T  2 2 E un t 1   n t 1 Page 14 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 to give the result  1  e  aT  T     aT    0     T    0   which indicates what the bump for any σ(T) should be to be consistent with a bump for σ(0). 2.7 Further Reading Reference “Options, Futures, and Other Derivatives” (Chapter 21 - Estimating volatilities and correlations), John C Hull “The illusions of dynamic replication”, Emmanuel Dermann and Nassim Taleb Page 15 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering 3 FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 The Implied Volatility Surface : Term Structure 3.1 Objectives for this Section 3.2 Option Markets 3.2.1 Observable Option Prices The following chart shows the option prices, in US$, observable at one moment in time (January 2010) on the CME, for a range of strikes and maturities. The options in this case are American-style call and put option contracts on WTI futures.3 Like Commodity options, Equity options are also exchange-traded (ETD), whilst the Currency options market is traded off-exchange or ‘over-the-counter’ (OTC) via interdealer brokers such as Tullett Prebon and GFI Group. 3 You can find similar quotes and definitions on the CME Group website: http://www.cmegroup.com/trading/energy/crude-oil/light-sweetcrude_quotes_globex_options.html?exchange=XNYM&foi=OPT&venue=G&productCd=CLX3 &underlyingContract=CL&floorContractCd=CLX3&prodid=#prodType=AME Page 16 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 The spot price is around 75 US$ per barrel. : strikes lower than this are quoted as Puts and strikes higher than this as Calls, hence the peak in the middle. The data is quite noisy as some of the option price data is slightly stale and does not reflect the very latest spot price. The data is shown as discreet columns rather than a continuous surface to reflect the fact that exchange contracts trade only with those strikes and maturities specified by the exchange, while an OTC market has no such constraints (market conventions aside). It is important to note that these option prices are determined entirely by supply and demand. There are various no-arbitrage constraints, which are covered later in the module, but the above ‘price surface’ has no relation whatsoever to Black-Scholes or volatility, and option markets were in existence long before 1973. 3.2.2 Option Markets by Asset Class Volatility Liquidity moderate volatility (10-15%) Demand Product OTC / ETD Complexity Tenor liquid symmetric moderate but short term from all OTC growing (0-2y) sectors mostly very high more liquid ‘bullish’ (buy medium Equity volatility in a rising calls /sell moderate mostly ETD term (2-5y) (40-60%) market puts) driven by retail mostly ‘bullish’ (buy high caps /sell long term Interest Rate very liquid very high mostly OTC volatility floors) from (3-10y) (20-40%) all market sectors symmetric demand correlationlong term Credit illiquid mostly from very high OTC driven (3-10y) institutional sector symmetric very high demand short term Commodity illiquid low ETD volatility mostly from (0-1y) (30%-100%) institutional sector Currency 3.3 Black-Scholes 3.3.1 The Black-Scholes Equation & Formula The BS stochastic partial differential equation is Page 17 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 V 1 2 2  2V V  2 S  rS  rV  0 2 t S S time decay convexity term drift term discounting term where V represents the option’s value. The solution of the PDE with the final condition VCall S , T   MaxS  K ,0  gives the Black-Scholes Formula VCall S , t   Se  D T t  N d1   Ke  r (T t ) N d 2  where N x   d1  d2  1 2  x e  1 2 2   K  r  D  ln S 1 2 d  2 T  t   T t  K  r  D  ln S 1 2  2 T  t   T t Model Assumptions The Black–Scholes model assumes the market consists of one risky asset (on which the option is written) and one riskless asset. Modelling assumptions are as follows:  the instantaneous log returns of the risky asset are a geometric Brownian motion (an infinitesimal random walk, each step is independent and identically distributed following a normal distribution) with constant drift and volatility  the risk-free interest rate is constant  the risky asset does not pay a dividend  the market is arbitrage-free (i.e. no