Pricing in discrete financial models Mnacho Echenim February 23, 2021 Contents 1 Generated subalgebras 2 1.1 Independence between a random variable and a subalgebra. 9 2 Filtrations 10 2.1 Basic definitions ......................... 10 2.2 Stochastic processes ....................... 13 2.2.1 Adapted stochastic processes .............. 13 2.2.2 Predictable stochastic processes ............. 15 2.3 Initially trivial filtrations .................... 20 2.4 Filtration-equivalent measure spaces .............. 22 3 Martingales 26 4 Discrete Conditional Expectation 29 4.1 Preliminary measurability results ................ 29 4.2 Definition of explicit conditional expectation ......... 34 5 Infinite coin toss space 62 5.1 Preliminary results ........................ 62 5.2 Bernoulli streams ......................... 68 5.3 Natural filtration on the infinite coin toss space ........ 71 5.3.1 The projection function ................. 71 5.3.2 Natural filtration locale ................. 80 5.3.3 Probability component .................. 92 5.3.4 Filtration equivalence for the natural filtration .... 103 5.3.5 More results on the projection function ........ 106 5.3.6 Integrals and conditional expectations on the natural filtration .......................... 114 5.4 Images of stochastic processes by prefixes of streams ..... 143 5.4.1 Definitions ........................ 144 5.4.2 Induced filtration, relationship with filtration gener- ated by underlying stochastic process ......... 153 1 6 Geometric random walk 173 7 Fair Prices 178 7.1 Preliminary results ........................ 178 7.1.1 On the almost everywhere filter ............. 178 7.1.2 On conditional expectations ............... 179 7.2 Financial formalizations ..................... 182 7.2.1 Markets .......................... 182 7.2.2 Quantity processes and portfolios ............ 183 7.2.3 Trading strategies .................... 193 7.2.4 Self-financing portfolios ................. 200 7.2.5 Replicating portfolios .................. 216 7.2.6 Arbitrages ......................... 217 7.2.7 Fair prices ......................... 224 7.3 Risk-neutral probability space .................. 242 7.3.1 risk-free rate and discount factor processes ...... 242 7.3.2 Discounted value of a stochastic process ........ 245 7.3.3 Results on risk-neutral probability spaces ....... 247 8 The Cox Ross Rubinstein model 263 8.1 Preliminary results on the market ................ 263 8.2 Risk-neutral probability space for the geometric random walk 283 8.3 Existence of a replicating portfolio ............... 300 9 Effective computation definitions and results 344 9.1 Generation of lists of boolean elements ............. 344 9.2 Probability components for lists ................. 347 9.3 Geometric process applied to lists ................ 348 9.4 Effective computation of discounted values ........... 349 10 Pricing results on options 349 10.1 Call option ............................ 349 10.2 Put option ............................. 351 10.3 Lookback option ......................... 352 10.4 Asian option ........................... 356 1 Generated subalgebras This section contains definitions and properties related to generated subal- gebras. theory Generated-Subalgebra imports HOL−Probability:Probability begin 2 definition gen-subalgebra where gen-subalgebra M G = sigma (space M) G lemma gen-subalgebra-space: shows space (gen-subalgebra M G) = space M by (simp add: gen-subalgebra-def space-measure-of-conv) lemma gen-subalgebra-sets: assumes G ⊆ sets M and A 2 G shows A 2 sets (gen-subalgebra M G) by (metis assms gen-subalgebra-def sets:space-closed sets-measure-of sigma-sets:Basic subset-trans) lemma gen-subalgebra-sig-sets: assumes G ⊆ Pow (space M) shows sets (gen-subalgebra M G) = sigma-sets (space M) G unfolding gen-subalgebra-def by (metis assms gen-subalgebra-def sets-measure-of ) lemma gen-subalgebra-sigma-sets: assumes G ⊆ sets M and sigma-algebra (space M) G shows sets (gen-subalgebra M G) = G using assms by (simp add: gen-subalgebra-def sigma-algebra:sets-measure-of-eq) lemma gen-subalgebra-is-subalgebra: assumes sub: G ⊆ sets M and sigal:sigma-algebra (space M) G shows subalgebra M (gen-subalgebra M G)(is subalgebra M ?N) unfolding subalgebra-def proof (intro conjI ) show space ?N = space M using space-measure-of-conv[of (space M)] unfolding gen-subalgebra-def by simp have geqn: G = sets ?N using assms by (simp add:gen-subalgebra-sigma-sets) thus sets ?N ⊆ sets M using assms by simp qed definition fct-gen-subalgebra :: 0a measure ) 0b measure ) ( 0a ) 0b) ) 0a measure where fct-gen-subalgebra M N X = gen-subalgebra M (sigma-sets (space M) fX −‘B \ (space M) j B: B 2 sets Ng) 3 lemma fct-gen-subalgebra-sets: shows sets (fct-gen-subalgebra M N X) = sigma-sets (space M) fX −‘B \ space M jB: B 2 sets Ng unfolding fct-gen-subalgebra-def gen-subalgebra-def proof − have fX −‘B \ space M jB: B 2 sets Ng ⊆ Pow (space M) by blast then show sets (sigma (space M)(sigma-sets (space M) fX −‘B \ space M jB: B 2 sets Ng)) = sigma-sets (space M) fX −‘B \ space M jB: B 2 sets Ng by (meson sigma-algebra:sets-measure-of-eq sigma-algebra-sigma-sets) qed lemma fct-gen-subalgebra-space: shows space (fct-gen-subalgebra M N X) = space M unfolding fct-gen-subalgebra-def by (simp add: gen-subalgebra-space) lemma fct-gen-subalgebra-eq-sets: assumes sets M = sets P shows fct-gen-subalgebra M N X = fct-gen-subalgebra P N X proof − have space M = space P using sets-eq-imp-space-eq assms by auto thus ?thesis unfolding fct-gen-subalgebra-def gen-subalgebra-def by simp qed lemma fct-gen-subalgebra-sets-mem: assumes B2 sets N shows X −‘B \ (space M) 2 sets (fct-gen-subalgebra M N X) unfolding fct-gen-subalgebra-def proof − have f1: fX −‘A \ space M jA: A 2 sets Ng ⊆ Pow (space M) by blast have 9 A: X −‘B \ space M = X −‘A \ space M ^ A 2 sets N by (metis assms) then show X −‘B \ space M 2 sets (gen-subalgebra M (sigma-sets (space M) fX −‘A \ space M jA: A 2 sets Ng)) using f1 by (simp add: gen-subalgebra-def sigma-algebra:sets-measure-of-eq sigma-algebra-sigma-sets) qed lemma fct-gen-subalgebra-is-subalgebra: assumes X2 measurable M N shows subalgebra M (fct-gen-subalgebra M N X) unfolding fct-gen-subalgebra-def proof (rule gen-subalgebra-is-subalgebra) show sigma-sets (space M) fX −‘B \ space M jB: B 2 sets Ng ⊆ sets M (is ?L ⊆ ?R) proof (rule sigma-algebra:sigma-sets-subset) 4 show fX −‘B \ space M jB: B 2 sets Ng ⊆ sets M proof fix a assume a 2 fX −‘B \ (space M) j B: B 2 sets Ng then obtain B where B 2 sets N and a = X −‘B \ (space M) by auto thus a 2 sets M using measurable-sets assms by simp qed show sigma-algebra (space M)(sets M) using measure-space by (auto simp add: measure-space-def ) qed show sigma-algebra (space M) ?L proof (rule sigma-algebra-sigma-sets) let ?preimages = fX −‘B \ (space M) j B: B 2 sets Ng show ?preimages ≤ Pow (space M) using assms by auto qed qed lemma fct-gen-subalgebra-fct-measurable: assumes X 2 space M ! space N shows X2 measurable (fct-gen-subalgebra M N X) N unfolding measurable-def proof ((intro CollectI ); (intro conjI )) have speq: space M = space (fct-gen-subalgebra M N X) by (simp add: fct-gen-subalgebra-space) show X 2 space (fct-gen-subalgebra M N X) ! space N proof − have X 2 space M ! space N using assms by simp thus ?thesis using speq by simp qed show 8 y2sets N: X −‘ y \ space (fct-gen-subalgebra M N X) 2 sets (fct-gen-subalgebra M N X) using fct-gen-subalgebra-sets-mem speq by metis qed lemma fct-gen-subalgebra-min: assumes subalgebra M P and f 2 measurable P N shows subalgebra P (fct-gen-subalgebra M N f ) unfolding subalgebra-def proof (intro conjI ) let ?Mf = fct-gen-subalgebra M N f show space ?Mf = space P using assms by (simp add: fct-gen-subalgebra-def gen-subalgebra-space subalgebra-def ) show inc: sets ?Mf ⊆ sets P proof − 5 have space M = space P using assms by (simp add:subalgebra-def ) have f 2 measurable M N using assms using measurable-from-subalg by blast have sigma-algebra (space P)(sets P) using assms measure-space mea- sure-space-def by auto have 8 A 2 sets N: f −‘A \ space P 2 sets P using assms by simp hence ff −‘A \ (space M) j A: A 2 sets Ng ⊆ sets P using hspace M = space P i by auto hence sigma-sets (space M) ff −‘A \ (space M) j A: A 2 sets Ng ⊆ sets P by (simp add: hsigma-algebra (space P)(sets P)i hspace M = space P i sigma-algebra:sigma-sets-subset) thus ?thesis using fct-gen-subalgebra-sets hf 2 M !M N i hspace M = space P i assms(2) measurable-sets mem-Collect-eq sets:sigma-sets-subset subsetI by blast qed qed lemma fct-preimage-sigma-sets: assumes X2 space M ! space N shows sigma-sets (space M) fX −‘B \ space M jB: B 2 sets Ng = fX −‘B \ space M jB: B 2 sets Ng (is ?L = ?R) proof show ?R⊆ ?L by blast show ?L⊆ ?R proof fix A assume A2 ?L thus A2 ?R proof (induct rule:sigma-sets:induct; auto) { fix B assume B2 sets N let ?cB = space N − B have ?cB 2 sets N by (simp add: hB 2 sets N i sets:compl-sets) have space M − X −‘B \ space M = X −‘ ?cB \ space M proof show space M − X −‘B \ space M ⊆ X −‘ (space N − B) \ space M proof fix w assume w 2 space M − X −‘B \ space M hence X w 2 (space N − B) using assms by blast thus w2 X −‘ (space N − B) \ space M using hw 2 space M − X −‘ B \ space M i by blast qed show X −‘ (space N − B) \ space M ⊆ space M − X −‘B \ space M proof fix w assume w2 X −‘ (space N − B) \ space M thus w 2 space M − X −‘B \ space M by blast qed 6 qed thus 9 Ba: space M − X −‘B \ space M = X −‘ Ba \ space M ^ Ba 2 sets N using h?cB 2 sets N i by auto } { fix S::nat ) 0a set assume (Vi: 9 B: S i = X −‘B \ space M ^ B 2 sets N)