Arbitrage-free market models for interest rate options and future options: the multi-strike case

Transcription

1 Technical report, IDE022, November, 200 Arbitrage-free market models for interest rate options and future options: the multi-strike case Master s Thesis in Financial Mathematics Anastasia Ellanskaya, Hui Ye School of Information Science, Computer and Electrical Engineering Halmstad University

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3 Arbitrage-free market models for interest rate options and future options: the multi-strike case Anastasia Ellanskaya, Hui Ye Halmstad University Project Report IDE022 Master s thesis in Financial Mathematics, 5 ECTS credits Supervisor: Prof. Mikhail Nechaev Examiner: Prof. Ljudmila A. Bordag External referee: Prof. Vladimir Roubtsov November, 200 Department of Mathematics, Physics and Electrical engineering School of Information Science, Computer and Electrical Engineering Halmstad University

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5 Preface The investors concern the most, when it comes to the financial derivatives market, is a fair and reasonable pricing of instruments. People interpret this as the underlying fundamental properties that should be possessed by the pricing models. They can be described mainly as the following: the models should be arbitrage-free; the model should be able to reproduce any initial option price data collected in the market; the model should incorporate characteristic features of joint dynamics of stock and options. One of such models is constructed and verified to be possessing the above properties by Martin Schweizer and Johannes Wissel in their paper [0] published in Inspired by their work, we take one step forward in this direction and extend the original model to be applicable for pricing options on interest rate indexes and options on futures. We draw a periodic end to our research by demonstrating the practical use of Schweizer-Wissel model in dealing with investment portfolios. During our work, we get some tremendous help from our professors, and we would like to thank all of them sincerely. It is a honor to be working under the supervision of Professor Mikhail L. Nechaev. We appreciate many helpful hints and discussions from him. We are also grateful for valuable comments and help with some technical difficulties from Professor Ljudmila A. Bordag. We also would like to thank our external referees and other professors from Financial Mathematics group for their sincere, honest and instructive comments and advices, from which we benefitted a lot. We are really looking forward to working under their supervisions again some day.

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7 Abstract This work mainly studies modeling and existence issues for martingale models of option markets with one stock and a collection of European call options for one fixed maturity and infinetely many strikes. In particular, we study Dupire s and Schweizer-Wissel s models, especially the latter one. These two types of models have two completely different pricing approachs, one of which is martingale approach in Dupire s model, and other one is a market approach in Schweizer-Wissel s model. After arguing that Dupire s model suffers from the several lacks comparing to Schweizer-Wissel s model, we extend the latter one to get the variations for the case of options on interest rate indexes and futures options. Our models are based on the newly introduced definitions of local implied volatilities and a price level proposed by Schweizer and Wissel. We get explicit expressions of option prices as functions of the local implied volatilities and the price levels in our variations of models. Afterwards, the absence of the dynamic arbitrage in the market for such models can be described in terms of the drift restrictions on the models coefficients. Finally we demonstrate the application of such models by a simple example of an investment portfolio to show how Schweizer-Wissel s model works generally. iii

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9 Contents Introduction. Motivation Objectives The Chapter review The Literature Review 3 2. The Martingale models Dupire s model The market models The Schweizer-Wissel model Conclusions and comments The extensions of the Schweizer-Wissel model 9 3. An option on interest rate indexes The new parameterization The arbitrage-free dynamics of the local implied volatilities An option on Futures The new parameterization The arbitrage-free dynamics of the local implied volatilities Applications 53 5 Conclusions 59 Notation 63 Appendix 65 Bibliography 67 v

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11 Chapter Introduction. Motivation Since the famous Black-Scholes model was published by Fischer Black and Myron Scholes in 973, the evolution of pricing derivative investment instruments has taken place. While being widely used in the financial instruments pricing, there are still some defects found during its application in the real financial market. Especially after the Black Tuesday of year 987, a new phenomenon was found in the structure of options prices. Before crisis, it is commonly believed that the volatility of the option should not be depending on its strike and expiration. Then the fact that the observed volatility surface became skewed was contradicted to the classic understanding of the volatility. Since the Black-Scholes model can not account for this phenomenon, which had been called the volatility smile, traders had to invent some new tricks to incorporate this effect into pricing models. One of the most popular solutions is to connect this effect to the notion of Stochastic Local Volatility, which was presented in the paper Pricing with a smile by B. Dupire, 994. The idea of the Stochastic Local Volatility was a real improvement of the pricing models, but still it was not consistent enough with the structure of the option prices. Therefore, other pricing models were proposed for the whole option series pricing instead of describing an evolution of a single option. One of such models was introduced by M. Schweizer and J. Wissel in their paper Arbitrage-free market models for option prices: the multi-strike case. Our study here is mainly based on the Schweizer-Wissel model and takes one step forward in the proposed direction.

