^{1}

^{*}

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This paper proposes a new efficient algorithm for the computation of Greeks for options using the binomial tree. We also show that
*Greeks* for European options introduced in this article are asymptotically equivalent to the discrete version of Malliavin Greeks. This fact enables us to show that our Greeks converge to Malliavin Greeks in the continuous time model. The computation algorithm of Greeks for American options using the binomial tree is also given in this article. There are three advantageous points to use binomial tree approach for the computation of Greeks. First, mathematics is much simpler than using the continuous time Malliavin calculus approach. Second, we can construct a simple algorithm to obtain the Greeks for American options. Third, this algorithm is very efficient because one can compute the price and Greeks (delta, gamma, vega, and rho) at once. In spite of its importance, only a few previous studies on the computation of Greeks for American options exist, because performing sensitivity analysis for the optimal stopping problem is difficult. We believe that our method will become one of the popular ways to compute Greeks for options.

Greeks are quantities that represent the sensitivity of the price of derivative securities with respect to changes in the price of underlying assets or parameters. They are defined by derivatives of the option price function with respect to parameters such as the price of underlying assets, volatility level, and spot interest rate. The computation of these sensitivities is very important for risk management, for an example on this and similar topics, refer to Hull [

Muroi and Suda [

The remainder of this paper is organized as follows. In Section 2, we give brief explanations on the binomial tree methods of Cox et al. [

In this section, we briefly explain the binomial tree model of Cox et al. [

Introduce independently and identically distributed random variables { ϵ i } i = 1, ⋯ , N on the probability space ( Ω , F , Q ) , where ϵ i is a random variable with probability Q [ ϵ i = Δ t ] = p , Q [ ϵ i = − Δ t ] = 1 − p . The time step Δ t is fixed as T = N Δ t . We introduce a random walk process, { W i Δ t } i = 1 , ⋯ , N , given by

W i Δ t = ϵ 1 + ⋯ + ϵ i .

One can regard the stochastic process { W i Δ t } i = 1 , ⋯ , N with p = 1 / 2 as an ap- proximation of the standard Brownian motion. On the other hand, the stocha- stic process { W i Δ t } i = 1 , ⋯ , N is generally regarded as an approximation of the

Brownian motion with drift 2 p − 1 Δ t in the general case ( p ≠ 1 2 ).

The binomial tree is a computational method for pricing options on securities whose price process is governed by the geometric Brownian motion

d P t = P t ( r d t + σ d Z t ) , P 0 = s , (1)

where { Z t } t ≥ 0 is a standard Brownian motion under the risk-neutral measure Q. A binomial tree is constructed in the following manner. We consider a model with N periods and assume that the maturity date of the options is fixed as T = N Δ t . If the price of underlying assets at time i Δ t ( i = 0 , 1 , ⋯ , N − 1 ) is given by S i Δ t , the price moves to u S i Δ t or d S i Δ t ( d < 1 < u ) in the next time period ( i + 1 ) Δ t . The probability of the price moving upward to ( u S i Δ t ) is p and downward to ( d S i Δ t ) is 1 − p . In order to construct the binomial tree so that the expectation and variance are consistent with the geometric Brownian motion (1), we fix the parameters u, d, p as

u = e σ Δ t , d = e − σ Δ t , p = e r Δ t − d u − d

where we assume an additional relationship, u = 1 / d . Consider European options with a pay-off function Φ ( ⋅ ) and a maturity date T. The price of options at time i Δ t is denoted by E ( x , i Δ t ) if the price of underlying assets is x at time i Δ t . The price is given by the backward induction algorithm

E ( x , i Δ t ) = e − r Δ t ( p E ( x u , ( i + 1 ) Δ t ) + ( 1 − p ) E ( x d , ( i + 1 ) Δ t ) )

E ( x , N Δ t ) = Φ ( x ) . (2)

One can derive the closed-form formulas for Greeks for European options using the discrete Malliavin calculus (See Muroi and Suda [

New computational methods of Greeks for European options are presented in this section. Although computational methods of delta and gamma using the binomial tree have already been proposed by Pelsser and Vorst [

Because the option price is given by the weighted sum of the pay-off function, we can compute Greeks if we assume that the pay-off function is smooth. However, the pay-off function for, say, European call options is not smooth. Therefore, the smoothness of the pay-off function is too strong to be assumed for practical purposes. In this section, however, we compute Greeks under the following assumption. We will show that the obtained formulas in this section converge to Greeks in the continuous time model under the milder conditions. It is discussed in Section 7.2.

Assumption 1. We assume that the pay-off function Φ ( ⋅ ) is a smooth function.

In this subsection, we calculate delta, which is used to measure the sensitivity of the option price with respect to changes in the price of underlying assets. Delta is given by the first derivative of the option value function with respect to the price of underlying assets. Delta for European options is computed as

Δ E ( s , 0 ) = e − r Δ t { p ∂ ∂ s E ( s e σ Δ t , Δ t ) + ( 1 − p ) ∂ ∂ s E ( s e − σ Δ t , Δ t ) } = e − r Δ t { p d d x E ( s e σ Δ t , Δ t ) e σ Δ t + ( 1 − p ) d d x E ( s e − σ Δ t , Δ t ) e − σ Δ t } . (3)

Now, we have two approaches to compute ∂ ∂ z E ( s e σ z , Δ t ) | z = ± Δ t . The first one is given by

∂ ∂ z E ( s e σ z , Δ t ) | z = ± Δ t = E ( s e σ Δ t , Δ t ) − E ( s e − σ Δ t , Δ t ) 2 Δ t + O ( Δ t ) , (4)

and the second one is given by

∂ ∂ z E ( s e σ z , Δ t ) | z = ± Δ t = d d x E ( x , Δ t ) | x = s e ± σ Δ t × σ s e σ z | z = ± Δ t .

