Daniel Jahn. Implied Volatility Modelling of Options

Transcription

1 BACHELOR THESIS Daniel Jahn Implied Volatility Modelling of Options Department of Probability and Mathematical Statistics Supervisor of the bachelor thesis: Study programme: Study branch: doc. RNDr. Ing. Miloš Kopa, Ph.D. Mathematics Financial Mathematics Prague 16

2 I declare that I carried out this bachelor thesis independently, and only with the cited sources, literature and other professional sources. I understand that my work relates to the rights and obligations under the Act No. 11/ Sb., the Copyright Act, as amended, in particular the fact that the Charles University has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 6 subsection 1 of the Copyright Act. In... date... signature of the author i

3 Title: Implied Volatility Modelling of Options Author: Daniel Jahn Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Ing. Miloš Kopa, Ph.D., Department of Probability and Mathematical Statistics Abstract: This text presents an analysis of constrained local polynomial estimation used to extract the implied volatility smile from options data. The optimization constraint derived from the state price density ensures the no arbitrage condition. The analysis contains an evaluation of the role of different parameters, such as the degree of the polynomial, kernel type and bandwidth, on the resulting IV smile. Two main approaches are suggested, one attempting to reflect the problematic case of the out-of-the-money options, the other focusing on producing a smooth state price density and a well-fitting IV smile. Keywords: implied volatility, local estimation, options, state price density Název práce: Modelování implikované volatility opcí Autor: Daniel Jahn Katedra: Katedra pravděpodobnosti a matematické statistiky Vedoucí bakalářské práce: Ing. Miloš Kopa, Ph.D., katedra pravděpodobnosti a matematické statistiky Abstrakt: Tato práce prezentuje analýzu podmíněného lokálního polynomiálního odhadu použitého za účelem získání tzv. implied volatility smile z opčních dat. Optimalizační podmínka, odvozená ze,,state price density, zaručuje splnění bezarbitrážní podmínky. Analýza obsahuje popis a vyhodnocení role různých parametrů, například stupně polynomu, typu použitého jádra a jeho šířky (bandwidth) na výsledný tvar IV smile. Práce navrhuje dva různé postupy. První postup se pokouší zachytit složitější chování opcí tzv. mimo peníze (,,out-ofthe-money ). Druhý postup se zaměřuje na kvalitní proložení křivky daty a na vytvoření hladké state price density. Klíčová slova: implikovaná volatilita, local estimation, opce, state price density ii

4 My thanks to doc. RNDr. Ing. Miloš Kopa, Ph.D. for guiding me through the process of writing this thesis and for finding time for me even when time was scarce. I thank Colin Beet for helping me with the linguistic aspects of this thesis, despite being somewhat perplexed as to why would anyone want to cut off the tails of something that has an infinite support. Without his infinite support this thesis would have been much more difficult to write, not to mention read. iii

5 Contents Introduction 1 Mathematical Preliminaries Probability Theory Stochastic Integration References Options and The Implied Volatility 1.1 Introduction To Derivatives Properties of options The Black-Scholes Formula Implied Volatility Volatility Smile and Volatility Surface State Price Density Local Estimation The Kernel Truncated Gaussian Kernel Local Estimation in the Context of Options Data Moneyness The Optimization Problem Expressed Through Moneyness References Results and Discussion The Data and Methods Used Notes on The Integral Condition Degree of the Polynomial deg = deg = deg = Summary of the Polynomial Degree Constraint Low Moneyness Kernel Bandwidth with deg = 1 vs deg = Kernel Bandwidth with deg = Kernel type Conclusion References Conclusion 36 Bibliography 37 List of Figures 39 List of Symbols and Abbreviations 4 1

6 Introduction Options as a financial instrument have been used in various forms ever since the 15th century. Only with the development of modern mathematical tools in the th cenutry, such as the Itô calculus, has it become possible to describe options in the language of stochastic analysis and thus develop more accurate tools for their understanding. This development eventually led to the Black-Scholes option pricing model (Black and Scholes [197]) which yielded an explicit formula for pricing European options. While the assumptions, especially that of a constant volatility, are generally recognized as far from realistic, the Black-Scholes model still remains an often used tool, particularly for obtaining the so-called implied volatility. Since the implied volatility is a topic of major interest in the financial world, an extensive literature on methods of constructing the volatility smile and volatility surface exists. The following text builds on three in particular. The most general is Homescu [11], a paper summarizing various methods of computing the implied volatility surface. The topic of implied volatility is then further elaborated in Fengler [1] and finally, a single method - the local quadratic estimation - is explored in Benko et al. [7], the paper this text directly builds on and extends. This text is divided into three chapters. The first will introduce the mathematical concepts behind the Black-Scholes option pricing model. The chapter opens with basics of probability theory, continues onto defining the Itô integral and finally touches upon the Itô lemma, a crucial tool in deriving the Black Scholes formula. The second chapter then leads the reader to an understanding of this text s topic: the implied volatility. The first part explains the basic principles of options - the Black Scholes formula and finaly implied volatility itself. The second part then focuses on explaining the concepts behind the aim of this text - using local estimation to obtain the implied volatility smile. This is also where the paper this text builds on, Benko et al. [7], is introduced. The third chapter then takes the paper s work on estimating the implied volatility smile with a local quadratic estimator as a starting point and builds on it. Variations to the method are introduced, including different polynomial degrees and kernels used during the estimation. Results of these variations are then presented and discussed, and possible alterations and improvements to the method are suggested.

