Modern Methods of Option Pricing

Modern Methods of Option Pricing Denis Belomestny Weierstraß Institute Berlin Motzen, 14 June 2007 Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 1 / 30

Overview 1 Introduction Contingent claims Options and Risks 2 Bermudan pricing problem Bermudan derivatives Snell Envelope Process Backward Dynamic Programming Exercise policies and lower bounds 3 Upper Bounds Dual Upper Bounds Riesz upper bounds 4 Fast upper bounds Doob-Meyer Martingale Martingale Representation Projection Estimator 5 Applications Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 2 / 30

Introduction Contingent claims Contingent claims Definition A derivative security or a contingent claim is a financial asset whose payment depends on the value of some underlying variable (stock, interest rate and so on). Definition An option is derivative security that gives the right to buy or sell the underlying asset, at (European option) or not later than (American or Bermudan option) some maturity date T, for a prespecified price K, called strike. A call (put) option is a right to buy (sell). These plain vanilla options have been introduced in 1973. Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 3 / 30

Introduction Options and Risks Options and Risks Risk reduction Any option is a kind of insurance policy that put floor on the losses related to the specific behavior of the underlyings. Example The seller of a call exchanges his upside risk (gains above the strike price) for the certainty of cash in hand (the premium). The buyer of a put limits his downside risk for a price - like buying fire insurance for a house. The buyer of a straddle limits both risks. Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 4 / 30

Options and Risks Introduction Options and Risks Call Option Put Option Straddle Option Payoff 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Payoff 0.0 0.2 0.4 0.6 0.8 1.0 Payoff 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 x 0 1 2 3 4 x 0 1 2 3 4 x Stock 0 1 2 3 4 Stock 0 1 2 3 4 Stock 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Time 0.0 0.2 0.4 0.6 0.8 1.0 Time 0.0 0.2 0.4 0.6 0.8 1.0 Time Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 5 / 30

Bermudan pricing problem Bermudan derivatives Bermudan derivatives Let L t R D be an underlying and T := {T 0, T 1,..., T J } be a set of exercise dates. Bermudan derivative: An option to exercise a cashflow C(T τ, L(T τ )) at a future time T τ T, to be decided by the option holder. Example The callable snowball swap pays semi-annually a constant rate I over the first year and in the forthcoming years (Previous coupon + A Libor) +, semi-annually, where A is given in the contract. Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 6 / 30

Bermudan pricing problem Bermudan derivatives Bermudan derivatives Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 7 / 30

Valuation Bermudan pricing problem Bermudan derivatives If N, with N(0) = 1, is a numeraire and P is the associated pricing measure, then with Z τ := C(T τ, L(T τ ))/N(T τ ), the t = 0 price of the derivative is given by the solution of an optimal stopping problem V 0 = sup E F 0 Z τ, τ {0,...,J } where the supremum runs over all stopping indexes τ with respect to {F Tj, 0 j J }, where (F t ) t 0 is the usual filtration generated by L. Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 8 / 30

Optimal stopping Bermudan pricing problem Bermudan derivatives Mathematical problem: Optimal stopping (calling) of a reward (cash-flow) process Z depending on an underlying (e.g. interest rate) process L Typical difficulties: L is usually high dimensional, for Libor interest rate models, D = 10 and up, so PDE methods do not work in general Z may only be virtually known, e.g. Z i = E F i j i C(L j) for some pay-off function C, rather than simply Z i = C(L i ) Z may be path-dependent Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 9 / 30

Bermudan pricing problem Snell Envelope Process Snell Envelope Process At a future time point t, when the option is not exercised before t, the Bermudan option value is given by V t = N(t) with κ(t) := min{m : T m t}. The process sup E F t Z τ τ {κ(t),...,j } Y t := V t N(t) is called the Snell-envelope process and is a supermartingale, i.e. E Fs Y t Y s. Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 10 / 30

Bermudan pricing problem Backward Dynamic Programming Backward Dynamic Programming Set Yj := Y (T j ), L j = L(T j ) F j := F Tj. At the last exercise date T J we have and for 0 j < J, Y j Y J = Z J ( ) = max Z j, E F j Yj+1. Observation Nested Monte Carlo simulation of the price Y0 would require Nk samples when conditional expectations are computed with N samples Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 11 / 30

Bermudan pricing problem Backward Dynamic Programming Backward Dynamic Programming Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 12 / 30

