Q1. What is pricing by replication?

Pricing by replication involves trading in financial instruments to match the payoff of another instrument, likely exotic.

One benefit of pricing exotics by (static) replication with vanillas --> exotic priced consistently with volatility smile.

2 main types of replication methods: static and dynamic.

- Static replication involves setting up a ptf today and not having to do any further trading.

- Dynamic replication involves continuously trading in the underlying.

Q2. A derivative pays with K1 < K2

Derive a model independent hedge in terms of a ptf of vanilla options.

If you have identified the payoff as being a straight line which changes gradient at various points.

Then every time you need to change the gradient by x you add x call options.

If there are jumps, then you use call spreads.

If there is value when spot equals zero, then you use put options or ZC bonds.

When trying to replicate a derivative the best way to start is by sketching its payoff.

Denominator:

- K1 for ST < K1

- ST for K1 < ST < K2

- K2 for ST > K2

The actual payoff will be the inverse of this.

The payoff is not linear between K1 and K2, therefore we will not be able to exactly replicate it with vanilla options.

However, we can get close.

If we make the approximation that the payoff is linear between K1 and K2 and the techniques above, our ptf contains:

- 1/K1 ZC bonds expiring at T

- a call option struck at K1 with notional: - a call option struck at K2 with notional: -a

This approximation will be good for K1 close to K2, but will deteriorate as the 2 values of K are further apart.

There are 2 ways to improve our replication.

1. We can either select a finite number of points in the interior and then do piecewise straightlines between them

2. We can write the payoff as a continuous super-position of options.

In the second case, we have an uncountably infinite number of options but can replicate precisely.

The key is that the second derivative of a call option w.r.t. strike is a delta function.