How to hedge Asian options in fractional Black-Scholes model

Transcription

1 How to hedge Asian options in fractional Black-Scholes model Heikki ikanmäki Jena, March 29, 211 Fractional Lévy processes 1/36

2 Outline of the talk 1. Introduction 2. Main results 3. Methodology 4. Conclusions Fractional Lévy processes 2/36

3 1. Introduction Asian options Fractional Brownian motion Fractional Black-Scholes Pathwise stochastic integration Hedging problem Case of European options Fractional Lévy processes 3/36

4 Introduction Change of variables formulas [Itô formulas] for both arithmetic and geometric averages of geometric fractional Brownian motion. Valid for all convex functions, not only for smooth ones. Can be used for obtaining hedges (but not prices) for Asian options in fractional Black-Scholes model. Explicit hedges in some cases where hedges are not known explicitely even in the ordinary Black-Scholes model. Fractional Lévy processes 4/36

5 Asian options Let S(t) be the price of the underlying asset. Asian options depend on the time average of the underlying. he payoff of the arithmetic Asian option is ( 1 ) f S(s)ds and the payoff of the geometric Asian option ( ( 1 )) f exp log S(s)ds. Fractional Lévy processes 5/36

6 Asian options (continued) Arithmetic Asian options are important in practise. Used for example in commodity markets. Geometric Asian options are easier to consider analytically e.g. in ordinary Black-Scholes model. he problem with arithmetic Asian options is that sum of lognormals is not lognormal. Here we overcome this problem by using pathwise methods. Fractional Lévy processes 6/36

7 Fractional Brownian motion Fractional Brownian motion (fbm) B H with Hurst index H (, 1) is a Gaussian process satisfying EB H (t) = B H () and having the following covariance structure Cov(B H (t), B H (s)) = 1 2 ( t 2H + s 2H t s 2H). If H = 1 2, we are in the case of ordinary BM. Fractional Lévy processes 7/36

8 FBM (continued) For H > 1 2 the process has long range dependence property and for H < 1 2 the increments are negatively correlated. FBM is self-similar with parameter H. FBM is not semi-martingale nor Markov process (unless H = 1 2 ). B H has Hölder continuous sample paths of any order δ (, H). For H > 1 2, fbm BH has zero quadratic variation over a sequence of subdivisions where the mesh goes to zero. Fractional Lévy processes 8/36

9 Fractional Black-Scholes he price of the underlying is modeled as S(t) = exp B H (t), where 1 2 < H < 1. S has Hölder continuous sample paths of any order δ (, H). S has zero quadratic variation property. he hedging results remain true if we add any deterministic drift to B H as long as the path properties (Hölder continuity and quadratic variation) are not changed. Fractional Lévy processes 9/36

10 Pathwise integration he stochastic integrals considered here are pathwise: Riemann-Stieltjes integrals (RS) generalized Lebesgue-Stieltjes integrals (gls) If not mentioned otherwise the integrals are gls. Fractional Lévy processes 1/36

11 Hedging problem Given claim F(S) = f ((S(s) s [, ])). Find adapted (H(s)) s [, ] such that F(S) = A + H(s)dS(s). Integral should be economically justified. Separate problem of mathematical finance compared to pricing. Fractional Lévy processes 11/36

12 Case of European options heorem Let f be convex function. hen f (S( )) = f (S()) + in generalized Lebesgue-Stieltjes sense. Note that ds(t) = S(t)dB H (t). (Azmoodeh-Mishura-Valkeila 29). f (S(t))S(t)dB H (t) Fractional Lévy processes 12/36

13 2. Main results Replication of the averages Options depending on geometric average Options depending on arithmetic average Fractional Bachelier model Fractional Lévy processes 13/36

14 Replication of geometric average Proposition G(t) = exp ( 1 t ) log S(s)ds S(t) t. For all t [, ] it holds almost surely that t G(t) = S() + in Riemann-Stieltjes sense. s G(s)dBH (s), Fractional Lévy processes 14/36

15 Replication of geometric average (continued) Corollary In particular ( 1 exp =S() + B H (s)ds s exp ) ( 1 s B H (u)du + s ) BH (s) db H (s). Fractional Lévy processes 15/36

16 Replication of arithmetic average Proposition For all t [, ] it holds almost surely that t S(t) + 1 t t S(s)ds = S() + in Riemann-Stieltjes sense. Corollary In particular 1 S(s)ds = S() + s S(s)dBH (s), s S(s)dBH (s). Fractional Lévy processes 16/36

17 Geometric Asian options G(t) = exp ( 1 t ) B H (s)ds S(t) t. heorem Let f be a convex function. hen it holds almost surely that t f (G(t)) = f (S()) + s f (G(s)) G(s)dB H (s), where the stochastic integral in the right side is understood in generalized Lebesgue-Stieltjes sense. Fractional Lévy processes 17/36

18 Geometric Asian options (continued) Corollary In particular, ( ( 1 f exp =f (S()) + B H (s)ds )) s f (G(s)) G(s)dB H (s). Fractional Lévy processes 18/36

19 Arithmetic Asian options heorem Let f be a convex function. hen it holds almost surely that ( t f S(t) + 1 t ) S(s)ds t ( s =f (S()) + f S(s) + 1 s ) s S(u)du S(s)dBH (s), where the stochastic integral in the right side is understood in the sense of generalized Lebesgue-Stieltjes integral. Fractional Lévy processes 19/36

20 Arithmetic Asian options (continued) Corollary In particular, ( 1 ) f S(s)ds ( s =f (S()) + f S(s) + 1 s ) s S(u)du S(s)dBH (s). Fractional Lévy processes 2/36

