Degree Project Regularized Calibration of Jump-Diffusion Option Pricing Models Hiba Nassar 2010-10-21 Subject: Mathematics Level: Master Course code: 5MA11E Regularized Calibration of Jump-Diffusion Option Pricing Models Hiba Nassar October 21, 2010 1 Contents 1 Introduction to stochastics 4 1.1 Stochasticprocess .......................... 4 1.2 Martingale............................... 5 1.3 L´evymodels.............................. 5 1.4 EquivalenceMeasure ......................... 6 1.5 Pricingoptions ............................ 7 1.5.1 Pricingcalloption ...................... 8 1.5.2 Option pricing for the Black-Scholes model . 9 1.5.3 OptionpricingfortheMertonmodel . 11 1.6 Relativeentropyfunction . 14 1.6.1 Properties: .......................... 15 1.7 Directproblemandinverseproblem . 18 2 Inverseproblemandregularizationmethods 19 2.1 Howtosolvetheinverseproblem? . 19 2.2 Regularizationmethod. 20 2.3 TikhonovRegularization. 21 3 Calibration problem 22 3.1 Non-linearLeastSquares(NLS). 22 3.1.1 Choice of weights wi:..................... 23 3.1.2 WhatiswrongwithNLS?. 23 3.2 Regularizationbyrelativeentropy . 23 3.2.1 Choiceofregularizationparameter . 25 4 Numerical implementation 27 5 Somecommentsofpossibleextensions 29 A Matlab Codes 30 References 33 2 Acknowledgment I am deeply indebted to associate professor Irina Asekritova and associate pro- fessor Roger Pettersson. Without their guidance and support, I would never been able to complete this work. It is a pleasure to thank everyone who has helped me along the way. I would like to say many thanks to my family without whose support none of this would have been possible. Lastly, the most special thanks goes to my partner and friend, my fianc´eRani Basna. 3 Abstract An important issue in finance is model calibration. The calibration problem is the inverse of the option pricing problem. Calibration is per- formed on a set of option prices generated from a given exponential L´evy model. By numerical examples, it is shown that the usual formulation of the inverse problem via Non-linear Least Squares is an ill-posed problem. To achieve well-posedness of the problem, some regularization is needed. Therefore a regularization method based on relative entropy is applied. 1 Introduction to stochastics It is useful to start with some definitions and basic concepts in stochastics. The definitions and theorems presented below can be found, for example, in [13] and [15]. 1.1 Stochastic process A stochastic process is an extension of deterministic processes expressed in terms of probability theory, instead of dealing with only one possible ’reality’ of how the process might evolve under time. For a stochastic process there is some in- determinacy in its future evolution described by probability distributions. This means that even if the initial condition (or starting point) is known, there are many possibilities the process might go to. Definition 1.1.1. A stochastic process is a collection of random variables (Xt; t )=(Xt(w); t ; w Ω); ∈T ∈T ∈ defined on some probability space (Ω, , P). We call X a continuous-time processF if is an interval, such as = [0,T ]; and we call it a discrete-time process if Tis a finite or countably infiniteT set, such as = 0, 1, 2,... Such processesT are also called time series. T { } Remark 1.1.2. A stochastic process is a function of two variables t and w, where the first variable represents time and the second uncertinity. For a fixed time t, Xt(w) is a random variable. • Xt = Xt(w); w Ω ∈ Here w is a number. For a fixed w, Xt(w) is a function of time: • Xt(w)= w(t); t ; ∈T which is called a realization, or a sample path of the process X. Here w is a function. 4 1.2 Martingale An increasing sequence of sub σ-algebras • ( t) , s t for s<t F t≥0 F ⊂F is said to be a filtration. t can be interpreted as the information until time t. F If for each t, Xt is t-measurable, it is said to be adapted to the filtra- • F tion t F Definition 1.2.1. A process Xt is a martingale with respect to a filtration t if: F 1. E [ Xt ] < for each t. | | ∞ 2. Xt is adapted to the filtration t. F 3. E [Xt s]= Xs for s<t. |F 1.3 L´evy models Definition 1.3.1. A stochastic process (Xt)t≥0 is a L´evy Process if: 1. It has independent increments: for different times t0,...,tn the random variables Xt ,Xt Xt ,Xt Xt ,...,Xt Xt − are independent. 