The Evaluation Of Barrier Option Prices Under Stochastic Volatility And .

The Evaluation of Barrier Option Prices Under StochasticVolatilityCarl Chiarella†, Boda Kang† and Gunter H. Meyer?†School of Finance and EconomicsUniversity of Technology, Sydney?School of MathematicsGeorgia Institute of Technology, Atlanta.BFS 2010Hilton, TorontoJune 24, 2010

Plan of Talk Barrier Options: Discretely and Continously monitored PDE setup for Barrier option prices under Heston Model Allow for early exercise Method of Lines Approach Numerical Examples ConclusionChiarella, Kang and MeyerBFS 20101

1Barrier Option Path-dependent options, very popular in foreign exchange markets. Thepurchaser uses them to hedge very specific cash flows with similar properties but pays a cheaper price than regular options. Payoff is dependent on the realized asset path via its level. See Figure 1for up-and-out call option payoff. Apart from “out” options, there are also “in” options which only receivea payoff if a certain level is reached, otherwise they expire worthless. In-Out-Parity for Barrier options: Knock-in Knock-out Vanilla. Put-Call-Symmetry for Barrier Options: Up-and-out Call(S, K, H, r, q, ρ) Down-and-out Put(K, S, SK/H, q, r, ρ). We consider up-and-out call options in the following.Chiarella, Kang and MeyerBFS 20102

Up and out Call145140Payoff 0 (European)orPayoff H K (American)135130Payoff max(S K,0)125S1201151101051009590tFigure 1: Diagram: Payoff of an up and out call options.Chiarella, Kang and MeyerBFS 20103

2Literature Review Merton (1973): the derivation of the pricing formula for barrier options; Rich (1994) and Wong & Kwok (2003): a list of pricing formulas forone and multi-asset barrier options both under the GBM framework; Gao, Huang & Subrahmanyam (2000): option contracts under GBMwith both knock-out barrier and American early exercise features; Zvan, Vetzal & Forsyth (2000): discuss the oscillatory behavior of theCrank - Nicolson method for pricing barrier options. The backward Euler method is applied to avoid unwanted oscillations; Griebsch (2008): discusses evaluation of barrier option prices under theHeston model with Fourier transform approach; Yousuf (2008, 2009): develops a higher order smoothing scheme forpricing barrier options under SV without early exercise.Chiarella, Kang and MeyerBFS 20104

3Barrier Options - Evaluation under SV We follow Heston (1993) assuming the dynamics for S under RN measure governed by dS (r q)Sdt vSdZ1 , dv (κv θv (κv λ)v)dt σ vdZ2 . Here S and v are correlated with E(dZ1 dZ2 ) ρdt. Assumes market price of vol. risk λ v.Chiarella, Kang and MeyerBFS 20105

The price of a barrier option C(S, v, τ ) at time to maturity τ is thesolution to a partial differential equation (PDE) problem. We need to solve the PDE C τ KC rC,on the interval 0 τ T , where the Kolmogorov operator KK vS 2 2 S v (κv (θv v) λv). vChiarella, Kang and Meyer2 S 2 ρσvS 2BFS 2010 σ2 v 22 v 2 (r q) S S 6

3.1Continuously monitored barrier options Continuously monitored barrier option C(S, v, τ ): an option which ismonitored all the time between the current time t and the maturity of theoption at time T . Note that τ T t. The option, has the terminal conditionC(S, v, 0) (S K) . The domain for the up and out call option is0 S H, 0 v , 0 τ T . The boundary conditions for the barrier option without the early exercisefeatures are:C(0, v, τ ) 0; C(H, v, τ ) 0; lim Cv (S, v, τ ) 0.v Chiarella, Kang and MeyerBFS 20107

The option with early exercise features has the free boundary conditionC(b(v, τ ), v, τ ) b(v, τ ) K, when b(v, τ ) Hwhere S b(v, τ ) is the early exercise boundary for the barrier optionat time to maturity τ and variance v. There also hold the smooth-pasting conditionslim CS b(v,τ ) S 1,limS b(v,τ ) C v 0. In the above case,C(S, v, τ ) S K, b(v, τ ) S H. However, if we cannot find a b(v, τ ) H thenC(H, v, τ ) H K.Chiarella, Kang and MeyerBFS 20108