These two formulas yield the approximation formula for the derivative d d x E ( x , Δ t ) | x = s e ± σ Δ t :

d d x E ( x , Δ t ) | x = s e ± σ Δ t = 1 σ s e ± σ Δ t E ( s e σ Δ t , Δ t ) − E ( s e − σ Δ t , Δ t ) 2 Δ t + O ( Δ t ) . (5)

Substituting this formula in (3) leads to the following relation

#Math_42# (6)

Taylor expansion yields the approximation formula for p, and it is given by

p = 1 2 + 1 2 μ Δ t + O ( Δ t 3 / 2 ) , (7)

where μ is a constant given by μ = 1 σ ( r − σ 2 2 ) . Let us compute the expec- tation E [ E ( s e σ ϵ 1 , Δ t ) ( ϵ 1 − μ Δ t ) ] . The above approximation formula leads to

E [ E ( s e σ ϵ 1 , Δ t ) ( ϵ 1 − μ Δ t ) ] = p E ( s e σ Δ t , Δ t ) ( Δ t − μ Δ t ) + ( 1 − p ) E ( s e − σ Δ t , Δ t ) ( − Δ t − μ Δ t ) = [ Δ t 2 E ( s e σ Δ t , Δ t ) + − Δ t 2 E ( s e − σ Δ t , Δ t ) ] + O ( Δ t 2 ) . (8)

Note that we have used the identities

1 2 ( E ( s e σ Δ t , Δ t ) + E ( s e − σ Δ t , Δ t ) ) = E ( s , Δ t ) + O ( Δ t )

E ( s e σ Δ t , Δ t ) − E ( s e − σ Δ t , Δ t ) = 2 Δ t ∂ ∂ z E ( s e σ z , Δ t ) | z = 0 + O ( Δ t ) = O ( Δ t )

to derive the last equality. This formula allows further computation of (6). It is given by

Δ E ( s , 0 ) = e − r Δ t s σ Δ t E [ E ( s e σ ϵ 1 , Δ t ) ( ϵ 1 − μ Δ t ) ] + O ( Δ t ) ≈ e − r Δ t s σ Δ t E [ E ( s e σ ϵ 1 , Δ t ) ( ϵ 1 − μ Δ t ) ] ≡ Δ E M S ( s , 0 ) (9)

We call this MS delta, which is the one-step version of discrete Malliavin delta given in Muroi and Suda [

Δ E M S ( s , 0 ) = Δ E ( s , 0 ) + O ( Δ t ) , (10)

if we assume smoothness to the pay-off function. This is organized as the following theorem, because we use the Formula (10) to show the accuracy of Vega.

Theorem 1. We assume that the pay-off function Φ ( ⋅ ) as a smooth function. We can estimate the accuracy of MS delta as

Δ E M S ( s ,0 ) = Δ E ( s ,0 ) + O ( Δ t ) .

[Proof] First we compute a higher order term for the error term for MS delta. Taylor expansion,

E ( s e ∓ σ Δ t , Δ t ) = E ( s e ± σ Δ t , Δ t ) + ∂ ∂ z E ( s e σ z , Δ t ) ( ∓ 2 Δ t ) | z = ± Δ t + 1 2 ∂ 2 ∂ z 2 E ( s e σ z , Δ t ) ( ∓ 2 Δ t ) 2 | z = ± Δ t + O ( Δ t 3 / 2 ) ,

yields the higher order term for equality (4). It is given by

∂ ∂ z E ( s e σ z , Δ t ) | z = ± Δ t = E ( s e σ Δ t , Δ t ) − E ( s e − σ Δ t , Δ t ) 2 Δ t ± ∂ 2 ∂ z 2 E ( s e σ z , Δ t ) Δ t | z = ± Δ t + O ( Δ t ) .

This leads to the higher order expansion formula for the delta given by (6):

Δ E ( s , 0 ) = e − r Δ t s σ Δ t E ( s e σ Δ t , Δ t ) Δ t + E ( s e − σ Δ t , Δ t ) ( − Δ t ) 2 + e − r Δ t s σ ( p ∂ 2 ∂ z 2 E ( s e σ z , Δ t ) | z = Δ t − ( 1 − p ) ∂ 2 ∂ z 2 E ( s e σ z , Δ t ) | z = − Δ t ) Δ t + O ( Δ t ) . (11)

The second term in (11) is calculated as

p ∂ 2 ∂ z 2 E ( s e σ z , Δ t ) | z = Δ t − ( 1 − p ) ∂ 2 ∂ z 2 E ( s e σ z , Δ t ) | z = − Δ t = ( μ ∂ 2 ∂ z 2 E ( s e σ z , Δ t ) | z = 0 + ∂ 3 ∂ z 3 E ( s e σ z , Δ t ) | z = 0 ) Δ t + O ( Δ t ) = O ( Δ t ) ,

where we have used the approximation (7). This result enables us to compute delta given by (11). It is computed as

Δ E ( s , 0 ) = e − r Δ t s σ Δ t E ( s e σ Δ t , Δ t ) Δ t + E ( s e − σ Δ t , Δ t ) ( − Δ t ) 2 + O ( Δ t ) .

Applying the Formula (8), the higher order expansion for delta is finally given by

Δ E ( s ,0 ) = Δ E M S ( s ,0 ) + O ( Δ t ) . (12)

This shows the theorem.

Discrete Malliavin Greeks for European options using an N-steps binomial tree were obtained by Muroi and Suda [

Δ E D = e − r N Δ t s σ N Δ t E [ Φ ( s e σ W N Δ t ) ( W N Δ t − μ N Δ t ) ] .

As discussed in Appendix, MS delta and discrete Malliavin delta is actually equivalent. Because Muroi and Suda [

If we exploit another approximation formula for the derivative d d x E ( x , Δ t ) | x = s e ± σ Δ t , we can obtain another approximation formula for delta. If we use a more direct formula for the derivative,

d d x E ( s e ± σ Δ t , Δ t ) ≈ E ( s e ∓ σ Δ t , Δ t ) − E ( s e ± σ Δ t , Δ t ) s e ∓ σ Δ t − s e ± σ Δ t ≈ E ( s e σ Δ t , Δ t ) − E ( s e − σ Δ t , Δ t ) 2 s σ Δ t , (13)

we get another approximation formula for delta for European options in the one-period time model. Substituting (13) in (3) yields Δ E H u l l , and it is given as

Δ E H u l l ( s ,0 ) ≈ E ( s e σ Δ t , Δ t ) − E ( s e − σ Δ t , Δ t ) 2 s σ Δ t .

This is the delta introduced by Hull [

different approximation formulas for delta^{1}. Our computational results indicate that MS delta converges a little bit faster than delta introduced by Hull [

Lastly, we can show the convergence of MS delta to delta in Black and Scholes model, even if we do not assume the smoothness to the pay-off function. This is shown in Appendix.

^{1}Actually, MS delta is deeply related to Hull’s delta. We have a relation, Δ E M S ( s ,0 ) = e − r Δ t Δ E H u l l ( s ,0 ) + O ( Δ t ) , because we have Formulas (8) and (9).

