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Valuing a Derivative Using a One-Period Binomial Model

Derivatives

Pricing a European Call Option

Learning Outcome Statement:

explain how to value a derivative using a one-period binomial model

Summary:

The learning outcome statement focuses on explaining the valuation of a European call option using a one-period binomial model. This model involves determining the possible future prices of the underlying asset and calculating the option's value based on these potential outcomes. The process includes constructing a risk-free portfolio by replicating the option's payoff and ensuring no-arbitrage conditions are met.

Key Concepts:

Binomial Model

A binomial model for option pricing uses a discrete time framework to model the possible future values of the underlying asset. It assumes the asset price can move up or down by specific factors, Ru and Rd, over a single period.

Call Option Payoff

The payoff of a European call option at expiration is the maximum of zero or the difference between the underlying asset's price and the strike price (X). This is calculated for both up and down movements in the asset price.

Risk-free Portfolio

A risk-free portfolio is constructed by combining positions in the underlying asset and the option to eliminate risk. The portfolio's value should be the same in all future scenarios under the no-arbitrage condition, leading to the determination of the hedge ratio (h).

No-arbitrage Pricing

No-arbitrage pricing ensures that the option's price is set such that no arbitrage opportunities exist, meaning the expected return on the option equals the risk-free rate. This is achieved by discounting the expected payoff of the option at the risk-free rate.

Hedge Ratio

The hedge ratio (h) is the proportion of the underlying asset needed to hedge against the risk of the option. It is calculated based on the difference in option payoffs across the up and down states divided by the difference in the underlying asset prices in these states.

Formulas:

Initial Portfolio Value

V0=hS0−c0V_0 = hS_0 - c_0

Represents the initial investment required to set up the risk-free portfolio.

Variables:
V0V_0:
Initial value of the portfolio
hh:
Hedge ratio
S0S_0:
Initial stock price
c0c_0:
Initial call option price
Units: currency units

Portfolio Value in Up State

V1u=hS1u−c1uV_{1u} = hS_{1u} - c_{1u}

Calculates the portfolio value if the underlying asset price increases.

Variables:
V1uV_{1u}:
Portfolio value if the price moves up
hh:
Hedge ratio
S1uS_{1u}:
Stock price in up state
c1uc_{1u}:
Call option value in up state
Units: currency units

Portfolio Value in Down State

V1d=hS1d−c1dV_{1d} = hS_{1d} - c_{1d}

Calculates the portfolio value if the underlying asset price decreases.

Variables:
V1dV_{1d}:
Portfolio value if the price moves down
hh:
Hedge ratio
S1dS_{1d}:
Stock price in down state
c1dc_{1d}:
Call option value in down state
Units: currency units

Hedge Ratio

h∗=c1u−c1dS1u−S1dh^* = \frac{c_{1u} - c_{1d}}{S_{1u} - S_{1d}}

Determines the proportion of the underlying asset needed to hedge the option position.

Variables:
h∗h^*:
Optimal hedge ratio
c1uc_{1u}:
Call option value in up state
c1dc_{1d}:
Call option value in down state
S1uS_{1u}:
Stock price in up state
S1dS_{1d}:
Stock price in down state
Units: unitless

Call Option Price

c0=h×S0−V11+rc_0 = h \times S_0 - \frac{V_1}{1 + r}

Calculates the initial price of the call option based on the hedged portfolio and the risk-free rate.

Variables:
c0c_0:
Initial call option price
hh:
Hedge ratio
S0S_0:
Initial stock price
V1V_1:
Value of the portfolio at time 1
rr:
Risk-free rate
Units: currency units

Risk Neutrality

Learning Outcome Statement:

describe the concept of risk neutrality in derivatives pricing

Summary:

Risk neutrality in derivatives pricing refers to the valuation of derivatives such as options using probabilities that are adjusted to reflect a world where investors are indifferent to risk. This method uses the risk-free rate to discount expected future payoffs from the derivative, rather than actual probabilities and risk preferences of investors. The concept simplifies the pricing of derivatives by focusing only on the expected volatility and the risk-free rate, allowing for consistent pricing across different market conditions.

Key Concepts:

Risk-neutral probabilities

Risk-neutral probabilities are used in the binomial model to calculate the expected payoff of derivatives. These probabilities are adjusted so that the expected return on the underlying asset equals the risk-free rate, reflecting a scenario where all investors are risk-neutral.

No-arbitrage condition

The no-arbitrage condition ensures that the derivative pricing model does not allow for arbitrage opportunities, meaning that the price of the derivative must be set such that no risk-free profits can be made through simple trading strategies.

Put-call parity

Put-call parity is a financial principle that defines a price relationship between a call option, a put option, and the underlying stock. Under risk-neutral pricing, both call and put options must satisfy this relationship, ensuring that their prices are consistent with one another given the same strike price and expiration.

Formulas:

Risk-neutral probability formula

π=1+r−RdRu−Rd\pi = \frac{1 + r - R_d}{R_u - R_d}

This formula calculates the risk-neutral probability of an upward price move in the underlying asset, which is used to price derivatives in a risk-neutral world.

