Pricing Derivatives with Barriers in a Stochastic Interest Rate - PowerPoint PPT Presentation

Pricing Derivatives with Barriers in a Stochastic Interest Rate Environment Carole Bernard (University of Waterloo) Olivier Le Courtois (E.M. Lyon) François Quittard-Pinon (University of Lyon) 1

Introduction Our aim : pricing barrier products when interest rates are stochastic . In order to illustrate our methodology, we study a particular barrier option : the Shark Option . Yet, our approach is very general , and has applications in lots of other fields. 2

Relevance 26% of Equity Linked Products currently traded on the American Stock Exchange are Barrier Products . Applications : Extension of Rubinstein and Reiner (1991) Formulas for Barrier Options, Defaultable Bonds, Structured Products, Index Linked Derivatives, Real Options, Stock Options, Portfolio Allocation Market Value of Life Insurance Contracts. Maturity of traded barrier products can be quite long , up to 5 years (among Index Linked Notes). ⇒ Stochastic Interest Rates . 3

Outline of the Talk ➠ Description of the Shark Option ➠ Underlying and Interest Rate Model ➠ Two Types of Barriers : 1. Constant Barrier 2. Stochastic Barrier ⇒ Two Valuation Methods ➠ Numerical Analysis 4

Shark Options A Shark Option is : an up and out Barrier Option with a Rebate . The optionholder receives at expiry T :  1 + ( S T − S 0 ) +   if S max < H  S 0    otherwise β ☞ H is the barrier level, β is the rebate. ☞ S T is the underlying price at time T , ☞ S max is the maximum of S over [0 , T ] . 5

Maximum Payoff of a Shark Option Nominal = S 0 = 100 , β = 1 . 1 , H = 135 140 135 130 Maximum Payoff 125 120 115 110 105 100 100 110 120 130 140 150 160 170 180 190 200 S T Payoff w.r.t. S T . ( S max ≥ S T ) thus this payoff is the maximum the holder might receive. because one might have S max > H and S T < H . 6

Shark Options Using standard results from arbitrage pricing theory, we can express the option price (at time 0 ) under the risk-neutral probability Q :        1 + ( S T − S 0 ) + e − � T 0 r s ds  1 { S max � H } + β 1 { S max >H }     C 0 = E Q S 0 7

Interest Rate Modeling The term structure is given through the default-free zero-coupon bonds P ( t, T ) which dynamics under Q are : dP ( t, T ) P ( t, T ) = r t dt − σ P ( t, T ) dZ Q 1 ( t ) We assume an exponential volatility : � 1 − e − a ( T − t ) � σ P ( t, T ) = ν a 8

Underlying Dynamics The index dynamics under the risk-neutral probability Q are : dS t = r t dt + σdZ Q ( t ) S t where Z Q and Z Q 1 are correlated Q -Brownian motions. ( dZ Q .dZ Q 1 = ρdt ). 9

Two Steps Decorrelation . Let Z Q 2 be independent from Z Q ☞ 1 : � dZ Q ( t ) = ρdZ Q 1 − ρ 2 dZ Q 1 ( t ) + 2 ( t ) ☞ Change of Measure. Let Q T be the T -forward-neutral measure. From Girsanov theorem, Z Q T and Z Q T are independent 1 2 Q T -Brownian motions when defined by : dZ Q T 1 + σ P ( t, T ) dt , dZ Q T = dZ Q = dZ Q 1 2 2 10

Option Valuation at t = 0 The Shark option’s price (at time 0) is equal to : � � � � 1 + S T C 0 = P (0 , T ) E Q T 1 { S max <H } + β 1 { S max ≥ H } S 0 We obtain the following option price : � � C 0 S T P (0 , T ) = β Q T ( γ � T ) + Q T ( S T < S 0 , γ > T )+ E Q T 1 { S T >S 0 , γ>T } S 0 where γ is the first-passage time of S to the level H 11

Two Types of Barriers 1. Constant Barrier : H Semi-closed-form Formulae can be obtained. Methodology : Extended Fortet’s Approximation 2. Discounted Barrier : HP ( t, T ) Closed-form Formulae can be obtained. 12

Shark Options : Constant Barrier Problem : We need to know the law of γ , first passage time of S above H . ➠ Longstaff and Schwartz (1995) use Fortet’s results to approximate the density of γ in a problem similar to ours. ➠ Collin-Dufresne and Goldstein (2001) correct the pre- vious method. 13

