OM-W10

Exotic Options Overview

Types of Exotic Options

• Packages

• Perpetual American Calls and Puts

• Nonstandard American Options

  • Exercisable only on specific dates (Bermudan options)

  • Early exercise allowed during only part of life (initial "lock out" period)

  • Strike price may change over the life (e.g., warrants, convertible options)

• Barrier Options

  • Option comes into existence only if the stock price hits a barrier before maturity.

  • Two types:

  • 'In' Options: Option is effective if the barrier has not been hit.

  • 'Out' Options: Option dies if the price reaches the barrier before maturity.

  • Subcategories include:

  • 'Up' Options: Stock price must hit the barrier from above.

  • 'Down' Options: Stock price must hit the barrier from below.

  • There are eight possible combinations of options.

• Binary Options

• Lookback Options

• Shout Options

• Asian Options

  • Options whose payoff is related to the average price of the stock.

• Chooser Options

• Compound Options

• Options to Exchange One Asset for Another

• Options Involving Several Assets

• Volatility and Variance Swaps

Non-Standard American Options

• These options differ from standard American options as they have restrictions on their exercisability:

  • Exercisable only on specific dates (Bermudan).

  • Early exercise is allowable only during part of the option's life; there exists an initial "lock out" period.

  • The strike price can change during the life of the option, typical in warrants and convertibles.

Barrier Options

• Defined by their dependence on the underlying asset's price crossing a predetermined barrier.

  • 'In' Options: Activate if the underlying asset reaches the barrier.

  • 'Out' Options: Become inactive if the asset price crosses the barrier beforehand.

Types of Barrier Options

• 'Up' Options: Require the stock price to hit the barrier from above.

• 'Down' Options: Require the stock price to hit the barrier from below.

• Options can be put or call types, and they present eight possible combinations based on activation conditions.

Asian Options

• These options are available in two types based on payoff calculations:

  • Average Price Options

  • Call Payoff: $ext{Payoff} = ext{max}( ext{Save} - K, 0)$

  • Put Payoff: $ext{Payoff} = ext{max}(K - ext{Save}, 0)$

  • Average Strike Options

  • Call Payoff: $ext{Payoff} = ext{max}(S_T - ext{Save}, 0)$

  • Put Payoff: $ext{Payoff} = ext{max}( ext{Save} - S_T, 0)$

• It is noted that exact analytic valuation does not exist; proximity in valuation can be realized by assuming the average stock price follows a lognormal distribution.

Monte Carlo Simulation in Option Pricing

• Introduced by Boyle (1977) as a method for simulating asset returns under risk-neutral conditions.

• Define methodology:

  1. Identify the probability distribution for input variables.

  2. Imitate movement of the input variables through random number generation to match the underlying distribution.

  3. Simulate the underlying variable by aggregating inputs logically.

  4. Repeat the simulation process numerous times for improved accuracy.

  5. Use variance reduction techniques to enhance precision.

Advantages and Disadvantages of Monte Carlo Simulation (MCS)

Advantages:

• Flexible: The method can be easily extended to accommodate diverse scenarios.

• Understandable: The approach is intuitive and accessible for users.

Disadvantages:

• Computational Intensity: Operations might take significantly longer compared to analytical models.

• Approximation Accuracy: Results are based on multiple simulations and may not yield exact solutions.

Generalized Wiener Process for Stock Price

• The discrete version is presented as:

  • ext{In}(S(t + riangle t)) - ext{In}(S(t)) = igg( rac{ ext{μ}- rac{σ^2}{2}}{1}igg) riangle t + σ ext{√} riangle t

  • S(t + riangle t) = S(t) e^{igg(igg( ext{μ}- rac{σ^2}{2}igg) riangle t+ ext{σ} ext{√} riangle tigg)}

• It is often cited that $S(t)$ follows a lognormal distribution, described with geometric Brownian motion.