riskless profits)  it is possible to borrow and lend any amount, even fractional, of the riskless asset at the risk-free rate Page 18 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering  FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 it is possible to buy and sell any amount, even fractional, of the risky asset (including short selling)  there are no transaction fees, costs or taxes Interpretation of the formula (i) the first and second terms on the rhs can be considered binary option values respectively; where N(d1) is used instead of N(d2) for the first term due to the different numeraire. (ii) alternatively, the first term on the rhs can be considered the present value of the expected underlying price given that exercise occurs (a conditional probability) (i.e. the expected proceeds of exercise), and the second term can be considered the present value of the strike given that exercise occurs (i.e. the expected cost of exercise). (iii) the returns are normally distributed, thus the underlying prices are lognormally distributed (iv) the σ2T term is an adjustment to the return due to the convex relationship between the underlying price and the return (ref. Jensen’s Inequality and Ito’s Lemma). (v) expected values are calculated using risk-neutral probabilities (no risk premia) and discounted using the risk-free rate. 3.3.2 Moneyness The moneyness of an option measures how far away from the current market is its strike. There are various measures used, and various interpretations. The key ones are set out below. Note that conventional usage of the term in financial markets is not entirely consistent. Time Value and Intrinsic Value The present value (or premium) of an option is said to consist of its intrinsic value plus its time value. The intrinsic value is equal to the present value of for a Call or MaxF  K ,0 MaxK  F ,0 for a Put. The time value is then the difference between the market value of the option and its intrinsic value. Page 19 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 OTM, ATM and ITM An option is said to be in-the-money (‘itm’) if its intrinsic value is positive. An option is said to be at-the-money (‘atm’) if ... (see below - multiple definitions). An option is said to be out-of-the-money (‘otm’) if its intrinsic value is otherwise zero. Standardized Moneyness Standardized moneyness m is defined as : m  K d  d Ln F  T 1 2 2 which is zero when K=F. An option’s strike K is referred to as at-the-money-forward (‘atmf’) when it is equal to the underlying asset’s forward price F. An ATMF strike has a moneyness of zero. Probability of Exercise The standard cumulative normal distribution function N x  gives the probability of a random variable with mean 0 and standard deviation 1 being above x. N d1  is the sensitivity of a Call option’s value to a change in the underlying spot price. This is also known as the delta. N d 2  is the probability of a Call option expiring in the money. This is also known as the dual delta. Since: d2  K  Ln F m 1 2  2T  T and  K Ln F  T d1  K  Ln F 1 2  2T  T N 0  50% , then neither an option with m=0 (‘atm’, K=F), nor an option with a delta of 50%, will have a 50% chance of being exercised. However, the three probabilities are close enough that they are often conflated in market usage. 3.3.3 Put Call Parity Page 20 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 3.3.4 Volatility as a Parameter The BS formula is used to calculate an option price based on a set of inputs : – spot, interest rates (market data) – strike, tenor (contract terms) – volatility (model parameter) Volatility is not observable, while option prices are. 3.3.5 Implied Volatilities by Strike We can use the BS formula to invert an option price to give an implied volatility. By doing this to the entire set of option prices shown in the previous chart, a ‘volatility surface’ is obtained. Implied Volatilities by Strike and Tenor 21.00% Implied Volatility 19.00% 17.00% 15.00% 13.00% 11.00% 24M 9M 3M 1M 102.4975 O/N 98.3976 94.2977 90.1978 86.0979 81.9980 77.8981 73.7982 Strike 69.6983 61.4985 65.5984 9.00% Tenor It can be seen that this volatility surface is somewhat flatter than the price surface – it is more easily ‘parametrized’. The implied