Library/CFA (Chartered Financial Analyst)/JuiceNotes 2024: Derivatives - Chartered Financial Analyst Level I/Valuing a Derivative Using a One-Period Binomial Model

Valuing a Derivative Using a One-Period Binomial Model

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The Binomial Model Structure5 min

The binomial model simplifies asset price movements into a tree structure where the price starts at $S$ and moves to either $S_u$ (up) or $S_d$ (down) over one period. The factors $u$ and $d$ determine the magnitude of these moves. A common assumption is that $d = 1/u$. The model uses these potential future prices to determine the option's intrinsic value at expiration (exercise value) for both scenarios. For a call option, this is the maximum of zero or the stock price minus the strike price.

Key Points

One-period model assumes two possible outcomes: up or down.
Uptick factor ($u$) and downtick factor ($d$) define future prices.
Downtick factor is often the reciprocal of the uptick factor ($d = 1/u$).
Option payoffs are calculated at the terminal nodes ($S_u$ and $S_d$).

Risk-Neutral Valuation and Hedge Ratio6 min

To value the option today, the model calculates the expected payoff using risk-neutral probabilities, not actual market probabilities. The risk-neutral probability of an up move ($π$) ensures the asset earns the risk-free rate on average. The formula is $π = (1 + RFR - d) / (u - d)$. The expected payoff is then discounted at the risk-free rate. The hedge ratio, calculated as $(C_u - C_d) / (S_u - S_d)$, indicates the sensitivity of the option's price to the stock price and is used to construct a risk-free portfolio.

Key Points

Risk-neutral probability ($π$) formula: $(1 + RFR - d) / (u - d)$.
Option Value = [$π imes C_u + (1-π) imes C_d$] / $(1 + RFR)$.
Hedge Ratio (Delta) = Change in Option Value / Change in Stock Value.
Hedge Ratio allows for replication and hedging of the option.

American vs. European Options and Investor Types5 min

The text discusses the equivalence of American and European option prices when early exercise is optimal. For non-dividend-paying stocks, American calls are valued identically to European calls. Early exercise is relevant for American calls with dividends and deep in-the-money American puts. The valuation framework assumes a 'risk-neutral' world where investors require no risk premium, allowing the use of the risk-free rate for discounting. This contrasts with risk-averse investors who dominate the real world but whose preferences are mathematically bypassed in this pricing model.

Key Points

American Call = European Call if no dividends.
Early exercise is valuable for American Calls with dividends.
Early exercise is valuable for Deep ITM American Puts.
Risk-neutral pricing yields the no-arbitrage price.
Risk-averse investors prefer certainty; risk-neutral investors focus on expected value.

Questions

Question 1

What is the primary assumption regarding asset price movements in a one-period binomial model?

Valuing a Derivative Using a One-Period Binomial Model

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