Stock Price Models in Risk Neutral World

• Presented processes for the underlying asset in a risk-neutral framework:

• Process 1:

  • $dS = σS dz + uSdt$

• Process 2:

  • $S(t + riangle t) - S(t) = ilde{u}S(t) riangle t + σS(t) ext{ɛ} ext{√} riangle t$

Monte Carlo Simulation Steps for Derivative Valuation

1. Simulate one path for the stock price in a risk-neutral world.

2. Calculate the payoff from the stock option.

3. Repeat simulations for a large enough sample to get multiple payoffs.

4. Calculate the mean payoff from all simulations.

5. Discount the mean payoff at the risk-free rate to derive the option's value.

Example Scenario and Simulation Results

• Example Assumptions:

  • Asset: S = 100

  • Drift: 5%

  • Step: 0.01

  • Strike Price: X = 105

  • Volatility: σ = 20%

• Process note:

  • Using relatively few paths with governed simulations (e.g., 250 for a payoff of one year).

  • Conducting subsequent analyses of call and put payoffs based on MAX payoff calculations.

  • Example of payoff calculations generated across multiple simulated paths.

• For instance:

  • Call Payoff: $ext{Payoff} = ext{MAX}(S_T - X, 0)$

  • Put Payoff: $ext{Payoff} = ext{MAX}(X - S_T, 0)$

Path Dependent Options

• Examples demonstrate that Asian and Lookback options have distinct payoff structures based on averages and historical price tracking over the path.

• Various techniques propose grading for average calculations (e.g., arithmetic and geometric averages).

Asian Options Pricing Methods

• Arithmetic Averages:

  • Payoff for average rate call:
    ext{Payoff} = ext{max} igg ( rac{1}{N} igg[ ext{Σ} S(t_i) igg] - K, 0 igg)

• Geometric Averages: Valued based on the closed form equations (formulated by Kemna & Vorst, 1990).

  • Involves cumulative normal distribution functions.

Standard Errors in Monte Carlo Simulation

• The standard error of the estimate of the option price based on sample size ( M ) is given by:
  $SE^2(M) = rac{S^2}{M}$

• As the sample size ( M ) increases, the estimate accuracy also increases.

Confidence Interval Representation

• The confidence interval at level (1 - ( α )) calculated using:
  rac{S^2(M)}{X(M) - Z{1 - \frac{α}{2}} \frac{S^2(M)}{M} < \mu < X(M) + Z{1 - \frac{α}{2}} \frac{S^2(M)}{M}}

• For a 95% confidence interval specifically:
  X(M) - 1.96 \frac{S^2(M)}{M} < \mu < X(M) + 1.96 \frac{S^2(M)}{M}

Case Examples

European Call Option Example

• Parameters:

  • S(0) = $50, E = $52, T = 5 months, r = 10%, σ = 40%

  • Black-Scholes Price computed: $5.1911

  • Monte Carlo (MC) Price with 1000 simulations: $5.4445, Confidence Interval [4.8776, 6.0115]

  • MC Price with 200,000 simulations: $5.1780, CI [5.1393, 5.2167]

Butterfly Spread Example

• Structure of a Butterfly Spread:

  • Buy a European call with exercise price K₁, buy another with K₃, and sell two with K₂, where K₂ is the average of K₁ and K₃.

• Parameters:

  • S(0) = $50, K₁ = $55, K₂ = $60, K₃ = $65, T = 5 months, r = 10%, σ = 40%

  • Exact Price: $0.6124

  • MC Price with 100,000 simulations: $0.6095, CI [0.6017, 0.6173]

Asian Average Rate Call Example

• Payoff structured based on arithmetic averaging, facing identical pricing parameters as previously discussed.

Greek Letters Estimation Methods

• For options like Asian options, follow these steps to calculate:

  1. Make a small alteration to asset price.

  2. Duplicate the simulation using the same random number streams.

  3. Assess the Greek letter by the change in option price divided by the change in asset price, elucidating the underlying risk sensitivity.

Conclusion

• The variances in Monte Carlo simulations applied to exotic options present robust methodologies to derive precise evaluations of complex financial instruments, where traditional analytical models become cumbersome or infeasible. The properties of exotic options, bolstered by techniques such as variance reduction and generalized stochastic models, lay fundamental ground for risk assessment and financial strategy.