Analysis of valuation formulae and Valuation Examples of - PowerPoint PPT Presentation

The model Analysis of valuation formulae and Valuation Examples of applications to option pricing in L´ evy models payoff functions L´ evy processes Ernst Eberlein 1 , Kathrin Glau 1 , and Exotic options Antonis Papapantoleon 2 Interest rate derivatives References 1 Department of Mathematical Stochastics and Center for Data Analysis and Modeling (FDM) University of Freiburg 2 Financial and Actuarial Mathematics, Vienna University of Technology Advanced Modeling in Finance and Insurance; Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz/Austria; Sept. 22–26, 2008 0

Volatility smile and surface The model Valuation Examples of payoff functions 14 30.0 13.5 L´ evy 28.0 processes 13 26.0 24.0 12.5 implied vol (%) Exotic options 22.0 12 20.0 Interest rate 11.5 18.0 derivatives 16.0 11 14.0 10.5 References 12.0 10 0 10.0 10 20 2.5 2 30 10 40 9 8 4.0 4 50 7 60 6 5 6.0 6 70 4 80 3 delta (%) or strike 2 8.0 8 90 1 maturity 10 10.0 Maturity (in years) Strike rate (in %) Volatility surfaces of foreign exchange and interest rate options • Volatilities vary in strike ( smile ) • Volatilities vary in time to maturity ( term structure ) • Volatility clustering 1

Exponential semimartingale model The model Valuation Examples of B T = (Ω , F , F , P ) stochastic basis, where F = F T and F = ( F t ) 0 ≤ t ≤ T . payoff Price process of a financial asset as exponential semimartingale functions L´ evy S t = S 0 e H t , 0 ≤ t ≤ T . (1) processes Exotic options H = ( H t ) 0 ≤ t ≤ T semimartingale with canonical representation Interest rate derivatives H = B + H c + h ( x ) ∗ ( µ H − ν ) + ( x − h ( x )) ∗ µ H . (2) References For the processes B , C = � H c � , and the measure ν we use the notation T ( H | P ) = ( B , C , ν ) which is called the triplet of predictable characteristics of H . 2

Alternative model description The model Valuation E ( X ) = ( E ( X ) t ) 0 ≤ t ≤ T stochastic exponential Examples of payoff functions S t = E ( e H ) t , 0 ≤ t ≤ T L´ evy processes dS t = S t − d e H t Exotic options Interest rate where Z t Z derivatives H t = H t + 1 ( e x − 1 − x ) µ H ( d s , d x ) e 2 � H c � t + References 0 R Note “ ” Y H t − 1 E ( e e 2 � e ( 1 + ∆ e H s ) exp ( − ∆ e H c � t H ) t = exp H s ) 0 < s ≤ t Asset price positive only if ∆ e H > − 1. 3

Martingale modeling The model Valuation Let M loc ( P ) be the class of local martingales. Examples of payoff Assumption ( ES ) functions L´ evy The process 1 { x > 1 } e x ∗ ν has bounded variation. processes Exotic options Then Interest rate derivatives S = S 0 e H ∈ M loc ( P ) ⇔ B + C 2 + ( e x − 1 − h ( x )) ∗ ν = 0 . (3) References Throughout, we assume that P is an equivalent martingale measure for S . By the Fundamental Theorem of Asset Pricing , the value of an option on S equals the discounted expected payoff under this martingale measure. We assume zero interest rates. 4

Supremum and infimum processes The model Valuation Let X = ( X t ) 0 ≤ t ≤ T be a stochastic process. Denote by Examples of payoff functions X t = sup X u X t = 0 ≤ u ≤ t X u inf and L´ evy 0 ≤ u ≤ t processes the supremum and infimum process of X respectively. Since the Exotic options exponential function is monotone and increasing Interest rate derivatives “ S 0 e H t ” = S 0 e sup 0 ≤ t ≤ T H t = S 0 e H T . S T = sup S t = sup (4) References 0 ≤ t ≤ T 0 ≤ t ≤ T Similarly S T = S 0 e H T . (5) 5

Valuation formulae – payoff functional The model We want to price an option with payoff Φ( S t , 0 ≤ t ≤ T ) , where Φ is a Valuation measurable, non-negative functional. Examples of payoff Separation of payoff function from the underlying process: functions L´ evy processes Example Exotic options Interest rate Fixed strike lookback option derivatives ` ´ ( S T − K ) + = ( S 0 e H T − K ) + = References e H T + log S 0 − K The payoff function is an arbitrary function f : R → R + ; for example 1 f ( x ) = ( e x − K ) + or f ( x ) = 1 { e x > B } , for K , B ∈ R + . The underlying process denoted by X , can be the log-asset price 2 process or the supremum/infimum or an average of the log-asset price process (e.g. X = H or X = H ). 6

Valuation formulae The model Consider the option price as a function of S 0 or better of s = − log S 0 Valuation Examples of X driving process ( X = H , H , H , etc.) payoff functions Φ( S 0 e H t , 0 ≤ t ≤ T ) = f ( X T − s ) ⇒ L´ evy processes Exotic options Time-0 price of the option (assuming r ≡ 0) Interest rate derivatives ˆ ˜ V f ( X ; s ) = E Φ( S t , 0 ≤ t ≤ T ) = E [ f ( X T − s )] References Valuation formulae based on Fourier and Laplace transforms Carr and Madan (1999) plain vanilla options Raible (2000) general payoffs, Lebesgue densities Borovkov and Novikov (2002) plain vanilla and lookback options In these approaches: Some sort of continuity assumption (payoff or random variable) 7