7 1. Mathematical Preliminaries In this chapter, the necessary mathematical preliminaries that lead to the Black- Scholes (BSM) 1 formula will be introduced. The aim is not to build up the BSM model fully rigorously which would be an impossible feat given the breadth and depth of the material and the scope of this chapter, but rather to illuminate the mathematical apparatus behind it and give sufficient intuition for its understanding with an emphasis on the financial interpretation. The main goal of this chapter is to familiarize the reader with the construction of the stochastic integral and subsequently the notion of stochastic differential equations (SDE) 1.1 Probability Theory This section will introduce the basic terms needed to build the theory of stochastic integration. The text assumes that the reader is familiar with measure theoretic concepts such as (Borel) σ-algebra (denoted B) and the product σ-algebra, measurable space and measurable functions, Lebesgue and Stieltjes integration, some basic probability theory concepts such as the (conditional) expected value, and finally basic topological terms such as a dense subset and a complete space. Definition 1. Let (Ω, F, P ) be a probability space. X : (Ω, F) (R, B) is called a random variable if it is F-measurable, that is if X 1 (A) F for any A B Equivalently instead of taking A B in the definition, simply open sets could be used instead, thanks to the properties of the preimage X 1 which preserves all operations needed to generate a σ-algebra. In case of R with the natural (Euclidean) topology this can be further reduced to open intervals due to the fact that R is separable. Definition. Let I R. A family of random variables X = {X t, t I} on a probability space (Ω, F, P ) with values in (R, B) is called a stochastic process with index set I and state space S. If we fix ω Ω, the map t X t (ω) is called trajectory of the stochastic process. The trajectory gives us a useful interpretation of the stochastic process - for a given ω Ω, we obtain a function of the variable t, usually interpreted as time. Definition 3. A real-valued stochastic process W = {W t, t [, )} is called Wiener process if: 1. W = a.s.. for any t < t 1 < the random variables (W tk W tk 1 ), k = 1,,... are independent (independent increments) 3. (W t W s ) N(, t s) for all s < t (normally distributed, stationary increments) 1 The M stands for Robert C. Merton to reflect his contribution. X 1 (A) = {ω Ω : X(ω) A}, often denoted as the random event [X A] 3

8 4. W has continuous trajectories. 3 Wiener process 4 is also frequently called Brownian motion 5 due to its roots in the research by the botanist Robert Brown who first observed the erratic motion of microscopic particles in a fluid. Brownian motion is of utmost importance in the financial world and in stochastic calculus in general. Many variants of the Brownian motion as well as processes derived from it are used - Brownian motion with a drift, Brownian bridge (figure 1.1), and geometric Brownian motion, to name a few. Figure 1.1: Example path of a Brownian bridge (Zemyla [6]) The last terms to be defined are filtration and adapted process. With them, the notion of the martingale comes quite naturally and will also be defined. The reason the Itô integral enjoys such wide use is partially because it is a martingale. Definition 4. Let {F t } t I be a family of σ-algebras with F t F for all t I. {F t } is called a filtration if F s F t for all s, t I with s t. 6 Intuitively, filtration can be viewed as the information we have about the market up to time t, naturally increasing with time. From now on, any filtration will be the natural filtration generated by the Wiener process, i.e. F t = σ(w s, < s t) 3 In the language of stochastic processes, this usually means trajectories are continuous for almost all ω Ω. It is also possible to simply set the potentially discontinuous trajectories to zero. In either case the theory is not affected. 4 Called after the American mathematician Norbert Wiener ( ) 5 This text will use both terms interchangeably, to reflect the different preferences in the mathematical and financial literature. 6 If the σ-algebra has this property, it is sometimes called an increasing σ-algebra 4

9 Definition 5. A stochastic process X = (X t, t I) is adapted to the filtration {F t } if for all t I the function ω X t (ω) is F t -measurable for all t I. If we see filtration as information available, then saying the process X is adapted to the filtration F T means that we have the full information about X t, t T at the time T. This notion is naturally very important in the financial interpretation of probabiltity theory. Consider developing a trading strategy which would attempt to find the optimal time of selling a derivative with the price process X t. Such a model could at time t only use the information contained in the filtration {F t } and thus it would be crucial for the strategy to be {F t }-adapted. 7 Definition 6. Let {F t } be a filtration in the probability space (Ω, F, P ), I R. Let X = (X t ) t I be a real-valued, F t -adapted stochastic process with E [ X t ] < for all t I. Then X is called a martingale if E [X t F s ] a.s. = X s for all s, t I with t > s. 8 Intuitively, this would mean that at any fixed time, the expected value of the variable X at any point in the future is the current value. Martingales transform the notion of a fair unbiased game (e.g. gambling dice) into mathematical rigour and as such have become an important tool in finance. 1. Stochastic Integration When Weierstrass first published his famous example of a everywhere continuous, but nowhere differentiable function in 187, 9 the construct was deemed too exotic by some mathematicians to perhaps even be considered a function; consider Leibniz (who introduced the term function into mathematics in the first place), whose notion of a function was what we now understand as a differentiable function. Brownian motion is, like the Weierstrass function, nowhere differentiable. Furthermore the total variation is infinite on any interval. Modern mathematics, however, accepts even more peculiar constructs and thus it is no wonder that even the Brownian motion with its properties has become a widely and frequently used tool. However, the properties do not yield themselves very well to the methods of the standard calculus and so it is not unexpected that the standard integral theory in the Riemann-Stieltjes sense does not work for Brownian motion 1. Therefore the theory of stochastic integration needs to be introduced. 7 From now on the braces will be dropped, i.e. the term will be F t -adapted 8 The last property often called the martingale property 9 Although the Czech mathematician Bernard Bolzano constructed a function with the same property forty years before Weierstrass did, his work was published posthumously in 19, making the so-called Weierstrass function the first known published example. 1 This, interestingly, is not entirely true. The integral of the form T f dw t for some function f and Wiener process W t can be defined in the Stieltjes sense, however not for all continuous functions - only for those with a bounded p-variation for some p < Mikosch [1999], Young [1936] 5