Technically, for the knock-out event and the exercise date to be welldefined, the option contract is defined in a way such that when the asset price firsttouches the barrier, the option holder has the option to either exercise orlet the option be knocked out. Since in this paper we assume the rebate is equal to zero, the optionshould be exercised once the asset price touches the barrier.Chiarella, Kang and MeyerBFS 20109

3.2Discretely monitored barrier options A discretely monitored barrier option is an option which is monitoredonly at discrete dates t t1 t2 · · · tN T . The option has the terminal conditionC(S, v, 0) (S K) . The domain for the up and out call option is:S (0, H),τ {T tN , T tN 1 , · · · , T t1 },(0, ),otherwise,and0 v , 0 τ T .Chiarella, Kang and MeyerBFS 201010

The boundary conditions for the barrier option without early exercisefeatures are:C(0, v, τ ) 0;C(H, v, τ ) 0, τ {T tN , · · · , T t1 };lim C(S, v, τ ) 0, τ / {T tN , · · · , T t1 };S lim Cv (S, v, τ ) 0.v A discretely monitored barrier option with the early exercise feature, atthe monitoring times τ {T tN , · · · , T t1 }, has the free (earlyexercise) boundary conditionC(b(v, τ ), v, τ ) b(v, τ ) K, when b(v, τ ) H.Chiarella, Kang and MeyerBFS 201011

Here b(v, τ ) is the early exercise boundary for the barrier option at timeto maturity τ and variance v, and satisfies the smooth-pasting conditionslim CS b(v,τ ) S 1,limS b(v,τ ) C v 0. In the above case, we haveC(S, v, τ ) S K, b(v, τ ) S Hso that C(S, v, τ ) is known over 0 S H. If there is no such b(v, τ ) then for the same reason as the case for thecontinuously monitored option, C(S, v, τ ) must satisfyC(H, v, τ ) H K. At all other times τ / {T tN , · · · , T t1 }, standard Americanoption free boundary conditions apply.Chiarella, Kang and MeyerBFS 201012

4Method of Lines (MOL) Approach The method of lines has several strengths when dealing with Barrieroptions, especially when allowing early exercise features: The price, free boundary, delta and gamma are all found as part ofthe computation. The method discretises the PDE in an intuitive manner, and is readilyadapted to be second order accurate in time. The key idea behind the method of lines is to replace a PDE with anequivalent system of one-dimensional ODEs. The system of ODEs is developed by discretising the time derivativeand the derivative terms involving the variance, v.Chiarella, Kang and MeyerBFS 201013

The PDE to be solved is C τ vS 2 2 C2 S 2 ρσvS (r q)S C S 2C S v σ2 v 2 C2 v 2 (κv (θv v) λv) C v. The computational domain for the problem will depend on the specificBarrier option, for example, for a continuously monitored up and out call option, we would have:0 S0 S H, 0 v , 0 τ T .Chiarella, Kang and MeyerBFS 201014

We discretise according to τn n τ and vm m v, where n 1, . . . , N ; m 1, . . . , M.n C(S, vm , τn ) Cm(S),V (S, vm , τn ) C(S, vm , τn ) Sn Vm(S). We use the standard central difference schemennnCm 1 2Cm Cm 1 2C 2C ,22 v( v) S v nnVm 1 Vm 12 v. We use an upwinding finite difference scheme for the first order derivative term nnCm 1 Cmα ifv , C vβ nnC Cmm 1 vif v α . vChiarella, Kang and MeyerBFS 2010β15

A second order approximation for the time derivative, C τ nn 13 Cm Cm2 τ n 1n 21 Cm Cm2 τ. After taking the boundary conditions into consideration, we must solvea system of (M 1) second order ODEs in S along the line segment(vm , τn ), S [S0 , H] or S [S0 , Smax ] depending on the properties of the barrier option for m 1, ., M 1 and fixed τn . We then solve the ODEs for increasing values of v, using the latest availnnnnable estimates for Cm 1, Cm 1, Vm 1and Vm 1. We iterate until the price profile converges to a desired level of accuracy.Chiarella, Kang and MeyerBFS 201016

vBoundary condition (v vM )InitialCondition(Payoff)τBoundary condition v 0Figure 2: One sweep of the solution scheme on the v τ grid. The stencilfor the typical point o is displayed in Figure 3.Chiarella, Kang and MeyerBFS 201017

nCm 1bn 2Cmbn 1CmbobnCmnCm 1Figure 3: Stencil for the typical grid point o of Figure 2. The stencil fornnnnn 1n 2Cmdepends on (Cm 1, Cm, Cm 1, Cm, Cm).Chiarella, Kang and MeyerBFS 201018

