Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.1/29

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1 Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing Peter Duck School of Mathematics University of Manchester Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p./29

2 Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.2/29

3 A crash course in singular perturbation theory Consider the ordinary differential equation: ɛ d2 y dx 2 y = subject to y() =, y() =. Standard (undergraduate) methods lead to the exact solution: y = + e 2/ ɛ e 2/ ɛ x/ ɛ e x/ ɛ Consider another approach if ɛ becomes small. Setting ɛ = in the ODE leads immediately to y = This clearly satisfies the boundary condition at x = This clearly violates the boundary condition at x = Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.3/29

4 Exact solution variation with ɛ PSfrag replacements y -.4 ɛ = x Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.4/29

5 Boundary-layer concept Exact results suggest, as ɛ : y everywhere, except Close to x =, steep solution gradient Suggests the concept of a boundary layer - the derivative term ( d2 y dx 2 ) term can no longer be neglected. Suggests need d2 dx 2 = O(ɛ ), i.e. x = O( ɛ) Define X = x/ ɛ, then differential equation becomes subject to y(x = ) =, y(x ) d 2 y dx 2 y = Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.5/29

6 The solution (continued) Solution (marginally ) simpler than full problem, leads to y = + e X i.e. y = + e x/ ɛ This agrees with exact solution as ɛ Singular perturbation problems typically arise from the neglect of the highest order derivative, which leads to difficulties in thin zones - boundary layers, shear layers but perfectly acceptable solutions outside these zones If y(), then another boundary layer exists, near x =. This is a Mickey Mouse example Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.6/29

7 THE LIMIT OF SMALL VOLATILITY V t + 2 σ2 S 2 2 V V + rs S2 S rv = Typical value of volatility σ..4 (per (annum) 2 ) Typical value of interest rates r.. (per annum) BSE has σ 2 multiplying highest order (S) derivative - conditions ripe for SINGULAR PERTURBATION PROBLEM Consider a European (call) option Simply setting σ = leads to C t + rs C S rc = Final condition: C(S, T ) = max(s E, ) Solution is C if S Ee r(t t) < C = S Ee r(t t) if S Ee r(t t) > Discontinuity (in slope) at S = Ee r(t t) (at the forward money) Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.7/29

8 The shear layer Asymptotic balancing suggests shear layer to smooth out discontinuity - along S = Ee r(t t) - at the forward money Thickness O(σ), so that σ 2 2 S 2 t Define Ŝ = (S Ee r(t t) )/σ = O(), Ĉ = C/σ = O() O(σ) equation: L{ Ĉ} rŝ Ĉ + Ĉ Ŝ t + 2 Ĉ as Ŝ, Ĉ Ŝ as Ŝ. Ee r(t t) 2 2 Ĉ rĉ = Ŝ2 Corresponding put option: ˆP as Ŝ, ˆP Ŝ as Ŝ. A thin region separating two mis-matching (outer) regions Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.8/29

9 PSfrag replacements 8 P = C = S Ee r(t t) 6 S S = Ee r(t t) 4 2 O(σ) C = P = Ee r(t t) S t Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.9/29

10 ˆP 5 4 PSfrag replacements 3 2 t = T =.5 t = Ŝ Put option, E = 7, r =.5, T =.5 Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p./29

11 Can continue asymptotic expansion ad nauseam by V = n= σn ˆVn (Ŝ, t) For n 2: PSfrag replacements. L{ ˆV n } = ŜEe r(t t) 2 ˆVn Ŝ 2 2 Ŝ 2 2 ˆVn 2 Ŝ 2. error. e-5 e Call option, E =, r =.6, T =.5, S = E rt, σ =.2 n Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p./29

