MALLIAVIN CALCULUS APPROACH TO PRICING AND HEDGING OF OPTIONS WITH MORE THAN ONE UNDERLYING ASSETS

Abstract

The problems of pricing and hedging in nancial market are fundamental because of uncertainties in the market which are measured by the sensitivities of the underlying assets. Ito calculus has been used to develop several models that deal with the problems of pricing and hedging of options with smooth payo functions. However, Ito calculus becomes ine ective when dealing with options with multiple underlying assets, whose payo s are non-smooth functions. Therefore, this study was designed to consider the sensitivities of options with multiple underlying assets whose payo are non-smooth function. The Malliavin integral calculus given by the Skorohod integral and the integra tion by part technique for stochastic variation were used to derive weight functions of the Greeks for Best of Asset Option (BAO) and Asian Option (AO). The Clark Ocone formula was used to derive an extension of the Malliavin derivative chain rule to nite dimensional vector form. This, together with the weight functions were used to derive expressions for the Greeks which represent the sensitivities of the options with respect to the parameters; price of the underlying asset at initial time S0, second derivative of the option with respect to S0, volatility σ, expiration time T, interest rate µ, namely: δ, γ, ρ, θ and ν respectively. Randomly generated data was used to compute the sensitivities. The weight functions obtained were ω ∆ = Wt S0σT , ω Γ = 1 (σT) 2 1 2S 2 0 (W2 T − T − WT σT ), ω ρ = WT σ , ω Θ = (µ− σ 2 2 ) σT)WT and ω ν = W2 T −T −2WT 2σT . The Malliavin derivative chain rule obtained was D(g(F j k )) = Pn j=1 g 0 (F j k )DFj k , k ≥ 1 and the Greek expression were obtained as: ∆ BAO = e −rT S0σT EQ(max(Si − K)ISi>Sj , i 6= j, i, j = 1, 2...nWT ), Γ BAO = −e −rT S 2 0 EQ[max(Si − K)ISi>Sj , i 6= j, i, j = 1, 2...n 1 (σT) 2 W2 T − T 2 − WT σT ], Θ BAO = −e −rTEQ[max(Si − K)ISi>Sj , i 6= j, i, j = 1, 2...n( µ − σ 2 2 σT )WT ], ρ BAO = e −rT σ EQ[max(Si − K)ISi>Sj , i 6= j, i, j = 1, 2...n]WT ], i ν BAO = e −rT 2σT EQ[max(Si − K)ISi>Sj , i 6= j, i, j = 1, 2...n(W2 T − T − 2WT )], and ∆ AO = e −rTEQ[( 1 T Z T 0 Stdt − k)( Wt S0σT )], Γ AO = e −rT S 2 0 EQ[( 1 T Z T 0 Stdt − k) 1 (σT) 2 W2 T − T 2 − WT σT ], ρ AO = e −rT σ EQ[( 1 T Z T 0 Stdt − k)WT ], Θ AO = −e rTEQ[( 1 T Z T 0 Stdt − k)(µ − σ 2 2 σT )WT ], ν AO = e −rT 2σT EQ[( 1 T Z T 0 Stdt − k)(W2 T − T − 2WT )] where EQ represent the expectation with respect to the equivalent martingale mea sure, WT is the standard Brownian motion at time T, ST is the price of the under lying asset at time T and K is the strike price. The computed sensitivities showed that the risk associated with the model was minimal when there were more than one underlying asset. The sensitivities of options with multiple underlying assets with non-smooth payo s was obtained, and these can be applied in nancial market to monitor and minimise risk.