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Pricing and Valuation of Options

Derivatives

Factors Affecting Option Value

Learning Outcome Statement:

identify the factors that determine the value of an option and describe how each factor affects the value of an option

Summary:

The value of an option is influenced by several factors including the value of the underlying asset, the exercise price, time to expiration, the risk-free interest rate, the volatility of the underlying, and any income or costs related to owning the underlying asset. Each of these factors affects the option's value in specific ways, either increasing or decreasing it based on the type of option (call or put) and the nature of the change in the factor.

Key Concepts:

Value of the Underlying

Changes in the value of the underlying asset affect the value of options. For call options, an increase in the underlying's value increases the option's value, while for put options, it decreases.

Exercise Price

The exercise price sets a threshold that determines the profitability of exercising an option. Lower exercise prices increase the value of call options and decrease the value of put options.

Time to Expiration

Longer time to expiration generally increases the value of options by providing more time for the underlying asset's price to move favorably. However, for deep-in-the-money put options, a longer time can reduce the option's value due to the lower present value of the exercise payoff.

Risk-Free Interest Rate

Higher risk-free rates decrease the present value of the exercise price, thus increasing the value of call options and decreasing the value of put options.

Volatility of the Underlying

Higher volatility increases the potential for the underlying's price to reach favorable levels, thus increasing the value of both call and put options.

Formulas:

Call Option Exercise Value

Max(0,(St−PV(X)))\text{Max}(0, (S_t - \text{PV}(X)))

Calculates the exercise value of a call option, which is the maximum of zero or the difference between the spot price and the present value of the exercise price.

Variables:
StS_t:
Spot price of the underlying at time t
PV(X)PV(X):
Present value of the exercise price
Units: currency

Put Option Exercise Value

Max(0,(PV(X)−St))\text{Max}(0, (\text{PV}(X) - S_t))

Calculates the exercise value of a put option, which is the maximum of zero or the difference between the present value of the exercise price and the spot price.

Variables:
StS_t:
Spot price of the underlying at time t
PV(X)PV(X):
Present value of the exercise price
Units: currency

Option Time Value

Learning Outcome Statement:

explain the exercise value, moneyness, and time value of an option

Summary:

The learning outcome focuses on understanding the components of an option's price, specifically the exercise value, moneyness, and time value. The exercise value is the intrinsic payoff of an option, moneyness describes whether an option is in, at, or out of the money, and the time value is the additional value from the possibility of favorable price movements until expiration. The time value is always positive and decreases to zero at maturity.

Key Concepts:

Exercise Value

The exercise value of an option is its intrinsic value, calculated as the difference between the spot price of the underlying asset and the strike price of the option, adjusted for the time value of money.

Moneyness

Moneyness describes the position of the spot price relative to the strike price. An option is 'in the money' if exercising it results in a positive payoff, 'at the money' if the spot price equals the strike price, and 'out of the money' if exercising it does not result in a payoff.

Time Value

The time value of an option is the part of the option's price that exceeds its exercise value. It represents the potential for further beneficial movement in the price of the underlying asset. Time value decreases as the option approaches maturity.

Formulas:

Call Option Time Value

ct=max⁡(0,St−X(1+r)−(T−t))+Time Valuec_t = \max(0, S_t - X(1 + r)^{-(T-t)}) + \text{Time Value}

This formula calculates the time value of a call option as the difference between the current option price and its intrinsic value.

Variables:
ctc_t:
price of the call option at time t
StS_t:
spot price of the underlying asset at time t
XX:
exercise price of the option
rr:
risk-free interest rate
TT:
time of option maturity
tt:
current time
Units: currency

Put Option Time Value

pt=max⁡(0,X(1+r)−(T−t)−St)+Time Valuep_t = \max(0, X(1 + r)^{-(T-t)} - S_t) + \text{Time Value}

This formula calculates the time value of a put option as the difference between the current option price and its intrinsic value.

Variables:
ptp_t:
price of the put option at time t
StS_t:
spot price of the underlying asset at time t
XX:
exercise price of the option
rr:
risk-free interest rate
TT:
time of option maturity
tt:
current time
Units: currency

Replication

Learning Outcome Statement:

contrast the use of arbitrage and replication concepts in pricing forward commitments and contingent claims

Summary:

The learning outcome statement focuses on contrasting the use of arbitrage and replication in pricing mechanisms for forward commitments and contingent claims. It explores how replication strategies ensure the law of one price and prevent arbitrage opportunities, and how these strategies differ between options and forwards due to the non-linear payoff profiles of options.

Key Concepts:

Replication Strategy

Replication involves creating a position in an underlying asset combined with borrowing or lending to mimic the cash flows of another financial instrument, ensuring no arbitrage opportunities.

Law of One Price

This economic rule states that in an efficient market, all identical goods must have only one price to prevent arbitrage opportunities.

Non-linear Payoff Profile

Options have payoffs that are not directly proportional to the movements in the underlying asset's price, requiring adjustments in the replication strategy as the likelihood of exercise changes.

Arbitrage

Arbitrage involves taking advantage of price differences between markets or instruments to make risk-free profits, which replication strategies aim to eliminate.

Formulas:

Put Option Lower Bound

pt,Lower bound=max⁡(0,X(1+r)−(T−t)−St)p_{t, \text{Lower bound}} = \max(0, X(1 + r)^{-(T-t)} - S_t)

This formula calculates the minimum value of a put option at any given time before expiration, considering the time value of money and the current price of the underlying asset.

Variables:
pt,Lower boundp_{t, \text{Lower bound}}:
Lower bound of the put option's price at time t
XX:
Exercise price
rr:
Risk-free rate
TT:
Time at maturity
tt:
Current time
StS_t:
Spot price of the underlying asset at time t
Units: currency

Call Option Lower Bound

ct,Lower bound=max⁡(0,St−X(1+r)−(T−t))c_{t, \text{Lower bound}} = \max(0, S_t - X(1 + r)^{-(T-t)})

This formula determines the minimum value of a call option, factoring in the present value of the exercise price and the current market price of the underlying asset.

Variables:
ct,Lower boundc_{t, \text{Lower bound}}:
Lower bound of the call option's price at time t
StS_t:
Spot price of the underlying asset at time t
XX:
Exercise price
rr:
Risk-free rate
TT:
Time at maturity
tt:
Current time
Units: currency

Option Moneyness

Learning Outcome Statement:

Explain the exercise value, moneyness, and time value of an option

Summary:

The learning outcome focuses on understanding the different components of an option's value: exercise value, moneyness, and time value. Exercise value is the immediate payoff of an option, moneyness describes whether an option is in, at, or out of the money based on the relationship between the spot price and the exercise price, and time value is the additional value derived from the time remaining until the option's maturity.

Key Concepts:

Exercise Value

The exercise value of an option is its intrinsic value, calculated as the difference between the spot price and the exercise price, depending on whether it's a call or put option. It represents the profit that could be realized if the option were exercised immediately.

Moneyness

Moneyness indicates the position of the spot price relative to the exercise price. An option is 'in the money' (ITM) if exercising it would result in a positive payoff, 'at the money' (ATM) if the spot price equals the exercise price, and 'out of the money' (OTM) if exercising it would not be profitable.

Time Value

The time value of an option is the part of the option's price that exceeds its intrinsic value. This component reflects the potential for the option to become profitable (or more profitable) before expiration due to changes in the underlying asset's price. It is influenced by the time remaining until expiration and the volatility of the underlying asset.

Formulas:

Exercise Value of a Put Option

PV(X)−St=EUR 1,000×(1.01)−0.5−EUR 950PV(X) - S_t = \text{EUR } 1,000 \times (1.01)^{-0.5} - \text{EUR } 950

This formula calculates the exercise value of a put option by discounting the exercise price back to its present value and subtracting the current spot price.

Variables:
PV(X)PV(X):
Present value of the exercise price
StS_t:
Spot price at time t
rr:
Risk-free rate
T−tT-t:
Time until expiration
Units: EUR

Option Exercise Value

Learning Outcome Statement:

Explain the exercise value, moneyness, and time value of an option

Summary:

The lesson focuses on understanding the exercise value, moneyness, and time value of European options, which are critical measures used to evaluate an option's potential payoff and its likelihood of being exercised. The exercise value is determined by the difference between the spot price and the exercise price, adjusted for the time value of money. Moneyness describes whether an option is in-the-money, at-the-money, or out-of-the-money, influencing the option's price sensitivity to changes in the underlying asset's price.

Key Concepts:

Option Exercise Value

The exercise value of an option is the value it would have if it were exercised at a specific time before maturity. For call options, it is the maximum of zero or the difference between the spot price and the present value of the exercise price. For put options, it is the maximum of zero or the difference between the present value of the exercise price and the spot price.