12 2 Chapter. Introduction.2 Objectives Our main objective is to get the possible extensions of the Schweizer-Wissel model, for the cases of two other derivatives, which are an option on futures and an option on interest rate indexes. Moreover, our by-product here is hopefully doing some practical financial analyses of the investment portfolios based on the two variations of the Schweizer-Wissel model..3 The Chapter review Chapter, Introduction. This chapter intends to presents the readers with some general ideas about the study background of introducing Schweizer- Wissel s model, as well as our motivation and objectives of this study. Chapter 2, Literature Review. In this chapter, we shortly introduce two main pricing approaches and models. They are the traditional martingale approaches and the market models. After reviewing some main literature concerning this subject, we discuss our ideas of these two different approaches and models, and describe them more precisely in two specific models chosen from each approach s category, the Dupire s model and the Schweizer- Wissel s model. Chapter 3, The extensions of Schweizer-Wissel s model. After an introduction of two underlying assets, i.e. futures and interest rate indexes which our extensions of Schweizer-Wissel s model are based on, we derive the extentions of the original Schweizer-Wissel s model in both of the scenarios, focusing on the new parameterizations, i.e. definitions and basic properties, and thus, present our results including the arbitrage-free dynamics of the local implied volatilities. Chapter 4, Applications. We start this chapter with demonstrations of some simple investment portfolios, to show how Schweizer-Wissel model can be possibly used in applications. We end it by verifying our assertment that under some special circumstances, Schweizer-Wissel s model can be unified with the Black-Scholes model which provides us the consistency between pricing models. Chapter 5, Conclusions. Finally, in the last chapter, we complete the study by drawing some conclusions of our work, summarizing what we have achieved in this area and proposing some thoughts about a further possible research direction.

13 Chapter 2 The Literature Review In this chapter, we describe the two main approaches to the option pricing and review them by the comparison of the Dupire model and the Schweizer Wissel model. Our ultimate goal of the financial instrument pricing, as started out in [2], is to establish a framework for the pricing and hedging of derivatives possibly exotic ones in an arbitrage-free way, using all the liquid tradables as potential hedging instruments. In order to achieve this goal, the class of models for bond, stock and options should at least possess the following features: 0 Fundamentally, these models should be arbitrage-free. Any initial option price data from the market can be reproduced by these models; this is called perfect calibration or smile-consistency. 2 The empirically observed stylized facts from the market time series, i.e. characteristic features of the joint dynamics of the stock and options, can be incorporated in these models. This implies that the explicit expressions for the option price processes and their dynamics should be available. After a thorough study of literature concerning the option pricing, one can easily categorize all approaches into two basic classes. The overwhelming majority of the literature uses the martingale approach, where one specifies the dynamics of the underlying S under some pricing i.e. martingale measure Q and defines the option prices C t, T by C t, T := E Q [S T + F t ], 3

14 4 Chapter 2. The Literature Review where F t is a sub-σ-field of F and Ω, F, P is a stochastic basis. Obviously, the assumption 0 holds, and for so-called smile-consistent models, a perfect fit as described in the assumption is available. However, this approach is still not perfect in the sense that the assumption 2 is usually not feasible, or comes with a loss with assumption to some extent. An alternative approach is the use of market models where one specifies the joint dynamics of all tradable assets, stock and options, for instance. This gives and 2 instantly by construction, but the absence of the arbitragefree requirement in the assumption 0 still remains to be satisfied. As explained in paragraph 2.2., the absence of the dynamic arbitrage corresponds to drift conditions for the joint dynamics of the stock S and the option prices C, T, and additionally the absence of the static arbitrage enforces a number of relations between the various C, T and S, which means the state space is constrained as well. Thus, to obtain such a tractable model, one must therefore reparameterize the tradables in such a way that the parameterizing processes have a simple state space and yet still capture all the static arbitrage constraints. Generally speaking, most of the literature with actual results on the arbitragefree market models for option prices, can be summarized in terms of the families and T. For the case = {}, T = {T } of one single call option available for trade, there are both an existence result and some explicit examples for models. For models with = {}, T = 0, one fixed strike, all maturities, the drift restrictions are well known, but the existence of the models has been proved only very recently. The other extreme = 0,, T = {T } all strikes, one fixed maturity is the focus of the Schweizer Wissel model. It is more difficult and has to the best of our knowledge no precursors in the terms of the parameterization or results. Finally, the case = 0,, T = 0, of the full surface of strikes and maturities is still open despite the recent work by Carmona and Nadtochiy []. 2. The Martingale models The overwhleming majority of the models for the option pricing uses the martingale approach, where the dynamics of the underlying asset S under some martingale measure Q and option prices are specified by

15 Arbitrage-free market models for IRO and future options 5 C t, T := E Q [ ST + Ft ], 0 t T, for 0,, and T T 0,. These models satisfy the assumption 0 by construction. The calibration to the known market option prices in the assumption is admissable in the stochastic volatility models or in models with jumps. But the calibration is restricted by the requirement that there only exists a finite amount of parameters to be fitted. Most of the famous models which achieved such a perfect fit for the whole option surface C 0, T for = 0,, T = 0, are the smile-consistent models, for instance, the Local Volatility Model of Dupire [2], and other local volatility models, like the model developed by Carmona and Nadtochiy []. But these models have one mutual fundamental defect that the joint dynamics of the option prices C, T and the stock prices S are not available. As we mentioned earlier, in the beginning of this chapter, an option pricing model should possess one of the fundamental properties such that empirically observed stylized facts from the market time series can be incorporated in the model. This leads us to the requirements that explicit expressions for the option price processes and their dynamics can be attained by some regular procedures. Otherwise we will not be able to express drift restrictions by terms of the coefficients of the joint dynamics of the price and volatility s evolutions. 2.. Dupire s model Dupire s model was a breakthrough in 994 for the option pricing theory. This model is such a type that in it we can avoid the complex calculations like in stochastic volatility models and difficulties by the fitting parameters to the current prices of options. The main idea of Dupire s model is the following that under risk-neutrality there is a unique diffusion process which is consistent with stock prices distributions. The corresponding coefficient σ L S, t of the unique diffusion process is known as local volatility function which is also in consistence with the current option prices. Indeed, the local volatility models do not represent a separate class of models, the basic idea of them is mainly to simplify assumptions so that practitioners can price exotic options consistently with the known prices of vanilla options. Dupire showed that there is a unique risk-neutral diffusion process which generates distributions of the final spot prices S T for each time T conditional