The generic first order form of the ODEDeltaGammandCmdSndVmdSn Vm,nnn Am (S)Cm Bm (S)Vm Pm(S),nnnnnwhere Pm(S) is also a function of Cm 1, Cm 1, Vm 1, Vm 1,n 1n 2Cm, Cm. We solve the above system using the Riccati transform. The Riccati transformationnnnCm(S) Rm (S)Vm(S) Wm(S).Chiarella, Kang and MeyerBFS 201019

Where R and W are solutions to the initial value problemsdRmdSndWmdS 1 Bm (S)Rm Am (S)(Rm )2 ,Rm (S0 ) 0,nn Am (S)Rm (S)Wm Rm (S)Pm(S),nWm(S0 ) 0.n Given R and W we try to find Vmby solvingndVmdSnnnn Am (S)(Rm (S)Vm Wm(S)) Bm (S)Vm Pm(S),backward subject to an terminal condition which depends on the properties and the specifications of the barrier options.Chiarella, Kang and MeyerBFS 201020

Figure 4: Solving for the option prices along a (vm , τn ) line.Chiarella, Kang and MeyerBFS 201021

Continuously monitored barrier options without early exercise opporntunities, using the fact that Cm(H) 0 we obtain from the Riccatitransform that the terminal condition isnVm(H) nWm(H)Rm (H),nand then integrate the equation for Vmfrom S H to S S0 . Continuously monitored barrier option with early exercise opportunitynwe integrate the equation for Wmand Rm from S0 to Smax and monitor the functionnφ(S) Rm (S) Wm(S) (S K).Chiarella, Kang and MeyerBFS 201022

If φ(S ) 0 for some S (S0 , H) then S is the early exerciseboundary b(vm , τn ) bnm at the grid point (vm , τn ).nn Once bnm is found we integrate the equation for Vm backward from bmtoward S0 subject to the terminal conditionV (bnm ) 1. If φ(S) has no zero in [S0 , H) then there is no early exercise belownthe barrier and we solve the equation for Vmsubject tonVm(H) nH K Wm(H)Rm (H). In fact, at any time to maturity τ , if the asset hits the barrier H, then theoption will be exercised, namely, C(H, v, τ ) H K, according tothe Riccati transform we havennnCm(H) Rm (H)Vm(H) Wm(H) H K.Chiarella, Kang and MeyerBFS 201023

5Numerical ExamplesParameterValueSV �v0.00ρ 0.50H130Table 1: Parameter values used for the barrier option. The stochastic volatility (SV) parameters are those used in Heston’s original paper.Chiarella, Kang and MeyerBFS 201024

ρ 0.50, v 0.1Method (N, M, Spts )S8090100110120MOL (50,100,1140)0.90451.88072.59782.48591.4858MOL (100,200,6400)0.90441.87812.59082.47691.4782FD (200, 100, 200)0.90291.87782.59032.47601.4775MC (400, 20)0.93551.95792.74072.67061.6773MC upper bound0.93891.96282.74642.67621.6820MC lower bound0.93211.95302.73512.66491.6726Table 2: Prices of the continuously monitored barrier option without earlyexercise features computed using method of lines (MOL), finite difference(FD) and Monte Carlo simulation (MC). Parameter values are given in Table1, with ρ 0.50 and v 0.1.Chiarella, Kang and MeyerBFS 201025

ρ 0.50, v 0.1Method (N, M, Spts )S8090100110120MOL (50,150,1140)1.40093.93508.298114.401521.8229MOL (100,200,2440)1.40123.93648.300314.403321.8219MOL (100,200,6400)1.40123.93638.300314.403221.8216MOL (200,400,9100)1.40153.93718.301414.403721.8201MC (100, 20, 50)1.39943.92388.230214.108620.9401MC upper bound1.40583.93478.245414.126120.9568MC lower bound1.39303.91298.215114.090920.9234Table 3: Prices of the continuously monitored barrier option with early exercise features computed using method of lines (MOL) and Monte Carlosimulation (MC). Parameter values are given in Table 1, with ρ 0.50and v 0.1.Chiarella, Kang and MeyerBFS 201026