IS

Corporate finance, L5

1. Options Overview

  • Options are financial derivatives representing contracts that grant the buyer the right, but not the obligation, to purchase (call) or sell (put) an underlying asset before or at a specific date (maturity).

1.1 Key Definitions

  • Call Option: Right to buy an underlying asset.

  • Put Option: Right to sell an underlying asset.

  • European Option: Can be exercised only at maturity.(discussed in this chapter)

  • American Option: Can be exercised any time before expiration.(discussed in the next chapter)

  • Exercise Price (K): The predetermined price at which the underlying asset can be bought or sold.

  • Premium: The cost paid by the buyer to the seller for the options contract.

2. Terminal Payoff of Options

2.1 European Call Terminal Payoff

  • meaning CF at maturity

  • Exercise Condition: If the stock price (ST) at maturity is greater than the exercise price (K)

  • If exercise P=100, me buyer of the call, I have the option to buy it at value 100 at maturity,

  • Now if it 60=price market, in this case, I don’t use my call option bc it would be cheaper directly in the market(opposite if it is 120)

  • Now the bold line, is the profit

  • Formula:

    • Call value at maturity: CT = MAX(ST - K, 0)

    • If ST > K, CT = ST - K(So if P>K, then value of call) ; otherwise, CT = 0.

  • Seller of Call

    • As a seller receives premium, but can potentially loose an infinite amount

2.2 European Put Terminal Payoff

  • Exercise Condition: If the stock price (ST) at maturity is less than the exercise price (K).

  • If K=100, want to sell at the highest value, so will not exercise on the RHS, ?

  • the profit is lower bc need to deduct the premium(buyer of put)

  • Buyer Formula:

    • Put value at maturity: PT = MAX(K - ST, 0)

    • If ST < K, PT = K - ST; otherwise, PT = 0.

3. The Put-Call Parity

3.1 Concept

  • The Put-Call Parity establishes a relationship between the prices of European call and put options with the same exercise price and maturity on the same stock.

    →It serves as a fundamental relationship that traders use to identify mispricings in the market and take advantage of arbitrage opportunities

  • Strategy 1: Buy 1 share of stock and 1 put option:

    • If ST < K: Value = ST + (K - ST) = K

    • If ST > K: Value = ST + 0 = ST

      → By having portfolio of stock, when you buy a put on the specific stock then it is for protection(red line would be profit when protected, comes at a price)

  • Strategy 2: Buy a call option and invest PV(K) (the present value of the strike price):

    • If ST < K: Value = 0 + K = K

    • If ST > K: Value = (ST - K) + K = ST

  • Both strategies yield the same terminal value.

    →(stock + put= Call + exercise price)

  • Current Value Equation

    • S + P = C + PV(K)

    • Where S = current stock price, P = current put value, C = current call value.

    • A call is equivalent to a purchase of stock and a put financed by borrowing the PV(K)


4. Valuing Options with the Binomial Model

4.1 Binomial Model Basics

  • Two state option model: Stock price can either increase (uS) or decrease (dS) over a set period.

  • S= non-dividend paying stock

    • Example: Current stock price (S) = 100. If u = 1.25, then uS = 125; if d = 0.80, then dS = 80.

  • For a call option at K = 100:

    • Call values can be derived as Cu = MAX(125 - 100, 0) = 25 and Cd = MAX(80 - 100, 0) = 0.

      → By increasing the S, we increase value of call

4.2 No Arbitrage Condition

  • In a perfect market, the call option's value must match that of its synthetic counterpart to avoid arbitrage. (Idea is to replicate, the 25 and 0, though this synthetic call)

    • Call value C = δ * S - B

  • Calculate δ and B using pricing equations:

    • Solve for δ using differences in stock prices and corresponding values.

4.3 Risk-Neutral Pricing

  • Assumption:In a risk-neutral world, the expected return equals the risk-free rate.

  • risk is not numerated, bc every asset earns rf

  • To calculate probabilities based on potential stock price movements:

    • Formula: p = 0.5 for equal chances of upward and downward movements.

5. Key Parameters and Pricing Dynamics

5.1 Influencing Factors on Call Option Value

  • Current Stock Price (S) Call: up; Put:doxn

  • Exercise Price (K) Call:down; Put:up

  • Time to Expiry (T) call:up; Put: up

  • Risk-Free Interest Rate (r) call:up; Put: down

  • Volatility of Asset (σ) call: up; Put: up

  • Increased volatility typically raises option values due to heightened potential for returns without linear losses. (volatility good for options, won’t impact losses)

5.2 Black-Scholes Model

  • For European call options(only), represented as:

    • C = S N(d1) - PV(K) N(d2)

    • d1 and d2 calculations depend on stock price, present value of K, volatility, and time to maturity.

  • Black-Scholes provides a closed-form solution limiting conditions of the binomial approach, allowing for performance analytics in non-dividend stocks with constant volatility.

  • Moneyness of an option= will copute the gap btw S and present value K, ie. if the gap today will prbly be also pretty small in the future

6. Cumulative Normal Distribution

  • Utilized for calculating N(d1) and N(d2): Probabilities in standardized normal distribution contexts.

  • Example values can be derived using statistical functions or probability tables.