volatility surface can be further re-scaled by plotting the strike (K) axis in terms of LN(K/F) and the tenor (T) axis in terms of √T. The result is much flatter still, since these terms closely reflect what is in the BS formula – but still not completely flat. The volatility surface exhibits both a term structure and a ‘smile’. Page 21 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 3.3.6 Characteristics of a Long Call Option The characteristics of a Call Option are reviewed, as an introduction to discussion of the drivers behind the term structure of implied volatility. Portfolio Value of an ATM Call Option 14000 12000 10000 8000 6000 4000 11.11% 9.18% 7.28% 5.41% 3.57% 1.77% 0.00% -1.74% -3.45% -5.13% -6.78% -8.41% -19.90% -10.00% -13.27% -16.58% -6.63% -9.95% 0.00% -3.32% 6.63% 3.32% 13.27% Vol Bump (absolute) 9.95% 0 16.58% 19.90% 2000 Spot Bump (relative) Portfolio Delta of an ATM Call Option 100000 90000 80000 70000 60000 50000 40000 30000 20000 Page 22 of 68 11.11% 9.18% 7.28% 5.41% 3.57% 1.77% 0.00% -1.74% -3.45% -5.13% -6.78% -8.41% -19.90% -10.00% -13.27% -16.58% -6.63% -9.95% 0.00% -3.32% 6.63% 13.27% 3.32% Vol Bump (absolute) 9.95% 0 16.58% 19.90% 10000 Spot Bump (relative) © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 Portfolio Gamma of an ATM Call Option 5000000 4500000 4000000 3500000 3000000 2500000 2000000 1500000 1000000 11.11% 9.18% 7.28% 5.41% 3.57% 1.77% 0.00% -1.74% -3.45% -5.13% -6.78% -8.41% -17.00% -10.00% -11.33% -14.17% -5.67% -8.50% 0.00% -2.83% 5.67% 11.33% 2.83% Vol Bump (absolute) 8.50% 0 14.17% 17.00% 500000 Spot Bump (relative) Portfolio Theta of an ATM Call Option 0 -5000 -10000 -15000 Note that both axes have been reversed for this graph for easy viewing -20000 -25000 Page 23 of 68 -10.00% -7.60% -5.13% -2.60% 0.00% 2.67% 5.41% 8.22% 16.58% 19.90% 11.11% 9.95% 13.27% 3.32% 6.63% 0.00% -6.63% -3.32% Vol Bump (absolute) -9.95% -19.90% -16.58% -13.27% -30000 Spot Bump (relative) © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 Portfolio Vega of an ATM Call Option 12000 10000 8000 6000 4000 19.90% 2000 14.93% 9.95% 4.98% 9.18% 7.28% 5.41% 3.57% 1.77% -1.74% -5.13% -6.78% -10.00% -19.90% -8.41% -14.93% -3.45% -9.95% 0.00% -4.98% 11.11% 0 0.00% Vol Bump (absolute) Spot Bump (relative) Portfolio Volga of an ATM Call Option 450000 400000 350000 300000 250000 200000 150000 100000 50000 19.90% 14.93% 0 9.95% 4.98% Page 24 of 68 -5.13% -6.78% -10.00% -19.90% -8.41% 9.18% 7.28% 5.41% 3.57% 1.77% -14.93% -1.74% -9.95% 0.00% -4.98% -3.45% Vol Bump (absolute) 11.11% -50000 0.00% Spot Bump (relative) © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 Portfolio Vanna of an ATM Call Option 1000000 800000 600000 400000 19.90% 16.58% 13.27% 9.95% 6.63% 3.32% Vol Bump 0.00% (absolute) -3.32% 200000 0 -200000 -400000 -6.63% -600000 -9.95% -13.27% -800000 -16.58% 9.18% 11.11% 5.41% 3.57% 1.77% 0.00% -1.74% -3.45% -5.13% -6.78% -8.41% -10.00% 7.28% -1000000 -19.90% Spot Bump (relative) Theta and Gamma Page 25 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 3.4 Term Structure of Implied Volatility 3.4.1 Volatility Cones ATM Implied Volatility by Tenor 25.00% 23.00% ATM Implied Volatility 21.00% 19.00% 17.00% 15.00% 13.00% 11.00% 9.00% 7.00% 1 101 201 301 401 501 601 701 Tenor in Days 3.4.2 Term Structure - Drivers • Vols show mean reversion • Short Maturities: – do not have any vega exposures – driven by the relative benefits of gamma versus the costs of theta – a long option position is long gamma and profits from actual volatility through delta rebalancing; but is also short theta and loses from the passage of time (and vice versa) • Long Maturities: – – • do not have any gamma or theta exposures driven by supply and demand for vega (exposure to volatility moves) For both long and short maturities, implied volatility is traded using Delta Neutral Straddles Page 26 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 3.4.3 Strategy Definition : Delta Neutral Straddle Buying a DELTA NEUTRAL STRADDLE involves: – buy 1x Call strike K – buy 1x Put strike K – K such that net delta (premium included) is zero  K  K1   K  K1       DNSTDL  C  Delta - Premium%  X %   P Delta - Premium%  X %          ATM ATM     The strike of a Delta