10 The definition of the Itô integral 11 will follow a similar path to the Lebesgue integral. We will first define the integral for a simpler class of functions (called elementary) and then extend the definition by approximation in the limit for a richer class of functions. Definition 7. A process X is called elementary if X t = e i I (ti t<t i+1 ), i= where e i is F ti -measurable i {, 1,... } and = t < t 1 < < t n as n. The class of elementary processes is denoted E. Furthermore, denote E E the class of elementary processes such that E e i < i. 1 Imagine now a stock modelled by the Wiener process, 13 W t, t. Imagine the trading strategy follows an elementary process, that is we hold e i stocks at times t [t i, t i+1 ). The integral we will define now would gives us the value of that investment, i.e. the value e i (W ti+1 W ti ) for each of the intervals. With this rationale, the following definition comes very naturally. Definition 8. Let X be an elementary process and W a Wiener process, T. Define T k 1 X dw = e i (W ti+1 W ti ) + e k (W T W tk ) for t k t < t k+1. i= Furthermore, define X dw = { T X dw, T } and T X dw = T X dw S S X dw Note that we are taking the integral at the leftmost point of the interval. Unlike in the case of the Rieman-Stieltjes integral, this makes a difference. This is useful in terms of mathematical properties (the integral then has the martingale property) and financial interpretation - if we consider s t to be a trading strategy, then the integral - the valuing function - does not require us to anticipate the future. It is possible to define the stochastic integral in other ways by choosing a value anywhere else in the interval, such as the middle, which yields the Stratonovich integral which is useful in solving stochastic differential equations as it is often easier to manipulate; for example, the ordinary chain rule applies with the Stratonovich integral, whereas the Itô integral requires its equivalent: the Itô formula. To fully develop the theory and define the Itô integral for a richer class of F t -adapted processes we would require a significantly longer text. We will only 11 As will be discussed later, the Itô integral is not the only integral of stochastic integration, but is the most common, especially in the financial setting. 1 This property will be hereinafter called square-integrable. 13 In reality this would have to be a non-negative version of the Wiener process or a process derived from it, such as the geometric Brownian motion. 6

11 outline some of the basic properties and the construction. Full treatment can then be found in much of the referenced literature, most notably in Oksendal [14] and Klenke [14]. 14 First we will introduce two important spaces. 1. M, the space of square-integrable continuous martingales with the metric m(m, N) = where a b = min(a, b) [1 n n=1 ( E (M n N n ) ) 1/ ],. L, the space of F t -adapted processes for which E t X s ds < t 15 with the metric l(x, Y ) = [1 n n=1 ( E n ) 1/ ] (X s Y s ) ds. Theorem 1. Properties of the Itô integral for elementary processes. Let X E be an elementary process and W the Wiener process. Then the following holds: 1. E T X dw =.. The map X T X dw is linear. 3. T X dw is F t-measurable. 4. The map T T 5. (Itô isometry) 6. T X dw is continuous. ( T E ) T X dw = E Xs ds. X dw is a square-integrable martingale. Proof. 1. follows directly from the definition and the fact that W t N(, t).. to 4. follow directly from the definition and properties of the Wiener process, namely F t -measurability and continuity. 5. Without loss of generality assume t k = T E ( k 1 ) ( k 1 ) k 1 e i (W ti+1 W ti ) = E e i (W ti+1 W ti ) = E e i (t i+1 t i ), i= i= using the fact that E(W t W s ) = (t s) and Ee i e j (W ti+1 W ti )(W tj+1 W tj ) = for i j thanks to independency of increments. 14 Oksendal assumes a good deal of mathematical maturity on the part of the reader and omits a great number of steps, Klenke offers a much more detailed approach. 15 Note that for elementary processes this is equivalent with square-integrability i= 7

12 6. Without loss of generality assume t k = T, t l = S [ T ] E X dw F S = E = = S S k 1 ] e i (W ti+1 W ti ) + e i (W ti+1 W ti ) F S [ l 1 i= X dw + E X dw [ k 1 i=l ] e i (W ti+1 W ti ) F S Where we are using the towering property of the conditional expectation to prove E ] e i (W ti+1 W ti ) F S = E [ k 1 i=l [ k 1 i=l E [ e i (W ti+1 W ti ) ] ] Fti FS =. Square-integrability follows from the Itô isommetry and from the fact that X E. i=l While the Itô isommetry might seem to be a simple consequence of the properties of the Wiener process, its real importance lies in connecting the two spaces M and L - the Itô isommetry can be restated for X, Y E L to m( X dw, ) Y dw = l(x, Y ). This, as will be shown shortly, is a crucial tool in defining the Itô integral. Using these properties, the following steps are then taken to define the Itô integral: 1. Show that E is dense in L, that is any X L is a limit of X n E.. Show that M is complete. 3. Show that for any X L and for any X n E such that l(x n, X), there exists a unique limit X n dw in M. Step three can be quickly demonstrated: 1. l(x n, X), thus X n is Cauchy in L.. m( X n dw, X m dw ) = l(x n, X m ) for n, m, that is X n dw is Cauchy in M. 3. (existence) M is complete there exists a limit M M of X n dw. 4. (uniqueness) m( X n dw, ) Y n dw = l(x n, Y n ) l(x n, X) + l(y n, X). Note that the Itô isommetry was used in steps. and 4. These steps together justify the following definition. 8