16 6 Chapter 2. The Literature Review on some starting spot price S 0. The set of all European option prices is known and it is possible to determine the functional form of the diffusion parameter local volatility of the unique risk-neutral diffusion process which generates these prices. In general, the local volatility will be a function of the current stock price S 0 and this stock price random process is described by the following equation, ds = µ t dt + σ S, t; S 0 dz. 2. Now we demonstrate the derivation of Dupire s equation lying underneath the model. Suppose for the given maturity time T and the stock price S 0, the set {CS 0,, T ; 0, } of undiscounted option prices with different strikes produces the risk-neutral density function ϕ of the final spot price S T and is defined by the relationship CS 0,, T = Differentiating 2.2 with respect to gives C = ds t ϕ S T, T ; S 0 S T. 2.2 ds t ϕs T, T ; S 0, 2 C 2 = ϕ, T ; S 0. In particular, the pseudo probability densities ϕ, T ; S 0 = 2 C must satisfy 2 the Fokker-Planck equation [5], and in the current situation it is consequently that 2 σ 2 ST 2 ϕ µs T ϕ = ϕ 2 S T T. S 2 T Now, differentiation 2.2 with respect to T gives = C T = ds t 2 2 S 2 T Integrating it by parts twice gives ds t T ϕs T, T ; S 0 S T σ 2 S 2 T ϕ S S T µs T ϕ S T.

17 Arbitrage-free market models for IRO and future options 7 C T = σ2 2 ϕ + 2 ds t µs T ϕ = σ C + µt C C Equation 2.3 is the Dupire equation if the underlying stock has a riskneutral drift µt. If µt = rt DT, where rt is the risk-free rate, Dt is the dividend yield and C is the short notation for CS 0,, T, then equation 2.3 acquires a form C T = σ2 2 2 C + rt DT C C The forward price of the current stock at time T is given by F T = S 0 2 T exp µtdt 0 such that one get the same expression of 2.3 minus the drift term C T = σ2 2 2 C 2, 2 where C now represents CF T,, T. Inverting this formula gives σ 2, T, S 0 = C T C The right hand side of equation 2.5 can be computed directly from the known European option prices. This means that the local volatilities defined by equation 2.3 are determined uniquely, meanwhile equation 2.5 can be represented as a definition of the local volatility function under Dupire s model, regardless what kind of the process actually governs the evolution of the volatilities. 2.2 The market models An alternative approach for the options pricing uses the market models, in which one can define the arbitrage-free joint dynamics of the stock prices and the option prices. The market models always satisfy assumptions and 2 by construction, but one other problem remains to be solved is how to show the absence of arbitrage, i.e. the model has to admit the assumption 0. In the interest rate models, it leads to the HJM drift conditions, but in the case

18 8 Chapter 2. The Literature Review of the options it is much more complicated to ensure the absence of arbitrage. One has to ensure that the option price models have to satisfy not only the HJM drift conditions, but also the static arbitrage bound restrictions on the state space of the quantities which describe the model. In this sense, the selection of an appropriate parameterization becomes a key issue. Indeed, the absence of the dynamic arbitrage is connected with the drift conditions on the joint dynamics of S and C, T. However, the absence of the static arbitrage also ensures a number of relations between the various C, T and S. It follows that the state space of the processes C, T and S is constrained also. To obtain a well-constructed model, the reparameterization should be done in such a way that after parameterizing processes, one get a simple state space which captures all the static arbitrage constraints, as we discussed above. Study of some special cases has been done before and can be easily found in the literature. If the option set comprises only a single call C = C, T, one has the static arbitrage bounds S t + C t S t as well as the terminal condition C T = S T. To define S and C dynamics directly which admit these constraints on the state space is not an easy procedure. It is much more easier to reparameterize the option price C, using the implied volatility ˆσ via C t = cs t,, ˆσ 2 t where c is the Black-Scholes formula defined by 2.6. Then the values of S and ˆσ can take any value in 0,. After this procedure, the static arbitrage conditions and the terminal condition are fulfilled. It means that one can proceed to define the joint dynamics of S, ˆσ. Such market models of implied volatilities for the pricing a single option was firstly proposed by Schönbucher [7]. Even in the case of one single option as simple as it is the construction of such model is still not completely straightforward: the drifts are essentially specified by the volatilities of S and ˆσ. Additionally, if we take these nonlinear drift restrictions, the resulting two-dimensional SDE system for S and ˆσ admits a non-trivial solution. For the option set consists of not only one single call, the situation becomes much more intricate. Indeed, for the resulting model to be an arbitrage-free, several versions of the necessary conditions on the implied volatility dynamics are described in the literature one of them was introduced by Schönbucher. But these models didn t incorporate the case of options with multi-strikes or multi-maturities. It means that the sufficient conditions are not given, in other words, the existence of such models with defined dynamics is an open question. The main problem is that in the case of options with different strikes and maturities, the realization of the static no-arbitrage conditions has