In this subsection, we calculate gamma, which is used to measure the sensitivity of delta with respect to changes in the price of underlying assets. Gamma is given by the second derivative of the option value function with respect to the price of underlying assets. The pricing algorithm for European options (2) yields delta for European options:

Δ E ( s ,0 ) = e − r Δ t { p ∂ ∂ x E ( s e σ Δ t , Δ t ) e σ Δ t + ( 1 − p ) ∂ ∂ x E ( s e − σ Δ t , Δ t ) e − σ Δ t } = e − r Δ t { p Δ E ( s e σ Δ t , Δ t ) e σ Δ t + ( 1 − p ) Δ E ( s e − σ Δ t , Δ t ) e − σ Δ t } .

Applying chain rule to E ( s e σ z , Δ t ) leads to

Δ E ( s e ± σ Δ t , Δ t ) = ∂ E ∂ z ( s e σ z , Δ t ) σ s e σ z | z = ± Δ t .

Delta for European options is given by

Δ E ( s , 0 ) = e − r Δ t σ s { p [ ∂ E ∂ z ( s e σ z , Δ t ) ] z = Δ t + ( 1 − p ) [ ∂ E ∂ z ( s e σ z , Δ t ) ] z = − Δ t } .

Taking derivative with respect to the price of underlying assets yields gamma for European options as

Γ E ( s , 0 ) = − 1 s Δ E ( s , 0 ) + e − r Δ t σ s { p ∂ ∂ z ( Δ E ( s e σ z , Δ t ) e σ z ) | z = Δ t + ( 1 − p ) ∂ ∂ z ( Δ E ( s e σ z , Δ t ) e σ z ) | z = − Δ t } .

Using the approximation formula

∂ ∂ z Δ E ( s e σ z , Δ t ) e σ z | z = ± Δ t = Δ E ( s e σ Δ t , Δ t ) e σ Δ t − Δ E ( s e − σ Δ t , Δ t ) e − σ Δ t 2 Δ t + O ( Δ t )

allows further calculation of gamma as

Γ E ( s , 0 ) = − e − r Δ t s 2 σ Δ t E [ E ( s e σ ϵ 1 , Δ t ) ( ε 1 − μ Δ t ) ] + e − r Δ t σ s Δ t E [ Δ E ( s e σ ϵ 1 , Δ t ) e σ ϵ 1 ( ϵ 1 − μ Δ t ) ] + O ( Δ t ) = − e − r Δ t s 2 σ Δ t E [ E ( s e σ ϵ 1 , Δ t ) ( ϵ 1 − μ Δ t ) ] + e − r Δ t σ s Δ t E [ Δ E M S ( s e σ ϵ 1 , Δ t ) e σ ϵ 1 ( ϵ 1 − μ Δ t ) ] + O ( Δ t ) ≈ − e − r Δ t s 2 σ Δ t E [ E ( s e σ ϵ 1 , Δ t ) ( ϵ 1 − μ Δ t ) ] + e − r Δ t σ s Δ t E [ Δ E M S ( s e σ ϵ 1 , Δ t ) e σ ϵ 1 ( ϵ 1 − μ Δ t ) ] ≡ Γ E M S ( s , 0 ) (14)

We call this formula MS gamma. This formula is valid only if the pay-off function Φ ( ⋅ ) is a smooth function. In this case, the order of the error term is O ( Δ t ) , i.e. Γ E ( s ,0 ) = Γ E M S ( s ,0 ) + O ( Δ t ) . Second equality in (14) is shown by using the Formula (10). We will also prove that MS gamma is asymptotically equivalent to the Gamma in the Black and Scholes model in the Appendix.

The computational method of vega for European options is presented in this subsection. Vega is the sensitivity of the option pricing formula with respect to changes in volatility level, σ . It appears that the computational methods of vega and rho using the binomial tree have not yet been considered seriously, except for the finite difference approach. See Hull [

V E ( s i Δ t , i Δ t ; σ ) = e − r Δ t ∂ ∂ σ { p E ( s i Δ t e σ Δ t , ( i + 1 ) Δ t ; σ ) + ( 1 − p ) E ( s i Δ t e − σ Δ t , ( i + 1 ) Δ t ; σ ) } = V 1 i + V 2 i + V 3 i ,

where V 1 i , V 2 i , V 3 i are given by

V 1 i = e − r Δ t { ∂ p ∂ σ E ( s i Δ t e σ Δ t , ( i + 1 ) Δ t ; σ ) + ∂ ( 1 − p ) ∂ σ E ( s i Δ t e − σ Δ t , ( i + 1 ) Δ t ; σ ) }

V 2 i = e − r Δ t { p ∂ ∂ x E ( s i Δ t e σ Δ t , ( i + 1 ) Δ t ; σ ) s i Δ t e σ Δ t Δ t + ( 1 − p ) ∂ ∂ x E ( s i Δ t e − σ Δ t , ( i + 1 ) Δ t ; σ ) s i Δ t e − σ Δ t ( − Δ t ) } = e − r Δ t E [ Δ E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; σ ) s i Δ t e σ ϵ i + 1 ϵ i + 1 ]

V 3 i = e − r Δ t E [ V E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; σ ) ] .

The derivative of p with respect to σ is given by

∂ ∂ σ p = 2 − e r Δ t ( e σ Δ t + e − σ Δ t ) ( e σ Δ t − e − σ Δ t ) 2 Δ t = − 1 4 ( 1 + 2 r σ 2 ) Δ t + O ( Δ t 3 / 2 ) .

This result and the approximation Formula (8) lead to the approximation formula for V 1 i :

V 1 i = − 1 2 ( 1 + 2 r σ 2 ) e − r Δ t × E ( s i Δ t e σ Δ t , ( i + 1 ) Δ t ; σ ) Δ t + E ( s i Δ t e − σ Δ t , ( i + 1 ) Δ t ; σ ) ( − Δ t ) 2 + O ( Δ t 3 / 2 ) = E [ e − r Δ t { − ( 1 + 2 r σ 2 ) } ϵ i + 1 − μ Δ t 2 E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; σ ) ] + O ( Δ t 3 / 2 ) .

