A Note on the Pricing of Basket Options Using Taylor Approximations
AA NOTE ON THE PRICING OF BASKET OPTIONSUSING TAYLOR APPROXIMATIONS
PABLO OLIVARES, ALEXANDER ALVAREZ
Abstract.
In this paper we propose a closed-form approximation forthe price of basket options under a multivariate Black-Scholes model,based on Taylor expansions and the calculation of mixed exponential-power moments of a Gaussian distribution. Our numerical results showthat a second order expansion provides accurate prices of spread optionswith low computational costs, even for out-of-the-money contracts. Introduction
The objective of the paper is the pricing of basket options using Taylorapproximations under diffusion multivariate models with constant covari-ance. Basket options are multivariate extensions of European calls or puts.A basket option takes the weighted average of a group of d stocks as theunderlying, and produces a payoff equal to the maximum of zero and thedifference between the weighted average and the strike (or the opposite dif-ference for the case of a put). Index options, whose value depends on themovement of an equity or other financial index such as the S&P500, areexamples of basket options.For the particular case of spread options, several approximations have beenpreviously considered in the works of Kirk(1995), Carmona and Durrle-man(2003), Li, Deng and Zhou(2008, 2010), Venkatramanan and Alexan-der(2011) where different ad-hoc approaches are studied.As an alternative Fast Fourier Transform methods have been successfullyimplemented to compute spread prices under more general Levy processes,see Hurd and Zhou(2009) and Cane and Olivares(2014) and under stochasticvolatility models in Depmster and Hong(2000).The approach to pricing by Taylor expansions can be traced back to Hulland White(1987), where the price of a one dimensional derivative is cal-culated. On the other hand, following an idea in Pearson(1995) it can beextended to multidimensional contracts by conditioning on the remaining d − Key words and phrases.
Taylor approximations, basket options, spread options. a r X i v : . [ q -f i n . P R ] A p r OLIVARES, ALVAREZ proving to be effective and accurate for most values in the parametric space.Although in the same spirit, in our case the expansion is done on the func-tion resulting from the conditional price, as opposed to a development basedon the conditional strike price, as previously considered by the authorscited above. Moreover, our method hinges on the calculation of mixedexponential-power moments of a Gaussian distribution, it is extended toexpansions about any point and higher dimensions. Our point of view mayallow for a better control on the approximation, particularly for out-of-the-money options. On a related paper, see Alvarez, Escobar and Olivares(2011),we apply a similar technique to the price of a spread option when correlationis stochastic, by expanding on the correlation matrix.The organization of the paper is the following, in section 2 we introducesome notations, the model and derive the Taylor approximation for basketoptions. In section 3 we specialize the formula for spread options and com-pute the mixed exponential-power moments of a Gaussian law. In section 4we discuss our numerical results.2.