13 Definition 9. Let X L. Then the Itô integral of X is defined by: X dw = lim X n dw (Limit in M ), n where X n E and l(x n, X) for n. All the previously stated properties stay true for the limiting case - the Itô integral. 16. With this definition, we have concluded the construction of the Itô integral. The final part of this chapter will focus on a short introduction to the notation of stochastic differential equations and on the chain rule for the Itô integral, the Itô formula - a crucial tool in deriving the BSM formula. First however, we will introduce a special type of a process. Definition 1. An Itô process is an F t -adapted process which can be written in the form X t = X + t U s ds + t V s dw s, where U t, V t L. This can be restated in the differential form: dx t = U t dt + V t dw t. The reason for this definition is twofold. First, it introduces the differential notation used for stochastic differential equations, 17 but more importantly it defines a type of a process that is stable under smooth maps, a property the general L process does not have. With this definition we are now finally able to state the chain rule for Itô integral - the Itô formula. Theorem. Itô formula. Let X t be an Itô process as in the previous definition. Let g(x) C (R) and g(x t ) L. Then g(x t ) is again an Itô process and ( dg(x t ) = g (X t )U t + 1 ) g (X t )Vt dt + g (X t )V t dw t. Proof. The proof (of a more general version) can be found in Oksendal [14] 1.3 References Section 1.1 was composed using Klenke [14], Lachout [1998], and from lecture notes from lectures on probability by RNDr. Jitka Zichova Dr. Section 1. closely followed the construction of the Itô integral in Oksendal [14], but was shortened and simplified using lecture notes on Stochastic Analysis by doc. RNDr. Daniel Hlubinka, Ph.D. However, a large amount of additional literature was also used, most notably Mikosch [1999], Klenke [14], Stefanica [11], Baxter and Rennie [1996], Hirsa and Neftci [13], Dupačová et al. [], Haugh [1], and Wilmott [7] 16 Some of the properties are easily obtained through the limiting process, for others, refer to Oksendal [14] or Klenke [14] for proofs 17 Note that the symbols dx t and dw t really only make sense in connection with the integral form and are not meaningful by themselves - there is no such thing as an actual derivative of a stochastic process. 9

14 . Options and The Implied Volatility.1 Introduction To Derivatives An option is a type of a financial derivative, meaning its price is derived from the fluctuation and performance of the underlying asset. The holder of the contract typically agrees to buy or sell the underlying asset for a stipulated price, called the strike price, at a specific date, called the settlement date or maturity date. The underlying assets include commodities, stocks, currencies, and even assets that have no intrinsic value, such as weather. (The so-called weather derivatives are often used to reduce the risk associated with unfavourable changes in weather.) Derivatives are held for two main reasons: 1. Speculation, where one or both parties expect the market to develop in a certain manner and use derivative to profit from such a development. 1. Hedging, where the financial instrument is used to diminish risks the individual is facing. A typical example is a corn farmer wishing to insure against unfavourable movement in the corn market, such as an unexpected price drop due to one year s abundance of harvested crops and subsequent oversupply, using a derivative to fix the price. Hedging can then naturally lead to a loss as the unexpected movement could occur in the other direction (for instance due to a flood). However, that does not render the hedging strategy a failure as its purpose was to reduce risk, not to profit. The main financial derivatives are composed of futures, forwards, options, and swaps (but also include the infamous CDO and MBS 3 ).. Properties of options Contrary to futures and forwards, where the two parties within the contract are obliged to make the transaction with the underlying asset at the settlement date, the holder of an option is merely buying a right to do so. As a consequence, while in the futures contract both parties are theoretically balanced and neither has an advantage over the other (assuming neither of them 1 Development in the market does not necessarily mean only rising or falling of prices. The speculation can also be of a different nature, such as simply speculating on the prices to move in either direction by a sufficient amount (straddle), or alternatively expecting the price to remain on its current level (strangle). Through a combination of options and/or by choosing less standard underlying assets, one can arrive at e.g. a derivative contract with payoff of either the originally stipulated price, or zero, dependent on whether the weather in a specific area drops below a certain temperature (a binary option on weather). Collateralized debt obligation 3 Mortgage-backed security 1