12 Barrier/down and out (put) options - CASE I Option worthless if underlying hits/drops below prescribed value B At expiry, P (S, T ) = max(e S, ) but P (S B, t) = First assume Ee rt > B (shear layer does not hit barrier) P if S > Ee r(t t), P = Ee r(t t) S if B < S < Ee r(t t) Close to S = Ee r(t t), P = ˆP /σ, Ŝ = (S E r(t t) )/σ P (Ŝ, t) similar to European option already described (shear layer) Boundary layer close to S = B: S = (S B)/σ 2 (VERY THIN); P = O() (to match with above), σ 2 S 2 2 S 2 S S : 2 B2 2 P S 2 + rb P S = Subject to P (S =, t) =, P Ee r(t t) B as S Quasisteady, solution P (S, t) = Ee r(t t) B ( e 2r B S ) No interaction between boundary and shear layers Breakdown when τ = T t σ 2 P E B as S = O() - can be fixed up - additional (τ) derivative, Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.2/29

13 8 P = PSfrag replacements S 6 S = Ee r(t t) 4 2 O(σ) P = Ee r(t t) S O(σ 2 ) t Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.3/29

14 Barrier/down and out (put) options - CASE II Collision of shear and boundary layers at tc, when Ee r(t t c) = B Define τ = t t c σ 2 leads to, S = S B σ 2 (thin, as case I), P = P/σ 2 P τ + B2 2 2 P P + rb S 2 S = Subject to P as S, and, as τ : P Brτ ( e 2r B S ) S = O() P τ B ηn( η) + N ( η), η = S Brτ B τ = O() P = Brτ S, S Brτ Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.4/29

15 8 P = 6 ag replacements S 4 O(σ) 2 S = Ee r(t t) P = Ee r(t t) S O(σ 2 ) t Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.5/29

16 American Options, limit of small volatility (σ ) Proceed as before - consider American put Set r = σr, - small interest rates, comparable to volatility = exercise boundary lies inside shear layer P =, S > E, S E = O() - worthless at asset prices O() above exercise price (c.f. Europeans) P = E S, S < E, S E = O() - exercise below strike price Discontinuity along S = E, c.f. Europeans - key regime Ŝ = S E σ Exercise along S = Sf = E + σŝf(t) + σ 2 Ŝ f (t) +... P = σ ˆP (Ŝ, t) + σ 2 ˆP (Ŝ, t) +... = O() O(σ ): L{ ˆP } 2 E2 2 ˆP Ŝ2 + RE ˆP Ŝ + ˆP t = ˆP (Ŝf, t) = S f, ˆP Ŝ (Ŝf, t) = O(σ ): L{ ˆP } = EŜ 2 ˆP Ŝ2 RŜ ˆP Ŝ + R ˆP ˆP (Ŝf, t) =, ˆP Ŝ (Ŝf, t) = 2R E Ŝf etc. Can scale E and R out of problem ( Ŝ f, S ) E R (Ŝf, S ), t t R 2 and if t T then universal set of results obtained (Widdicks et al, 25) Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.6/29

17 American Options - comparison 4 P (S = E, t = ) exact 2-term asymptotic PSfrag replacements σ Comparison of exact and asymptotic results, S = E =, r = σ, T =.5 (body-fitted coordinates used in both cases) Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.7/29

18 Multiple underlyings In the case of an option involving two underlyings V t + 2 V 2 σ2 S2 S 2 V + rs + 2 V S 2 σ2 2 S2 2 S2 2 V 2 V + rs 2 + σ σ 2 ρs S 2 rv = S 2 S S 2 Setting σ = σ 2 = leads to the hyperbolic PDE Many payoff scenarios exist. If V t + rs V V + rs 2 rv =. S S 2 V (S, S 2, t = T ) = max(s E, S 2 E 2, ) V (S, S 2, t = T ) = max(e S, E 2 S 2, ) (calls) (puts). Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.8/29

19 Calls (iii) V = S 2 E 2 e r(t t) S2 4 3 placements 2 (i) worthless (ii) V = S E e r(t t) S Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.9/29

20 E =, E 2 =.5, r =., ρ =.4, T = call value call value ag replacements PSfrag replacements error S S error S S (a) σ =. (b) σ =.25 PSfrag replacements call value error S S (c) σ =.5 Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.2/29

21 Puts (E > E 2 ) 2 (i) worthless 8 (ii) V = E e r(t t) S S2 6 4 ements 2 (iii) V = E 2 e r(t t) S S If E < E 2 the 45 o line would intersect with the S 2 -axis Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.2/29