Basket Derivatives and Taylor expansions
We introduce some notations. Let (Ω , F , {F t } t> , P ) be a filtered prob-ability space. We define the filtration F X t := σ ( X s , ≤ s ≤ t ) as the σ -algebra generated by the random variables { X s , ≤ s ≤ t } completed inthe usual way. Denote by Q an equivalent martingale risk neutral measureand E Q the expectation under Q .By r we denote the (constant) interest rate, A (cid:48) represents the transposeof matrix A = ( a ij ) ≤ i,j ≤ d while diag ( A ) is a vector with components( a ii ) ≤ i ≤ d . The d-dimensional column vector of ones is denoted by 1 d .For a l -times differentiable function f in R d and a vector L = ( l , l , . . . , l d )with l k ∈ N such that (cid:80) nk =1 l k = l , D L f represents its mixed partial deriv-ative of order l differentiated l k times with respect to the variable y k .The process of spot prices is denoted by S t = (cid:16) S (1) t , S (2) t , . . . , S ( d ) t (cid:17) (cid:48) ≤ t ≤ T and Y t = ( Y (1) t , Y (2) t , . . . , Y ( d ) t ) (cid:48) ≤ t ≤ T are the asset log-returns related by:(1) S ( j ) t = S ( j )0 exp( Y ( j ) t ) for j = 1 , , . . . , d We analyze European Basket options whose payoff at maturity T , for astrike price K , is given by:(2) h ( S T ) = d (cid:88) j =1 w j S ( j ) T − K + where ( w j ) ≤ j ≤ d are some deterministic weights and x + = max ( x, d = 2 with payoff:(3) h ( S T ) = ( S (1) T − S (2) T − K ) +RICING OF BASKET OPTIONS 3 Also, we have 3:2:1 crack spreads with d = 3 and payoff:(4) h ( S T ) = (cid:18) S (1) T − S (2) T − S (3) T − K (cid:19) + where S (1) t , S (2) t and S (3) t are respectively the spot prices of gasoline, heatingoil and crude oil.Exchange options are derivatives whose payoff is a particular case of (3)when K = 0. Exact formulas are available in the case of a diffusion, seeMargrabe(1978).We assume a multidimensional Black-Scholes dynamics under the risk neu-tral probability following:(5) dS t = rS t dt + Σ S t dB t where ( B t ) t ≥ is a d-dimensional vector of Brownian motions such that d < B ( l ) t , B ( m ) t > = ρ lm dt , for j, m = 1 , , . . . , d and Σ is a positive defi-nite symmetric matrix with components ( σ ij ) i,j =1 , ,...,d and σ ii = σ i .We denote by ˜ Y t = ( Y (2) t , Y (3) t , . . . , Y ( d ) t ) the vector of log-returns, excludingthe first component. The price of a basket option with maturity at T > h ( S T ) is: p = e − rT E Q h ( S T ) = E Q (cid:16) e − rT E Q (cid:104) h ( S T ) |F ˜ Y T (cid:105)(cid:17) = E Q (cid:104) C ( ˜ Y T ) (cid:105) (6)where: C ( y ) := E Q (cid:104) h ( S T ) |F ˜ Y T (cid:105) | ˜ Y T = y Assuming C ( y ) is smooth enough, we denote the n-th order Taylor develop-ment of C around the point y ∗ ∈ R d − as ˆ C n ( y ). It is given by:ˆ C n ( y ) = n (cid:88) l =0 (cid:88) R l D L C ( y ∗ ) l ! l ! . . . l d − ! d − (cid:89) k =1 ( y k − y ∗ k ) l k (7)where: L = ( l , l , . . . , l d − ) and R l = { L ∈ N d − /l + l + . . . + l d − = l, ≤ l k ≤ l } .The next proposition provides the Taylor approximation for the price p of abasket option: Proposition 1.
