OM-W5

Binomial Trees

Lei Yu

Simple Binomial Model

• Current Stock Price (S_t): $20

• Binomial Model Assumption: In 3 months, stock price could be either:

  • Up: $22

  • Down: $18

• Dividends: None considered at this stage.

• Continuously Compounded Interest Rate (r): 12% for 3-month maturity (noted as an unusual number from the book).

• Observations:

  • The actual probabilities of reaching either node (up or down) are irrelevant for option pricing due to the assumptions made.

  • It’s vital for derivative pricing to understand lists of possible scenarios without needing actual probabilities.

3-Month Call Option

• Consideration of a Call Option:

  • Strike Price (K): $21

  • Backward Induction: Compute option payoff based on terminal stock price (S_T):

  • Payoff Formula:
    $(S_T - K)^+$

  • Zero-Coupon Bond Price with $1 par value:
    $B_t = 1 imes e^{-0.12 imes 0.25} = 0.9704$

  • Options at Terminal Outcomes:

  • For S_T = 22:

    • Bond Price (B_T): $1

    • Call Option Payoff (c_T): $1

  • For S_T = 18:

    • Bond Price (B_T): $1

    • Call Option Payoff (c_T): $0

  • Connecting Points:

  • Replicating: Ability to replicate the payoff using stock and bond, which must equal the call option value.

  • Hedging: Assessing risk and computing present value using the risk-free rate.

Replicating Strategy

• From Assumed Values:

  • Bond Price: $0.9704

  • Current Stock Price: $20

  • Unknown Call Option Price (c_t): ?

• Replication Condition:

  • $1 = riangle (22) + D(1)$

  • $0 = riangle (18) + D(1)$

• Solving for ∆ (Delta):

  • $riangle = rac{(1 - 0)}{(22 - 18)} = rac{1}{4}$

  • Definition: ∆ represents the sensitivity, change in Call Price (C) over change in Stock Price (S).

• Debt (D):

  • $D = - rac{1}{4} imes 18 = -4.5$

• Final Option Value:

  • $Option = rac{1}{4} imes 20 - 4.5 imes 0.9704 = 0.633$

Hedging Strategy

• Using the Same Values:

  • Bond Price: $0.9704

  • Current Stock Price: $20

  • Determine the Hedged Portfolio:

  • Need to short  share of stock to hedge risk.

• Hedged Portfolio Condition:

  • $1 - riangle(22) = 0 - riangle(18)$

• Solving for ∆ (Delta):

  • The value remains consistent with the replication method; find Delta as:

  • $riangle = rac{(1 - 0)}{(22 - 18)} = rac{1}{4}$

• Present Value of Hedged Portfolio Calculation:

  • $PV = - riangle(22) + riangle(20)$ results in:

  • $ct = -4.3668 + rac{1}{4}(20) = 0.633$

Underlying Principles of Replication and Hedging

• If replication is possible, hedging can also be achieved:

  • Long Position: In an option contract while shorting the replicating portfolio consisting of stock and bonds.

• Key Insight:

  • Bonds generate stable returns but do not hedge risks; stock assumes the primary hedging role.

• Hedge Ratio (Delta):

  • Defined as the ratio of changes in option payoff to changes in stock payoff.

• Delta Hedging:

  • Hedging derivative risks with the underlying security (e.g., stock, currency).

Limitations of Delta Hedging

• Risk Eradication through Delta Hedging:

  • Achieves risk elimination under the assumption of only two possible stock values.

  • Stocks and bonds can replicate the payoff of single options.

  • Stock and options can replicate bond payoffs.

• If Stock Could Take on More Values:

  • Introduction of a third stock price option would make perfect hedging or replication unattainable.

Pricing Multiple Options Using the Same Tree

• Initial Values:

  • Bond Price: $0.9704

  • Current Stock Price: $20

  • Call Option Pricing Example:

  • Call with strike price of $21 is valued at $0.633.

• Further Options Priced:

  • Call options at strikes of $20, 22, 23, 24, 25, 26,

  • Put options at strikes of $21, 18, 17, 16…

• Note on One-Step Binomial Tree:

  • Pointing out the limitations and exploring how to overcome them through altered methodologies.

Risk-Neutral Valuation Approach

• Setting Up a Risk-Neutral Framework:

  • Assigning Artificial Probabilities (p and 1-p) to stock price movements to value options:

  • Formula Relation:
    $20 = 0.9704(22p + 18(1 - p))$

  • Solve for p:
    $p = rac{20}{0.9704} - rac{18}{22 - 18} = 0.6523$

• Implication:

  • Using risk-neutral probabilities allows for discounting of risky payoffs via the risk-free rate. Creates an artificial market scenario where participants are risk-neutral.