19 Arbitrage-free market models for IRO and future options 9 an unpleasant consequence that quite complicated relations will arise among the implied volatilities of options. The illustration of this situation is given in [0], Chapter 3 in details. The main idea we are trying to point out here is the existence of a crucial problem that the classical implied volatilities are not appropriate for the modeling call option prices in a multi-strike case or in a multi-maturity case, and they give the wrong parameterization, despite their importance in quoting option prices on the market in standard way. The idea of replacing the implied volatilities by other quantities such as parameterization of all call option prices in the market models is not quite new. In the case of the family with one fixed strike and all maturities T > 0, = {}, T = 0,, Schöbucher has specified the forward implied volatilities. But in the case of the other extreme of the spectrum, i.e. the case of a family of options with one fixed maturity and all strikes T > 0, it is much more complicated to reparameterize the model because the assets in the interest rate market models don t have a strike structure. The breakthrough in this direction was done by Schweizer and Wissel [5]. They considered one extreme case = 0,, T = T and constructed arbitrage-free market models for the set of call options, which have one fixed maturity T and all strikes > 0. Schweizer and Wissel introduced a new parameterization by specifying the new quantities called local implied volatilities of the option prices for the multi-strike case, so that the arbitrage-free dynamic modeling will actually be easier to perform. These quantities have no comparable predecessors or analogues in the interest rate theory. The crucial feature of these parameters is that they have a simple state space and capture all static arbitrage restrictions. In spite of that Schweizer and Wissel provided a new parametrization and defined an arbitrage-free dynamic of the local implied volatility, but the dynamic arbitrage conditions and existence results for the dynamic option models still need to be studied The Schweizer-Wissel model The classical implied volatilities models suffer from one disadvantage. For the market models with more than one strike, one stock, several call options and squared implied volatilities satisfying the arbitrage-free conditions, it is not clear how to choose the volatility coefficients in the diffusion processes of the stock price and squared implied volatilities in order to determine the value of the drift terms, which implies the absence of arbitrage. The Schweizer-Wissel model uses market models where the stock and option

20 0 Chapter 2. The Literature Review price processes are constructed simultaneously, that is why this model fulfills the assumption about the perfect calibration and the assumption 2 that implies that this model has at the same time joint dynamics for the stock and options prices. The main idea of the Schweizer-Wissel model is the choice of a good parameterization, which should be done in such a way that the static arbitrage restrictions do result in a not complicated state space for the quantities which describe the model. Note also that the option prices are not automatically conditional expectations, as in the martingale approach, and the absence of the dynamic arbitrage now transfers into the drift conditions on the new modeled quantities to guarantee the local martingale property. Thus Schweizer and Wissel proposed the new concept of the local implied volatilities and price levels to overcome difficulty described above. These new quantities can be used to construct, and prove the existence of the arbitragefree multi-strike market models with the specified volatilities for the option prices See Schweizer and Wissel [0], Chapter 4.3 and Chapter 5. In this chapter we make a brief review of the Schweizer-Wissel work and introduce the main definitions and properties underlying the model and describe in details the new parametrizaton which provides the joint arbitrage-free dynamics of the stock and option prices. The new parameterization: definitions and basic properties Throughout this paragraph, we work with the following setup. Let Ω, F, P be a probability space and T > 0 a fixed maturity. Let S t 0 t T be a positive process modeling the discounted stock price and B t 0 t T be a positive process with B T, P a.s., modeling the discounted price of a zero-coupon bond with maturity T. Let C t 0 t T be a non-negative process modeling the price of the European call options on S with one fixed maturity T > 0 and all strikes > 0. Also, let cs,, γ be the Black- Scholes function log S cs,, γ = SN + γ 2 log S N γ 2 γ 2 γ 2 γ > 0, 2.6 cs,, 0 = S +, 2.7 where N is the standard normal distribution function. The value C t satisfies

21 Arbitrage-free market models for IRO and future options S t B t + C t S t, for all 0 t T. 2.8 According to , the implied volatility of the price C t is the unique parameter ˆσ t 0 satisfying the condition cs t, B t, ˆσ 2 t = C t. 2.9 Equation 2.9 can be rewritten, for every positive numeraire process M as cs t /M t, B t /M t, ˆσ 2 t = C t /M t. We would like to point out that the bond B always uses from now on as a numeraire, and therefore, all price processes B, S, C are denoted as B-discounted price processes, so that B. By setting C t 0 = S t, the model is specified through the processes C, 0, on the interval [0, T ]. Definition. A function Γ : [0, [0, is called a price curve. A price curve is called statically arbitrage-free if it is convex and satisfies Γ + 0 for all 0. This definition is motivated by the following proposition. Proposition. If the market where 0 does not admit an elementary arbitrage opportunity, then for each t [0, T, the price curve Γ := C t is statically arbitrage-free. Proof: See Davis and Hobson [3], Theorem 3.. Definition 2. An option model C, 0 is called admissible if the price curve Γ := C t has an absolutely continuous derivative with Γ > 0 for all > 0, < Γ < 0 for all > 0, and lim Γ = 0 for each t [0, T, and if we have C T = S T + for all 0, P -a.s. Let N denote the quantile function and n = N the density function of the standard normal distribution. The first and second partial derivatives with respect to the strike of Black- Scholes function cs,, σ 2 are given by c S,, σ 2 = Nd 2,

22 2 Chapter 2. The Literature Review with d 2 = log S T tσ 2 /2 T tσ. Hence the identity is c S,, σ 2 = nd 2 σ, σ = nn c S,, σ 2 c S,, σ 2. Now a new set of fundamental quantities, which allow a straightforward parameterization of the admissible option model, can be defined in such form: Definition 3. Let C t 0 t T be admissible. The local implied volatility of the price curve at time t [0, T is the measurable function X t given by X t := N C t C t for a.e. > 0, 2.0 and the price level of the price curve at time t [0, t for a fixed 0 0, is defined by Y t := N C t Proposition 2. Let X and Y be the local implied volatilities and the price level of an admissible model C, 0. Suppose that, for a small interval I = [a, b] 0, and a fixed t < T, we have X t = X t a for all I. Then there exists a unique pair x t, z t 0, 0, such that cz t,, x 2 t cz t, 2, x 2 t = C t C t 2 holds for all, 2 I. It is given by x t = X t a, z t = exp X t ay t X t a a 0 + log a + X X thh 2 ta 2. Proof: See Schweizer and Wissel [0], Proposition 4.4.