The n-th order Taylor approximation around y ∗ = ( y ∗ , y ∗ , . . . , y ∗ d − ) of the price p of a basket option with payoff h ( S T ) , defined as ˆ p n := e − rT E Q ˆ C n ( ˜ Y T ) ,under model (5), is given by: (8)ˆ p n = w n (cid:88) l =0 (cid:88) R l D L C ( y ∗ ) l ! l ! . . . l d − ! E Q (cid:34) e − ( r − σ Y (1) T / ˜ YT ) T + µ Y (1) T / ˜ YT d − (cid:89) k =1 ( Y ( k +1) T − y ∗ k ) l k (cid:35) OLIVARES, ALVAREZ where for y ∈ R d − : (9) C ( y ) := C BS ( K ( y ) , σ Y (1) T / ˜ Y T = y , S (1)0 ) is the Black-Scholes price of a call option with strike price K ( y ) , maturityat T > , volatility σ Y (1) T / ˜ Y T = y ∗ , spot price S (1)0 and strike price: (10) K ( y ) = 1 w e ( r − σ Y (1) T / ˜ YT = y ) T − µ Y (1) T / ˜ YT = y K − d (cid:88) j =2 w j S ( j )0 e y ∗ ( j ) with: (11) µ Y (1) T / ˜ Y T = ( r − σ ) T + Σ Y Σ − Y ( ˜ Y − r + 12 diag (Σ ˜ Y )) T (12) σ Y (1) T / ˜ Y T = σ − Σ Y Σ − Y Σ (cid:48) Y Σ Y = ( σ , σ , . . . , σ ,d − ) (cid:48) and Σ ˜ Y is the covariance matrix of the vector ˜ Y T .Proof. From equation (5) a straightforward application of Ito formula leadsto:(13) Y T = ( r d − diag (Σ)) T + Σ √ T Z d in law, where Z d is a random variable with a multivariate normal distribu-tion in R d with zero mean and covariance matrix I d . Hence Y T has alsoa multivariate normal distribution. Also conditionally on ˜ Y T , the randomvariable Y (1) T has a univariate normal distribution. Thus, we can write:(14) Y (1) T = µ Y (1) T / ˜ Y T + σ Y (1) T / ˜ Y T √ T Z (1) in law, where Z (1) is independent of Y T and it has, conditionally on ˜ Y T , astandard univariate normal distribution. Moreover it is well known, see forexample Tong (1989), that µ Y (1) T / ˜ Y T and σ Y (1) T / ˜ Y T are given by equations (11)and (12) respectively.Next, from equation (6) we have: p = e − rT E Q (cid:16) E Q (cid:16) h ( S T ) |F ˜ Y T (cid:17)(cid:17) = w e − rT E Q E Q S (1)0 e Y (1) T − Kw − d (cid:88) j =2 w j w S ( j )0 e Y ( j ) T + |F ˜ Y T = w e − rT E Q (cid:18) E Q (cid:20)(cid:16) S (1)0 e Y (1) T − K (cid:48) ( ˜ Y T ) (cid:17) + |F ˜ Y T (cid:21)(cid:19) (15) RICING OF BASKET OPTIONS 5 where K (cid:48) ( y ) = Kw − (cid:80) dj =2 w j w S ( j )0 e y ( j ) .Moreover, substituting equation (14) into (15) we have: p = w e − rT E Q (cid:34) E Q (cid:32)(cid:18) S (1)0 e µ Y (1) T / ˜ YT + σ Y (1) T / ˜ YT √ T Z (1) − K (cid:48) ( ˜ Y T ) (cid:19) + |F ˜ Y T (cid:33)(cid:35) = w e − rT E Q (cid:20) e − rT + σ Y (1) T / ˜ YT T + µ Y (1) T / ˜ YT E Q (cid:32)(cid:18) S (1)0 e rT − σ Y (1) T / ˜ YT T + σ Y (1) T / ˜ YT √ T Z (1) − K (cid:48) ( ˜ Y T ) (cid:19) + |F ˜ Y T (cid:33)(cid:35) = w E Q (cid:20) e − rT + σ Y (1) T / ˜ YT T + µ Y (1) T / ˜ YT C ( ˜ Y T ) (cid:21) where: C ( ˜ Y T ) := C BS ( K ( ˜ Y T ) , σ Y (1) T / ˜ Y T , S (1)0 )= e − rT E Q (cid:34)(cid:18) S (1)0 e ( r − σ Y (1) T / ˜ YT ) T + σ Y (1) T / ˜ YT √ T Z (1) − K ( ˜ Y T ) (cid:19) + |F ˜ Y T (cid:35) Applying a n-th order Taylor development around y ∗ = ( y ∗ , y ∗ , . . . , y ∗ d − ) ∈ R d − to C ( y ) we compute the approximated conditional price based on the first underlyingand conditional on the remaining d − C n ( ˜ Y T ) = n (cid:88) l =0 (cid:88) R l D L C ( y ∗ ) (cid:81) d − k =1 l k ! d − (cid:89) k =1 ( Y ( k +1) T − y ∗ k ) l k After replacing equation (16) into the expression for p above we get immediatelyequation (8) in Proposition 1. (cid:3) Remark 2.