For further computation of V 2 i , one needs delta at time ( i + 1 ) Δ t . Moreover, it is necessary to compute vega for European options at time ( i + 1 ) Δ t to evaluate V 3 i . This implies that one has to use the backward induction algorithm to compute vega for European options. Vega at the maturity date is given by V E ( s N Δ t , N Δ t ; σ ) = 0 . These results are combined to form the computational formulas for vega:

V E ( s i Δ t , i Δ t ; σ ) = E [ e − r Δ t { − ( 1 + 2 r σ 2 ) } ϵ i + 1 − μ Δ t 2 E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ) ] + E [ e − r Δ t Δ E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; σ ) s i Δ t e σ ϵ i + 1 ϵ i + 1 ] + E [ e − r Δ t V E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; σ ) ] + O ( Δ t 3 / 2 ) . (15)

Delta and vega at time ( i + 1 ) Δ t are substituted by MS delta and vega to get MS vega at time i Δ t , respectively. In other words, MS vega at time i Δ t is defined recursively by the backward algorithm,

V E M S ( s i Δ t , i Δ t ; σ ) ≡ E [ e − r Δ t { − ( 1 + 2 r σ 2 ) } ϵ i + 1 − μ Δ t 2 E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ) ] + E [ e − r Δ t Δ E M S ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; σ ) s i Δ t e σ ϵ i + 1 ϵ i + 1 ] + E [ e − r Δ t V E M S ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; σ ) ] . (16)

We have the following theorem on MS Vega.

Theorem 2 Let us assume that the pay-off function Φ ( ⋅ ) is a smooth function. The accuracy of the MS vega is given by

V E ( s ,0 ; σ ) = V E M S ( s ,0 ; σ ) + O ( Δ t ) .

We need a little bit more effort show this fact and it is shown in Appendix. We also show the convergence of MS vega to the vega in the Black and Scholes model even if we do not assume the smoothness of the pay-off function in the Appendix. It should be noted that in order to compute Vega, we formally assume V 2 N − 1 = 0 if the pay-off function Φ ( ⋅ ) is not smooth one. As an alter- native approach, the finite difference approach is used to obtain vega for prac- tical purposes. In many cases, it works well: however, in some it does not, for example one cannot obtain a stable estimator of vega for digital options.

The computational method of rho for European options is discussed in this subsection. Rho measures the sensitivity of the option price with respect to changes in the spot interest rate level, r. It is defined by the first derivative of the option value function with respect to the spot interest rate, r. The price and rho for European options are denoted by E ( s i Δ t , i Δ t ; r ) and ρ E ( s i Δ t , i Δ t ; r ) , respec- tively, if the price of underlying assets at time i Δ t is given by s i Δ t . Simple cal- culation yields

ρ E ( s i Δ t , i Δ t ; r ) = ρ 1 i + ρ 2 i + ρ 3 i ,

where ρ 1 i , ρ 2 i , ρ 3 i are given by

ρ 1 i = − Δ t e − r Δ t E [ E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; r ) ]

ρ 2 i = e − r Δ t { ∂ p ∂ r E ( s i Δ t e σ Δ t , ( i + 1 ) Δ t ; r ) + ∂ ( 1 − p ) ∂ r E ( s i Δ t e − σ Δ t , ( i + 1 ) Δ t ; r ) }

ρ 3 i = e − r Δ t E [ ρ E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; r ) ] .

The derivative ∂ p ∂ r in ρ 2 i is given by

∂ ∂ r p = e r Δ t e σ Δ t − e − σ Δ t Δ t = Δ t 2 σ + O ( Δ t 3 / 2 ) .

This relation and the Formula (8) leads to the further calculation,

ρ 2 i = e − r Δ t σ { Δ t 2 E ( s i Δ t e σ Δ t , ( i + 1 ) Δ t ; r ) + − Δ t 2 E ( s i Δ t e − σ Δ t , ( i + 1 ) Δ t ; r ) } + O ( Δ t 3 / 2 ) = e − r Δ t σ E [ E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; r ) ( ϵ i + 1 − μ Δ t ) ] + O ( Δ t 3 / 2 ) .

In order to compute ρ 3 i , one needs to compute rho at time ( i + 1 ) Δ t , i.e., ρ E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; r ) . This implies that one has to use the backward induc- tion approach to evaluate rho. These results are combined to form the computa- tional formulas of rho for European options:

ρ E ( s i Δ t , i Δ t ; r ) = e − r Δ t E [ ( ϵ i + 1 σ − μ Δ t σ − Δ t ) E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; r ) + ρ E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; r ) ] + O ( Δ t 3 / 2 ) . (17)

In order to compute MS rho at time i Δ t , rho at time ( i + 1 ) Δ t is substituted by MS rho at time ( i + 1 ) Δ t . In other words, MS rho is defined recursively by the backward algorithm,

ρ E M S ( s i Δ t , i Δ t ; r ) ≡ e − r Δ t E [ ( ϵ i + 1 σ − μ Δ t σ − Δ t ) E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; r ) + ρ E M S ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; r ) ] .

Because rho at the maturity date is equal to 0, rho given by the Formula (17) is further calculated as

ρ E ( s , 0 ; r ) = e − r Δ t E [ ( ϵ 1 − μ Δ t σ − Δ t ) E ( s e σ ϵ 1 , Δ t ; r ) + ρ E ( s e σ ϵ 1 , Δ t ; r ) + O ( ( Δ t ) 3 / 2 ) ] = ⋯ = ∑ i = 1 N e − r i Δ t E [ ( ϵ i − μ Δ t σ − Δ t ) E ( s e σ W i Δ t , Δ t ; r ) ] + O ( Δ t ) = ρ E M S ( s , 0 ; r ) + O ( Δ t ) . (18)

This shows the final result,

ρ E ( s ,0 ; r ) = ρ E M S ( s ,0 ; r ) + O ( Δ t ) .

MS rho is also asymptotically equivalent to the rho in the Black and Scholes model.

