Dynamic pricing with diffusion models

Dynamic pricing with diffusion models INFORMS revenue management & pricing conference 2017, Amsterdam Asbjørn Nilsen Riseth Supervisors: Jeff Dewynne, Chris Farmer June 29, 2017 OCIAM, University of Oxford 1

Pricing challenge Continuous time pricing challenge Given an initial amount of stock for a product, a termination time. Maximise revenue and minimise cost of unsold items. 2

Mathematical challenge Hamilton-Jacobi-Bellman (HJB) equations Find v : [0, T] D such that { } σ(t, x, a) 2 v t + max v xx + b(t, x, a)v x + f(t, x, a) = 0 a A 2 v(t, x) = g(x) 3

Outline Pricing problem modelling uncertainty Problem with negative sales HJB solutions Closed-form, approximate pricing policy 4

Demand modelling

Continuum approximation Large inventory limit 4

Sales dynamics Deterministic pricing model Pricing policy α(t) Stock levels S α (t) over period t [0, T], S α (0) = s 0 > 0 Expected demand per time, q(a) ds α (t) = q(α(t))dt, when S α (t) > 0. Include uncertainty Add noise [2, 3]: Volatility σ(t, s, a), Brownian motion W(t) ds α (t) = q(α(t))dt + σ(t, S α (t), α(t))dw(t) 5

1 0.8 ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 0.6 S(t) 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t Figure 1: Demand q(a) = 3 2 (1 a), volatility σ(t, s, a) = 0.08q(a), constant price α(t) = 1/3.

1 0.99 ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 0.98 S(t) 0.97 0.96 0.95 0.94 0 0.5 1 1.5 2 2.5 t 3 3.5 4 4.5 10 2 Figure 1: Demand q(a) = 3 2 (1 a), volatility σ(t, s, a) = 0.08q(a), constant price α(t) = 1/3.

High probability of negative sales over small timescales 6

Negative sales Volatility constant, σ(t, s, a) = σ Brownian motion O(dW(t)) = O( dt) Constant price α(s) = a on [t, t + t), S(t + t) = S(t) t+ t t q(a)dt + t+ t t σdw(t) 7

Negative sales Volatility constant, σ(t, s, a) = σ Brownian motion O(dW(t)) = O( dt) Constant price α(s) = a on [t, t + t), S(t + t) = S(t) q(a) t + σ t Z, Z N (0, 1) 7

Negative sales Volatility constant, σ(t, s, a) = σ Brownian motion O(dW(t)) = O( dt) Constant price α(s) = a on [t, t + t), S(t + t) = S(t) q(a) t + σ t Z, Z N (0, 1) Probability of negative sales, as t 0 ( P q(a) t + σ ) ( t Z < 0 = P Z < ) t q(a)/σ 0.5 7

Model uncertainty in parameters 7

Modelling uncertainty Multiplicative uncertainty Geometric Brownian motion starting at 1 Uncertainty parameter 0 < γ 1 dg(t) = G(t)γ dw(t), ds α (t) = q(α(t))g(t) dt, when S α (t) > 0. G(t) = e γ2 2 t+γw(t), E[G(t)] = 1, Var[G(t)] = e γ2t 1. 8

1 0.8 ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 0.6 S(t) 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t Figure 2: Demand q(a) = 3 2 (1 a), volatility γ = 0.08, constant price α(t) = 1/3.

Optimal control and Hamilton-Jacobi-Bellman

Pricing objective Pricing objective Hitting time T h T for S α (t) = 0 Revenue T h 0 α(u) q(α(u))g(u) du Cost per unit of remaining stock C > 0 Total value T h 0 α(u) q(α(u))g(u) du C S α (T) 10

Pricing objective Pricing objective Hitting time T h T for S α (t) = 0 Revenue T h 0 α(u) q(α(u))g(u) du Cost per unit of remaining stock C > 0 Total value T h 0 α(u) q(α(u))g(u) du C S α (T), a random variable! Maximise expected value [ ] T h E α(u) q(α(u))g(u) du C S α (T) 0 10

Stochastic control problem Pricing challenge Allowed prices A = [a min, a max ] Pricing policies of the form α(t) = a(t, S(t), G(t)) A, taken from collection A Given initial stock s 0, solve the optimisation problem [ ] T h max E α(u) q(α(u))g(u) du C S α (T) α A 0 11