Variables:
π\pi:
risk-neutral probability of an upward move in the underlying asset price
rr:
risk-free rate
RuR_u:
up gross return on the underlying asset
RdR_d:
down gross return on the underlying asset
Units: dimensionless

Option pricing formula

c0=(πc1u+(1−π)c1d)(1+r)Tc_0 = \frac{(\pi c_1u + (1 - \pi) c_1d)}{(1 + r)^T}

This formula calculates the current price of a call option based on the expected payoff at expiration, discounted at the risk-free rate, using risk-neutral probabilities.

Variables:
c0c_0:
current price of the call option
c1uc_1u:
value of the call option if the underlying asset price goes up
c1dc_1d:
value of the call option if the underlying asset price goes down
π\pi:
risk-neutral probability of an upward move
rr:
risk-free rate
TT:
time to expiration of the option in years
Units: currency (e.g., USD, EUR)

Binomial Valuation

Learning Outcome Statement:

explain how to value a derivative using a one-period binomial model

Summary:

The one-period binomial model is used to value derivatives by assuming the underlying asset's price can either increase or decrease in the next period. This model helps in determining the no-arbitrage value of options by considering different scenarios of price movements and their probabilities. The impact of changes in probabilities and the spread between upward and downward price movements on the put option price are analyzed.

Key Concepts:

Impact of Probability of Upward Movement (q)

An increase in the probability of an upward price movement (q) does not affect the value of a put option in the one-period binomial model.

Impact of Spread between Up and Down Factors (Ru - Rd)

Increasing the spread between the up and down factors (Ru - Rd) increases the range of potential prices, enhancing the likelihood of the option being in the money, thereby increasing the put option price.

Impact of Risk-Neutral Probability (π)

An increase in the risk-neutral probability (π) of a price increase reduces the likelihood of the put option ending up in the money, thus decreasing its price.

Risk-Neutral Probability Calculation

The risk-neutral probability (π) is calculated based on the risk-free rate and the up and down factors, providing a measure of the expected price movement under a risk-neutral world.

Formulas:

Risk-Neutral Probability (π)

π=1+r−RdRu−Rd\pi = \frac{1 + r - R_d}{R_u - R_d}

This formula calculates the probability of an upward price movement in a risk-neutral world, considering the risk-free rate and the potential returns from upward and downward movements.

Variables:
ππ:
Risk-neutral probability of a price increase
rr:
Risk-free rate of return
RuR_u:
Gross return from an upward price movement
RdR_d:
Gross return from a downward price movement
Units: dimensionless (probability)

The Binomial Model

Learning Outcome Statement:

explain how to value a derivative using a one-period binomial model

Summary:

The binomial model is a fundamental tool used in financial derivatives valuation, particularly for options. It simplifies the complex payoff profiles of options by assuming that the underlying asset price can only move up or down by specific factors within one period. This model uses a no-arbitrage framework and does not require the knowledge of the exact probability of upward or downward movement but focuses on the magnitude of these movements to capture the asset's volatility.

Key Concepts:

Binomial Model Basics

The binomial model posits that in a given time period, the price of an asset will either increase to a higher price (S1u) or decrease to a lower price (S1d). These movements are akin to the outcomes of a Bernoulli trial, where the asset price moves up with a probability 'q' and down with a probability '1-q'.

Gross Return Calculation

The gross return from an upward or downward price movement is calculated as Ru = S1u/S0 and Rd = S1d/S0, respectively. These returns help in determining the potential future values of the asset based on its initial price S0.

Valuing European Call Option

In the context of a European call option, the binomial model helps determine the option's value at expiration by considering the possible up and down scenarios of the underlying asset's price relative to the strike price. The value of the option is the maximum of zero or the difference between the underlying asset's price and the strike price in each scenario.

Formulas:

Gross Return Up

Ru=S1uS0R_u = \frac{S1u}{S0}

Calculates the gross return ratio for an upward price movement of the asset.

Variables:
RuR_u:
Gross return from an upward price movement
S1uS1u:
Price after an upward movement
S0S0:
Initial price of the underlying asset
Units: dimensionless

Gross Return Down

Rd=S1dS0R_d = \frac{S1d}{S0}

Calculates the gross return ratio for a downward price movement of the asset.

Variables:
RdR_d:
Gross return from a downward price movement
S1dS1d:
Price after a downward movement
S0S0:
Initial price of the underlying asset
Units: dimensionless

Option Value at Expiration

c1u=max⁡(0,S1u−X)andc1d=max⁡(0,S1d−X)c_{1u} = \max(0, S1u - X) \quad \text{and} \quad c_{1d} = \max(0, S1d - X)

Calculates the value of a European call option at expiration under both upward and downward scenarios.

Variables:
c1uc_{1u}:
Value of the call option if the price goes up
c1dc_{1d}:
Value of the call option if the price goes down
S1uS1u:
Price after an upward movement
S1dS1d:
Price after a downward movement
XX:
Exercise price of the option
Units: currency
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