23 Arbitrage-free market models for IRO and future options 3 This property justifies the therminoligy local implied volatility by demonstrating that there exists such a unique implied volatility parameter x t inside the third argument x 2 t of Black-Scholes formula for all call options, that the differences C t C t 2 of the call option prices are consistent with strikes, 2 I = [a, b]. It means that, with the same implied volatility parameter x t and the same implied stock price z t, the result in Proposition 2 holds only locally. It remaims to give the exact form for the option prices under the new parameters, using the one-to-one corresponding relation between the local implied volatilities and the price levels. It is shown in the following theorem. Theorem. Let X, Y be the local implied volatilities and the price level of the admissible model C. Then C t = N Yt k 0 X thh dk, [0,, 2.2 Yt k C t = N 0 X thh, [0,, 2.3 Yt k C t = n 0 X thh X t, [0,. 2.4 Conversely, for the continuous adapted processes X > 0, Y on [0, T ] for which the right-hand side of 2.2 is finite P-a.s., define C t by 2.2. Then C, 0 is an admissible model having the local implied volatilities X and the price level Y. Proof: See Schweizer and Wissel [0], Theorem 4.6. We also present, for the finiteness of the integral in 2.2, a sufficient criterion which is a condition on X. Proposition 3. If there exists > 0 such that X t for a.e., then the outer integral in 2.2 is finite. Proof: See Schweizer and Wissel [0], Proposition ln T t

24 4 Chapter 2. The Literature Review The arbitrage-free dynamics of the local implied volatilities Since Schweizer and Wissel have introduced a new parametrization and defined the explicit formulas for the prices of options with the strikes > 0 and one maturity T, it is possible to derive an arbitrage-free dynamics of the local implied volatilities. To achieve this goal, we need to add some additional assumptions. Let W be an m-dimensional Brownian motion on Ω, F, P, F = F t 0 t T the P -augmented filtration generated by W, and F = F T. Suppose that there is a positive process X t for a.e. > 0, satisfying the conditions in Proposition 3, and a real valued process Y t with P -dynamics dx t = u t X t dt + v t X t dw t, 0 t T, 2.5 dy t = β t dt + γ t dw t, 0 t T, 2.6 where β, u L loc R, and γ, v L2 loc Rm for a.e. > 0 and u, v are uniformly bounded in ω, t, such that the initial local implied volatility curve satisfies 0 X 0 < for all > 0. The process C h 2 t, 0 is defined by Theorem, X t and Y t are the local implied volatilities and the price level of the option prices C t, 0. Note that S t = C t 0 and the values X t 0 for defining C t 0 via 2.2 are not necessary. Now the aim is to show the existence of a common equivalent local martingale measure for the set of the option prices C for 0,. Theorem 2. a If there exists a common equivalent local martingale measure Q for all C 0, then there exists a market price of the risk process b L 2 loc Rm such that the drift restrictions 2.7, 2.8 for a.e. > 0 hold for a.e. t [0, T ], P -a.s. + Y t 0 Y t β t = 2 u t = [ X t hh γt 2 γ t b t, γ t + γ t v t h 0 X t hh ] v t h X t hh v t + v t 2 v t b t, 2.8

25 Arbitrage-free market models for IRO and future options 5 b Conversely, suppose that the coefficients β, γ, u and v satisfy, as functions of Y t and X t, relations 2.7, 2.8 for a.e. > 0 for a.e. t [0, T ], P -a.s. for some bounded uniformly bounded in t, ω process b L 2 loc Rm. Also suppose that there exists a family of the continuous adapted processes X > 0, Y satisfying the system 2.5 for a.e. > 0 and 2.6. Then there exists a common equivalent local martingale measure Q on F T for C, 0. One such measure is given by dq dp := E bdw, 2.9 T where E is the stochastic exponential. c In the situation of a or b, the dynamics of C under Q are given by dc t = Yt k n 0 X thh k γ t + 0 v t h X t hh dkd W t, 2.20 for 0 and a Q-Brownian motion W = W b s ds. Proof: See Schweizer and Wissel [0], Theorem 4.2. Note that the free input parameters are the market price of the risk process b and the volatilities of the state variables Y and X, such as γ and the family of the processes v for all. Since S t = C t 0, then the volatility σ t of the stock price process ds t = σ t S t d W t can be easily derived from 2.2 and 2.20 as σ t = 0 Yt k n 0 X thh k γ t + 0 Yt k N 0 X thh dk 0 v t h X t hh dk.