We suggest a new algorithm for computation of Greeks for American options. An American option is a contingent claim that its holder can exercise at any time before its maturity date. Consider an American option with a pay-off function Φ ( ⋅ ) and a maturity date T = N Δ t . If the price of underlying assets at time i Δ t is given by x, the price of these options at time i Δ t is denoted by A ( x , i Δ t ) . The price of American options is given by the backward induction algorithm:

A ( x , i Δ t ) = m a x { e − r Δ t ( p A ( x u , ( i + 1 ) Δ t ) + ( 1 − p ) A ( x d , ( i + 1 ) Δ t ) ) , Φ ( x ) } ,

where the price of options at the maturity date is given by A ( x , N Δ t ) = Φ ( x ) . Introduce new sets,

S i = { x | x ∈ ( 0, ∞ ) , A ( x , i Δ t ) = Φ ( x ) } and C i = ( 0 , ∞ ) \ S i ( i = 0 , 1 , ⋯ , N )

The continuous region and stopping region for American options are defined by C = ∪ i = 0 N C i and S = ∪ i = 0 N S i , respectively. We will present an intuitive way for the computation of Greeks for American options. If the price of underlying assets at the initial time satisfies s ∈ S 0 , Greeks are simply the derivative of the pay-off function. Sensitivity for American options at the initial time respect to a parameter θ , is computed as

∂ θ A ( s ,0 ) = ∂ θ Φ ( s )

if the function Φ ( ⋅ ) is differentiable at s ∈ S 0 . (Second derivatives such as gamma is given by ∂ s s A ( s ,0 ) = ∂ s s Φ ( s ) .) If the price of underlying assets at the initial time satisfies s ∈ C 0 , Greeks are also given by the derivative of the price function of American options, and it is given by^{2}

∂ θ A ( s ,0 ) = ∂ ∂ θ [ e − r Δ t { p A ( s u , Δ t ) + ( 1 − p ) A ( s d , Δ t ) } ] . (19)

Notice that the Formula (19) has a same functional form to the European options Greeks. Delta, for example, is given by

Δ A M S ( s ,0 ) = { ∂ s Φ ( s ) if s ∈ S 0 e − r Δ t s σ Δ t E [ A ( s e σ ϵ 1 , Δ t ) ( ϵ 1 − μ Δ t ) ] if s ∈ C 0 .

^{2}Strictly speaking, one cannot use our computational formula for delta if the initial price is on the early exercise boundary, and one cannot use our gamma, vega, and rho formulas if the nodes on the binomial tree are on the early exercise boundary. However, numerical results show that our formula works very well when we compute Greeks for American put options with the pay-off Φ ( x ) = ( K − x ) + if the delta is fixed at Δ A ( s , 0 ) = − 1 on the stopping region and the boundary. This is because we have a smooth-fit condition in the continuous model, and our model is an approximation of the Black and Scholes model.

One can compute other Greeks using same method. Numerical demonstra- tions are shown in Section 5. The extended binomial tree of Pelsser and Vorst [

In this section, we demonstrate the numerical results for the new computational methods of Greeks that were introduced in previous sections. In order to check the effectiveness of our approach, Greeks for American put options are com- puted by the newly proposed approach and the finite difference approach. We also demonstrate the extended binomial tree approach of Pelsser and Vorst [

Figures 2-4 and

Δ A E B = A ( s ( 2 , 0 ) , 0 ) − A ( s ( − 2 , 0 ) , 0 ) s ( 2 , 0 ) − s ( − 2 , 0 )

Γ A E B = A ( s ( 2 , 0 ) , 0 ) − A ( s ( 0 , 0 ) , 0 ) s ( 2 , 0 ) − s ( 0 , 0 ) − A ( s ( 0 , 0 ) , 0 ) − A ( s ( − 2 , 0 ) , 0 ) s ( 0 , 0 ) − s ( − 2 , 0 ) s ( 2 , 0 ) − s ( − 2 , 0 )

where s ( i , j ) = s ∗ u i . These results are computed using binomial trees with N = 4 , 8 , ⋯ , 100 steps for one year. The parameter values assumed for these numerical studies were

s = 100 , σ = 0.3 , r = 0.05 , K = 100 , T = 1 ( year ) .

The MS Greeks (delta, gamma, vega, and rho) are represented by the real lines and the dotted line (without any mark) represents the finite difference Greeks (FD Greeks). Two other kinds of dotted lines, dotted lines with a circle and square, represent EB Greeks and Hull’s Greeks (delta and gamma), respectively. The horizontal lines in

results are expected to be very accurate. If the initial underlying asset price s is in the continuous region, i.e. s ∈ C 0 , the FD delta and gamma are given by

Δ A F D = A ( s + Δ s , 0 ) − A ( s − Δ s , 0 ) 2 Δ s

Γ A F D = A ( s + Δ s , 0 ) − 2 A ( s , 0 ) + A ( s − Δ s , 0 ) ( Δ s ) 2

and the FD vega and rho are given by

V A F D ( s ,0 ; σ ) = A ( s ,0 ; σ + Δ σ ) − A ( s ,0 ; σ − Δ σ ) 2 Δ σ

ρ A F D ( s ,0 ; r ) = A ( s ,0 ; r + Δ r ) − A ( s ,0 ; r − Δ r ) 2 Δ r

where Δ s , Δ σ , and Δ r are small parameters. We take Δ s = s × Δ h , Δ σ = σ Δ h , and Δ r = r × Δ h , ( Δ h = 10 − 3 ) for our computations.

^{3}It is enough to use binomial trees with 100 steps to obtain Greeks. Then, one can compute in an instant.

time for MS rho is expected to be shorter than that for FD rho. ^{3}. It was found that the computational time for MS rho was about 20% shorter even though the computational results obtained were almost the same. Hence, computing MS rho rather than the FD rho is more advanta- geous.

Figures 7-10 present Greeks (delta, gamma, vega, and rho, respectively) for

MS rho | FD rho | |
---|---|---|

Value | −34.8461 | −34.8460 |

Time | 14.77 (s) | 18.78 (s) |

American put options as a function of the price of underlying assets. The price range of underlying assets is from 50 to 200. Other parameters used for these numerical studies are same as those used in the previous numerical studies.

and EB Greeks are very smooth, whereas those of FD delta and FD gamma are unstable.

Finally, we compared our new methods with other existing tree methods. We compare MS delta with delta for European options computed by other kinds of binomial tree, namely tree methods introduced by Chung and Shackelton [

This paper presented new computational methods of Greeks using the binomial tree. There are two important results in this paper. First, we obtain a very efficient algorithm to evaluate Greeks. It is especially efficient to compute Greeks for American options. Although many studies have been conducted for the computation of Greeks for European options, few papers have examined the

computation of Greeks for American options. We introduce the binomial tree approach to overcome these problems and confirm its effectiveness for comput- ing Greeks for American options very quickly and accurately. Numerical results indicate that Greeks converge faster when computed using our method than when computed using the extended binomial tree approach of Pelsser and Vorst [

This work was supported by Japan Society for the Promotion of Science. Grant- in-Aid for Scientific Research (C) 16K03731.This paper was written while Shintaro Suda was a graduate student at Tohoku University. The views ex- pressed in this paper are those of the authors and do not necessarily reflect the official views of MTEC.

