The intuition behind mean square hedging process for an exotic option

In a simplified mathematical form, the mean square hedging process for an exotic option could be represented as follows:

Let P(T) be the payoff of the exotic option at maturity time T.

The value of the hedging portfolio at time t, V(t), is composed of holdings in the risky asset, φ(t), and the risk-free asset, ψ(t), such that:

V(t) = φ(t) * S(t) + ψ(t) * B(t) (*) 

Here, S(t) is the price of the risky asset and B(t) is the value of the risk-free asset at time t.

The objective is to minimize the expected squared hedging error at maturity:

E[(V(T) - P(T))^2]

The portfolio is dynamically adjusted, meaning φ(t) and ψ(t) are continually updated over time to maintain an optimal hedge.

To find the optimal hedge, we take the derivative of the expected squared error with respect to φ(t) and set it equal to zero, since we want to minimize this error:

dE[(V(T) - P(T))^2]/dφ(t) = 0


This results in an equation that we solve for φ(t).

The hedge ratio, or delta (Δ), is the partial derivative of the option price with respect to the price of the underlying asset, and it informs the adjustments to φ(t).

In continuous time, the problem often results in a stochastic differential equation that must be solved to find the optimal hedge.

Since the Black-Scholes-Merton model may not be appropriate for exotic options, more sophisticated models like local volatility or stochastic volatility models may be used.

For exotic options, we may need to consider higher-order Greeks or scenario-based sensitivities, which involve partial derivatives of the option price with respect to other risk factors.

Oftentimes, numerical methods like Monte Carlo simulations or finite difference methods are employed to estimate the necessary Greeks and solve the stochastic differential equation.

The Implementation is typically done using sophisticated algorithms and software due to the complexity of the calculations.


(*)

A risk-free bond is included in a hedging portfolio for an exotic option to provide stability and predictability. It serves as a reliable source of funds, helps reduce overall portfolio risk, counters interest rate sensitivities, and aids in managing the dynamic hedging process. The bond's steady growth due to known interest rates helps ensure that the hedging portfolio can meet the future obligations of the option payoff. 

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