The space A is infinite-dimensional 11

HJB: optimality condition Strategy Define a value function v(t, s, g) Function v(t, s, g) satisfies the HJB equation Find optimal a(t, s, g) in terms of v(t, s, g) 12

HJB: optimality condition Value function Expected value of having stock S(t) = s > 0 at time t [ ] T h v(t, s, g) = max E t α(u)q(α(u))g(u) du C S α (T) α A Subscript t: conditioned on S α (t) = s, G(t) = g. t 13

HJB for pricing problem Dynamic programming principle + Itô s lemma [1]: Find v : [0, T] [0, ) 2 such that v t + γ2 2 g2 v gg + g max a A {q(a)(a v s)} = 0 v(t, s, g) = C s v(t, s, 0) = C s v(t, 0, g) = 0 14

HJB optimal control function Optimality result The value function is the unique viscosity solution to HJB The optimal pricing function a B (t, s, g) is given by a B (t, s, g) = arg max {q(a)(a s v(t, s, g))} a A 15

HJB pricing solution Example q(a) = 3 (1 a), γ = 0.08, C = 1/3. 2 16

HJB pricing solution Figure 3: Optimal pricing function, ab (1 τ, s, 1.0). 17

HJB pricing solution S(t) 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 0.6 0.5 α(t) 0.4 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t Figure 3: Optimal sales dynamics 17

HJB pricing solution S(t) S(t) 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 Figure 3: α(t) = a B (t, s, g) versus α(t) = 1/3 17

Estimating parameters

Estimating G(t) Assume S(t) > 0 Constant price α(s) = a for s [t t, t) Ĝ(t) = S(t t) S(t) q(a) Good approximation over small timesteps Var[Ĝ(t) G(t)] = O(eσ2 t 1) 5 10 5 18

80 70 60 Density 50 40 30 20 10 0 2 1.5 1 0.5 0 0.5 1 1.5 2 Relative error 10 2 Figure 4: Distance between Ĝ and G, with t = 0.01 and γ = 0.08

Certainty Equivalent Control policy

Certainty Equivalent Control Assume no uncertainty Solution when γ = 0 γ 2 v t + 2 g2 v gg + g max {q(a)(a v s)} = 0 a A 20

Certainty Equivalent Control Assume no uncertainty Solution when γ = 0 γ 2 v t + 2 g2 v gg + g max {q(a)(a v s)} = 0 a A For q(a) = q 1 q 2 a q 1 x a C q (t, x) = 2 q 2 g(t t), 0 x g(t t) min{q 1, 1 2 (q 1 + q 2 C)}, 1 2 max(0, q 1 q 2 C), otherwise. 20

Certainty Equivalent Control Assume no uncertainty Solution when γ = 0 γ 2 v t + 2 g2 v gg + g max {q(a)(a v s)} = 0 a A For q(a) = q 1 q 2 a q 1 x a C q (t, x) = 2 q 2 g(t t), 0 x g(t t) min{q 1, 1 2 (q 1 + q 2 C)}, 1 2 max(0, q 1 q 2 C), otherwise. Generalises to other demand functions 20

Compare Bellman to CEC solution 20

Figure 5: HJB pricing function ab (1 τ, s, 1.0)

Figure 5: CEC pricing function ac (1 τ, s, 1.0)

Figure 5: Difference (a C a B )(1 τ, s, 1.0)

0.6 0.5 α(t) 0.4 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t 0.6 0.5 α(t) 0.4 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t Figure 5: Sample paths, α B (t) and α C (t).

Wrapping up

Summary and future work Summary Modelling of uncertainty Data-assimilation Pricing policies from HJB equation Closed-form CEC policy function Future work Performs better than numerical approximation More general models for demand function Each parameter Seasonality, trends Impact of pricing Asymptotics improved closed-form pricing policies 22

References i References [1] H. Pham. Continuous-time stochastic control and optimization with financial applications, volume 61. Springer Science & Business Media, 2009. [2] K. Raman and R. Chatterjee. Optimal monopolist pricing under demand uncertainty in dynamic markets. Management Science, 41(1):144 162, 1995. 23

References ii [3] L.-L. B. Wu and D. Wu. Dynamic pricing and risk analytics under competition and stochastic reference price effects. IEEE Transactions on Industrial Informatics, 12(3):1282 1293, 2016. 24