22 Shear layers along S = E e r(t t) and S 2 = E 2 e r(t t) similar to -D case generally Somewhat different shear layer along S E e r(t t) = S 2 E 2 e r(t t) but can be reduced to a D calculation Transition point at S = E e r(t t), S 2 = E 2 e r(t t). Can be extended to incorporate early exercise Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.22/29

23 Calculation of implied volatilities Can use this analysis to more efficiently back out volatilities and correlation coefficients - by choosing best regions of parameter space. Using asymptotic form along S 2 = E 2 and comparing with exact calculation (American put, E =, E 2 =.5, T = ; in each case r = σ = σ 2 = σ, ρ =.4) -e-4 σ =.2 PSfrag replacements σ2 error σ = σ = S Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.23/29

24 Calculation of implied correlation coefficients Error in calculation of ρ 2, results along E S = E 2 S 2.2 PSfrag replacements ρ2 error σ =. σ = σ = S Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.24/29

25 Pricing bonds A canonical equation form for pricing bonds (using several popular interest-rate models) leads to PDEs of the form (arising from stochastic modelling of interest rates) 2 σ2 (r, t)r 2β 2 F r 2 + f(r, t) + σ(r, t)rβ λ(r, t) F r rf + F t = For a bond, the final condition is F (r, t = T ) = F f (r). For example, for the CIR model f(r, t) = κ(θ r), β = 2, λ = ˆλ r/σ. This has a (messy) exact analytic solution. Let us use small perturbation theory instead. Setting σ = in the full equation, leads to (first-order) equations of the generic form (τ = T t): (A B r) F r rf F τ = Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.25/29

26 Solution of reduced problem Equation can be solved very easily using the method of characteristics, i.e. F (r, τ) = F f A B ( A B r)e B τ exp B 2 (A B r)( e B τ ) A B τ In the case of options on these bonds, (horrible) exact solutions do exist, but the small perturbation method is much simpler - and surprisingly accurate even for quite big σ s. For a call (maturing at t = T, τ = T T ) on a bond (maturing at t = T, τ = T ) C (r, τ) = max F f exp B 2 (A B r )( e B (T T ) ) A B (T T ) X, exp B 2 (A B r)( e B τ ) A B τ, where r = A B ( A B r)e B τ, τ = τ (T T ) Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.26/29

27 Illiquid markets Modified Black-Scholes model (c.f. Liu & Yong, 25, without fudge factor), with feedback - a seriously nonlinear PDE - an example where insight gleaned from asymptotics is invaluable. P τ + 2 σ2 S 2 2 P S 2 ( ρ 2 P S 2 ) 2 + rs P S rp = Consider the European put version of this option (using my favourite Crank-Nicolson method).5.5 PSfrag replacements -.5 τ =. τ = S Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.27/29

28 Asymptotics close to expiry (τ ) Solution takes the form P τ 2 ŜH( Ŝ) + τ ˆP (Ŝ) +..., where Ŝ = (S E)/τ 2 is the key (shear-layer) scaling, close to strike price (H( Ŝ) the Heaviside function) ˆP + 2 Ŝ ˆP Ŝ + 2 σ2 X 2 2 ˆP Ŝ2 ( ρ 2 ˆP rh( Ŝ) = )2 Ŝ2 where ˆP and ˆP continuous at Ŝ =, and ˆP + H( Ŝ)rE as Ŝ. Ŝ Discontinuity in the at S = E of + for all time - can build jump condition into full calculation (using Keller-Cebeci box scheme).2 τ = τ = PSfrag replacements S Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.28/29

29 Conclusions Singular perturbation techniques can provide quick, simple solutions across wide regions of parameter space. Along (thin) zones, discontinuities may occur, but can be resolved by blending (i.e. boundary or shear) layers. Gives insight into financial product pricing Potentially quite universal tool for rapid (and surprisingly accurate) pricing of financial products described by Black-Scholes-like PDEs. Singular perturbation problems arising in mathematical finance: fluid dynamics concepts in option pricing p.29/29