Moneyness

Moneyness describes the relationship between the spot price of the underlying asset and the exercise price of the option. It indicates whether an option is in-the-money (ITM), at-the-money (ATM), or out-of-the-money (OTM), affecting the likelihood of the option being exercised and its sensitivity to price changes of the underlying asset.

Time Value of Money

In the context of options, the time value of money is considered when calculating the present value of the exercise price. This adjustment reflects the value of money at different points in time due to factors like interest rates.

Formulas:

Call Option Exercise Value

Max(0,St−X(1+r)−(T−t))\text{Max}(0, S_t - X(1 + r)^{-(T-t)})

Calculates the exercise value of a call option at any time before maturity, considering the time value of money.

Variables:
StS_t:
Spot price of the underlying asset at time t
XX:
Exercise price of the option
rr:
Risk-free interest rate
TT:
Time of option maturity
tt:
Current time
Units: Currency

Put Option Exercise Value

Max(0,X(1+r)−(T−t)−St)\text{Max}(0, X(1 + r)^{-(T-t)} - S_t)

Calculates the exercise value of a put option at any time before maturity, considering the time value of money.

Variables:
StS_t:
Spot price of the underlying asset at time t
XX:
Exercise price of the option
rr:
Risk-free interest rate
TT:
Time of option maturity
tt:
Current time
Units: Currency

Arbitrage

Learning Outcome Statement:

contrast the use of arbitrage and replication concepts in pricing forward commitments and contingent claims

Summary:

This LOS explores the concepts of arbitrage and replication in the context of pricing forward commitments and contingent claims. Arbitrage involves exploiting price discrepancies to make risk-free profits, while replication involves creating a portfolio that mimics the cash flows of another asset. Both concepts are crucial in ensuring that the law of one price holds, preventing riskless arbitrage opportunities and helping to determine the no-arbitrage price bounds for options.

Key Concepts:

Arbitrage

Arbitrage refers to the practice of taking advantage of a price difference between two or more markets, striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices.

Replication

Replication involves constructing a portfolio that replicates the cash flows of another asset. This is often used in derivative pricing, where a derivative's payoff is replicated using a combination of other financial instruments.

No-arbitrage condition

This condition asserts that if no riskless profit is to be made, the price of a derivative must be such that it does not allow for arbitrage opportunities. This is used to establish price bounds for derivatives.

Forward commitments

Forward commitments, such as forward contracts, involve agreeing to buy or sell an asset at a future date for a price agreed upon today. They have symmetric payoff profiles.

Contingent claims

Contingent claims, such as options, have asymmetric payoff profiles and their payoff depends on the underlying asset's price at maturity relative to the exercise price.

Formulas:

Call Option Lower Bound

ct,Lower bound=max⁡(0,St−X(1+r)−(T−t))c_{t,\text{Lower bound}} = \max(0, S_t - X(1 + r)^{-(T-t)})

This formula calculates the minimum price of a call option at any given time before maturity, ensuring no arbitrage opportunities.

Variables:
ct,Lower boundc_{t,\text{Lower bound}}:
Lower bound of the call option price at time t
StS_t:
Spot price of the underlying asset at time t
XX:
Exercise price of the option
rr:
Risk-free interest rate
TT:
Time of option maturity
tt:
Current time
Units: Currency

Call Option Upper Bound

ct,Upper bound=Stc_{t,\text{Upper bound}} = S_t

This formula states that the call option price should not exceed the current spot price of the underlying asset, as paying more would not be rational.

Variables:
ct,Upper boundc_{t,\text{Upper bound}}:
Upper bound of the call option price at time t
StS_t:
Spot price of the underlying asset at time t
Units: Currency

Put Option Lower Bound

pt,Lower bound=max⁡(0,X(1+r)−(T−t)−St)p_{t,\text{Lower bound}} = \max(0, X(1 + r)^{-(T-t)} - S_t)

This formula calculates the minimum price of a put option at any given time before maturity, ensuring no arbitrage opportunities.

Variables:
pt,Lower boundp_{t,\text{Lower bound}}:
Lower bound of the put option price at time t
XX:
Exercise price of the option
StS_t:
Spot price of the underlying asset at time t
rr:
Risk-free interest rate
TT:
Time of option maturity
tt:
Current time
Units: Currency

Put Option Upper Bound

pt,Upper bound=Xp_{t,\text{Upper bound}} = X

This formula states that the put option price should not exceed the exercise price, as paying more would not be rational.

Variables:
pt,Upper boundp_{t,\text{Upper bound}}:
Upper bound of the put option price at time t
XX:
Exercise price of the option
Units: Currency
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