The Evaluation of American Compound Option Prices under Stochastic Volatility Carl Chiarella and Boda Kang School of Finance and Economics University of Technology, Sydney CNR-IMATI Finance Day Wednesday, 8th April, 9
Plan of Talk Compound options A two pass PDE free BVP Sparse grid implementation Monte Carlo simulation implementation Numerical examples Conclusions Chiarella and Kang CNR-IMATI 9
Compound Option Compound option is an option on an option. Allow for large leverage. Often used in currency and fixed-income markets. Problem: the tender for a contract that needs 4 years of financing. If company wins tender, could be exposed to an interest rate rise by the time the contract starts (say months). Solution: purchase a -month call option on a 4 year cap. Properties: easily leverage a position; with very little upfront premium but possible to take a substantial position. Chiarella and Kang CNR-IMATI 9
$ Price of the underlying asset K D Price of the daughter option K M T M T D time Figure : The components of a compound option Chiarella and Kang CNR-IMATI 9
Literature on the Evaluation of Compound Options Geske (979)-the first closed-form solution for the price of a vanilla European call on a European call. Han in his thesis () and Fouque and Han (5): derive a fast, efficient and robust approximation to compute the prices of compound options within the context of multiscale stochastic volatility models; they only consider the case of European option on European option; their method relies on certain expansions so its range of validity is not entirely clear. Buchen (8): Exotic compound options in BS-world. Chiarella and Kang CNR-IMATI 9 4
Compound Options - Evaluation under Stochastic Volatility We follow Heston assuming the dynamics for S under RN measure governed by ds = (r q)sdt + vsdz, () dv = (κ v θ v (κ v + λ)v)dt + σ vdz. () Here S and v are correlated with E(dZ dz ) = ρdt. Assumes market price of vol. risk = λ v. Chiarella and Kang CNR-IMATI 9 5
The price of an American compound option under SV can be formulated as the solution to a two-pass Free-BVP with the following Kolmogorov operator: K = vs + ρσvs S S v + σ v v + (r q) S S + (κ v(θ v v) λv) v. () Chiarella and Kang CNR-IMATI 9 6
First the PDE for the value of the daughter option D(S, v, t) with the Kolmogorov operator K: KD rd + D t =. (4) Solve on t T D s.t. D(S, v, T D ) = (S K D ) +, and free (early exercise) boundary and smooth pasting conditions: D(d(v, t), v, t) = d(v, t) K D ; (5) lim S d(v,t) D S =, lim S d(v,t) D v =. (6) Here S = d(v, t) is the early exercise boundary for the daughter option at time t and variance v. Chiarella and Kang CNR-IMATI 9 7
Next, the PDE for the mother option M(S, v, t): KM rm + M t =. (7) Solve on t T M with terminal condition M(S, v, T M ) = (D(S, v, T M ) K M ) +, (8) and free (early exercise) boundary and smooth pasting conditions: M(m(v, t), v, t) = D(m(v, t), v, t) K M, (9) lim S m(v,t) M S = D S, lim S m(v,t) M v = D v. () Here S = m(v, t) is the early exercise boundary for the mother option at time t and variance v. Chiarella and Kang CNR-IMATI 9 8
Sparse Grid Implementation It is computationally demanding to solve the two nested PDEs (4) and (7). Hence we apply the sparse grid approach. We implement the sparse grid combination technique of Reisinger and Wittum (7) to solve these PDEs. The technique relies on a combination technique requiring solution of the original equation only on several specific grids and a subsequent extrapolation step. In Figure, those grids are dense in one direction but sparse in the other direction. Solve the two PDEs on each of the grids in parallel and combine the results from different grids. Chiarella and Kang CNR-IMATI 9 9