Neutral Straddle can be formulated as: K  F e  1  2 T 2 + for counter ccy premium - for base ccy premium The  ATM term is referred to as the ‘at-the-money (atm) volatility’. It is used to describe the implied volatility used to price a Delta Neutral Straddle, whose strike is often referred to as the ‘atm strike’. We will see later that the market parametrizes the implied volatility smile by using just three volatilities:  ATM ,  RR ,  BF . Premium included delta The net delta for a straddle is calculated as the sum of the two option deltas less the sum of the two option premia (i.e. ‘premium included delta’), since the premium for a straddle is relatively large. The two premiums must be converted to %den by dividing the premium from the BS formula by spot, since the delta from the BS formula is in %den units. Terminology for units (DEN and NUM) Value and Greeks The following graphs show the value and greeks for a Delta Neutral Straddle, for a range of spot rates and implied volatilities: Page 27 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 Portfolio Value of a DN Straddle 14000 12000 10000 8000 6000 4000 Page 28 of 68 11.11% 9.18% 7.28% 5.41% 3.57% 1.77% 0.00% -1.74% -3.45% -5.13% -6.78% -8.41% -10.00% -10.00% -6.67% -8.33% -3.33% -5.00% 0.00% Vol Bump (absolute) 0 -1.67% 10.00% 8.33% 6.67% 5.00% 3.33% 1.67% 2000 Spot Bump (relative) © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 Portfolio Delta of a DN Straddle 100000 80000 60000 40000 20000 0 -20000 -40000 -60000 11.11% 9.18% 7.28% 5.41% 3.57% 1.77% 0.00% -1.74% -3.45% -5.13% -6.78% -8.41% -10.00% -10.00% -6.67% -8.33% -3.33% -5.00% 0.00% Vol Bump (absolute) -100000 -1.67% 10.00% 8.33% 6.67% 5.00% 3.33% 1.67% -80000 Spot Bump (relative) Portfolio Gamma of a DN Straddle 3000000 2500000 2000000 1500000 1000000 Page 29 of 68 11.11% 9.18% 7.28% 5.41% 3.57% 1.77% 0.00% -1.74% -3.45% -5.13% -6.78% -8.41% -10.00% -10.00% -6.67% -8.33% -3.33% -5.00% 0.00% Vol Bump (absolute) 0 -1.67% 10.00% 8.33% 6.67% 5.00% 3.33% 1.67% 500000 Spot Bump (relative) © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 Portfolio Theta of a DN Straddle 0 -5000 -10000 -15000 -20000 -25000 -30000 -35000 11.11% 9.18% 7.28% 5.41% 3.57% 1.77% 0.00% -1.74% -3.45% -5.13% -6.78% -8.41% -10.00% -10.00% -6.67% -8.33% -3.33% -5.00% 0.00% Vol Bump (absolute) -45000 -1.67% 10.00% 8.33% 6.67% 5.00% 3.33% 1.67% -40000 Spot Bump (relative) Portfolio Vega of a DN Straddle 25000 20000 15000 10000 Page 30 of 68 11.11% 9.18% 7.28% 5.41% 3.57% 1.77% 0.00% -1.74% -3.45% -5.13% -6.78% -8.41% -10.00% -10.00% -6.67% -8.33% -3.33% -5.00% 0.00% Vol Bump (absolute) 0 -1.67% 10.00% 8.33% 6.67% 5.00% 3.33% 1.67% 5000 Spot Bump (relative) © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 Portfolio Volga of a DN Straddle 180000 160000 140000 120000 100000 80000 60000 40000 20000 11.11% 9.18% 7.28% 5.41% 3.57% 1.77% 0.00% -1.74% -3.45% -5.13% -6.78% -8.41% -10.00% -10.00% -6.67% -8.33% -3.33% -5.00% 0.00% Vol Bump (absolute) -20000 -1.67% 10.00% 8.33% 6.67% 5.00% 3.33% 1.67% 0 Spot Bump (relative) Portfolio Vanna of a DN Straddle 500000 400000 300000 200000 100000 0 -100000 -200000 -300000 11.11% 9.18% 7.28% 5.41% 3.57% 1.77% 0.00% -1.74% -3.45% -5.13% -6.78% -8.41% -10.00% -10.00% -6.67% -8.33% -3.33% -5.00% 0.00% Vol Bump (absolute) -500000 -1.67% 10.00% 8.33% 6.67% 5.00% 3.33% 1.67% -400000 Spot Bump (relative) 3.4.4 Strategy Definition : Calendar Spread • Buying a CALENDAR SPREAD involves: – – Page 31 of 68 buy 1x STDL maturity T1 sell 1x STDL maturity T2 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 – where T2 > T1 – strikes often ATM for each tenor, or numerically equal 3.4.5 Forward volatility and no-arbitrage constraints The variance of a random variable X is its second central moment, the expected value of the squared deviation from the mean. In the GARCH spreadsheet, we calculated a variance for the daily returns (the daily variance) by taking an average of the squared returns for the sample: 1 n  S   2     ln i 1  n i  0  Si    2 The daily variance is an average, calculated by summing the realized squared daily returns and dividing by the number of returns. Variance measures how far a set of numbers (here, the returns) is spread out. Assume