26 6 Chapter 2. The Literature Review This implies that if γ or v t are random, then for the stock price S can be obtained a model with a certain quite specific stochastic volatility and it implies also that a class of the arbitrage-free local implied volatility models can be constructed like Schweizer and Wissel did in [0], Chapter Conclusions and comments In this chapter we make a parallel comparison between two models described above. The discrepancies between Dupire s model and Schweizer-Wissel s model and the motivations for the development of the latter one will become more clearer after a slight review of the multiple sources of research including the one done by Carmona and Nadtochiy [], which is one of the most successful studies in this local implied volatilities direction. The main attribute of the Dupire s model is the abibility to be suitable for any initial option price surface which fulfills the static arbitrage bounds. Despite this significant property, the fact that it is a one-factor model reduces the usage of itself, because of consideration of one-factor model can not incorporate multiple sources of randomness into stochastic processes. Thus, it leads to the unrealistic price dynamics which is ensured by the absence of the possibility to recalibrate the option price surface since all sources of uncertainty are reduced to the one comes from the stock price evolution St, which is driven by a one-dimensional fractional Brownian motion as demonstrated in 2.. To overcome this drawback Carmona and Nadtochiy [] tried to incorporate additional stochastic factors into a local volatility model. In the initial setup for their model, they also changed coordinates from strike price to the log-strike price and assumed that the dynamics is dst = StσtdW t, dˆα 2 x,τt = α x,τ tdt + β x,τ tdw t, for a multi-dimensional Brownian motion W under the risk-neutral measure, where ˆα x,τ t = σ expx,τ+t t. According to the Carmona and Nadtochiy[], the local volatility surface is given by σ,t t := 2 T C,T t 2 C,T t. 2.2

27 Arbitrage-free market models for IRO and future options 7 This expression coincides with the expression 2.5 of the local volatility in the Dupire s model and it is expected that this local volatility surface in principle can be a market-observable quantity. Under the regularity conditions on σ,t t, the unique classical solution of the PDE 2.2 for all > 0 and T > t with the initial condition C,T t = St + for all > 0 and T = t are the option prices C,T t. As for Schweizer-Wissel s model, the absence of the dynamic arbitrage in this model leads to the drift restrictions on α x,τ Therem in []. Although Carmona and Nadtochiy did eliminate the vulnerabilities of Dupire s model regarding the limitation of the usage under a one-dimensional fractional Brownian motion model, they couldn t express the drift restrictions independently from the volatility coefficients of the option prices because of the option prices C,T are not given explicitly. Moreover, in contrast to the local volatility model, the Schweizer-Wissel s model has an explicit form of solutions for the option prices 2.2 in the terms of the new parameters X and Y. So, in conclusion, it is obviously true that Schweizer-Wissel s model has a great advantage over all other models because without loss in other properties, it represents the drift restrictions in a closed form as shown in 2.7 and 2.8 and also allows to provide the joint dynamics for the option prices and the stock prices.

28 8 Chapter 2. The Literature Review

29 Chapter 3 The extensions of the Schweizer-Wissel model In our work, we studied both Dupire s and Schweizer-Wissel s model. They are two different models for a stock S and a set of the European call options with all strikes > 0 and one fixed maturity T. In the previous chapter, we have already shown the main differences between them, made some comments on the main ideas, analyzed their advantages and disadvantages. We verified that the Schweizer-Wissel model admits all basic assumptions 0-2, which should be satisfied by a well-constructed option pricing model. Now let us explore the possible extensions of Schweizer-Wissel s model applicable to other types of options. In this chapter we provide an extension of this original model to the cases of an option on interest rate indexes and an option on futures. As the Schweizer-Wissel approach based on the introduction of a new parameterization allows us to provide an explicit expression for the option prices, so, consequently, our new extensions of this model will be based on this idea of parameterization too. In Chapter 3., we discuss an option on interest rate indexes and proceed to introduce the basic definitions and properties using the concept of a new parameterization, and finally derive the arbitrage-free dynamics of the local implied volatilities. In Chapter 3.2, we follow the similar procedure in the case of an option on futures. 9

30 20 Chapter 3. The extensions of the Schweizer-Wissel model 3. An option on interest rate indexes Options on interest indexes, for example, options on the 3-month LIBOR indexes, can be regarded as options on interest rate futures. To be exact, an interest rate cap is a derivative in which the buyer receives payments at the end of each period if the interest rate exceeds the agreed strike price. An example of a cap would be an agreement to receive a payment for each month the LIBOR exceeds 2.5 percent. The interest rate cap can be analyzed as a series of European call options or caplets, which exist for each period of the cap agreement. A cap payoff on the rate L struck at is N α maxl, 0, where α stands for the day count fraction, and N is the principal for entering such option contracts. An interest rate floor is a series of European put options on the floorlets on a specified reference rate, usually LIBOR. The buyer of the interest rate floor will receive an amount of payments if the interest rate falls below the agreed strike price. An example of a floor would be an agreement to receive a payment for each month in which the LIBOR falls below 2.5 percent. Correspondingly, the floor payoff on the rate L struck at is N α max L, 0, where α stands for the day count fraction, and N will be the principal for entering such option contracts. Now, let s consider the statistic behavior of the interest rate r. It s reasonable to trust that the interest rate changes over time, and at the time interest rate is affected by multiple macroeconomic factors, such as an inflation rate, the monetary reasons, the international forces and so on. Thus we suppose the stochastic behavior of the interest rate is given by dr = σdw + µdt, where σ stands for the volatility of the interest rate, and µ stands for the drift of the indexes, for instance, the drift of 3-month LIBOR, or U.S. 3-month treasury bill indexes, and W t would be the Brownian motion which affects the interest rate indexes stochastically. Since options on the 3-month LIBOR or the 3-month treasury bill rate can be regarded as options on the interest rate futures, which is a zero-cost security. It follows that the modified Black-Scholes equation for the so-called zero-cost security can be expressed in the following way V t + 2 V 2 S 2 σ2 S 2 rv = 0,

