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Questions and Answers
What fundamental concept, key to the Black-Scholes model, does the Binomial Option Pricing Model utilize?
What fundamental concept, key to the Black-Scholes model, does the Binomial Option Pricing Model utilize?
- Discounting expected payoffs at the risk-free rate.
- Replicating option payoffs by constructing a portfolio of a risk-free bond and the underlying stock. (correct)
- The assumption of constant volatility.
- The use of stochastic calculus to model stock price movements.
The price of an option derived from the Binomial Option Pricing Model is solely dependent on the expected return of the underlying asset.
The price of an option derived from the Binomial Option Pricing Model is solely dependent on the expected return of the underlying asset.
False (B)
In the context of option pricing, what does 'replicating portfolio' refer to?
In the context of option pricing, what does 'replicating portfolio' refer to?
A combination of assets that duplicates the payoff of an option.
In a two-state single-period binomial model, the stock price can either go ______ or go down.
In a two-state single-period binomial model, the stock price can either go ______ or go down.
Considering a European call option with an exercise price of $52 and a current stock price (S0) of $52, which of the following describes the stock price movement in one period according to the example used in the lecture?
Considering a European call option with an exercise price of $52 and a current stock price (S0) of $52, which of the following describes the stock price movement in one period according to the example used in the lecture?
In the binomial option pricing model, to replicate the value of an option you need to combine the stock with a risk-free investment.
In the binomial option pricing model, to replicate the value of an option you need to combine the stock with a risk-free investment.
In the context of replicating an option with a portfolio, if Δ (Delta) represents the number of shares of stock, what does B represent?
In the context of replicating an option with a portfolio, if Δ (Delta) represents the number of shares of stock, what does B represent?
If the value of the portfolio in the 'up' state is $10 to match the CU payoff of the call option, then the equation is A62 + 1.05B = ______.
If the value of the portfolio in the 'up' state is $10 to match the CU payoff of the call option, then the equation is A62 + 1.05B = ______.
In the 'down' state, according to the binomial option pricing model, what is the value of the portfolio equal to?
In the 'down' state, according to the binomial option pricing model, what is the value of the portfolio equal to?
The binomial pricing formula can be used to value a security whose payoff depends on the stock price.
The binomial pricing formula can be used to value a security whose payoff depends on the stock price.
In valuing a put option using a replicating portfolio, if Delta is negative 0.3 (i.e., -0.3), what does this indicate about the portfolio's position in the stock?
In valuing a put option using a replicating portfolio, if Delta is negative 0.3 (i.e., -0.3), what does this indicate about the portfolio's position in the stock?
According to the content, the Black-Scholes formula calculates the price of a ______ option.
According to the content, the Black-Scholes formula calculates the price of a ______ option.
Match the terms found in the Black-Scholes Option Pricing Formula with their corresponding description.
Match the terms found in the Black-Scholes Option Pricing Formula with their corresponding description.
According to the formula presented, what adjustment is made to d1 in the Black-Scholes Option Pricing formula to arrive at d2?
According to the formula presented, what adjustment is made to d1 in the Black-Scholes Option Pricing formula to arrive at d2?
The Black-Scholes formula assumes that the option can only be exercised at expiration.
The Black-Scholes formula assumes that the option can only be exercised at expiration.
In the context of options, what does 'Put-Call Parity' allow you to determine?
In the context of options, what does 'Put-Call Parity' allow you to determine?
According to put-call parity, purchasing a put option on a stock you already own is known as a ______ put.
According to put-call parity, purchasing a put option on a stock you already own is known as a ______ put.
According to put-call parity, what two components create a payoff that is identical to purchasing a stock and a put option?
According to put-call parity, what two components create a payoff that is identical to purchasing a stock and a put option?
The Put-Call Parity relationship applies exclusively to American-style options due to their early exercise feature.
The Put-Call Parity relationship applies exclusively to American-style options due to their early exercise feature.
According to the 'Using Put-Call Parity example', if you find that the cost of creating a portfolio that replicates the payoff of a call option is $1.632, what does this imply for the price at which you can sell the call option to guarantee a profit?
According to the 'Using Put-Call Parity example', if you find that the cost of creating a portfolio that replicates the payoff of a call option is $1.632, what does this imply for the price at which you can sell the call option to guarantee a profit?
Flashcards
Binomial Option Pricing Model
Binomial Option Pricing Model
Options are priced by assuming the stock price has two possible values at the end of the next period.
Key insight of Black Scholes
Key insight of Black Scholes
Option payoffs can be replicated by constructing a portfolio out of a risk-free bond and underlying stock.