Bermudan pricing problem Exercise policies and lower bounds Construction of exercise policies Any stopping family (policy) (τ j ) satisfies j τ j J, τ J = J, τ j > j τ j = τ j+1, 0 j < J. A lower bound Y for the Snell envelope Y, Y i := E F i Z τi Y i Example The policy τ i := inf{j i : L(T j ) G R D } exercises when the underlying process L enters a certain region G. Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 13 / 30

Bermudan pricing problem Exercise policies and lower bounds Regression Methods An exercise policy τ can be constructed via τ J = J, τ j = jχ {Cj (L j ) Z j } + τ j+1 χ {Cj (L j )>Z j }, j < J, where a continuation value C j (L j ) := E F j Yj+1 can be approximated as C j (x) R β jr ψ r (x), j = 0, 1,..., J 1, r=1 using a sample from (L j, Z τ j ) and a set of basis functions {ψ r }. Question Is the policy τ a good one? Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 14 / 30

Upper Bounds Dual Upper Bounds Dual upper bounds Consider a discrete martingale ( M j )j=0,...,j with M 0 = 0 with respect to the filtration ( F j. Following Rogers, Haugh and Kogan, we )j=0,...,j observe that Y 0 = sup E F 0 [Z τ M τ ] E F 0 τ {0,,...,J } [ ] max Zj M j. 0 j J Hence the r.h.s. with an arbitrary martingale gives an upper bound for the Bermudan price Y 0. Question What martingale does lead to equality? Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 15 / 30

Dual upper bounds Upper Bounds Dual Upper Bounds Theorem (Rogers (2001), Haugh & Kogan (2001)) ( Let M be the (unique) Doob-Meyer martingale part of M j is an ( F j ) -martingale which satisfies Yj = Y0 + M j A j, j = 0,..., J, Y j ) 0 j J, i.e. with M0 := A 0 := 0 and A being such that A j is F j 1 measurable for j = 1,..., J. Then we have [ ] Y0 = E F 0 max Z j Mj. 0 j J Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 16 / 30

Riesz upper bounds Upper Bounds Riesz upper bounds Doob-Meyer decomposition Yj = Y0 + M j A j, j = 0,..., J, and Y J = Z J imply Riesz decomposition Y j = E F j Z J + E F j (A J A j ) Since A i+1 A i = Y i E F i Y i+1 = [Z i E F i Y i+1 ]+, Y j J 1 = E F j Z J + E F j [Z i E F i Yi+1 ]+. i=j Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 17 / 30

Riesz upper bounds Upper Bounds Riesz upper bounds Theorem (Belomestny & Milstein (2005)) If Y i is a lower approximation for Y i, then Y up j J 1 = E F j Z J + E F j [Z i E F i Y i+1 ] + i=j is an upper approximation for Y j, that is Y j Y j Y up j, j = 0,..., J. Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 18 / 30

Riesz upper bounds Upper Bounds Riesz upper bounds Properties Monotonicity Ỹ i Y i Ỹ up i Y up i Locality Let {Y α i, α I i } be a family of local lower bounds at i, then Y α,up j is an upper bound. J 1 = E F j Z J + E F j [Z i max E F i Yi+1 α α I ]+ i+1 i=j Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 19 / 30

Fast upper bounds Doob-Meyer Martingale Doob-Meyer Martingale For any martingale M Tj with respect to the filtration (F Tj ; 0 j J ) starting at M 0 = 0 [ ] Y up 0 (M) := E F 0 max (Z T j M Tj ) 0 j J is an upper bound for the price of the Bermudan option with the discounted cash-flow Z Tj. Exact Bermudan price is attained at the martingale part M of the Snell envelope, Y T j = Y T 0 + M T j A T j, M T 0 = A T 0 = 0 and A T j is F Tj 1 measurable. Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 20 / 30

Fast upper bounds Doob-Meyer Martingale Doob-Meyer Martingale Assume that Y Tj = u(t j, L(T j )) is an approximation of the Snell envelope YT j, 0 j J, with Doob decomposition Y Tj = Y T0 + M Tj A Tj, where M T0 = A T0 = 0 and A Tj It then holds: is F Tj 1 measurable. M Tj+1 M Tj = Y Tj+1 E T j [Y Tj+1 ] Observation The computation of M Tj by MC leads to quadratic Monte Carlo for Y up 0 Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 21 / 30