15 Figure.1: Call option payoff Figure.: Put option payoff holds the knowledge about the future market s development), in an option contract the holder is in an advantageous position as he may choose not to make the exchange (exercise the option) in case the price movement was not favourable for him. Thus, in order to enter the advantageous options contract as a holder, the individual has to buy the option and pay the option premium. Options divide into two groups - call and put - based on whether the holder wishes to buy or sell the underlying asset. This differs from the language of futures and forwards, where the terms long and short are used. In options language, long and short relate to the actual contract itself, not the underlying asset, thus for instance if one enters a put option contract in a long position, one is buying the right to sell the underlying asset at the settlement date. If we then take K, the strike price and S t the underlying asset price at time t, we can express the option s payoff function (depicted in figures.1 and. 4 ) as call option payoff: max(s t K, ), put option payoff: max(k S t, ). Lastly, while this text has hitherto assumed that the settlement can only occur at the settlement date, this is only the case with European options - in case of American options, 5 the option holder has the right to exercise the option at any point of time before the option s maturity date. This complicates the mathematics of the situation greatly by introducing the additional problem of when the option should be exercised. 6 In such case the Black-Scholes differential equation does not even yield a closed form solution and has to be estimated numerically..3 The Black-Scholes Formula With the mathematical apparatus from the first chapter, the BSM model for option pricing will now be introduced. Let (Ω, F, P ) be a probability space equipped with the filtration F t generated by the Wiener pocoess W t. Assume that 4 Gxti [9a], Gxti [9b] 5 The etymology of the name being simply historical, not referring to the actual place where these respective options are traded. 6 Which, coincidentally, often is at the maturity date, in any case. The problem of the option s exercise time is called the optimal stopping problem. 11

16 the F t -adapted stock price S t is modelled by the Geometric Brownian motion; that is S t satisfies the SDE: 7 ds t = µs t dt + σs t dw t, where µ is called the drift parameter of S t and σ is the variance of the return process ln(s t ). At the maturity date T, the call option payoff is equal to max(s T K), where K is the strike price and the put option payoff is equal to max(k S T ). If we introduce a riskless interest rate r, it can be shown, using the Itô formula, that the call option price C t = C(S t, t) is twice differentiable in S and once in t and that it satisfies the PDE: C t C + rs S + 1 σ S C rc =, S with the boundary condition C(S T, T ) = max (S T K, ). Through a lengthy derivation, 8 this partial differential equation can be solved by the well known Black-Scholes formula: where C t (S t, K, τ, r, σ) = S t Φ(d 1 ) Ke rt Φ(d ), (.1) d 1 = ln(s t/k) + (r + 1 σ )τ σ τ d = d 1 σ τ. Φ(y) = y φ(x)dx, the cdf of the standard normal distribution with the pdf φ(x) = 1 π e x /, S t is the stock price at time t, K is the strike price, r is a riskless interest rate, τ = T t, where T is the maturity date, and finally σ is the constant volatility parameter. While this is only the price for the call option, we can easily derive the put option price from the obtained result using the put-call parity, the relationship that holds between the call and put option prices (under the no-arbitrage assumption): P t = C t S t + e rτ K, where P t is the price of the put option. 7 In the integral form the equation would be, t S s ds = S + t µs s ds + t σs s dw s. 8 Multiple approaches exist, refer to, for example, Wilmott [7] 1

17 .4 Implied Volatility As the risk-free rate r can be estimated from the available market data, 9 the only parameter that is not directly observable on the market is the volatility of the underlying asset σ. Furthermore, the Black-Scholes model assumes the volatility of the option to be a constant term during the whole period of the option s lifetime. As is now widely believed, this is an erroneous assumption. One of the methods of obtaining an estimate of the actual volatility is the implied volatility (IV). Instead of using all the required parameters for the Black-Scholes model to obtain the option price, we use the price actually observed on the market, C t, and find the parameter σ, called the implied volatility, which, if inputted into the Black Scholes model, yields the observed price C t. If we denote the Black-Scholes price as Ct BS, we can define the implied volatility σ as the volatility σ, such that the following holds: C t = Ct BS (S t, K, τ, r, σ). Due to the monotonicity of the BSM price in σ, a unique solution σ [, ) exists..5 Volatility Smile and Volatility Surface As mentioned above, the volatility of the option is not a constant term with respect to both the parameters τ, the time-to-maturity, and K, the strike price. A smile-like pattern together with a skew are typical for the IV plotted against the strike price. Both indicate the non-normality of the underlying distribution; a smile suggests fat tails in the return distribution, whereas a skew suggests an asymmetry (as opposed to the symmetric normal distribution). 9 In practice, the risk-free rate is usually determined from the government securities using the popular bootstrap method Hull [11] 13

18 If we consider the mapping σ : (t, K) σ t (K), we obtain what is called the volatility smile. The name is due to the usual shape of the graph of this function, depicted in figure.3. Implied volatility If we consider the mapping Strike price Figure.3: The IV smile σ : (t, K, τ) σ t (K, τ), we obtain the volatility surface, represented as a three-dimensional graph depicted in figure.4 (Benko et al. [7]) Figure.4: The IV surface. κ is moneyness, defined in section.8. There exists a vast number of methods for obtaining the volatility smile and the volatility surface. For a comprehensive summary of various types as well as the 14