Valuation formulae – assumptions The model Valuation M X T moment generating function of X T Examples of payoff g ( x ) = e − Rx f ( x ) functions (for some R ∈ R ) dampened payoff function L´ evy L 1 bc ( R ) bounded, continuous functions in L 1 ( R ) processes Exotic options Interest rate Assumptions derivatives References g ∈ L 1 bc ( R ) (C1) (C2) M X T ( R ) exists g ∈ L 1 ( R ) b (C3) 8

Valuation formulae The model Valuation Theorem Examples of payoff Assume that (C1)–(C3) are in force. Then, the price V f ( X ; s ) of an option functions on S = ( S t ) 0 ≤ t ≤ T with payoff f ( X T ) is given by L´ evy Z processes V f ( X ; s ) = e − Rs e ius ϕ X T ( − u − iR ) b f ( u + iR ) d u , (6) Exotic options 2 π R Interest rate derivatives where ϕ X T denotes the extended characteristic function of X T and b f References denotes the Fourier transform of f . Proof Z Z f ( X T − s ) dP = e − Rs e Rx g ( x − s ) P X T ( d x ) . V f ( X ; s ) = (7) Ω R cont. next page 9

Proof (cont.) The model Under assumption (C1), g ∈ L 1 ( R ) and b g is well-defined. With (C3) Valuation g ∈ L 1 b bc ( R ) . Examples of Z payoff g ( x ) = 1 e − ixu b functions g ( u ) d u . (8) 2 π L´ evy R processes Exotic options Returning to the valuation problem (7) we get Interest rate ! Z Z derivatives 1 e − i ( x − s ) u b V f ( X ; s ) = e − Rs e Rx g ( u ) d u P X T ( d x ) References 2 π R R Z ! Z = e − Rs e ius e i ( − u − iR ) x P X T ( d x ) b g ( u ) d u 2 π R R Z = e − Rs e ius ϕ X T ( − u − iR ) b f ( u + iR ) d u . (9) 2 π R � 10

Examples of payoff functions The model Valuation Example (Call and put option) Examples of payoff functions Call payoff f ( x ) = ( e x − K ) + , K ∈ R + , L´ evy processes K 1 + iu − R Exotic options b f ( u + iR ) = ( iu − R )( 1 + iu − R ) , R ∈ I 1 = ( 1 , ∞ ) . (10) Interest rate derivatives References Similarly, if f ( x ) = ( K − e x ) + , K ∈ R + , K 1 + iu − R b f ( u + iR ) = ( iu − R )( 1 + iu − R ) , R ∈ I 1 = ( −∞ , 0 ) . (11) 11

Example (Digital option) The model Call payoff 1 { e x > B } , B ∈ R + . Valuation 1 Examples of b f ( u + iR ) = − B iu − R iu − R , R ∈ I 1 = ( 0 , ∞ ) . (12) payoff functions L´ evy Similarly, for the payoff f ( x ) = 1 { e x < B } , B ∈ R + , processes 1 Exotic options b f ( u + iR ) = B iu − R iu − R , R ∈ I 1 = ( −∞ , 0 ) . (13) Interest rate derivatives References Example (Double digital option) The payoff of a double digital call option is 1 { B < e x < B } , B, B ∈ R + . “ B iu − R − B iu − R ” 1 b f ( u + iR ) = , R ∈ I 1 = R \{ 0 } . (14) iu − R 12

Example (Asset-or-nothing digital) The model f ( x ) = e x 1 { e x > B } Call payoff Valuation Examples of f ( u + iR ) = − B 1 + iu − R payoff b 1 + iu − R , R ∈ I 1 = ( 1 , ∞ ) functions L´ evy f ( x ) = e x 1 { e x < B } Put payoff processes Exotic options B 1 + iu − R b f ( u + iR ) = 1 + iu − R , R ∈ I 1 = ( −∞ , 1 ) Interest rate derivatives References Example (Self-quanto option) f ( x ) = e x ( e x − K ) + Call payoff K 2 + iu − R b f ( u + iR ) = ( 1 + iu − R )( 2 + iu − R ) , R ∈ I 1 = ( 2 , ∞ ) 13

Non-path-dependent options The model European option on an asset with price process S t = e H t Valuation Examples of payoff Examples: call, put, digitals, asset-or-nothing, double functions digitals, self-quanto options L´ evy processes − → X T ≡ H T , i.e. we need ϕ H T Exotic options Interest rate derivatives Generalized hyperbolic model (GH model): References ` p α 2 − ( β + iu ) 2 ´ ϕ H 1 ( u ) = e iu µ “ α 2 − β 2 ” λ/ 2 K λ δ p ` α 2 − β 2 ´ α 2 − ( β + iu ) 2 K λ δ I 2 = ( − α − β, α − β ) ϕ H T ( u ) = ( ϕ H 1 ( u )) T similar: NIG, CGMY, Meixner 14