Muroi, Y. and Suda, S. (2017) Computation of Greeks Us- ing Binomial Tree. Journal of Mathematical Finance, 7, 597-623. https://doi.org/10.4236/jmf.2017.73031

We first show the error estimate of MS vega given by (16). This is presented in Section A. We also prove that MS Greeks converge to Greeks for continuous time Black and Scholes model. This is shown in Section B. Closed-form expectation formulas for MS Greeks (delta, gamma, vega, rho) for European options are investigated in Appendix B. We found that MS Greeks are approxi- mations of the discrete version of the Malliavin Greeks in the continuous time model and these results indicate that MS Greeks converge to Greeks for a continuous time model (Black and Scholes model) when we take a limit, Δ t → 0 .

A. Error terms for MS vega

We present theorem 2 again as theorem 3.

Theorem 3. Let us assume that the pay-off function Φ ( ⋅ ) is a smooth function. The accuracy of the MS vega is given by

V E ( s ,0 ; σ ) = V E M S ( s ,0 ; σ ) + O ( Δ t ) .

[Proof]

Insert (12) to the Formula (15) leads to

V E ( s i Δ t , i Δ t ; σ ) = E [ e − r Δ t { − ( 1 + 2 r σ 2 ) } ε i + 1 − μ Δ t 2 E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ) ] + E [ e − r Δ t Δ E M S ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; σ ) s i Δ t e σ ϵ i + 1 ϵ i + 1 ] + E [ e − r Δ t V E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; σ ) ] + O ( Δ t 3 / 2 ) .

This formula yields

V E ( s i Δ t , i Δ t ; σ ) = V 1 i ( s i Δ t , i Δ t ; σ ) + V 2 i ( s i Δ t , i Δ t ; σ ) + E [ e − r Δ t V E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; σ ) ] + O ( Δ t 3 / 2 ) , (20)

where new variables V 1 i and V 2 i are defined by

V 1 i ( s i Δ t , i Δ t ; σ ) = E [ e − r Δ t { − ( 1 + 2 r σ 2 ) } ϵ i + 1 − μ Δ t 2 E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ) | F i Δ t ]

V 2 i ( s i Δ t , i Δ t ; σ ) = E [ e − r Δ t Δ E M S ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; σ ) s i Δ t e σ ϵ i + 1 ϵ i + 1 | F i Δ t ] ( i ≠ N − 1 )

V 2 N − 1 ( s ( N − 1 ) Δ t , ( N − 1 ) Δ t ; σ ) = E [ e − r Δ t { d d x Φ ( x ) | x = s ( N − 1 ) Δ t e σ ϵ N } s ( N − 1 ) Δ t e σ ϵ N ϵ N | F ( N − 1 ) Δ t ]

If the pay-off function Φ ( x ) is not smooth, we formally define V 2 N − 1 ( s ( N − 1 ) Δ t , ( N − 1 ) Δ t ; σ ) = 0, although we assume that the pay-off function Φ ( x ) is smooth in this subsection. If the pay-off function Φ ( x ) is smooth, vega given by (20) is further computed as

V E ( s i Δ t , i Δ t ; σ ) = V 1 i ( s i Δ t , i Δ t ; σ ) + V 2 i ( s i Δ t , i Δ t ; σ ) + E [ e − r Δ t V E ( s i Δ t e σ ϵ i + 1 , ( i + 1 ) Δ t ; σ ) | F i Δ t ] + O ( Δ t 3 / 2 ) .

Vega for the binomial tree model at the initial time is given by

V E ( s , 0 ; σ ) = V 1 0 ( s , 0 ; σ ) + V 2 0 ( s , 0 ; σ ) + O ( Δ t 3 / 2 ) + e − r Δ t E [ V E ( s e σ W Δ t , Δ t ; σ ) ] = V 1 0 ( s , 0 ; σ ) + V 2 0 ( s , 0 ; σ ) + O ( Δ t 3 / 2 ) + e − r Δ t E [ V 1 1 ( s e σ W Δ t , Δ t ; σ ) + V 2 1 ( s e σ W Δ t , Δ t ; σ ) + O ( Δ t 3 / 2 ) + e − 2 Δ t V E ( s e σ W 2 Δ t , 2 Δ t ; σ ) ] = ⋯ = ∑ i = 0 N − 1 e − r i Δ t E [ V 1 i ( s e σ W i Δ t , 0 ; σ ) + V 2 i ( s e σ W i Δ t , i Δ t ; σ ) ] + O ( Δ t ) = V E M S ( s , 0 ; σ ) + O ( Δ t ) . (21)

This shows the result.

B. Convergence of MS greeks to black and scholes model

In Section 3, we derive Greeks under the Assumption 1. As discussed in Section 3, the smoothness of the pay-off function is too strong to be assumed. Therefore, in this section, we assume that the pay-off function Φ ( ⋅ ) is not a necessarily smooth function. We show that MS Greeks obtained in Section 3 converge to Greeks in the continuous time model under the following assum- ption.

Assumption 2. We assume that the pay-off function, Φ ( ⋅ ) , is a function in the class K . K is the class of real-valued functions on R that satisfy the following conditions: (i) ϕ ( ⋅ ) is piecewise C ( 2 ) , (ii) at each x, the function

ϕ ( ⋅ ) satisfies ϕ ( x ) = 1 2 ( ϕ ( x + ) + ϕ ( x − ) ) , and (iii) ϕ , ϕ ′ , and ϕ ″ are poly-

nomial bounded. We assume that f ( ⋅ ) is a function in the class K .

For example, the pay-off function for European call/put options is included in class K .

Theorem 4. We assume that the pay-off function is in class K . We also assume the number of steps of the binomial tree give by N to be even. Then, MS delta, gamma vega and rho converge to delta, gamma, vega, and rho in Black and Scholes model.