19 specific methods themselves, refer to Homescu [11]. Amongst the main types are models based on the local stochastic volatility, Levy processes, and parametric and nonparametric representations. This text will focus on the method of local polynomial estimation by Benko et al. [7], a non-parametric method since it does not assume any particular model for the underlying stock price process..6 State Price Density If we consider only the strike price K, the observational model becomes σ i = σ(k i ) + ɛ i, where i = 1,..., n (n denotes the number of observed IVs), σ is the IV calculated from the observed prices, σ(k i ) is the true IV function, and ɛ i is the observational error. The estimation is achieved by the local polynomial estimator, however, the results do not necessarily need to satisfy the no-arbitrage condition, rendering the plain local polynomial estimation unfeasible. For this reason, an additional constraint has to be added. Here Benko et al. [7] utilizes the notion of state price density (SPD): q t,st (x, τ) def = e rτ C t (K, T ) K. (.) K=x (State-price-density in terms of the call-price: Breeden and Litzenberger [1978]) The SPD is based on the following idea. If the market does not obtain any arbitrage opportunities, there necessarily exists an equivalent martingale measure 1 Q that is (uniquely) determined by the state price density q t,st of the underlying asset price process. Therefore the price Π t (H) of a derivative with pay-off H(S T ) is given by the arbitrage-free formula: Π t (H) = e rτ E Q (H F t ) = e rτ H(s)q t,st (s, τ)ds for all t (, T ). Combining the expression. and the BSM formula (.1) with the above relationship, we obtain the state-price-density in terms of IV: ( q t,st (K, τ) = e rτ 1 d 1 S t τφ(d1 ) K στ Kσ σ τ K + d ( ) ) 1d σ + σ. (.3) σ K K.7 Local Estimation The method used in this text is the local polynomial estimator, a non-parametric regression method. This method builds on the classic least squares method, but instead of using a single function with appropriate parameters to fit the data, the method performs the regression locally on a lattice of points, with neighbouring 1 Also called risk-neutral measure. 15

20 (a) Uniform kernel (b) Epanechnikov kernel Figure.5: Comparison of the uniform and Epanechnikov kernel points having more weight on the final result of the local regression. The method thus performs the regression many times over and as such is quite computationally expensive. The exact specifics of the local polynomial estimator used in this text are best understood through a simple example. The local estimation method can be split into two separate parts - the function used for local fitting and the kernel. If we were to minimize the function min α n (y i α), i=1 we would obtain a least squares mean of all the y values, independent of their location on the x-axis. However, if we were to do the same only on localized segments of the data, it stands to reason that we could obtain a curve somewhat fitting the data..7.1 The Kernel The localization is done through multiplying the expression by the kernel. Kernel in this context typically is a symmetric density function on the interval [ 1, 1]. The simplest example being an indicator function (figure.6a) K(u) = 1 1 { u 1}, or the kernel used in this text, the Epanechnikov kernel (figure.6b) K(u) = 3 4 (1 u )1 { u 1}. Such a function would of course only return nonzero values on the interval [-1,1]. To this end, the kernel bandwidth is utilized. Bandwidth is a parameter h that divides both the argument and the kernel itself; ( ) 1 x h K xi h 16

21 (a) Bandwidth too small (b) Bandwidth too large Figure.6: Effect of the bandwidth for smaller and large values is thus nonzero only on the interval [x h, x + h]. Since the bandwidth controls the width of the interval where points still have an influence on the estimation, it functions as a trade-off between bias and variance. Its role is illustrated in figures.6 and.7. A small interval will fit the data extremely well but will introduce a large variance (as even small perturbations in the data will influence the resulting fitted value). A large interval will mean a small variance and a smooth resulting curve, but might ignore the local behaviour of the data. Figure.7: Bandwidth just right.7. Truncated Gaussian Kernel The only kernel that deserves a little more attention from the mathematical point of view is the truncated Gaussian kernel. As noted earlier, a kernel function is nonzero typically on [ 1, 1]. If we therefore want to use the standard Gaussian density function as a kernel (which has an infinite support), we have to cut off the tails, i.e. multiply the density function by 1 { u 1} and arrive at the so called truncated Gaussian kernel. In order for the kernel to be a density function the integral over its domain needs to equal zero. Thus the density ψ(u) of the 17

22 truncated Gaussian kernel for u [ 1, 1] has the form ψ(u) = φ(u) Ψ(1) Ψ( 1) = φ(u) 1 (1 + erf(1/ )) 1(1 + erf( 1/ )) = φ(u) erf(1/ ), where erf(u) = 1 u π u e t dt, the error function, which is very useful in many computations with the normal distribution. The fact that erf(x) = erf( x) was used in the computation and the identity Ψ(u) = 1(1 + erf(u/ )) is easily arrived at through substitution..7.3 Local Estimation in the Context of Options Data Hitherto we have only used a constant as the function, thus assuming the underlying generator of the data is locally constant. This may not be a valid assumption and thus different functions may be used. Typically the assumption is that the function is locally polynomial, and this is also the approach we will use for the option data. The method used in Benko et al. [7] is the local quadratic estimator together with the constraint of non-negative SPD. The objective function then becomes n τ ( ) min σ i α α 1 (K i K) α (K i K) K h (K K i ), α,α 1,α i=1 where n τ is the number of observations for a given time to maturity and σ i is the value of the ith observation, and by comparing the coefficients with the Taylor expansion of σ 11 σ(k) = σ(k i ) + σ (K i )(K K i ) + 1 σ (K i )(K K i ) + O(K 3 ), we obtain the relationships for the local estimator ˆσ of σ α = ˆσ(K i ), α 1 = ˆσ (K i ), α = ˆσ (K i ) and thus, after substituting into.3, the constraint becomes ( ) e rτ 1 S t τφ(d1 ) K α τ + d 1 α 1 + d 1d (α 1 ) + α. Kα τ α where d 1 = ln(st/k)+(r+ 1 (α ) )τ α, d τ = d 1 α τ, and Kh (K K i ) def K being the kernel function. = 1 h K ( K K i 11 The expansion is expressed only up to the second degree for the purposes of obtaining the estimators for the parameters. Since the analysis in chapter three does include the cubic polynomial we could continue with the term 1 6 σ (K i )(K K i ) 3 and obtain the relationship 6α 3 = σ (K i ). However, since the SPD does not feature any derivatives of third (or higher) order, deriving the relationship is unnecessary. h ), 18