there are 252 daily returns in each year (and that these daily returns are independent and identically distributed). Then the variance for the annual returns (the annual variance) would be:  2  252   2 where  is the annualized volatility. (Volatility is always shown as annualized, unless explicitly stated otherwise.) The annual variance is calculated by multiplying the average daily variance by the number of days in one year. So the volatility implies an average daily variance, and vice versa. Equally for some period of T years, the variance would be:  2 T   252  T   2  T If we know the volatilities for tenors T1 and T2>T1 (where T0 is today):  T 0T 1 2  T1  T 0  2 T 0  T1  252  T1  T 0  T 0T 2 2  T 2  T 0  2 T 0  T 2  252  T 2  T 0  T 1T 2 2  T 2  T1  2 T1  T 2  252  T 2  T1 Page 32 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 Since it is trivial to show the relationship between the average daily variances:  2 T 0  T 1  T 1  T 0   2 T 1  T 2  T 2  T 1   2 T 0  T 2  T 2  T 0 then  T 0T 1 2  T1  T 0   T 1T 2 2  T 2  T1   T 0T 2 2  T 2  T 0 and  T 1T 2 2  T 2  T1   T 0T 2 2  T 2  T 0   T 0T 1 2  T1  T 0 i.e. if we know the volatilities for two tenors T1 and T2, we can imply the volatility for the forward period between the two tenors. Furthermore, since variances cannot be negative, this introduces a constraint on the relationship between  T 0T 1  and  T 0T 2  :  T 0T 2 2  T 2  T 0   T 0T 1 2  T1  T 0 The constraint means that, while  T 0T 2  can be less than  T 0T 1  (i.e. the volatility term structure can have a negative slope), there is a limit to how much lower it can be. Noting that  2 T is a key term in the BS formula, this constraint is equivalent to saying that the cost of a calendar spread must be positive. It follows from this that the way to arbitrage a term structure whose slope is too negative is via the purchase of a calendar spread (buy the shorter tenor and sell the longer tenor) structured to result in a net receipt of premium. 3.4.6 Term Structure Interpolation See SS3. Page 33 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 31% 29% 27% 25% 23% 21% 19% 17% 15% 0 90 180 270 360 BS Vol 450 540 630 720 Fwd Vol 3.4.7 Accounting for holidays and events See SS3. Page 34 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 27.00% 26.50% 26.00% 25.50% 25.00% 24.50% 24.00% 23.50% 30 37 44 51 58 vol 3.5 Further Reading Reference “Why we have never used the BSM option pricing formula”, Espen Gaarder Haug and Nassim Taleb “040522 A note on sufficient conditions for no arbitrage”, Peter Carr and Dilip B. Madan “041102 Implied Vol Constraints”, Peter Carr “060801 Volatility Surfaces Theory, Rules of Thumb, and Empirical Evidence”, Toby Daglish, John Hull and Wulin Suo Page 35 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering 4 FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 The Implied Volatility Surface : Smile 4.1 Objectives for this Section 4.2 The Observed Smile The following graph shows a range of option prices computed using the Black-Scholes formula and a single volatility for all strikes and tenors. For strikes above the current market, the options are Calls, and for strikes below the current market, the options are Puts. Option Prices for Flat Vol (BS) 6.000 5.000 Price 4.000 3.000 2.000 1.000 24M 9M 3M 1M 102.4953 O/N 94.2957 98.3955 86.0961 90.1959 77.8965 81.9963 69.6968 Strike 73.7966 61.4972 65.5970 0.000 Tenor The next graph shows, by comparison, option prices for the same underlying as they are observed in the market. Option Prices in $ by Strike and Tenor 8.000 7.000 6.000 Price 5.000 4.000 3.000 2.000 1.000 Page 36 of 68 102.4975 O/N 1W 1M 2M 3M 6M 9M 12M 24M 98.3976 94.2977 86.0979 90.1978 77.8981 69.6983 81.9980 Strike 73.7982 61.4985 65.5984 0.000 Tenor © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 By visual comparison of the two price surfaces, the overall shape is at first sight very similar. While the two surfaces are clearly not the same, it is difficult to pinpoint any characteristic difference between the two. Nevertheless, the market is clearly not using the same volatility parameter for all strikes and tenors. How is this question to be approached? We can try to describe this set of observed option prices