Notice that the approximation ˆ p k depends only on the deriva-tives of the function C ( y ) with respect y , which in turn is computed asthe Black-Scholes price composed with the function K ( y ) and the mixedexponential-power moments of a Gaussian multivariate distribution. Remark 3.
Sensitivities to the parameters can be computed by a similarapproximation, as Greeks for a Black-Scholes option model are known. Forexample the delta with respect to the j-th asset can be approximated by: ˆ∆ ( j ) n = w n (cid:88) l =0 (cid:88) R l D L ∂C ( y ∗ ) ∂s ( j ) l ! l ! . . . l d − ! E Q (cid:34) e − ( r − σ Y (1) T / ˜ YT ) T + µ Y (1) T / ˜ YT d − (cid:89) k =1 ( Y ( k +1) T − y ∗ k ) l k (cid:35) Pricing spreads options by Taylor approximations
In order to illustrate the method studied in the previous section we con-sider the case of a bidimensional spread option under model (5) with covari-ance matrix: Σ = (cid:18) σ σ (cid:19) OLIVARES, ALVAREZ
We find the n-th Taylor approximation in this specific situation. Denotingby d < B (1) t , B (2) t > ρdt we have that:(17) Y T = ( Y (1) T , Y (2) T ) ∼ N (cid:18) ( r − diag (Σ)) T, T Σ ρ (cid:19) where:(18) Σ ρ = (cid:18) σ ρσ σ ρσ σ σ (cid:19) From equation (13) the conditional distribution of Y (1) T given Y (2) T is: Y (1) T /Y (2) T ∼ N (cid:18) r (1 − σ σ ρ ) T + 12 σ σ ρT + σ σ ρY (2) T − σ T, (1 − ρ ) σ T (cid:19) Thus we can write: Y (1) T = µ ( Y (2) T ) + σ √ T Z in law, where Z ∼ N (0 ,
1) independent of Y T , with(19) µ ( Y (2) T ) := µ Y (1) T / ˜ Y T = r (1 − σ σ ρ ) T + 12 σ ( σ ρ − σ ) T + σ σ ρY (2) T and σ := σ Y (1) T / ˜ Y T = (cid:112) (1 − ρ ) σ From Proposition 1 the n-th approximation simplifies to:(20) ˆ p n = n (cid:88) l =0 D l C ( y ∗ ) l ! E Q (cid:104) e − ( r − σ ) T + µ ( Y (2) T ) ( Y (2) T − y ∗ ) l (cid:105) Moreover: E Q (cid:104) e ( − r + σ ) T + µ ( Y (2) T ) ( Y (2) T − y ∗ ) l (cid:105) = e A E Q (cid:20) e σ σ ρY (2) T ( Y (2) T − y ∗ ) l (cid:21) where: A = ( − ( r − σ ) + r (1 − σ σ ) ρ + 12 σ ( σ ρ − σ )) T = − ( 12 ρ σ + r σ σ ρ − σ σ ρ ) T RICING OF BASKET OPTIONS 7
Now, from equation (17) we have that Y (2) T ∼ N (( r − σ ) T, T σ ), then theexponential-power moments can be calculated as follows: E Q (cid:20) e σ σ ρY (2) T ( Y (2) T − y ∗ ) l (cid:21) = l (cid:88) m =0 (cid:18) lm (cid:19) (cid:18) ( r − σ ) T − y ∗ (cid:19) l − m E Q (cid:20) e σ σ ρY (2) T ( Y (2) T − E Q ( Y (2) T )) m (cid:21) = l (cid:88) m =0 (cid:18) lm (cid:19) (cid:18) ( r − σ ) T − y ∗ (cid:19) l − m T m σ m e σ σ ρ ( r − σ ) T E Q (cid:104) e √ T σ ρZ Z m (cid:105) = e σ σ ρ ( r − σ ) T l (cid:88) m =0 (cid:18) lm (cid:19) (cid:16) √ T σ (cid:17) m B ( y ∗ ) l − m E Q (cid:104) e √ T σ ρZ Z m (cid:105) where: B ( y ∗ ) = ( r − σ ) T − y ∗ Next integrate by parts: E Q (cid:104) e √ T σ ρZ Z m (cid:105) = 1 √ π (cid:90) R e − ( x − σ ρ √ T x ) x m dx = e σ ρ T √ π (cid:90) R e − ( x − σ ρ √ T ) x m dx = e σ ρ T √ π (cid:90) R e − y ( y + σ ρ √ T ) m dy = e σ ρ T m (cid:88) ν =0 (cid:18) mν (cid:19) ( σ ρ √ T ) m − ν E ( Z ν )= e σ ρ T [ m ] (cid:88) ν =0 (cid:18) m ν (cid:19) ( σ ρ √ T ) m − ν √ π (cid:90) R e − y y ν dy = e σ ρ T [ m ] (cid:88) ν =0 (cid:18) m ν (cid:19) ( σ ρ √ T ) m − ν (2 ν − n !! is the double factorial defined as the product of all odd numbersbetween 1 and n including both. When the set is empty, by convention, theproduct is equal to one.Similarly for y ∗ = E Q ( Y (2) T ) we have: E Q (cid:20) e σ σ ρY (2) T ( Y (2) T − y ∗ ) l (cid:21) = T l σ l e σ σ ρ ( r − σ ) T e σ ρ T l (cid:88) ν =0 (cid:18) lν (cid:19) ( σ ρ √ T ) l − ν E ( Z ν )= T l σ l e − A [ l ] (cid:88) ν =0 (cid:18) l ν (cid:19) ( σ ρ √ T ) l − ν (2 ν − OLIVARES, ALVAREZ
After gathering all pieces and substituting in equation (20) we have thefollowing result:
Proposition 4.
The n-th Taylor approximation of a spread contract withmaturity at T and strike price K , under the model (5) is given by: ˆ p n = n (cid:88) l =0 l (cid:88) m =0 D l C ( y ∗ ) l ! (cid:18) lm (cid:19) (cid:16) √ T σ (cid:17) m B ( y ∗ ) l − m E ( m ) with: E ( m ) = m (cid:88) ν =0 (cid:18) mν (cid:19) ( σ ρ √ T ) m − ν E Q ( Z ν ) for m = 1 , , . . . , k and E (0) = 1 , where E Q Z ν = ( ν − if ν is even orzero if it is odd, and K ( y ) = e ( r − σ ) T − µ ( y ) ( K + S (2)0 e y ) = e − A (cid:16) Ke − σ σ ρy + S (2)0 e (1 − σ σ ρ ) y (cid:17) with µ ( y ) given by equation (19 ). Next, we compute the derivatives of the function C ( y ) with respect to y .From the Black-Scholes pricing formula: C ( y ) := C BS ( K ( y ) , σ, S (1)0 ) = S (1)0 N ( d ( K ( y )) − K ( y ) e − rT N ( d ( K ( y ))where: d ( K ( y )) = log (cid:18) S (1)0 K ( y ) (cid:19) + ( r + σ ) Tσ √ Td ( K ( y )) = d ( K ( y ) − σ √ T and N ( . ) is the cumulated distribution function of a standard normal dis-tribution.The first two derivatives are computed by elementary methods.First notice that: D K ( y ) = e − A (cid:18) − σ σ ρKe − σ σ ρy + S (2)0 (1 − σ σ ρ ) e (1 − σ σ ρy ) (cid:19) D K ( y ) = e − A (cid:18) ( σ σ ρ ) Ke − σ σ ρy + S (2)0 (1 − σ σ ρ ) e (1 − σ σ ρ ) y (cid:19) Also: D C BS ( y ) = S (1)0 f Z ( d ( K ( y ))) D d ( K ( y )) − e − rT D K ( y ) N ( d ( K ( y ))) − e − rT K ( y ) f Z ( d ( K ( y ))) D d ( K ( y ))= − D K ( y ) K ( y ) σ √ T A ( y )where f Z is the density function of a standard normal random variable and A ( y ) = S (1)0 f Z ( d ( K ( y )))+ σ √ T e − rT K ( y ) N ( d ( K ( y ))) − e − rT K ( y ) f Z ( d ( K ( y ))) RICING OF BASKET OPTIONS 9