31 Arbitrage-free market models for IRO and future options 2 where V = V S, t is the price of the option, r is the risk-free rate, and S is the price of the underlying asset. This equation was solved by Fischer Black to give the price of options on zero-cost securities. For the call option C = CF, t with the interest rate indexes as an underlying asset, its differential equation transforms to the following one C t + 2 C 2 C 2 σ2 F 2 rc = 0, where r is the risk-free rate and F is for the forward rate of the interest rate indexes. We introduce the discounted function F t for the time point t, so, the discounted function for the maturity time T is F T. Accordingly, from the basic definition meaning of the forward rate, at time point t, we get F = α t,t [ ] Ft, F T and with the risk-neutrality taken into consideration, the dynamics of F is df = σd W t. Thus, for a caplet, the solution given by Fischer Black becomes Ct, F = F T α t,t [F Nd Nd 2 ], where α t,t is simply a day-count fraction which transforms the number of trading days into equivalent years, and, the parameters in the cumulative distribution function for standard normal distribution are [ ] F d = σ log + σ2, 2 d 2 = d σ. After inserting the parameters above, and treating F, and σ 2 as our new arguments like we introduced before in Chapter 2, we obtain the following form of the value of the caplets cf,, σ 2 = F T α t,t [F Nd Nd 2 ]. When = 0 holds, naturally we have cf, 0, σ 2 = F T α t,t F.

32 22 Chapter 3. The extensions of the Schweizer-Wissel model 3.. The new parameterization In this section we provide the new parameterization of the local implied volatilities and the price level for the market model of the interest rate indexes options as options and the interest rate indexes forward rates as a price evolution, then introduce the underlying definitions and basic properties for our model. This new parameterization allows us in the future research to define an arbitrage-free joint dynamics of the option prices and the interest rate indexes. Our method for derivation of these new parameters is based on Schweizer-Wissel s approach, and instead of the treating stock as an underlying asset we take the interest rate indexes and the set of the European call options on the interest rate indexes to replace the set of vanilla European call options. Throughout this chapter, we work with the following setup. Let Ω, F, P be a probability space and T > 0 be a fixed maturity. Let F t 0 t T be a positive process modeling the forward rate of the interest rate indexes F and B t 0 t T a positive process, with B t = B 0 e T t rτdτ, B 0 =, modeling the discounted bond price evolution. For > 0, let C t 0 t T be a nonnegative process modeling the price of the European call options on the interest rate indexes F with one fixed maturity T > 0 and all strikes > 0. By setting C t 0 = F T α t,t F, the model is specified through the processes C t 0 t T, on the interval [0, T ] and the option model C t 0 t T is admissible and statically arbitrage-free. Now we can introduce a new set of the fundamental quantities which allows us to provide a straightforward parameterization of the admissible option model. Let N denote the quantile function and n = N denote the density function of the standard normal distribution. The first and the second partial derivatives with respect to the strike price of the Black-Scholes formula cf,, σ 2 are given by c F,, σ 2 = α t,t F T Nd 2, c F,, σ 2 = α t,t F T nd 2 σ, [ where d = σ log ] F T t σ 2 T t, F = F t 2 α t,t F T. So, we can easily get the identity

33 Arbitrage-free market models for IRO and future options 23 σ = α t,t F T nn α t,t F T c F,, σ 2 c F,, σ 2. Now we define the new parameters. Definition 4. Let C t 0 t T be admissible. The local implied volatility of the price curve at time t [0, T is the measurable function X t given by X t := N C t for a.e. > 0, C t α t,t F T 3. and the price level of the price curve at time t [0, t for a fixed constant 0 0, is defined by Y t := N C t α t,t F T The terminology the local implied volatillity in the case of an option on the interest rate indexes is justified by the following results. Proposition 4. Let X and Y be the local implied volatilities and the price level of an admissible model C, 0. Suppose that, for a small interval I = [a, b] 0, and fixed t < T, we have X t = X t a for all I. Then there exists a unique pair x t, z t 0, 0, such that cz t,, x 2 t cz t, 2, x 2 t = C t C t holds for all, 2 I. It is given by x t = α t,t F T X t a, z t = exp X t ay t X t a a 0 + log a + α X thh 2 t,t F T X t a 2. Proof: Proof is analogous to the proof of the Proposition 8 in paragraph 3.2., for the extension of the Schweizer-Wissel model with the future contracts as its underlying asset. By this proposition, there exists such an unique implied volatility parameter x t in the Black-Scholes formula for all call options on the interest rate indexes,

34 24 Chapter 3. The extensions of the Schweizer-Wissel model that the prices difference C t C t 2 is consistent with the strike prices, 2 I = [a, b]. It means that Proposition 4 holds locally only. Thus we find the exact form for the option prices under the new parameters, using the one-to-one corresponding relation between the local implied volatilities and the price level, as it is shown by the following. Theorem 3. Let X, Y be the local implied volatilities and the price level of the admissible model C. Then C t = α t,t F T N Yt k 0 X thh α t,t F T dk, 0,, 3.4 C t = α t,t F T N Yt k 0 X thh α t,t F T, 0,, 3.5 C t = n Yt k 0 X thh α t,t F T X t α t,t F T, 0,. 3.6 Conversely, for the continuous adapted processes X > 0, Y on [0, T ] for which the right-hand side of 3.4 is finite P-a.s., define C t by 3.4. Then C, 0 is an admissible model having the local implied volatilities X and the price level Y. Proof: The local implied volatility and the price level in this case are defined by formulas 3. and 3.2 correspondingly. So, we have to find an explicit form of C t, C t, C t expressed by n, N, X t and Y t. It is reasonable to assume that the option value C t has the following form C t = + fkdk, where fk is a differentiable function, and for the contingent claim C t has the form C t = + gkdk,