23 .8 Moneyness Since the data used are intra-day, the previously stated form of the constraint derived from the BSM model is not appropriate for this purpose. In the case of the intra-day data, the underlying prices, S t, are not constant. It is however possible to express the constraint as a function of the future moneyness κ = K S te and thus combine two variables into one - the strike price K and the stock rτ price. Apart from being a useful concept for making the intra-day case optimization possible, the broader concept of moneyness is also a frequently used concept in derivatives. The basic idea lies in relating the current price of the underlying asset to the strike price of the option. Moneyness, in general sense, answers the question Would the derivative be profitable if it were to expire today? The derivative is said to be: In the money if the derivative were to make money At the money: if the current price and strike price are equal Out of the money: if the derivative were not to make money For example, a call option on a $1 stock is in the money if the strike is $9, out of the money if it is $11 and at the money if it is $1. The actual function used to represent moneyness only needs to be monotone for both strike price and stock price. The simplest non-trivial example is simple moneyness, the ratio S t /K or the reciprocal K/S t, depending on whether the option is call or put respectively. Since the expression S t e rτ remains monotone in S t, the future moneyness used in this text is a valid moneyness function. Future moneyness compares the future value of the current stock price at the expiry date with the strike price. Simple moneyness only imagines the strike price to be present moment - in future moneyness, both variables actually are valued at the same date and time - the expiry date. In this sense, future moneyness can be considered more accurate..9 The Optimization Problem Expressed Through Moneyness Since the objective function does not utilize the stock price, it can simply be expressed as a function of the future moneyness in the form min α,α 1,α n τ i=1 { σ i α α 1 (κ i κ) α (κ i κ) } Kh (κ κ i ), where σ i are now the IVs expressed in terms of the moneyness. In the constraint the situation is slightly more difficult as both K and S t are used, as well as derivatives of σ. The derivation that follows is merely a more detailed version of Benko et al. [7]. 19

24 Using the fact that K = κf we compute the derivatives Using the chain rule, from we obtain and from σ κ = σ K K κ = F σ κ = σ K K κ σ K = σ κ 1 F, ( ) K + σ κ K K κ where K =, we obtain κ σ K = σ κ 1 F. Substituting κ into d 1 we obtain d 1 = ln(κerτ ) + (r +.5σ )τ σ τ Using the fact that ln(κe rτ ) = ln(κ) + rτ, this can be rewritten as d 1 =.5σ τ ln(κ) σ. τ Using the derived equations for the derivatives on.3 we obtain q(κ, τ) = F { 1 τφ(d 1 ) (κf ) στ + d 1 κf σ τ σ κ 1 F + d ( 1d σ σ κ 1 ) + σ F κ 1 }, F which simplifies to q(κ, τ) = 1 { 1 τφ(d1 ) F κ στ + d 1 κσ τ σ κ + d ( ) 1d σ + σ }, σ κ κ but because we have now exchanged variables, we need to rescale the SPD in order for the condition q(k)dk = 1 to hold. Since K = κf, we obtain (using substitution) q(k)dk = q(κ)f dκ. Therefore the F cancels out and finally we obtain q(κ, τ) = { 1 τφ(d 1 ) κ στ + d 1 κσ σ τ κ + d ( ) 1d σ + σ }. σ κ κ.1 References Sections.1 to.5 have been mostly based on Hull [11], Wilmott [7], and Black and Scholes [197], although nearly all previously stated literature has been used at some point too. Sections.6 to.9 closely follow Benko et al. [7], but are also based on Fengler [1] and Homescu [11]..

25 3. Results and Discussion The method used is local polynomial estimation, as described in section.7. Most estimation is done with the Epanechnikov kernel, but results with other kernels can be found in section 3.8. The optimization problems were calculated using the Sequential Least Squares Programming (SLSQP), except for the results with constant estimator which used the Constrained Optimization by Linear Approximation (COBYLA), both implemented in Python 3 using the packages NumPy (matrix operations), pandas (data handling), SciPy (optimization), matplotlib (plotting). The numerical integration of the calculated SPDs was done with SciPy using Simpson s rule. 3.1 The Data and Methods Used The data used are the same data as Benko et al. [7], that is the intra-day call and put option quotes from December 9, 3 with the underlying stock being the DAX (Deutsche Aktienindex). As seen in figure 3.1, the data show a distinct IV smile. The distribution of the moneyness values is quite uneven; most data are centered around at-the-money (ATM) values, the far left out-of-the-money (OTM) options have only a few values Figure 3.1: The intra-day data used for optimization. Vertical axis - IV, horizontal - moneyness. The optimization results will be judged according to the following criteria: 1. Visual fit of the curve. Values of the objective function (both local values and global sum) 3. Shape and smoothness of SPD 1

