By James Newbury

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**Additional resources for Applications of Malliavin calculus to the pricing and hedging of Bermudan options **

**Example text**

N }. 34 4. Repeat step 3 with k = 1. 2). , N }, compute the quantity w1 (Xδt ) defined by: (q) (q) (q) w1 (Xδt ) = ∂Φ(Xδt )1{v1 (X (q) )<Φ(X (q) )} +e−rδt D[v2 ](Xδt )1{v1 (X (q) )≥Φ(X (q) )} δt δt δt δt 5. Do the following: (a) Compute E0 [v1 ](x) = (b) Compute w0 (x) = 1 N 1 N N q=1 N q=1 (q) v1 (Xδt ) and set v0 (x) = max(Φ(x), e−rδt E0 [v1 ](x)). (q) w1 (Xδt ). The approximation of the price and the delta are then respectively given by the quantities v0 (x) and w0 (x). 3 We first look at the case of geometric Brownian motion.

This pricing and hedging algorithm is comprised of the five main following steps: 1. Set the parameters related to the underlying price process, set the payoff of the option Φ, its derivative ∂Φ, and the maturity of the option T . Set the number of samples N and the number of discretized subintervals for T time M . Define the time step δt = M . 2. , N }. , N }. 3. , N }. , N }. 1). , N }. 34 4. Repeat step 3 with k = 1. 2). , N }, compute the quantity w1 (Xδt ) defined by: (q) (q) (q) w1 (Xδt ) = ∂Φ(Xδt )1{v1 (X (q) )<Φ(X (q) )} +e−rδt D[v2 ](Xδt )1{v1 (X (q) )≥Φ(X (q) )} δt δt δt δt 5.

N }. 3. , N }. , N }. 1). , N }. 34 4. Repeat step 3 with k = 1. 2). , N }, compute the quantity w1 (Xδt ) defined by: (q) (q) (q) w1 (Xδt ) = ∂Φ(Xδt )1{v1 (X (q) )<Φ(X (q) )} +e−rδt D[v2 ](Xδt )1{v1 (X (q) )≥Φ(X (q) )} δt δt δt δt 5. Do the following: (a) Compute E0 [v1 ](x) = (b) Compute w0 (x) = 1 N 1 N N q=1 N q=1 (q) v1 (Xδt ) and set v0 (x) = max(Φ(x), e−rδt E0 [v1 ](x)). (q) w1 (Xδt ). The approximation of the price and the delta are then respectively given by the quantities v0 (x) and w0 (x).