ARCH 2014.1 Proceedings July 31-August 3, 2013 Option Pricing - - PDF document

Article from:

ARCH 2014.1 Proceedings

July 31-August 3, 2013

Option Pricing Without Tears: Valuing Equity-Linked Death Benefits

Elias S. W. Shiu Department of Statistics & Actuarial Science The University of Iowa U.S.A. Joint work with Hans U. Gerber & Hailiang Yang

Let Tx denote the time-until-death random variable for a life aged x. Let S(t) be the time-t price of a stock or mutual fund. Consider death benefits that depend on the value of S(Tx), i.e., consider b(S(Tx)) for some function b(.). Examples: b(s) = Max(s, K) b(s) = (s – K)+ Problem: Evaluate where the expectation is taken with respect to an appropriate probability distribution and δ is a continuously compounded interest rate.

x

T x

E[e b(S(T ))]

−δ

if Tx is independent of {S(t)}.

x x

T x T x x

E[ ] E[e b(S(T ))] E[ e b(S(T )) T ]

|

−δ −δ

=

x

t x T

e b(S(t)) T =t f (t) dt E[ ]

|

∞ −δ

= ∫

x

t T

e b(S(t)) E[ ] ) dt f (t

∞ −δ

= ∫

So we want to calculate If then

x j

j T j

f (t) f (t) c ,

τ

= ∑

x j

j t T t j

E[e b(S(t))]f (t)dt = E[e b(S(t))]f (t)dt c

∞ −δ ∞ τ −δ

∫ ∑ ∫

x

t T

E[e b(S(t))]f (t)dt.

∞ −δ

∫

j

j j j

= E[e b ) c (S( ) ].

− τ δ

τ

∑

The time-until-death density function can be approximated by linear combinations of exponential density functions Thus, our valuation problem becomes finding E[e−δτ b(S(τ))], where τ is an exponential random variable independent of {S(t)}. It turns out to be an elementary calculus exercise for geometric Brownian motion {S(t)}.

j j x

j j j j t T j

c c f (t) f (t) = e .

−λ τ

≈ × × λ

∑ ∑

Let S(t) = S(0)eµt + σZ(t) , t ≥ 0, where {Z(t)} is a standard Brownian motion. Let τ be an independent exponential random variable with mean 1/λ. Then, E[e−δτ b(S(τ), Max{S(t); 0 ≤ t ≤ τ})] where α < 0 and β > 0 are the solutions of ½σ2x2 + µx − (λ + δ) = 0.

m x m x ( )m 2

2 b S(0)e S(0)e e d ( , ) x e dm

∞ −α − β−α −∞

  λ =   σ  

∫ ∫

E[e−δτ b(S(τ), Max{S(t); 0 ≤ t ≤ τ})] Examples: b(s, u) = (s – K)+ call option b(s, u) = (K – s)+ put option b(s, u) = u high water mark payoff

m x m x ( )m 2

2 b S(0)e S(0)e e d ( , x e m. ) d

∞ −α − β−α −∞

  λ =   σ  

∫ ∫

Barrier Options Assume S(0) < B, a barrier. b(s, u) = I(u < B)×π(s) Up-and-out option b(s, u) = I(u ≥ B)×π(s) Up-and-in option Useful for incorporating lapses or surrenders.

Assume S(t) = S(0)eX(t) , t ≥ 0, where

N (t) N (t) j k j 1 k 1

X(t) t Z(t) J K

ν ω

= =

= µ + σ + −

∑ ∑

i i

m v x i i i 1 n K w x i i 1 J i

f (x) A v e , x f (x) B w e , x

− = − =

= > = >

∑ ∑

m n i i i 1 i 1

A 1, B 1

= =

= =

∑ ∑

S(t) = S(0)eX(t) , t ≥ 0 Running maximum M(t) := Max{X(u); 0 ≤ u ≤ t} Running minimum m(t) := Min{X(u); 0 ≤ u ≤ t} Because {X(u)} is a Levy process, (i) M(τ) and [X(τ) − M(τ)] are independent random variables, (ii) [X(τ) − Μ(τ)] has the same distribution as m(τ). (i) is hard to prove; (ii) is easy.

In fact, (ii) is true for each fixed t. X(t) – M(t) = X(t) – Max{X(s); 0 ≤ s ≤ t} = X(t) + Min{−X(s); 0 ≤ s ≤ t} = Min{X(t) − X(s); 0 ≤ s ≤ t} = Min{X(t − s); 0 ≤ s ≤ t} in distribution = Min{X(s); 0 ≤ s ≤ t} = m(t)

Running maximum M(t) := Max{X(u); 0 ≤ u ≤ t} Running minimum m(t) := Min{X(u); 0 ≤ u ≤ t} (i) M(τ) and [X(τ) − M(τ)] are independent r.v.’s. (ii) [X(τ) − M(τ)] and m(τ) have the same distribution. Then, E[ezX(τ)] = E[ez[X(τ)−Μ(τ)+M(τ)]] = E[ez[X(τ)−M(τ)]] × E[ezM(τ)] = E[ezm(τ)] × E[ezM(τ)], which is a version of Wiener-Hopf factorization.