.5.5.5.5.5.5.5 Figure : A typical S (vertical) v (horizontal) axes. A sparse grid with a level 6 with respect to the combinations (from left to right), (6,), (5,), (4,), (,), (,4), (,5), (,6). The (6, ) grid has 6 subintervals in the S direction and subintervals in the v direction, and so forth. Chiarella and Kang CNR-IMATI 9
.5.5.5.5.5.5 Figure : A typical S (vertical) v (horizontal) axes. A sparse grid with a level 5 with respect to the combinations (from left to right), (5,), (4,), (,), (,), (,4), (,5). The (, ) grid has subintervals in the S direction and subintervals in the v direction, and so forth. Chiarella and Kang CNR-IMATI 9
Following the combination technique, the solution c l (l sparse grid level) of the PDE is c l = l C(l n, n) l C(l n, n). () n= n= The combination gives a more accurate solution of the PDE. There are (l + ) PDE solvers running in parallel at the same time on each of the sparse grids for level l and level (l ) respectively. Because of different scale characteristics in S and v direction and also some bad behavior along the boundary we need to use a modified sparse grid. (see Figure 4) Chiarella and Kang CNR-IMATI 9
.5.5.5 Figure 4: A modified sparse grid with a initial level and total level 6 with respect to each combination. From left to right we see the combinations (,4), (,), (4,). Chiarella and Kang CNR-IMATI 9
Monte Carlo simulation implementation We need an alternative method to check the solution. Use the Method of Lines (MOL) to solve the PDE for the Daughter option and obtain the option prices with a range of maturities, and store the results. Implement Monte Carlo Simulation scheme of Ibanez & Zapatero (4) to find the price of the American mother option with suitable terminal condition (8) and free boundary condition (9). The data of the underlying daughter option are available from the previous results from MOL. Chiarella and Kang CNR-IMATI 9 4
S S max S * = d ( v τ ) m, n Early ex. condn. satisfied. τ n T τ v m v Figure 5: Solving for the free boundary point of the Daughter option along a (v m, τ n ) line using MOL. Chiarella and Kang CNR-IMATI 9 5
c Solve daughter option using MOL Early exercise along this line Solve mother option using MC v S Figure 6: Illustrating the MOL-MC scheme along one (S, v) line Chiarella and Kang CNR-IMATI 9 6
Monte Carlo Simulation for the Mother option. A finite number of exercise opportunities = t < t, < t N = T M are considered. The optimal exercise strategy at every point at time t n is characterized by a region in a two dimensional-space (v, t n ). Going backward in time, we solve the following equation for each n = N,..., at different variance levels v i : M(S, v i, t n ) = D(S, v i, t n ) K M ; for S using Newton s method to find the optimal exercise frontier S t n (v i ); Chiarella and Kang CNR-IMATI 9 7
S Χ Χ S * t N ( v) Χ S S * t N ( v) K M t N t N t N = T M t Figure 7: Monte Carlo Simulation for the Mother option for some fixed v Chiarella and Kang CNR-IMATI 9 8
Continuing to work backwards, we can find all optimal strategies at times t < t,, < t N for a certain number of variance levels. Finally implementing another MC simulation to generate paths for both the underlying prices and the variance forward in time starting from t to find the price of the mother option M(S, v, t ) based on all known optimal exercise strategies. Chiarella and Kang CNR-IMATI 9 9
Numerical Examples Parameter Value SV Parameter Value r. θ.4 q.5 κ v. T D. σ. K D λ v. T M.6 ρ ±.5 K M 7 Table : Parameter values used for the American call daughter option. The stochastic volatility (SV) parameters correspond to the Heston model. Chiarella and Kang CNR-IMATI 9