0 1 − 0 2 − 1 n − n 1 2. It has stationary increments: the distribution of Xt+h Xt does not de- pend on t. − 3. it is continuous in probability: lim P ( Xt+h Xt ε)=0 forall ε> 0. h→0 | − |≥ 4. Its sample paths are right-continuous with left limits (”cadlag”). Definition 1.3.2. Let Xt be a L´evy process. The L´evy measure v is defined by: v (A)= E [#t [0, 1] : ∆Xt A, ∆Xt = 0] , ∈ ∈ for any Borel set A, i.e. it is an expected number of jumps, in the time interval [0, 1], whose height belongs to A. Note: A Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Here our topological space is just the space (Xt)t>0 lives on, namely R. Example 1. A Poisson process Nt is a L´evy process with L´evy measure ν given by { } λ , 1 A ν(A)= ∈ 0 , 1 / A ∈ for Borel-sets A. It means that the jump heights are always one. The parameter λ is the mean number of jumps of Xt in the interval [0, 1], named intensity. 5 Example 2. A compound Poisson process is a L´evy process Nt Xt = Yi i=1 where Nt is a Poisson process and Yi are independent and identically dis- tributed{ random} variables with distribution function F . The L´evy measure of Xt is given by ν(A)= λ dF (x) A where λ is the intensity of Nt . { } Example 3. A standard Wiener process Wt is a L´evy process where the in- { } crements Wt+h Wt are normal distributions with mean zero and variance h. The L´evy measure− is identical to zero. Theorem 1.3.3 (L´evy-Ito decomposition). Let (Xt)t≥0 be a L´evy process and v its L´evy measure, Assume x v (dx) < . Then there exist a |x|<1 | | ∞ constant γ and a Brownian motion B = µt + σW such that: t t l ε Xt = γt + Bt + Xt + lim Xt (1) ε↓0 l ε where Xt denotes the sum of large jumps of size larger than 1 and Xt denotes the sum of small jumps of size between ε and 1 The triple σ2,v,γ is called the characteristic triple or the L´evy triplet of the process X . t Note: if x v (dx) < is not true then more careful decomposition |x|<1 | | ∞ is needed. 1.4 Equivalence Measure Let P and Q be two probability measures defined on Ω, equipped with σ-algebra . We say that P is absolutely continuous with respect to Q (P Q) if F ≪ Q (A) = 0 P (A) = 0 A . ⇒ ∀ ∈F If P Q and Q P, then we say that P and Q are equivalent measures (Q ∼ P). ≪ ≪ Theorem 1.4.1. Let P and Q be two measures on a probability space (Ω, ), such that Q is absolutely continuous with respect to P, i.e. Q P. Then thereF exists a unique non-negative function Z : Ω R such that: ≪ → Z is -measurable. • F Q (A)= Z (x) dP (x), A . • A ∀ ∈F Q (A) < . • ∞ In this case dQ dQ = ZdP i.e. Z = . dP Z is the Radon-Nikodym derivative of Q with respect to P. 6 Properties: 1. If Q P then ≪ d(aQ) dQ = a , a R. dP dP ∀ ∈ 2. If Q P and Q P then 1 ≪ 2 ≪ d (Q + Q ) dQ dQ 1 2 = 1 + 2 . dP dP dP Theorem 1.4.2 (Girsanov). Let (Ω, , P) be a probability space, and W (t) , 0 t T , be a Brownian motion. Let Θ(Ft) , 0 t T , be an adapted process,i.e.≤ ≤ ≤ ≤ Θ(t) t, t. Define Z (t) as: ∈F ∀ t 1 t Z (t) = exp Θ(u) dW (u) Θ2 (u) du − − 2 0 0 and t W (t)= W (t)+ Θ(u) du. 0 P 1 T Θ2(s)ds Let Z = Z (T ). Assume the Novikov condition i.e. E e 2 0 < . ∞ dQ R Then E (Z) = 1 and Q defined by Z = dP is a probability measure, and the process W (t) is a standard Wiener process under the probability measure Q. In many applications of the Girsanov Theorem, Θ is just a constant = 0 for which the Novikov condition is true. In finance, a probability measure Q is called a risk-neutral measure if −rt Sˆt = e St is a Q-martingale, where r is the interest rate. 1.5 Pricing options Definition 1.5.1. An option is a financial contract that gives the owner the right, but not the obligation, to make a specified transaction for a specified time. Definition 1.5.2. European call option is an option that gives the owner the right, but not the obligation, to buy a share at a given price, known as strike K, at a certain time, known as maturity date T . Remark 1.5.3. An American call option is similar to European call option with the only difference that the owner has the right to exercise the option at any time before the maturity, not only at the expiry date.