in terms of as few model parameters as possible (i.e. a ‘parsimonious’ or minimalistic model of the market, similar to Nelson-Siegel for the yield curve). Therefore we switch from the price space to the implied volatility space to try and remove any ‘noise’ and focus on the residual marginal underlying differences between these computed and observed prices, bearing in mind that in this implied volatility space the set of computed option prices in the first graph would be entirely flat. The following graph shows the set of observed option prices in the second graph, in terms of their implied volatilities. The option price for each strike and tenor has been fed into the Black-Scholes formula, and a volatility parameter has been implied from this desired output (hence ‘implied volatility’). In this graph, it is now very obvious where this volatility surface deviates from an entirely flat surface; and also some structure is visible: versus tenors (term structure) and versus strikes (smile). Implied Volatilities by Strike and Tenor 21.00% Implied Volatility 19.00% 17.00% 15.00% 13.00% 11.00% Page 37 of 68 24M 9M 3M 1M 102.4975 O/N 98.3976 90.1978 94.2977 86.0979 77.8981 69.6983 81.9980 Strike 73.7982 61.4985 65.5984 9.00% Tenor © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 The Black-Scholes formula is not scaled in terms of the strike (K) or tenor (T) but rather in terms of LnK  and T , so re-scaling the x and y axes should remove any artefacts in the surface due to this scaling. The graph below shows the same volatilities versus LnK  and T . The result is much smoother, which is a good result as this means the surface will be easier to parametrize. Implied Volatilities by Delta and Tenor Implied Volatility 20.00% 18.00% 16.00% 14.00% 12.00% 10.00% 24M 9M 3M 1M 3.50% O/N 10.00% 30.00% 20.00% 50.00% 40.00% 40.00% 20.00% Delta 30.00% 3.50% 10.00% 8.00% Tenor 4.3 The Smile – Constraints The following section describes the three ways that option prices are constrained along the strike axis. 4.3.1 Put Spreads and Call Spreads Observation : Put Spreads must have a +ve value, otherwise there is an arbitrage. Observation : Call Spreads must have a +ve value, otherwise there is an arbitrage. +P(K+dK) – P(K) ≥ 0  -F(K+dK) + C(K+dK) - P(K) ≥ 0  -F(K) + dK + C(K+dK) - P(K) ≥ 0  -C(K) + dK + C(K+dK) ≥ 0  +C(K) - C(K+dK) ≤ dK +C(K) – C(K+dK) ≥ 0  +C(K) - F(K+dK) - P(K+dK) ≥ 0  +C(K) - F(K) + dK - P(K+dK) ≥ 0  +P(K) + dK - P(K+dK) ≥ 0  +P(K+dK) - P(K) ≤ dK This also tells us, via Put-Call Parity, that a Call Spread must be worth at most the difference between its strikes. This also tells us that, via Put-Call Parity, a Put Spread must be worth at most the difference between its strikes. These constraints are on the future value of the spreads. They also imply constraints on individual option values : Page 38 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 P(K+dK) ≥ P(K) As strike is increased, the premium for a Put option must increase. C(K) ≥ C(K+dK) As strike is decreased, the premium for a Call option must increase. And by Put-Call Parity : And by Put-Call Parity : C(K+dK) ≥ C(K) - dK As strike is increased, the premium for a Call option cannot fall faster than the strike is rising. P(K) ≥ P(K+dK) - dK As strike is decreased, the premium for a Put option cannot fall faster than the strike is falling. 4.3.2 Butterflies The second constraint is based on the observation that a Butterfly must have a positive future value. A Butterfly strategy is a spread of two Call spreads (or two Put spreads). In this case, the two Call spreads will have equal notionals and the Butterfly will be ‘symmetric’ (i.e. the three strikes will be equally spaced), but the constraint applies equally to ‘asymmetric’ Butterflies where the constituent Call spreads have different notionals and the three strikes are not equally spaced. The constraint on the minimum value of the Butterfly can be written as follows: Observation : Butterfly (of Put spreads) must have a +ve value, otherwise there is an arbitrage. Observation : Butterfly (of Call spreads) must have a +ve value, otherwise there is an arbitrage. {+P(K+dK)–P(K)} – {+P(K)-P(K-dK)} ≥0  + P(K+dK) – 2*P(K) + P(K-dK) ≥0  + P(K+dK) – P(K) ≥+ P(K) - P(K-dK) {+C(K-dK)–C(K)} – {+C(K)-C(K+dK)} ≥0  + C(K-dK) – 2*C(K) + C(K+dK) ≥0  + C(K-dK) – C(K) ≥+ C(K) - C(K+dK)   As strike is increased, the rate of increase of the premium for a Put option must increase. As strike is decreased, the rate of decrease of the premium for a Put option must decrease.   