Consider a down-and-out call option on a foreign currency....

Consider a down-and-out call option on a foreign currency. The initial exchange rate is 0.90 , the time to maturity is 2 years, the strike price is 1.00 , the barrier is 0.80 , the domestic risk-free interest rate is $5 \%$, the foreign risk-free interest rate is $6 \%$, and the volatility is $25 \%$ per annum. Use DerivaGem to develop a static option replication strategy involving five options.

Key Concepts

Volatility and Option Sensitivities

Volatility represents the degree of variation in the price of the underlying asset and is a critical input in option pricing models. It directly affects the option’s premium and the sensitivity measures (often called Greeks), which quantify how the option’s price responds to changes in underlying parameters such as price, time, and interest rates.

Risk-Neutral Pricing

Risk-neutral pricing is a fundamental valuation method where it is assumed that all investors are indifferent to risk; under this framework, the expected returns of the underlying asset equal the risk-free rate. This allows the pricing of derivatives by discounting their expected payoffs under a risk-neutral probability measure, simplifying the valuation process.

Foreign Currency Options

Foreign currency options give the holder the right, but not the obligation, to exchange one currency for another at a predetermined strike rate, making them essential tools for managing currency risk. Their pricing is influenced by the interest rate differential between the two currencies and reflects the dynamics of foreign exchange markets.

Static Option Replication

Static option replication is a technique used to approximate or exactly replicate the payoff of a complex derivative by constructing a portfolio of simpler, more liquid options that remains unchanged over the life of the derivative. This method helps in understanding and pricing complex instruments by decomposing them into elementary building blocks.

Down-and-out Options

Down-and-out options are a type of barrier option that become worthless if the underlying asset’s price falls below a specified barrier level at any time before expiration. This feature allows these options to be priced more affordably relative to their standard counterparts, but it requires careful analysis of the probability of barrier breach.

Barrier Options

Barrier options are a class of exotic options whose payoff depends not only on the final price of the underlying asset but also on whether the asset’s price has reached or breached a predetermined barrier level during the option’s life. They introduce path dependency, making their pricing and hedging more complex than standard European or American options.

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