A new European-style floating lookback call option on a...

A new European-style floating lookback call option on a stock index has a maturity of 9 months. The current level of the index is 400 , the risk-free rate is $6 \%$ per annum, the dividend yield on the index is $4 \%$ per annum, and the volatility of the index is $20 \%$. Use DerivaGem to value the option.

Key Concepts

European Options

European options are financial derivatives that can only be exercised at expiration. They form the basis for many option pricing models and are distinct from American options, which allow exercise at any time before maturity. This type of option is commonly used in theoretical pricing due to its simpler exercise feature and mathematical tractability.

Lookback Options

Lookback options are a class of exotic options where the payoff depends on the optimal value of the underlying asset's price over the life of the option. Instead of just the final asset price, these options use either the maximum or minimum price observed during the option’s term, making them inherently path-dependent and sensitive to the asset’s price history.

Floating Strike Option

A floating strike option, particularly in the context of lookback options, is one where the strike price is not set in advance but is determined based on the underlying asset's performance, often set to the extreme value (minimum for a call, maximum for a put) achieved during the option’s life. This feature typically makes such options more expensive due to the increased benefit for the option holder.

Option Pricing

Option pricing involves determining the fair value of derivative securities by assessing the probabilistic outcomes of their payoffs and discounting them to present value using a risk-neutral measure. This process incorporates various factors including the underlying asset's volatility, the time to maturity of the option, and prevailing interest rates.

Risk-Free Rate

The risk-free rate is the theoretical return on an investment with no default risk, standardly used as the discount rate in risk-neutral valuation frameworks. It is crucial in option pricing models because it provides the baseline for comparing the return of riskier assets.

Dividend Yield

Dividend yield is the annual dividend payment expressed as a percentage of the underlying stock's price. In the context of option pricing, particularly for equity options, the dividend yield affects the expected return and price dynamics of the underlying asset and thus impacts the valuation of the option.

Volatility

Volatility is a statistical measure of the dispersion of returns for a given asset, indicating the degree of variation in its price. In option pricing, volatility is a key input because it significantly influences the premium, as higher volatility increases the probability of the option finishing in the money.

Time to Maturity

Time to maturity represents the length of time until an option expires. It is a critical element in option pricing models as it determines the duration over which the underlying asset's price can fluctuate, impacting the option's risk and time value.

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