Similarly the second derivative is obtained as: D C ( y ) = − σ √ T (cid:20) A ( y ) K ( y ) D K ( y ) − ( D K ( y )) K ( y ) + D A ( y ) D K ( y ) K ( y ) (cid:21) with: D A ( y ) = − S (1)0 f Z ( d ( K ( y ))) d ( K ( y )) D d ( K ( y )) + σ √ T e − rT D K ( y ) N ( d ( K ( y )))+ σ √ T e − rT K ( y ) f Z ( d ( K ( y ))) D d ( K ( y ))+ e − rT f Z ( d ( K ( y ))) D d ( K ( y )) d ( K ( y )) K ( y )= D K ( y ) K ( y ) σ √ T (cid:104) S (1)0 f Z ( d ( K ( y ))) d ( K ( y )) + σ T e − rT K ( y ) N ( d ( K ( y ))) − σ √ T e − rT K ( y ) f Z ( d ( K ( y ))) − e − rT K ( y ) f Z ( d ( K ( y ))) d ( K ( y )) (cid:105) In particular when we develop around y mean = E Q ( Y (2) T ) = ( r − σ ) T wehave the first and second approximations given respectively by :ˆ p = C ( y mean ) + σ σ ρT D C ( y mean )ˆ p = ˆ p + 12 (cid:2) T σ (1 + σ ρ T ) (cid:3) D C ( y mean )More generally expanding around y ∗ we have the first two approximationsdenoted by ˆ p ( y ∗ ) and ˆ p ( y ∗ ) respectively and given by:ˆ p ( y ∗ ) = C ( y ∗ ) + D C ( y ∗ )( B ( y ∗ ) + √ T σ E (1))= C ( y ∗ ) + D C ( y ∗ )( B ( y ∗ ) + T σ σ ρ )ˆ p ( y ∗ ) = ˆ p ( y ∗ ) + 12 D C ( y ∗ ) (cid:2) B ( y ∗ ) + 2 T σ σ ρB ( y ∗ ) + T σ (1 + T σ ρ ) (cid:3) Pricing Spreads: numerical results
We consider spread options in the following benchmark numerical set: S (1)0 = 100, S (2)0 = 96, σ = 0 . σ = 0 . ρ = − . r = 0 . K = 1 and T = 1.In Figure 1 the graph of the conditional price C ( y ) given by equation (7) isshown (blue line), together with the first and second order Taylor approxi-mation around the mean, for the benchmark parameter set.Notice that the first approximation underestimates the price. Not surpris-ingly the second approximation estimates the price fairly well for valuesclose to the point y mean while is less accurate for values far from the mean.Although it seems a drawback of the method it does not constitutes a seri-ous problem as values far from the mean are unfrequent, thus the error incalculating the outer expected value by the Taylor approximation is small.In Figure 2 a histogram for simulated returns on asset 1 (blue rectangles)and asset 2(red rectangles) is shown. Notice that only a few values of thereturns lie outside the interval [ − , Figure 1.
The function C BS ( y ) is shown in blue, togetherwith the first and second order approximations around themean for the benchmark parameters.Next we compare Taylor approximations with Monte Carlo simulations.In Table 1(column 2) prices from Monte Carlo are shown for the bench-mark parameters, except the correlation parameter that takes values ρ = − . , − . , . , .