26 4. Value of SPD after integration, i.e. checking the condition q(κ)dκ = 1 The primary objective will be to compare the results to the Benko et al. [7] paper s 1 solution: local quadratic estimator using the Epanechnikov kernel with bandwidth h =.45 show in figure 3.. black: deg=, h =.45, kern = epanechnikov IV Figure 3.: Solution of the original paper. Left: fitted curve (legend is the sum of the objective function over the domain), middle: SPD (legend is q(κ) dκ 1), right: objective function. 3. Notes on The Integral Condition While the integral condition may seem unimportant as the obtained SPD can simply be divided by the value of the integral to yield a legitimate density function, it is still worth observing, as the methods often implicitly yield an SPD function that does fulfill the integral condition. However to ensure accuracy of the results, we have to estimate the area under the right tail of the SPD that is visibly cut off after 1.1 as it lies outside of the interval of the data. Thus to be able to assess the integral condition q(κ) dκ = 1 we need to estimate the area under the missing tail. The method used was to assume that the density drops to zero before reaching the value 1.15, that the speed of the decrease is similar as in the area between.88 and.9, and that the shape of the SPD generally resembles the shape of the original paper s solution. Depending on small changes in parameters the difference from the cutoff SPD was between.7 and.1. A better estimate could be developed, but this value serves as a rough idea. 3.3 Degree of the Polynomial The first parameter to be evaluated was the degree of the polynomial used in the local polynomial estimation. As the only parameter, the degree actually alters the constraint. This alteration however consists in simply setting the remaining parameters to zero. 1 From now on referred to as the original paper.

27 3.3.1 deg = By setting α 1 = α = the optimization problem becomes min α n τ i=1 { σi α } Kh (κ κ i ), subject to q(κ, τ) = { 1 } τφ(d 1 ), κ α τ that is a form of a moving average. It is worth noting that the SLSQP algorithm failed here possibly due to the shape of the constraint, which quickly approaches zero for a sufficiently large α. Therefore the COBYLA algorithm was used instead. black: deg =, h =.45, kern = epanechnikov red: deg =, h =.45, kern = epanechnikov IV Figure 3.3: Comparison with polynomial of degree. Lower row contains differences. As immediately obvious from the values of the objective function in figure 3.3, the constant approximation does not perform well in fitting the data. However it does improve the smoothness of the SPD but does not perform as well on the integral condition. Given the value of the objective function however (which does not improve much even for other bandwidth settings) the local constant estimator was deemed unfit for the task deg = 1 Through setting α = the optimization problem becomes min α,α 1 n τ i=1 { σ i α α 1 (κ i κ)} Kh (κ κ i ), 3

28 subject to q(κ, τ) = { 1 τφ(d 1 ) κ α τ + d 1 α 1 + d } 1d α κα τ 1. α black: deg =, h =.45, kern = epanechnikov red: deg = 1, h =.45, kern = epanechnikov IV Figure 3.4: Comparison with polynomial of degree 1. Lower row contains differences. Linear estimation reacts less to local changes in the data compared to the quadratic estimation (other parameters being equal). This leads to a smoother SPD, but performs worse in terms of the integral condition and overall data fit. More discussion will be needed with connection to bandwidth to assess the quality of linear estimation, which will be done in section deg = 3 Instead of setting parameters to zero as in previous cases, in the case of the cubic estimator we instead introduce a new coefficient α 3. Since the third derivative of σ does not appear in the SPD expression, the constraint remain unchanged. The optimization problem becomes min α,α 1,α,α 3 subject to n τ i=1 { σ i α α 1 (κ i κ) α (κ i κ) α 3 (κ i κ) 3 } Kh (κ κ i ), q(κ, τ) = { 1 τφ(d 1 ) κ α τ + d 1 α 1 + d } 1d α κα τ 1 + α. α 4

29 black: deg =, h =.45, kern = epanechnikov red: deg = 3, h =.45, kern = epanechnikov IV Figure 3.5: Comparison with polynomial of degree 3. Lower row contains differences. It is apparent at first sight that the third degree polynomial shows virtually no difference from the second degree, as do polynomials of any higher degree. Different kernels and bandwidths have been tried with the same result. This makes sense, as we are modelling the local behaviour of the curve and in most cases (where it is non-zero) the second degree Taylor polynomial is a good estimator on a small enough neighbourhood. Furthermore, the greatest strength of the cubic polynomial compared to quadratic arguably is the ability to change convexity which, again, is not a very useful property in the local sense. Combined with the fact that the third degree polynomial is computationally more expensive this renders the cubic estimator inferior to the quadratic Summary of the Polynomial Degree The constant estimator improves the smoothness of SPD but fails in every other area. Most notably the sum of the objective function values is five times higher than that of the quadratic estimator. The linear estimator has shown itself to be the only option worth exploring further as it both smoothes out the SPD and retains a reasonably low value for the objective function, although at the price of somewhat worse performance on the integral condition. Higher degree polynomials only bring higher demands on computation with virtually no effect on performance. The salient point of the analysis seems to be the interval around the value.83. Both the sparsity of the data points together with a more volatile behaviour of the IV values challenge the estimation methods. The low volume of options traded around that moneyness level will result in an exaggerated importance of each data point, especially of any outliers. More discussion on this will come 5

30 later. This means that a higher value of the objective function is perhaps not necessarily a worse result if the higher value is caused by higher smoothing in the.83 area. 3.4 Constraint The original paper compared the optimization results with and without the constraint in the quadratic estimator. We will now have a look at its role for the other degrees. Interestingly enough, the constraint does not influence the performance for both constant and linear estimators, as the SPD obtained through unconstrained optimization already remains nonnegative everywhere on its domain. Many other bandwidths have also been tried with the same result of no difference between unconstrained and constrained optimization. black: deg =, h =.45, kern = epanechnikov red (constraint off): deg =, h =.45, kern = epanechnikov IV Figure 3.6: Comparison with a solution without the constraint, degree Apart from small numeric differences caused by the mere presence of a constraint in the optimization algorithm. 6