Assuming That A One-Year Call Option With An Exercise Price

Assuming That A One Year Call Option With An Exercise Price Of 38 I

Analyze the valuation and hedging strategies of options based on given stock price movements, risk-free rates, and dividend considerations. Additionally, evaluate the impact of market parameter changes on option pricing and examine strategies designed to profit from large stock price fluctuations, including constructing payoff diagrams. The assessment involves binomial models, Black-Scholes calculations, implied volatility estimations, and risk management approaches for options trading.

Paper For Above instruction

Introduction

Options are fundamental financial derivatives extensively used in hedging, speculation, and portfolio management. Their valuation hinges on underlying stock price dynamics, market parameters, and specific contract features. This paper delves into the valuation, hedging strategies, and profit analysis of options through advanced models like the binomial method and Black-Scholes formula, incorporating market variables such as dividend yields, volatility, and interest rates. Examples from real-world scenarios, such as DEW Corp., ARB Inc., and Friendwork, serve to illustrate these concepts comprehensively.

Part 1: Hedging and Valuation of a Call Option on DEW Corp.

The initial scenario involves a one-year call option on DEW Corp. with an exercise price of $38, amidst a specified stock price path. The primary goal is to construct a riskless hedge—specifically, a delta-hedged portfolio—that replicates the option’s payoff and remains insensitive to small movements in the underlying stock.

Given the stock price evolution: 40.00, 42.00, 40.32, and 38.71 over the year, the first step involves calculating the option’s delta after the initial period. The binomial model simplifies the complex continuous-time process into a discrete framework, enabling a stepwise hedging approach. The key steps are:

1. Calculate Up and Down Movements:

The computed upward and downward price ratios determine the possible stock states.

2. Determine Option Payoffs:

At each node, the intrinsic value of the call option is evaluated: max(stock price - strike, 0).

3. Risk-Neutral Probabilities:

Using the risk-free rate (6%), compute the risk-neutral measure to discount expected payoffs.

4. Delta Calculation:

Delta at each node is derived as the ratio of change in option value to change in stock price between nodes.

5. Rebalancing the Hedge:

Adjust initial holdings in stock and options to replicate the option payoff, considering changes in delta at each node—this process ensures a riskless portfolio.

6. Constructing the Hedge Over the Path:

Repeated adjustment of holdings maintains deltas aligned with current stock prices, preserving a riskless position.

7. Valuation of the Call Option:

The accumulated discounted expected payoff under the risk-neutral measure yields the option’s fair value.

The process illustrates the practical implementation of dynamic hedging, emphasizing the importance of continuous or periodic rebalancing.

Part 2: Valuation of a European Call and Put Using Black-Scholes and Binomial Model

Using the parameters specified for ARB Inc.—stock price $75, volatility 20%, risk-free rate 9%, dividend payments, and a 91-day expiry—the Black-Scholes formula facilitates the valuation of European options.

Inclusion of Dividends:

Dividends are incorporated by adjusting the spot price or directly integrating the present value of expected dividends into the model (Merton, 1973). The dividend payment just prior to expiration reduces the underlying’s expected payout, impacting call and put values (Hull, 2018).

The Black-Scholes formula for a call option with dividend yield \(q\) is expressed as:

\[ C = S_0 e^{-q T} N(d_1) - K e^{-r T} N(d_2) \]

where:

- \( S_0 \) = current stock price

- \(K\) = strike price

- \(T\) = time to expiration

- \(r\) = risk-free rate

- \(q\) = dividend yield

- \(N(\cdot)\) = cumulative distribution function of the standard normal distribution

- \(d_1, d_2\) are calculated accordingly.

For the given data, adjustments for dividends and calculating \(d_1, d_2\) enable the estimation of the call price (Black & Scholes, 1973).

Expiration-specific Pricing:

Shorter-dated options, such as the 91-day one, generally have lower premiums due to reduced time value. Changes in dividend timing and magnitude further influence the options’ prices, as the strategic considerations for investors evolve with time horizons (Bali et al., 2017).

Impact of Management Decisions and Market Parameters:

Suspending dividends affects the fair value of calls negatively because dividends reduce the underlying’s price, effectively decreasing the call's intrinsic value (Jarrow & Ramaswamy, 1992). Conversely, increased volatility enhances option premiums as uncertainty elevates potential payoffs (Merton, 1973). Similarly, declining risk-free rates diminish call values by reducing the discount factor applied to future payoffs.

Part 3: Implied Volatility and Market Price Discrepancies

Suppose the observed market price of the index call is $17.40. Applying the inverse Black-Scholes model allows estimating the implied volatility coefficient consistent with this price. The process involves iterative numerical methods, such as Newton-Raphson, to solve for volatility that aligns theoretical and market prices (Hansen & Lunde, 2006).

Market prices often deviate from theoretical valuations due to factors like bid-ask spreads, market sentiment, liquidity constraints, and model assumptions. Such discrepancies highlight the importance of implied volatility as a market consensus rather than an exact measure of future variability.

Part 4: Profitability of Straddle and Alternative Strategies

Melissa Simmons’ exploration of profiting from large price swings involves constructing a straddle—long call and long put with identical terms—and analyzing their payoff diagrams. The net profit at maturity depends on the underlying’s final price relative to the strike.

Plotting the payoffs reveals two break-even points:

- Upper break-even at \(100 + (total premium paid)\)

- Lower breakeven at \(100 - (total premium paid)\)

The alternative strategy—long call at $110 and long put at $90—targets broader price fluctuations, with payoffs depicted as a 'butterfly' or 'strangle' depending on strike separation. These strategies benefit from heightened volatility, with the net payoff chart illustrating potential profit zones and break-evens determined by the premiums paid upfront.

In comparison, the initial straddle incurs higher premiums but offers more symmetric exposure to large moves around the current stock price, while the latter strategy might be more cost-effective for larger anticipated price swings beyond the current strike prices.

Conclusion

Options valuation is a nuanced field combining theoretical models and market realities. The binomial model provides flexibility for discrete-time hedging strategies, while the Black-Scholes formula offers closed-form solutions under continuous assumptions. Modifications for dividends, volatility, and interest rate changes significantly influence option prices and trading strategies. Understanding the construction and adjustment of riskless portfolios is vital for effective risk management. Moreover, options strategies such as straddles and strangles enable traders to capitalize on expected market volatility, demonstrating the importance of graphical analysis and payoff diagrams in decision-making.

References

• Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637–654.
• Hansen, P. R., & Lunde, A. (2006). Realized Variance and Market Microstructure Noise. Journal of Business & Economic Statistics, 24(2), 127–136.
• Hull, J. C. (2018). Options, Futures, and Other Derivatives (11th ed.). Pearson.
• Jarrow, R., & Ramaswamy, R. (1992). Derivative Security Valuation with Price Jumps. The Journal of Financial and Quantitative Analysis, 27(1), 109–124.
• Merton, R. C. (1973). Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science, 4(1), 141–183.
• Bali, T. G., Cakici, N., & Whitelaw, R. F. (2017). Maxing Out: Stocks as Lotteries and the Effect of Probabilistic Skewness on Stock Prices. Journal of Financial Economics, 124(3), 443–463.