5. The number of simulations is n = 10 , where a stabil-ity of order 10 − is attained. Partial Monte Carlo prices (shown in column3) are obtained by sampling directly the one dimensional conditional price C ( Y (2) T ) and taking the corresponding average of the payoff. It leads to amore efficient simulation algorithm as only one Brownian motion needs tobe simulated, as oppose to two correlated Brownian in the standard MonteCarlo approach. It is done though at the expense of an extra evaluation ofthe Black-Scholes formula in every step.Taylor prices of first and second order are shown in columns 4 and 5 of Table1. The expansions take place around y ∗ = 0. While in some cases the firstorder approximation reveals significant different with Monte Carlo, secondorder approximation shows an improved agreement with a relative error inthe order of 10 − for the parameter set considered. RICING OF BASKET OPTIONS 11
Figure 2.
Histogram of simulated returns on asset1 (bluerectangles) and asset 2 (red rectangles) for the benchmarkparameter.Correlation Monte Carlo Partial Monte Carlo First approx. Second approx. ρ = 0 . ρ = − . ρ = 0 . ρ = − . Table 1.
Spread prices for the benchmark parameters andseveral values of ρ , using monte Carlo, partial Monte Carloand first and second Taylor expansions around y ∗ = 0.For extreme values of the correlation coefficient ρ , e.g. larger than anabsolute value of 0 .
7, the Taylor expansions around y ∗ = 0 do not workwell. Nevertheless it is interesting to notice that the approximations arerather sensible to the point where the expansion is taken. Moreover, byslightly changing the latter the accuracy of the method can be considerablyimproved. In Table 2 spread prices for the benchmark parameters and ρ = − . Expansion point Monte Carlo Partial Monte Carlo First approx. Second approx. y ∗ = − .
015 16.2463 16.2540 12.3734 16.3011 y ∗ = − .
02 16.2463 16.2540 12.2966 15.8566 y ∗ = − .
05 16.2463 16.2540 11.8434 12.9761 y ∗ = 0 16.2463 16.2540 12.5208 17.5217 y ∗ = 0 .
01 16.2463 16.2540 12.5089 18.2168
Table 2.
Spread prices for the benchmark parameters, ex-cept ρ = − . y ∗ .Parameters Monte Carlo Taylor (first order) Taylor (second order) S (1) T = 90 , S (2) T = 100 7.040956 5.30281 7.0468998 K = 5 , y ∗ = 0 . S (1) T = 90 , S (2) T = 110 4.8015937 3.442070 4.800319 K = 5 , y ∗ = 0 . S (1) T = 90 , S (2) T = 100 5.7623 4.3248347 5.7726138 K = 10 , y ∗ = 0 . S (1) T = 90 , S (2) T = 110 3.89825 2.71934 3.89966 K = 10 , y ∗ = 0 . Table 3.
Prices of out-of-the-money spread contracts forselected strike and spot prices. Other parameters are keptwithin the benchmark set.We test the Taylor expansion method for out-of-the-money contracts andcompare with the price obtained via Monte Carlo with n = 10 repetitions.The results are shown in Table 3. The benchmark parameters are the same,except for the spot and strike prices that are changed accordingly. again asecond order Taylor expansion seem to capture the Monte Carlo prices.5. Conclusions
We present an efficient method to price basket options under a multi-dimensional Black-Scholes model, based on a Taylor expansion of the con-ditional one dimensional price resulting from fixing one of the underlyingassets. The formula is given in terms of exponential-power moments of amultivariate Gaussian law and the evaluation of certain derivatives in theBlack-Scholes price.We implement it numerically in the case of spread contracts. Within thebenchmark parametric set this approach is in closed agreement with theprice obtained via Monte Carlo, even for deep out-of-the-money contracts,at considerable lesser computational effort. A second order developmentseems to be sufficient to achieve a relative error around 10 − . RICING OF BASKET OPTIONS 13
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