First Passage Time Approximate Density Let us recall the definition of γ : γ = inf { t ∈ [0 , T ] / S t < H } Scheme’s Idea : Approximate the density of γ at any time t under Q T as a piecewise constant function. – The interval [0 , T ] is subdivided into n T subperiods : t 0 = 0 , ... , t j , ... , t n T = T – The interest rate is discretized between r min and r max into n r intervals. r i = r min + iδ r are the discretized values of the interest rate. 14

� � γ ∈ [ t j , t j +1 ] with r ∈ [ r i , r i +1 ] The probability of the event is denoted by : q ( i , j ) Collin-Dufresne and Goldstein give a recursive formula to com- pute these probabilities, starting with : q ( i, 0 ) = Φ( r i , t 0 ) where one first computes q ( i, 0 ) for each i , and then q ( i, j ) recursively for j ≥ 1 using : j − 1 n r � � q ( i, j ) = Φ( r i , t j ) − q ( u, v ) Ψ( r i , t j | r u , t v ) v =0 u =0 where Φ and Ψ are completely known. 15

Expressions of Φ and Ψ  � �  µ ( r t , l 0 , r 0 ) − h  Φ( r t , t ) = f r ( r t , t | l 0 , r 0 , 0) N �    Σ 2 ( r t , l 0 , r 0 )  � �    µ ( r t , l s = h, r s ) − h  Ψ( r t , t | r s , s ) = f r ( r t , t | l s = h, r s , s ) N �   Σ 2 ( r t , l s = h, r s ) where : 2 πv e − ( rt − m )2 1 ∗ f r ( r t , t | l s = h, r s , s ) = √ , m = E [ r t | r s ] , v = V ar [ r t | r s ] 2 v ∗ l is defined by : l t = ln S t , h = ln( H ) , ∗ µ and Σ are the conditional moments of l. 16

Shark Options : Constant Barrier The Shark’s price can therefore be expressed as : C 0 = P (0 , T ) [ βE 1 + E 2 ] + E 3 where the three components can be written in terms of such sums :  n T n r � �   E 1 = q ( i, j )     j =0 i =0       � � n T n r n r l 0 − � µ tj,T   � � � l 0 − M T √   = N − δ r f r ( r k | r i , t j , l t j ) N  q ( i, j ) E 2 �     V T  Σ 2 j =0 i =0 � k =0   tj,T     ... 17

Shark Options : Stochastic Barrier From now on, we suppose the barrier is discounted : D t = HP ( t , T ) γ = inf { t ∈ [0 , T ] / S t < HP ( t, T ) } The barrier is proportional to a zero-coupon bond ( P ( t, T ) is stochastic). 18

Shark Options : Discounted Barrier We use time change techniques in a similar way as Briys and de Varenne [1997] who extended the Black and Cox model [1976] by considering a stochastic default barrier. and the following well-known Tools : - Girsanov Theorem - Dubins-Schwarz Theorem 19

The Shark’s price can therefore be expressed as : C 0 = P (0 , T ) [ βE 1 + E 2 ] + E 3 where the three components can be written in closed-form :  � � � �     S 0 S 0 − τ ( T ) + τ ( T )   ln  ln   2 S 0 2 KP (0 ,T ) KP (0 ,T )  +  √ √ = N KP (0 ,T ) N  E 1    τ ( T ) τ ( T )        � �   S 2 � � + τ ( T ) 0 ln  ln( P (0 ,T ))+ τ ( T )   K 2 P (0 ,T )  2  S 0  √  √  2 E 2 = N − KP (0 ,T ) N      τ ( T ) τ ( T )        ... � � � T ( σ P ( u, t ) + ρσ ) 2 + σ 2 (1 − ρ 2 ) where τ ( T ) = du 0 20

Numerical Analysis Parameters Chosen Values : S 0 σ T H β a ν r 0 θ 100 20 % 1 135 1.1 0.46 0.007 0.015 0.05 where r 0 and θ give the initial term structure of interest rates. 21

Sensitivity to the Correlation ρ Discounted Barrier 1.04 Constant Barrier C(0,T) 1.02 1 0.98 0.96 0.94 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Correlation ρ Option Value w.r.t. the correlation ρ . 22

Conclusion The aim is to develop a methodology for the pricing of barrier options in closed form and with stochastic interest rates. When the barrier is constant , quasi-closed-form formulae can be found thanks to an Extended Fortet Methodology. When the derivative’s barrier is a discounted one, using time change techniques we obtain closed-form formulae. 23

Conclusion -> Beyond the chosen example, our article shows how we can price barrier options and compute all their Greeks, under stochastic interest rates. -> The method yields accurate results (for the prices and Greeks) much more quickly than Monte-Carlo simulations 24