Assume where Then, E[ezX(t)] = etΨ(z) for each t ≥ 0, with

N (t) N (t) j k j 1 k 1

X(t) t Z(t) J K

ν ω

= =

= µ + σ + −

∑ ∑

i i

m v x J i i i 1 n w x K i i i 1

f (x) A v e , x f (x) B w e , x

− = − =

= > = >

∑ ∑

m n 2 2 i i i i i 1 i 1

z z (z) z ½ z A B v z w z

= =

Ψ = µ + σ + ν − ω − +

∑ ∑

can be extended by analytic continuation. The moment-generating function of X(τ) is E[ezX(τ)] = E[E[ezX(τ)|τ]] = E[eΨ(z)τ] The zeros of the RHS are the poles of Ψ(z). The poles of the RHS are the zeros of λ − Ψ(z).

. (z) λ = λ − Ψ

m n 2 2 i i i i i 1 i 1

z z (z) z ½ z A B v z w z

= =

Ψ = µ + σ + ν − ω − +

∑ ∑

Label the parameters (the poles of Ψ(z)) such that v1 < v2 < … < vm w1 < w2 < … < wn If the weights A’s and B’s are positive, then −∞<αn+1< −wn<...<−w1<α1<0<β1< v1<...<vm<βm+1<∞

Label the parameters (the poles of Ψ(z)) such that v1 < v2 < … < vm w1 < w2 < … < wn If the weights A’s and B’s are positive, then −∞<αn+1< −wn<...<−w1<α1<0<β1< v1<...<vm<βm+1<∞ Wiener-Hopf: E[ezX(τ)] = E[ezm(τ)]×E[ezM(τ)]. For z > 0, 0 < E[ezm(τ)] ≤ 1. No positive zeros or poles. For z < 0, 0 < E[ezM(τ)] ≤ 1. No negative zeros or poles.

−∞<αn+1< −wn<...<−w1<α1<0<β1< v1<...<vm<βm+1<∞

n n 1 zm( ) j j 1 j 1 m m 1 zM( ) j j 1 j 1 j j

1 E[e ] (z w ) z 1 E[e ] (z v ) z

+ τ = = + τ = =

   ∝       −   α      ∝ −       −   β + 

∏ ∏ ∏ ∏

n n 1 j zm( ) j j 1 j 1 m m 1 j zM( ) j j 1 j j j j j 1

z w E[e ] w z v z E[e ] v z

+ τ = = + τ = =

   − =       −       − =       −   β +  α β α

∏ ∏ ∏ ∏

k

m 1 M( ) 1 m m 1 j j j j j 1 j 1,j k k k k k k x k

Thus, f ( ) e , 0, v where b b x x v

+ −β τ = + = = ≠

= >    −β β    = β    β −β   

∑ ∏ ∏

k M( ) m m 1 j j j j j 1 j 1 m m 1 m 1 j j j j k k k 1 j 1 j 1,j z k k

E[e ] v v v . v z z z

τ + = = + + = = = ≠

   − β =       β −       −β β β =       β −β β −   

∏ ∏ ∑ ∏ ∏

m( ) n n 1 j j j j j 1 j 1 n n 1 n 1 j j j j 1 k z j 1 j 1 k , k j k k k

E[e ] w w w z z w z

τ + = = + + = = = ≠

   + −α =       − α       α + −α −α =       α − α − α   

∏ ∏ ∑ ∏ ∏

k

n 1 m( ) 1 n n 1 j j j j j 1 j 1, k k k x k k j k k

Thus, f ( ) e , 0, w where ( ) w a a x x

+ −α τ = + = = ≠

= <    α + −α = −α       α − α   

∑ ∏ ∏

For y ≥ max(x, 0), fX(τ), M(τ)(x, y) = fM(τ), X(τ)−Μ(τ)(y, x − y)×1 = fM(τ)(y) fX(τ)−Μ(τ)(x − y) = fM(τ)(y) fm(τ)(x − y)

j k j k j

m 1 n 1 (x y) y k j k 1 j 1 m 1n 1 x ( )y j k k 1 j 1

b e a e a b e e

+ + −α − −β = = + + −α − β −α = =

    =         = ×

∑ ∑ ∑∑

Further Work

• 1. Use SOA CAE research grant to hire graduate

students to estimate the parameters and to program the formulas.

• 2. Binomial tree version.

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