21 Fractional Bachelier model he case of arithmetic average can be written also when the geometric price process S is replaced by a fractional Brownian motion B H with H ( 1 2, 1). In that case we obtain for a convex function f that ( t f BH (t) + 1 t ) B H (s)ds =f (B H ()) + t s f ( s BH (s) + 1 s B H (u)du almost surely as a generalized Lebesgue-Stieltjes integral. ) db H (s) Fractional Lévy processes 21/36

22 3. Methodology Pathwise philosophy Functional change of variables formula Fractional Besov space techniques and generalized Lebesgue-Stieltjes integral Fractional Lévy processes 22/36

23 Pathwise philosophy Stochastic integrals are pathwise. In practise one observes the path, not distribution. Drift can be added, if path properties do not change. In practise one does not observe the drift. Arithmetic Asian options can be considered. he problem in classical Black-Scholes is that sum of lognormals is not lognormal anymore. Here it does not matter. Fractional Lévy processes 23/36

24 Functional change of variables formula Let Bt H = (B H (s) s (, t)) be the whole path of B H up to t. hen by (Cont-Fournié 21) F (B H ) = F (B H ()) + D t F t (Bu H )du + x F u (Bu H )db H (u) ( ) x F t(bu H )d[b H ](u) in Riemann-Stieltjes sense. F t is called non-anticipative functional and (F t ) is so called non-anticipative flow. D t is horizontal and x vertical derivative. [ ] denotes the pathwise quadratic variation. Fractional Lévy processes 24/36

25 Functional change of variables formula and fbm In the fbm setup the quadratic variation term vanishes. In the case of geometric Asian options, change of variables formula is applied to non-anticipative functional ( 1 t ) F t (x t ) = exp x(s)ds e t x(t). And in the case of arithmetic Asian options F t (x t ) = t ex(t) + 1 t e x(s) ds. Fractional Lévy processes 25/36

26 Horizontal derivative he horizontal extension of x t for h > is defined as x t,h (u) = x(u), u [, t] and x t,h (u) = x(t), u (t, t + h]. Now the horizontal derivative of F at x C([, ]) is defined as F t+h (x t,h ) F t (x) D t F(x) = lim, h h if the limit exists. Fractional Lévy processes 26/36

27 Vertical derivative he vertical perturbation of path x t is defined for h R as x h t (u) = x(u), u [, t) and x h t (t) = x(t) + h. he vertical derivative is defined in the following way. A non-anticipative functional F is vertically differentiable at x C([, t]) if limit exists. x F t (x) = lim h F t (x h t ) F t(x) h Fractional Lévy processes 27/36

28 Fractional Besov spaces Definition Let f : [, ] R be measurable. hen f W β 1 ([, ]) if f 1,β = sup s<t ( f (t) f (s) t (t s) β + s ) f (u) f (s) du <. (u s) β+1 Definition Let f : [, ] R be measurable. hen f W β 2 ([, ]) if f 2,β = f (t) t t β dt + f (t) f (s) dsdt <. (t s) β+1 Fractional Lévy processes 28/36

29 Generalized Lebesgue-Stieltjes integral Let β (, 1). Generalized Lebesgue-Stieltjes integral is defined as t t ) ( ) fdg := (D β + f (x) (x)dx. D 1 β t g t Operators D β + and Dβ t are Riemann-Liouville fractional derivatives. f W β 1 β 2 ([, ]), g W1 ([, ]). g t (x) = (g(t ) g(x))1 (,t) (x). Fractional Lévy processes 29/36

30 Remarks Remark s t t + =. s Remark he integral is the same for all β for which it can be defined. Remark For < β < H the trajectories of fbm B H belong to W β 1 ([, ]) almost surely by Hölder continuity. Fractional Lévy processes 3/36

31 Convergence theorem heorem Let f, (f n ) n=1 W β 1 β 2 ([, ]) and g W1 ([, ]). If then f n f 2,β, t f n dg t fdg, in generalized Lebesgue-Stieltjes sense for all t (, ]. Fractional Lévy processes 31/36

32 4. Conclusions Extended the functional Itô formula of (Cont-Fournié 21) for non-smooth convex functions in the special case of driving gfbm or fbm and functional depending on the average of the driving process. Obtained hedging strategies for Asian options in fractional Black-Scholes model. We were able to find hedges also for Asian options depending on the ordinary arithmetic average. Explicit hedges for such options are not known even in the case of Black-Scholes model. Fractional Lévy processes 32/36

33 Conclusions (continued) In the case of Asian options, fbm behaves as continuous function of bounded variation. However, this is not the case for all path-dependent options: see for example the case of lookback options (Azmoodeh-ikanmäki-Valkeila 21). Some related models behave differently. For example in exponential mixed Brownian motion and fractional Brownian motion market model the hedges of Asian options are the same as in ordinary Black-Scholes model, (Bender-Sottinen-Valkeila 28). Fractional Lévy processes 33/36

34 References [1] E. Azmoodeh, Yu. Mishura, and E. Valkeila. On hedging European options in geometric fractional Brownian motion market model. Statist. Decisions, 27(2): , 29. [2] E. Azmoodeh, H. ikanmäki, and E. Valkeila. When does fractional Brownian motion not behave as a continuous function with bounded variation? Statist. Probab. Lett., 8: , 21. [3] C. Bender,. Sottinen, and E. Valkeila. Pricing by hedging and no-arbitrage beyond semimartingales. Finance Stoch., 12(4): , 28. Fractional Lévy processes 34/36

35 References (continued) [4] R. Cont, and D.-A. Fournié. Change of variable formulas for non-anticipative functionals on path space. J. Funct. Anal., 259(4): , 21. [5] H. ikanmäki. How to hedge Asian options in fractional Black-Scholes model arxiv: , 211. Fractional Lévy processes 35/36

36 hanks for your attention! Fractional Lévy processes 36/36