ρ =.5, v =.4 S Runtime Method 8 9 (sec) SG (4,6).769.69.96 7.754 4.799 5 MOL + MC (5,).758.6898.9567 7.69 4.789 997 std err.6.9.4.65.78 Lower Bound.747.686.9486 7.679 4.697 Upper Bound.77.695.9649 7.746 4.74 Table : Compound prices (American call on American call) computed using sparse grid (SG), Monte Carlo simulation (MC) together with method of lines (MOL). Parameter values are given in Table, with ρ =.5 and v =.4. Chiarella and Kang CNR-IMATI 9
ρ =.5, v =.4 S Runtime Method 8 9 (sec) SG (4,6).4.945.4 7.679 4.885 495 MOL + MC (5,).8.94.46 7.6646 4.76 95 std err.5..5.7.8 Lower Bound.99.88.964 7.659 4.558 Upper Bound.57.998.459 7.678 4.87 Table : Compound prices (American call on American call) computed using sparse grid (SG), Monte Carlo simulation (MC) together with method of lines (MOL). Parameter values are given in Table, with ρ =.5 and v =.4. Chiarella and Kang CNR-IMATI 9
Free surface of daughter option d(v,τ).4.5..5..5..5....4.5.6.7.8.9 v τ Free surface of mother option (MC Simulation) m(v,τ).4.5..5..5..5....4.5.6.7 v Free surface of mother option (PSOR) τ m(v,τ).4.5..5..5..5....4.5.6.7 v τ Figure 8: Free surfaces of both daughter option and mother option, with the parameters in Table and ρ =.5. Chiarella and Kang CNR-IMATI 9
Free surface of daughter option d(v,τ).4.5..5..5..5....4.5.6.7.8.9 v τ Free surface of mother option (MC Simulation) m(v,τ).4.5..5..5..5....4.5.6.7 v Free surface of mother option (PSOR) τ m(v,τ).4.5..5..5..5....4.5.6.7 v τ Figure 9: Free surfaces of both daughter option and mother option, with the parameters in Table and ρ =.5. Chiarella and Kang CNR-IMATI 9 4
5 Free surfaces of daughter option, ρ=.5 d(v,τ) τ=. 5 τ=.4 τ=.6 τ=.8.5..5..5..5.4 v 5 Free surfaces of daughter option, ρ=.5 d(v,τ) τ=. 5 τ=.4 τ=.6 τ=.8.5..5..5..5.4 v Figure : Free surfaces of daughter option with different ρ, the time to maturity τ and parameters in Table. Chiarella and Kang CNR-IMATI 9 5
5 Free surfaces of mother option, ρ=.5 m(v,τ) τ=. 5 τ=.4 τ=.6 τ=.48.5..5..5..5.4 v 5 Free surfaces of mother option, ρ=.5 m(v,τ) τ=. 5 τ=.4 τ=.6 τ=.48.5..5..5..5.4 v Figure : Free surfaces of mother option with different ρ, the time to maturity τ and parameters in Table. Chiarella and Kang CNR-IMATI 9 6
6 Free surfaces of daughter option, ρ=.5 5 4 d(v,τ)....4.5.6.7.8.9 τ v =.4 v =. 5 Free surfaces of daughter option, ρ=.5 4 d(v,τ)....4.5.6.7.8.9 τ v =.4 v =. Figure : Free surfaces of daughter option with different ρ, the variance v and parameters in Table. Chiarella and Kang CNR-IMATI 9 7
6 Free surfaces of mother option, ρ=.5 55 5 45 m(v,τ) 4 5 5....4.5.6 τ v =.4 v =. 5 Free surfaces of mother option, ρ=.5 m(v,τ) 45 4 5 5 5....4.5.6 τ v =.4 v =. Figure : Free surfaces of mother option with different ρ, the variance v and parameters in Table. Chiarella and Kang CNR-IMATI 9 8
.4 Compound Option Price Differences, ρ=.5. %Price Differences (=%)..4.6.8 zero price difference θ v =.4 θ v =.9 θ v =.6 4 6 8 4 6 8 Share Price (S) Figure 4: Percentage price differences between constant and stochastic volatility ρ <. Chiarella and Kang CNR-IMATI 9 9
%Price Differences (=%) 7 6 5 4 Compound Option Price Differences, ρ=.5 zero price difference θ v =.4 θ v =.9 θ v =.6 4 6 8 4 6 8 Share Price (S) Figure 5: Percentage price differences between constant and stochastic volatility ρ >. Chiarella and Kang CNR-IMATI 9
Conclusions Compound Options under Stochastic Volatility Allow for early exercise feature Set up as two pass PDE FBVP Solve using the sparse grid technique Benchmark against MOL/MC The sparse grid approach can be speeded up by using better PDE solvers, e.g. MOL, operator splitting Future work; apply to specific examples e.g. real options applications such as multi-stage investment projects Chiarella and Kang CNR-IMATI 9