77. One-Period Binomial Model

Valuing Options with the One-Period Binomial Model

• To value an option using the one-period binomial model, the risk-free rate, the volatility of the underlying asset's price, and the current asset price are needed.
• The risk-adjusted rate of return and the actual probability of an up-move are not required.

Exploiting Mispricing in European Put Options

• If a European put option trades at a higher price than the binomial model implies, investors can earn excess returns by selling the underlying asset, buying a call option, and investing at the risk-free rate.
• Use put-call parity and rearrange to isolate the put option: S0 + p0 = c0 + X(1 + r)–T, which leads to p0 = c0 – S0 + X(1 + r)–T.
• When a put is overpriced, it should be sold.
• The components of the right side of the rearranged equation should be transacted by buying a call, selling the underlying asset, and investing at the risk-free rate.

Risk-Free Rate and Option Valuation

• The risk-free rate can be used to value an option with a one-period binomial model.
• Combining options with the underlying asset in a specific ratio results in a risk-free future payment.
• A portfolio can be constructed with an option position and a position in the underlying asset.
• The portfolio value at option expiration is the same for both an up-move and a down-move.

Hedge Ratio Calculation and Portfolio Value

• Consider a stock with a future value of either 22 or 14 one year from now, with a risk-free rate of 5%.
• The ratio of shares to short call options with an exercise price of 18 for a risk-free portfolio at expiration is 0.50.
• With a stock price of 22 at expiration, the short call payoff is -4, and with a stock price of 14, the call payoff is 0.
• The hedge ratio is calculated as (4 – 0) / (22 – 14) = 0.5.
• The portfolio value is 0.5(22) – 4 = 0.5(14) = 7.
• A portfolio of 0.5 shares of stock to 1 short call option yields the same portfolio value whether the stock price at expiration is 22 or 14.

Valuing Call Options with the Binomial Model

• Valuing a call option with a one-period binomial model involves finding a combination of the call option and the underlying asset that will have the same value regardless of the underlying asset's price at expiration.
• A portfolio combining the call option with the underlying asset can have the same value at option expiration, regardless of an up-move or down-move in the asset price.
• The present value of this portfolio is the discounted present value of the certain future payment, used to value the option.
• Option valuation models based on risk neutrality use risk-neutral pseudo-probabilities of an up-move and a down-move, not actual probabilities.

Factors Affecting Option Value

• An option's value is influenced by expected probabilities of underlying price increases or decreases.
• The actual probabilities do not affect the value.

Impact of Risk-Free Rate on Call Option Price

• A decrease in the risk-free rate will decrease the value of a call option.
• Decreasing the risk-free rate increases the risk-neutral probability (π) of a price decrease and decreases the present value of the expected option payoff.
• An increased probability of a downward price move reduces the expected payoff from the call, decreasing the call option value.
• These effects reduce the call option value as the return on risk-free investments decreases.

Call Option Valuation Example

• A stock is priced at $30.
• At the end of three months, options expire, with the stock price expected to rise or fall by 10%.
• The value of a call option with a strike price of $31 is $1.30.

Risk-Neutral Pricing

• Risk-neutral pricing can be applied to any model using future underlying asset price movements.
• Risk-neutral pricing requires expected volatility, not expected return, to price an option.
• Risk-neutral probabilities are determined using the risk-free rate and assumed "up gross returns" and "down gross returns," not investor views on risk.