Hedging Fixed Indexed Annuities : describe the distribution of replicating errors.Make a plot of this distribution.

Hedging Fixed Indexed Annuities.

1. Simulate an arbitrary number of Geometric Brownian Motion pathways. You’ll
probably want to simulate at least 1,000 paths. Each path depends on a set of random
numbers, a constant volatility and default-free rates. This is a simulation for the
underlying stock index, S(t) up until a terminal time T. (No dividend yield.)

2. Along each pathway, model the value of a theoretical (Black-Scholes) call option on
S, with a strike K. Also model the delta of this option at each time step. This will
represent the embedded option in the FIA contract. We will be simulating the
performance of uncapped returns with a participation rate.

3. Now replicate the value of the long call by simulated trading the underlying. Also
make sure you keep track of cash. In other words, construct a replicating portfolio.
This is driven by the equation 𝑐 = Δ𝑆 + 𝐡. This is equivalent to:

π‘π‘Žπ‘™π‘™ π‘œπ‘π‘‘π‘–π‘œπ‘› = (#π‘œπ‘“ π‘ β„Žπ‘Žπ‘Ÿπ‘’π‘ )($ π‘π‘’π‘Ÿ π‘ β„Žπ‘Žπ‘Ÿπ‘’) + (π‘π‘Žπ‘ β„Ž π‘Žπ‘π‘π‘œπ‘’π‘›π‘‘)

At each time step, make an appropriate trade that serves to adjust your stock position,
all the while earning interest on credits or paying interest on debits.

4. For each simulated pathway, set aside the terminal value of the replicating portfolio,
the terminal value of the underlying and the terminal value of the theoretical call
option. The difference of the portfolio and the call option is β€œreplicating error.”

β€’ DescribeΒ  the distribution of replicating errors.Make a plot of this distribution.

β€’ DescribeΒ  the value of the replicating portfolio and the value of the theoretical call as a function of the underlying at time T.

β€’ Make a plot of the value of the replicating portfolio and the value of the theoretical call as a function of the underlying at time T.

β€’ Investigate the effect of changing the crediting rate, interest rates, volatility,