[Proof] We show that MS Greeks (delta, gamma, vega, and rho) are asympto- tically equivalent to the Malliavin Greeks. Malliavina Greeks are Greeks calcu- lated using the Malliavin calculus. See Kohatsu-Higa and Montero [

1. Delta

MS delta given by the Formula (9) is further calculated as

Δ E M S ( s , 0 ) = e − r Δ t s σ Δ t E [ E ( s e σ ϵ 1 , Δ t ) ( ϵ 1 − μ Δ t ) ] = e − r T s σ Δ t E [ Φ ( s e σ W T ) ( ϵ 1 − μ Δ t ) ) ] = e − r T s σ Δ t E [ Φ ( s e σ W T ) ( ϵ i − μ Δ t ) ] ,

where we used the fact that ϵ 1 , ⋯ , ϵ N is an i.i.d. sequence to deduce the last equality. This yields

Δ E M S ( s , 0 ) = 1 N ∑ i = 1 N e − r T s σ Δ t E [ Φ ( s e σ W T ) ( ϵ i − μ Δ t ) ] = e − r T s σ T E [ Φ ( s e σ W T ) ( W T − μ T ) ] = e − r T s σ T E [ Φ ( S T ) log ( S T / s ) − μ σ T σ ] , (22)

where S T is given by S T = s e σ W N Δ t . Note that this formula indicates that the MS delta is identical to the discrete Malliavin delta. See Kohatsu-Higa and Montero [

Δ E M S ( s , 0 ) = e − r T s σ T E [ Φ ( P T ) log ( P T / s ) − μ σ T σ ] + O ( 1 N ) = e − r T s σ T E [ Φ ( P T ) Z T ] + O ( 1 N ) = Δ E B S ( s , 0 ) + O ( 1 N )

^{4}The order of error term is O ( 1 N ) , if the discontinuity for the pay-off function Φ ( ⋅ ) is not on a lattice point. However, if all discontinuities are on lattice points, the order of the error term is O ( 1 N ) . See Corollary 4.2 in Walsh [

where the stochastic process P t is a geometric Brownian motion given in (1) and Δ E B S ( s ,0 ) is delta in a continuous time model (Black and Scholes model). Even though MS delta does not approximate delta for the binomial tree model, if the pay-off function Φ ( ⋅ ) is not smooth, it still is an approximation formula for the continuous time delta. Under Assumption 2, corollary 4.2 in Walsh [^{4}

Δ E M S ( s , 0 ) = e − r T s σ T E [ Φ ( P T ) log ( P T / s ) − μ σ T σ ] + O ( 1 N ) = Δ E B S ( s , 0 ) + O ( 1 N ) .

As previously discussed, MS delta does not approximate delta for the binomial tree model, if the pay-off function is not smooth. On the other hand, even if the pay-off function is not smooth, MS delta still is an approximation for continuous delta.

2. Gamma

MS gamma for European options is given by (14). Further computation of MS gamma yields,

Γ E M S ( s , 0 ) = − e − r Δ t s 2 σ Δ t E [ E ( s e σ ϵ 1 , Δ t ) ( ϵ 1 − μ Δ t ) ] + e − r Δ t σ s Δ t E [ Δ E M S ( s e σ ϵ 1 , Δ t ) e σ ϵ 1 ( ϵ 1 − μ Δ t ) ] . (23)

The first and second terms in (23) are denoted by G 1 and G 2 , i.e. Γ E M S ( s , 0 ) = G 1 + G 2 :

G 1 = − e − r Δ t s 2 σ Δ t E [ E ( s e σ ϵ 1 , Δ t ) ( ϵ 1 − μ Δ t ) ]

G 2 = e − r Δ t σ s Δ t E [ Δ E M S ( s e σ ϵ 1 , Δ t ) e σ ϵ 1 ( ϵ 1 − μ Δ t ) ] .

The first term is given by

G 1 = − e − r T s 2 σ Δ t E [ Φ ( s e σ W T ) ( ϵ 1 − μ Δ t ) ] = − e − r T s 2 σ Δ t N ∑ i = 1 N E [ Φ ( s e σ W T ) ( ϵ i − μ Δ t ) ] = − e − r T s 2 σ T E [ Φ ( s e σ W T ) ( W T − μ T ) ] = − 1 s Δ E M S ( s , 0 ) ,

and the second term is given by

G 2 = e − r Δ t σ s Δ t E [ e − r Δ t s e σ ϵ 1 σ Δ t E [ E ( s e σ ( ϵ 1 + ϵ 2 ) , 2 Δ t ) ( ϵ 2 − μ Δ t ) | F Δ t ] e σ ϵ 1 ( ϵ 1 − μ Δ t ) ] = 1 s 2 σ 2 ( Δ t ) 2 E [ e − r N Δ t Φ ( s e σ W T ) ( ϵ 1 − μ Δ t ) ( ϵ 2 − μ Δ t ) ] .

This formula is divided into three parts, G 2 = G 2 1 − 2 G 2 2 + G 2 3 , where G 2 1 , G 2 2 and G 2 3 are

G 2 1 = 1 s 2 σ 2 Δ t 2 E [ e − r T Φ ( s e σ W T ) ϵ 1 ϵ 2 ] = 1 s 2 σ 2 Δ t 2 E [ e − r T Φ ( s e σ W T ) ϵ i ϵ j ( ≠ i ) ] = 1 s 2 σ 2 T 2 N N − 1 E [ e − r T Φ ( s e σ W T ) { ( ∑ i ϵ i ) 2 − N Δ t } ] = 1 s 2 σ 2 T 2 E [ e − r T Φ ( s e σ W T ) ( W T 2 − T ) ] + O ( 1 / N )

G 2 2 = 1 s 2 σ 2 Δ t 2 E [ e − r T Φ ( s e σ W T ) ϵ 1 μ Δ t ] = μ s 2 σ 2 T E [ e − r T Φ ( s e σ W T ) W T ]

G 2 3 = 1 s 2 σ 2 Δ t 2 E [ e − r T Φ ( s e σ W T ) μ 2 Δ t 2 ] = μ 2 s 2 σ 2 E [ e − r N Δ t Φ ( s e σ W T ) ] .

These results are combined into the closed-form formulas for MS Gamma:

Γ E M S ( s , 0 ) = e − r T s 2 σ T E [ Φ ( s e σ W T ) { ( W T − μ T ) 2 σ T − ( W T − μ T ) − 1 σ } ] + O ( 1 / N ) .

As discussed in the delta case, Walsh [

Γ E M S ( s , 0 ) = e − r T s 2 σ T E [ Φ ( S T ) { ( log ( S T / s ) − μ σ T ) 2 σ 3 T − log ( S T / s ) − μ σ T σ − 1 σ } ] + O ( 1 / N ) = e − r T s 2 σ T E [ Φ ( P T ) { ( log ( P T / s ) − μ σ T ) 2 σ 3 T − log ( P T / s ) − μ σ T σ − 1 σ } ] + O ( 1 / N ) = e − r T s 2 σ T E [ Φ ( P T ) { Z T 2 σ T − Z T − 1 σ } ] + O ( 1 / N ) = Γ E B S ( s , 0 ) + O ( 1 / N ) ,

even if Φ ( ⋅ ) is not smooth.