Fast upper bounds Martingale Representation Martingale Representation If L is Markovian and F is generated by a D-dimensional Brownian motion W, then due to the martingale representation theorem M Tj =: =: Tj 0 Tj 0 H t dw t h(t, L(t))dW t, j = 0,..., J, where H t is a square integrable and previsible process. Observation For any square integrable function h(, ) we get a martingale Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 22 / 30

Projection Estimator Fast upper bounds Projection Estimator We are going to estimate H t on partition π = {t 0,..., t I } with t 0 = 0, t I = T, and {T 0,..., T J } π. Write formally, Y Tj+1 Y Tj t l π;t j t l <T j+1 H tl (W tl+1 W tl ) + A Tj+1 A Tj. By multiplying both sides with (Wt d i+1 Wt d i ), T j t i < T j+1, and taking F ti -conditional expectations, we get by the F Tj -measurability of A Tj+1, H d t i 1 [ ] E F t i (Wt d t i+1 t i+1 Wt d i ) Y Tj+1. i Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 23 / 30

Projection Estimator Fast upper bounds Projection Estimator The corresponding approximation of the martingale M is MT π j := Ht π i π W i, t i π;0 t i <T j with π W d i := W d t i+1 W d t i. Theorem (Belomestny, Bender, Schoenmakers 2006) [ ] lim E max π 0 0 j J Mπ T j M Tj 2 = 0, where π denotes the mesh of π. Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 24 / 30

Projection Estimator Fast upper bounds Projection Estimator In fact for T j t i < T j+1 H π t i = h π (t i, L(t i )) = 1 π i [ ] E i ( π W i ) u(t j+1, L(T j+1 )) and this expectation can be computed by regression. 1 Take basis functions ψ(t i, ) = (ψ r (t i, ), r = 1,..., R) 2 Simulate N independent samples (t i, n L(t i )), n = 1,..., N from L(t i ) constructed using the Brownian increments π nw i. Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 25 / 30

Regression Fast upper bounds Projection Estimator 3 Construct the matrix A t i := (A t i A ti ) 1 A t i, where 4 Define A ti = {(ψ r (t i, n L(t i ))), n = 1,..., N, r = 1,..., R}. ĥ π (t i, x) = ψ(t i, x) A t i ( π W i π i Y Tj+1 ) =: ψ(t i, x) β ti, where β ti is the R D matrix of estimated regression coefficients at time t i. Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 26 / 30

Fast upper bounds Fast MC Upper Bound Projection Estimator Finally we construct Ŷ up 0 = 1 Ñ Ñ max 0 j J n=1 [z(t j, n L(Tj )) M j ], with M j = t i π;0 t i <T j ĥ π (t i, L(T j )) ( π Wi ) by simulating new paths ( n L(Tj ), π n W i ), n = 1,..., Ñ. Observation is always a martingale, so the upper bound is true! Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 27 / 30

Max Call on D assets Black-Scholes model: Applications dx d t = (r δ)x d t dt + σx d t dw d t, d = 1,..., D, Pay-off: Z t := z(x t ) := (max(x 1 t,..., X D t ) κ) +. T J = 3yr, J = 9 (ex. dates), κ = 100, r = 0.05, σ = 0.2, δ = 0.1, D = 2 and different x 0 D x 0 Lower Bound Upper Bound A&B Price Y 0 Y up 0 ( M π ) Interval 90 8.0242±0.075 8.0891±0.068 [8.053, 8.082] 2 100 13.859±0.094 13.958±0.085 [13.892, 13.934] 110 21.330±0.109 21.459±0.097 [21.316, 21.359] Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 28 / 30

Max Call on D assets Applications Basis functions at (t, x) with T j 1 t < T { j } SI 1, X k EP(t,X; T j ), X k EP(t,X; T J ), k = 1,..., D { x k x k } SII 1, X k EP(t,X; T j ), k = 1,..., D x k SIII Pol 3 (X, EP(t, x; T j )) SIV Pol 3 (X) SV 1 Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 29 / 30

Max Call on D assets Applications Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 30 / 30

Literatur Belomestny, D. and Milstein, G. Monte Carlo evaluation of American options using consumption processes. International Journal of Theoretical and Applied Finance, 02(1):65 69, 2000. Belomestny, D., Milstein, G. and Spokoiny, V. Regression methods in pricing American and Bermudan options using consumption processes. Journal of Quantitative Finance, tentatively accepted. Belomestny, D., Bender, Ch. and Schoenmakers, J. True upper bounds for Bermudan products via non-nested Monte Carlo. Mathematical Finance, tentatively accepted. Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 30 / 30