Valuation Principle
Valuation Principle
Value of call option today must equal current market value of the replicating portfolio; value of shares less borrowed amount.
Binomial Pricing Formula
Binomial Pricing Formula
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Valuing a put option example
Valuing a put option example
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Black-Scholes Model
Black-Scholes Model
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Put-Call Parity
Put-Call Parity
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Protective Put
Protective Put
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Call/put option
Call/put option
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Study Notes
The Binomial Option Pricing Model
- Options are priced by assuming the stock price has two possible values at the end of the next period.
- A key element from Black Scholes allows option payoffs to be replicated by creating a portfolio of a risk-free bond and the underlying stock.
Two-State Single-Period Model Example
- Considers a European call option expiring in one period with an exercise price of $52.
- The price of the underlying non-dividend-paying stock is $52 today (S0 = $52).
- The one-period risk-free rate is 5%.
- In one period, the stock price can either increase to $62 (SU = $62) or decrease to $42 (SD = $42).
Replicating Option Value
- The goal is to replicate the option's value, by combining the stock with a risk-free investment into a portfolio with the same payoffs as the call option.
- Delta (A) represents the number of shares of stock to buy, and B represents the investment in the risk-free asset.
'Up'State Scenario
- The portfolio value must be $10 to match the CU payoff of the call option, resulting in the equation A62 + 1.05B = 10.
'Down' State Scenario
- The portfolio value must be $0, matching the call's value in the down state, leading to A42 + 1.05B = 0.
Determining Delta (A)
- Subtract one equation from another (A20 = 10), indicating that A = 0.5.
Determining Investment in Risk-Free Asset (B)
- B = -20
Valuation Principle
- The current market value of the replicating portfolio must equal the price of the call option today (C0)
- The replicating portfolio is valued at 0.5 shares less the amount borrowed (B is negative).
Call Option Price Today (C0)
- C0 = 0.5 x 52 – 20 = 6
- The price of the call today is $6.
Binomial Pricing Formula
- The payoff of the replicating portfolio must equal the option's value in both states of the economy.
- ASU + (1+rf)B = CU and ASD + (1+rf)B = CD
Delta (A) and B values
- A = (CU – CD) / (SU – SD)
- B = (CD - ASD) / (1 + rf)
Calculating the Value of C0
- Once the composition of the replicating portfolio is known.
Value of Replicating Portfolio
- C0 = AS0 + B
Put Option Example
- Valuing a put option through replicating portfolio considerations.
- A stock currently trades for $53. In one period, it will be either $63 or $43.
- Find the price of a European put option expiring in one period with an exercise price of $49, given one-period risk-free rate 5%.
Example Calculations
- Delta (A) calculation: A = (PU – PD) / (SU – SD) = (0-6) / (63-43) = -0.3.
- B calculation: B = (PD - ASD) / (1 + rf) = (6-(-0.3)43) / 1.05 = 18.
- The portfolio is short 0.3 shares of stock with $18 invested at the risk free rate
Portfolio Replication Check
- If stock price goes up: -0.3 x 63 +1.05 x 18 = 0 .
- If stock price goes down: -0.3 x 43 + 1.05 x 18 = 6.
Value of Put Option
- Po = AS0 + B = -0.3 x 53 + 18 = 2.1
Black-Scholes Call Option Formula
- Call Price = Stock Price ×N(d₁)- PV(Stock Price) ×N(d₂)
Determining d1 and d2 for Black-Scholes Pricing
- where: d₁ = [In[Stock Price/PV(Strike Price)] / σ√T] + σ√T / 2, and d₂ = d₁ - σ√T
Put-Call Parity
- Portfolio insurance = Protective Put
- Non-dividend-paying stock.
- Stock Price + Put Price = PV(Strike Price) + Call Price
- Call Price = Put Price + Stock Price – PV(Strike Price)
Put-Call Parity Example
- One client wants to purchase a one-year European call option on HAL Computer Systems stock with a strike price of $20.
- HAL pays no dividends and trades at $18/share, and the risk-free interest rate is 6%.
- Another dealer writes a one-year European put option on HAL stock with a strike of $20 for $2.50/share.
Put-Call Parity in determining price
- Call Price = Put Price + Stock Price – PV(Strike Price)
- Call Price = $2.50 + $18- $20 / 1.06 =$1.632
- Portfolio replication
- Buy one-year put option with strike price of $20, buy the stock, sell one-year, risk-free zero-coupon bond with a face value of $20.
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