3. Vega

MS Vega for the binomial tree model is given by (16). The expectation E [ V 1 i ] is given by

E [ V 1 i ] = e − r ( N − i ) Δ t { − ( 1 + 2 r σ 2 ) } E [ ϵ i + 1 − μ Δ t 2 Φ ( s e σ W T ) ] .

Under the condition i ≠ N − 1 , the expectation E [ V 2 i ] is given by

E [ V 2 i ] = E [ e − r Δ t e − r Δ t s e σ ( W i Δ t + ϵ i + 1 ) σ Δ t × E [ E ( s e σ W i Δ t + ϵ i + 1 e σ ϵ i + 2 , ( i + 2 ) Δ t ; σ ) ( ϵ i + 2 − μ Δ t ) | F ( i + 1 ) Δ t ] × s e σ ( W i Δ t + ϵ i + 1 ) ϵ i + 1 ] = e − r ( N − i ) Δ t σ Δ t { E [ Φ ( s e σ W T ) ϵ i + 2 ϵ i + 1 ] − E [ Φ ( s e σ W T ) ϵ i + 1 ] μ Δ t } .

Formulas

E [ Φ ( s e σ W T ) ϵ i + 2 ϵ i + 1 ] = E [ Φ ( s e σ W T ) ϵ i ϵ j ( ≠ i ) ] = 1 N 2 − N E [ Φ ( s e σ W T ) { ( ∑ i = 1 N ϵ i ) 2 − N Δ t } ] = 1 N 2 − N E [ Φ ( s e σ W T ) ( W T 2 − T ) ]

E [ Φ ( s e σ W T ) ϵ i + 1 ] = 1 N E [ Φ ( s e σ W T ) ∑ i = 1 N ϵ i ] = 1 N E [ Φ ( s e σ W T ) W T ]

lead to

E [ V 2 i ] = e − r ( N − i ) Δ t σ T { 1 N − 1 E [ Φ ( s e σ W T ) ( W T 2 − T ) ] − E [ Φ ( s e σ W T ) W T ] μ Δ t } ( i ≠ N − 1 )

If the pay-off function Φ ( ⋅ ) is a smooth one, the expectation e − r ( N − 1 ) Δ t E [ V 2 N − 1 ] is calculated as

e − r ( N − 1 ) Δ t E [ V 2 N − 1 ] = e − r T s e σ W ( N − 1 ) Δ t Δ t × [ p Φ ′ ( s e σ ( W ( N − 1 ) Δ t + Δ t ) ) e σ Δ t − ( 1 − p ) Φ ′ ( s e σ ( W ( N − 1 ) Δ t − Δ t ) ) e − σ Δ t ] = e − r T s e σ W ( N − 1 ) Δ t Δ t × ∂ ∂ z Φ ′ ( s e σ ( W ( N − 1 ) Δ t + z ) ) e σ z | z = 0 + O ( Δ t 3 / 2 ) = e − r T s σ e σ W ( N − 1 ) Δ t Δ t × ( Φ ″ ( s e σ W ( N − 1 ) Δ t ) s e σ W ( N − 1 ) Δ t + Φ ′ ( s e σ W ( N − 1 ) Δ t ) ) + O ( Δ t 3 / 2 ) = O ( 1 / N ) ,

If the pay-off function Φ ( x ) is not smooth, the relation

e − r ( N − 1 ) Δ t E [ V 2 N − 1 ] = 0 ( = O ( 1 / N ) ) ,

still satisfied, because we formally assumed V 2 N − 1 = 0 . These results are combi- ned into

V E M S ( s , 0 ; σ ) = ∑ i = 0 N − 1 e − r T { − ( 1 + 2 r σ 2 ) } E [ ϵ i + 1 − μ Δ t 2 Φ ( s e σ W T ) ] + ∑ i = 0 N − 2 e − r T σ T { 1 N − 1 E [ Φ ( s e σ W T ) ( W T 2 − T ) ] − E [ Φ ( s e σ W T ) W T ] μ Δ t } = e − r T E [ ( ( W T − μ T ) 2 σ T − ( W T − μ T ) − 1 σ ) Φ ( s e σ W T ) ] + O ( 1 N ) .

As discussed in delta case, the approximation

V E M S ( s , 0 ; σ ) = e − r T E [ ( ( log ( S T / s ) − μ σ T ) 2 σ 3 T − log ( S T / s ) − μ σ T σ − 1 σ ) Φ ( S T ) ] + O ( 1 N ) = e − r T E [ ( ( log ( P T / s ) − μ σ T ) 2 σ 3 T − log ( P T / s ) − μ σ T σ − 1 σ ) Φ ( P T ) ] + O ( 1 N ) = e − r T E [ ( Z T 2 σ T − Z T − 1 σ ) Φ ( P T ) ] + O ( 1 N ) = V E B S ( s , 0 ; σ ) + O ( 1 N )

is valid, even if the pay-off function Φ ( ⋅ ) is not smooth.

4. Rho

In this subsection, we derive the closed-form formula of rho for European options. We also show that MS rho converges to rho in the continuous time model. On the other hand, the formula

E [ ( ϵ i − μ Δ t σ − Δ t ) E ( s e σ W i Δ t , Δ t ; r ) ] = e − r ( N − i ) Δ t E [ ( ϵ i − μ Δ t σ − Δ t ) Φ ( s e σ W N Δ t ) ]

leads the further calculation of rho. This formula is plugged into the Formula (18) and we have

ρ E M S ( s , 0 ; r ) = e − r T E [ ( W T − μ T σ − T ) Φ ( S T ) ] = T ( s Δ E M S ( s , 0 ) − E ( s , 0 ; r ) ) .

The last equality comes from the closed-form formula for MS delta given by (9) (or (22)). As is the discussions in the previous cases, the formula

ρ E M S ( s , 0 ; r ) = T ( s Δ E B S ( s , 0 ) − E [ e − r T Φ ( P T ) ] ) + O ( 1 / N ) = ρ E B S ( s , 0 ) + O ( 1 / N )

must be satisfied under certain conditions given by Walsh [