35 Arbitrage-free market models for IRO and future options 25 where gk can be regarded as a regular density function, gk = { fk, k, 0, k <. Thus from 3.2, we get a so-called initial condition C t =0 = α t,t F T N Yt = f From 3. we get the following expression for X t X t = f n N f α t,t F T = f n N α t,t F T Using this expression we obtain N N α t,t F T f kdk = α t,t F T f kdk. f kdk = α t,t F T f. So if we denote N deduction. α t,t F T f kdk as Φ, we get the following Because and then it follows that dnφ d dnφ d = f = n N α t,t F T α t,t F T = nφ dφ d, df α t,t F T d = f, α t,t F T d N f kdk α t,t F T f kdk. d

36 26 Chapter 3. The extensions of the Schweizer-Wissel model It means that n N and we have that α t,t F T f α t,t F T n N d N = f kdk α t,t F T f kdk, d f = f kdk Xt. α t,t F T According to these formulas we obtain d N f α t,t F T kdk =, d Xt α t,t F T from which we have that d N α t,t F T then consequently N α t,t F T f kdk Adding 3.8 to the initial condition 3.7, we obtain α t,t F T N α t,t F T N α t,t F T X t hh T t Y t α t,t F T T t X t = d, α t,t F T f X kdk = t d. 3.8 α t,t F T = f kdk = fk = C t, = f 0 = C t =0 = C t 0. By solving this system we obtain the unique solution of the system above Yt C t = α t,t F T N 0 X thh. α t,t F T After differentiation and integration of C t with respect to, we also get Yt 0 X thh C t = n α t,t F T X t,

37 Arbitrage-free market models for IRO and future options 27 C t = k α t,t F T N Yt 0 X thh α t,t F T dk. The integral in 3.4 is finite. This property is ensured by the following proposition the suffcient condition. Proposition 5. If there exists > 0 such that X t for a.e., then the outer integral in 3.4 is finite. 2 ln T t Proof: Proof is analogous to the proof of Proposition 9 in paragraph 3.2., for the extension of the Schweizer-Wissel model with the future contracts as its underlying asset. In order to present the images of the new parameters more precisely, we express the local implied volatilities and the price level via the classical implied volatility ˆσ t. γt Proposition 6. Define γ t = ˆσ t 2 and d 2 = logft/ 2 γt. The local implied volatility and the price level can be represented by following expressions Y t = N α t,t F T N d 2 t, 0 2 n d 2t, 0 0 d γ t 0 ˆσ t 2 0 d ˆσ2 t 0, 3.9

38 28 Chapter 3. The extensions of the Schweizer-Wissel model X t = ˆσ t n N α t,t F T Nd 2 t, 2 n d 2t, d γ t ˆσ t 2 d ˆσ2 t α [ t,t F T Ft + log n d 2 t, + 2 γ d t ˆσ t 2 d ˆσ2 t + log 2 Ft γ t 4 2 d 4 γ2 t ˆσ t 2 d ˆσ2 t + ] 2 γ 2 d 2 t ˆσ t 2 d 2 ˆσ2 t. 3.0 Proof: Proof is analogous to the proof of Proposition 0 in Chapter 3.2. for the extension of the Schweizer-Wissel model with the future contracts as its underlying asset The arbitrage-free dynamics of the local implied volatilities In this paragraph, we derive the joint dynamics of option s price levels and the local implied volatilities, under the arbitrage-free condition. First of all, we do some preparation work for the deduction, which is similar to the original Schweizer-Wissel model. Let W be an m-dimensional Brownian motion on the probability space Ω, F, P, and F = F t 0 t T be the P -augmented filtration generated by W, and F = F T. Suppose that there exist the positive processes X t for a.e. > 0, satisfying the condition described in Proposition 5, and a real valued process Y t with the P -dynamics dx t = u t X t dt + v t X t dw t, 0 t T 3. dy t = β t dt + γ t dw t, 0 t T 3.2

39 Arbitrage-free market models for IRO and future options 29 where β, u L loc R, and γ, v L2 loc Rm for a.e.. Further, we suppose u, v are uniformly bounded in ω, t, and that the initial local implied volatility curve satisfies 0 <, > 0. X 0 h 2 Then, we define the process C t, 0 by Theorem 3, X t and Y t are the local implied volatilities and the price level of the option prices C t, 0, respectively. We notice that F = C t 0, so for defining C t 0 by 3.4, the values of X t 0 are not necessarily needed here. After all preparations here, it becomes more clear that if we have some conditions or restrictions in our case here, the drift restrictions imposed on the joint dynamics of the price level and the local implied volatilities, then there exists a common equivalent martingale measure for C, > 0. Our aim here is to show the existence of such equivalence between the existence of a common local martingale measure for C, > 0 and the drift restrictions of joint dynamics of the price levels and the local implied volatilities. We prove our arguments more precisely in the following theorem. Theorem 4. a If there exists a common equivalent local martingale measure Q for all C 0 then there exists a market price of the risk process b L 2 loc Rm such that 3.3, 3.4 for a.e. > 0 hold for a.e. t [0, T ], P a.s. Y t β t = 2 γ t 2 α t,t Y t dα t,t dt b t γ t, 3.3 α t,t u t k = + γ t γ t + 0 v th 2 X th h 2α 2 t,t F 2 T v t h Yt X t h h 0 α 2 t,t F 2 T X th h v t + v 2 t v t b t + ] dα t,t dt α t,t. 3.4