Pricing Options with Risk-Neutral Valuation

• Application of Risk-Neutral Probabilities:

  • Call Option at strike K = 21:

  • $c_t = 0.9704(1 imes 0.6523 + 0 imes (1 - 0.6523)) = 0.633$

  • Put Option at strike K = 21:

  • $p_t = 0.9704(0 imes 0.6523 + 3 imes (1 - 0.6523)) = 1.012$

  • Call Option at strike K = 20:

  • $c_t = 0.9704(2 imes 0.6523 + 0 imes (1 - 0.6523)) = 1.266$

  • Put Option at strike K = 20:

  • $p_t = 0.9704(0 imes 0.6523 + 2 imes (1 - 0.6523)) = 0.6748$

• Risk-Neutral Value Understanding:

  • Useful in estimating unit prices of state-dependent claims without engaging forecasting.

More on Risk-Neutral Probabilities

• Defining Actual vs. Risk-Neutral Probabilities:

  • Actual Work: Reflected through physical, statistical probabilities based on observable data (obtained historically).

  • Risk-Neutral: Mixture formed through observed prices of existent securities and subjective expectations.

• Estimating Procedures:

  • Actual probabilities based on historical data; risk-neutral derived from observed prices with discounting considerations.

• Conceptual Distinction:

  • Actual and risk-neutral probabilities provide insights but do not need to align. Usually aren't the same.

Multiple-Step Binomial Trees

• Static Example:

  • Starting stock price S_t = 20, Parameters set with proportional increases/decreases defined by (u, d).

• 6-Month Call Option Pricing Case:

  • Backward induction from expiry:

  • Payoffs at nodes (A, B, C) provided for calculations (3.2, 0, 0 respectively).

  • Using option values at nodes prior enables application of methods from single-step strategies for final pricing decisions.

Valuing 6-Month Call at Node D

• Price Calculation:

  • Current Stock Position at D:

  • $riangle D = rac{(3.2-0)}{(24.2-19.8)} = 0.7273$

  • Value Determination:

  • Portfolio Calculation: $P_D = 0.9704(1 imes 0 - 0.7273 imes 19.8) = -13.9738$

  • Option Value at D:

  • $c_D = -13.9738 + 0.7273 imes 22 = 2.026$

• Replication Method Equivalency:

  • Verification through replication yields similar results leading to robust conclusions.

Summary of Valuing 6-Month Call (K = 21)

• Continued Iteration through Node Calculations:

  • Process consistent through additional nodes based on existing calculated values.

  • All nodes should reflect parameter evaluation changes accordingly.

Complete Stock Price and Option Trees

• Final Stock Price Tree State:

  • Probability Movements through components lead towards defining risk-neutral evaluations consistently.

• Call Option Delta Values:

  • Each node provides the needed delta for effective hedging against risks depicted previously.

Pricing Other Options on Tree

• Application: Price a 6-month put with K = 21 based on earlier frameworks.

• Insistent Importance of Calculating Deltas:

  • Remains critical despite non-usage for immediate valuations.

Evaluating American Put Options

• Procedure: At each node, evaluate exercise value versus continuation value:

  • Reported Values: (p_t, ∆, exercise value, action needed)

  • For each case, make the decision in real-time to exercise if favorable.

General Setup and Notation for Binomial Trees

• Symbols Utilized:

  • S_t: Stock price at time t.

  • f_t: Price of derivative at time t.

  • u: Proportional increase in price at the up node.

  • d: Proportional decrease in price at the down node.

  • (fu, fd): Derivative values at respective nodes 'up' and 'down'.

Pricing Under the General Setup

• Calculation Strategy:

  • Trading off hedging metrics:

  • $riangle = rac{f<em>u - f</em>d}{S_t(u - d)}$

  • Determining Hedged Portfolio Price:

  • $P = e^{-r riangle t}(f<em>u - riangle S</em>t)$

• Final Derivative Value:

  • Usage of expected payoffs combined with discounting strategies yields final resultant values bilaterally approaching each node.

Calibrating the Binomial Tree

• Parameter Framework:

  • Using realized volatilities alongside u and d to inform stock returns:

  • $u = e^{ rac{ ext{volatility}}{ ext{√∆t}}}, ext{ } d = rac{1}{u}$

• Risk-Neutral Probability Setting:

  • Reflected through structures designed to accommodate dividends and leading into dynamic trading areas.

Estimating Return Volatility

• Definitive Approach:

  • Evaluate return volatility utilizing market data rationally derived and successful predictions towards their effective match:

  • ext{Volatility } ( ext{σ}) = rac{1}{∆t} imes rac{1}{N-1} imes igg( ext{Sum } (R_t - ar{R})^2igg)

  • Notably emphasizing implied volatility from matching options' data concerning theory when executed correctly.