As strike is increased, the rate of decrease of the premium for a Call option must decrease. As strike is decreased, the rate of increase of the premium for a Call option must increase. What this means is that the relationship between the option price and the strike is convex. Page 39 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 4.3.3 Option Price Asymptotes While the Butterfly price constraint suggests that, for example, a Put option’s premium is increasing endlessly for higher strikes, this does not mean that the premium tends to infinity for very high strikes; instead it tends towards an asymptote. For strikes tending to zero: Observation : a Call with a notional of 1 den and strike K of zero on an underlying with units num den must have a future value equal to 1 den. This Call option gives the owner the right to receive at maturity 1 den at no cost. Thus its ˆ future value C K  must be equal to 1 den, with the assumption that the probability of ST<0 is zero. Since F is the current forward price and the notional is 1 den, this future value is equivalent to a future value of 1 den  F num den or simply F. In num den 1 den  F num den units, using Put-Call Parity: ˆ  C 0  F ˆ ˆ   F 0  P0  F ˆ ˆ   P0  F  F 0  0   As strike tends to zero, the future value of a Call option tends to the forward price F. As strike tends to zero, the future value of a Put option tends to zero. For strikes tending to infinity: Observation : a Call with a notional of 1 num and a strike K * of zero on an underlying with units den num must have a future value equal to 1 num. This Call option gives the owner the right to receive at maturity 1 num at no cost. Thus its future value must be equal to 1 num, with the assumption that the probability of S*T<0 is zero. Since F *  1 F is the current forward price and the notional is 1 num, this future value is equivalent to den 1 num  F * num but we need to express this in terms of den F * num  Page 40 of 68 num num den den  F * num den : F * den  F 1 num den 1 num  K * num 1 num K * den © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering K * is zero, in and since num FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 den units the value will be ∞. This Call C*(K*) on underlying with units with units num den : and equally In   num den den num is equivalent to a Put P(K) on underlying + C*(K*=0) = +P(K=∞) + P*(K*=0) = +C(K=∞) units: ˆ  C * 0   1 ˆ   P  K* ˆ ˆ   F * 0   P* 0   1 * K ˆ ˆ ˆ   P* 0   1 *  F * 0   0  C   K As strike tends to ∞, the future value of a Call option tends to zero. As strike tends to ∞, the future value of a Put option tends to K. 4.3.4 Examples (from Hull) Page 41 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 4.4 The Smile - Drivers 4.4.1 Principal Component Analysis Page 42 of 68 © 2013 by Charles Brown

NUS Risk Management Institute Masters in Financial Engineering FE5101 Fixed Income & Derivatives Semester 1 AY2013/2014 4.4.2 Quoting Conventions The Smile can be quoted “sticky delta” or “sticky strike” basis according to liquidity and precedent (if rigidly interpreted, both would be open to arbitrage - real world is somewhere inbetween). 4.4.3 Smile Dynamics and Exotics Forward dynamics of the smile matter a lot for exotic options.  K , T  • The Vol Surface gives us: • But we would like to have: • For example, to calculate the option delta:  S, t  V S , T , K ,  S , K  S V S , T , K ,  S , K   S , K    h1     S , K  S  Deterministic Smiles • Deterministic Smile  det   det t , T , K , S t  • Sticky Smile (Sticky Strike) • Floating Smile (Sticky Delta) • Forward-Propagated Smile  fwd   fwd  T  t ,   stick   stick t , T , K    float   float  t , T ,     K St K St         4.4.4 Interpretations of the Smile As Probability Distribution As Risk Premia The Smile represents the marginal risk premium charged for risks not incorporated in Black-Scholes (principally, that vol

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