Asset pricing 1

Asset pricing: a comprehensive summary

Based on Lectures by Davide Petturiti

March 2025

Contents

1 Introduction to Derivative Contracts 3

1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 The Basic Structure of Derivatives . . . . . . . . . . . . . . . . . . . . . 3

1.3 Types of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 By Timing Structure . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Forward Contracts 4

2.1 Definition and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Long and Short Positions . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Mathematical Representation . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Market Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.5 Payoff of a Forward Contract . . . . . . . . . . . . . . . . . . . . . . . 6

2.6 Determining the Forward Price . . . . . . . . . . . . . . . . . . . . . . 6

2.6.1 Case without Dividends . . . . . . . . . . . . . . . . . . . . . . 6

2.6.2 Case with Dividends . . . . . . . . . . . . . . . . . . . . . . . . 7

2.7 Numerical Example: Forward on Oil . . . . . . . . . . . . . . . . . . . 7

2.8 Numerical Example: Forward on a Stock with Dividends . . . . . 8

3 European Options 8

3.1 Definition and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Long and Short Positions . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Mathematical Representation . . . . . . . . . . . . . . . . . . . . . . . 9

3.4 Payoff of European Call Options . . . . . . . . . . . . . . . . . . . . . 9

3.5 Payoff of European Put Options . . . . . . . . . . . . . . . . . . . . . 9

3.6 Moneyness of Options . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.7 Intrinsic Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.8 Put-Call Parity Relation . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.8.1 Case without Dividends . . . . . . . . . . . . . . . . . . . . . . 10

3.8.2 Case with Dividends . . . . . . . . . . . . . . . . . . . . . . . . 10

3.9 No-arbitrage Bounds for Option Prices . . . . . . . . . . . . . . . . . 10

3.9.1 For European Call Options . . . . . . . . . . . . . . . . . . . . 11

3.9.2 For European Put Options . . . . . . . . . . . . . . . . . . . . 11

4 American Options 11

4.1 Definition and Structure . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2 Pricing Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3 Non-optimality of Early Exercise for American Calls without Dividends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.4 No-arbitrage Bounds for American Options . . . . . . . . . . . . . . 12

4.4.1 For American Call Options . . . . . . . . . . . . . . . . . . . . 12

4.4.2 For American Put Options . . . . . . . . . . . . . . . . . . . . 12

5 Embedded Options 13

5.1 Corporate Zero-Coupon Bonds (ZCBs) . . . . . . . . . . . . . . . . . 13

5.2 Numerical Example: Corporate ZCB . . . . . . . . . . . . . . . . . . 13

5.3 Guaranteed Investment Fund . . . . . . . . . . . . . . . . . . . . . . 14

5.4 Put Decomposition of Guaranteed Investment Fund . . . . . . . . 14

5.5 Call Decomposition of Guaranteed Investment Fund . . . . . . . . 15

5.6 Numerical Example: Guaranteed Investment Fund . . . . . . . . . 15

6 Probability Concepts for Asset Pricing 16

6.1 Events and Probability Spaces . . . . . . . . . . . . . . . . . . . . . 16

6.2 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.3 Stochastic Independence . . . . . . . . . . . . . . . . . . . . . . . . 17

6.4 Numerical Example: Conditional Probability . . . . . . . . . . . . . 17

7 Random Variables and Stochastic Processes 17

7.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7.2 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7.3 Numerical Example: Expected Value and Variance . . . . . . . . . 18

8 The Binomial Model for Option Pricing 19

8.1 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8.2 Risk-Neutral Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8.3 Pricing a European Call Option . . . . . . . . . . . . . . . . . . . . 20

8.4 Multi-period Binomial Model . . . . . . . . . . . . . . . . . . . . . . 20

8.5 Numerical Example: One-period Binomial Model . . . . . . . . . . 20

9 The Black-Scholes Model 21

9.1 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

9.2 The Black-Scholes Formula . . . . . . . . . . . . . . . . . . . . . . 22

9.3 Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

9.4 Numerical Example: Black-Scholes Pricing . . . . . . . . . . . . . . 23

10 Interest Rate Models 23

10.1 General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

10.2 The Vasicek Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10.3 The Cox-Ingersoll-Ross (CIR) Model . . . . . . . . . . . . . . . . . 24

10.4 Numerical Example: Zero-Coupon Bond Pricing with Vasicek Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

11 Conclusion 25

1 Introduction to derivative contracts

1.1 Basic concepts

A derivative contract (also called a contingent claim) is a financial instrument whose value depends on one or more underlying variables. In essence, it’s a contract whose payoff is determined by the future value of some other asset.

This underlying asset could be:

• Stocks
• Stock indices
• Bonds
• Foreign currencies
• Commodities (oil, gold, wheat)
• Non-traded goods (interest rates, weather conditions)

In this course, we focus on derivatives with a single underlying variable, typically the price of a stock or an interest rate.

1.2 The basic structure of derivatives

The simplest class of derivatives involves two dates:

• t: valuation date (current date)
• ≤T : expiration date or maturity, where t T (future date)

We also have:

• S(t): price at time t of the underlying asset (known at time t)
• S(T ): price at time T of the underlying asset (random variable from the perspective of time t)
• Y (t): price at time t of the derivative (to be determined)
• Y (T ): payoff at time T of the derivative (determined by the contract function) 3

At maturity T, the derivative has payoff:

Y (T ) = ϕ(S(T )) (1)

Where ϕ is the contract function that characterizes the particular derivative contract and is known at valuation date t.

Our goal in derivative pricing is to find the fair value (or price) of the derivative contract at time t: Y (t) = V [t; Y (T )] (2)

This notation means “the value at time t of a contract that pays Y (T ) at time T.”

1.3 Types of derivatives

1.3.1 By timing structure

• Standard Derivatives: Only involve two fixed dates - the contract date and the expiration date.
• Early Exercise Derivatives: The contract can be exercised at a date′ ′≤ ≤t where t t T. In this case, we need to determine Y (t) considering′ the optimal early exercise time t (which is a random variable). American options are examples of this type.
• Path-dependent Derivatives: The payoff at time T depends on the ≤ ≤history of the underlying asset up to time T, i.e., Y (T ) = ϕ(S(u), t uT ). Asiatic options are examples of this type.

2 Forward contracts

2.1 Definition and structure

A forward contract is an agreement made at time t to buy or sell an asset at ≤a future date T, with t T. In contrast, a spot contract is an agreement to buy or sell an asset immediately (which can be thought of as a forward contract where t = T).

A forward contract is characterized by the forward price F (t, T ), which is the price decided at time t and paid at time T to receive the underlying asset.

The life of the contract is:

• At time t, all the details of the contract are fixed, in particular, the forward price F (t, T )
• At time T, the exchange is made 4

2.2 Long and short positions

• Long position: The party that buys the underlying asset at time T by paying F (t, T )
• Short position: The party that sells the underlying asset at time T by receiving F (t, T )

Both sides of the contract have made a binding commitment (mutual obligations).

2.3 Mathematical representation

For a forward contract we have:

• t: valuation date (current date)
• ≤T : delivery date with t T (future date)
• F (t, T ): forward price fixed at time t for delivery at time T
• S(t): price at time t of the underlying (known at time t)
• S(T ): price at time T of the underlying (random variable)
• D(t, T ): market value at time t of dividends paid by the asset in [t, T ]

If t = T, then F (t, t) = S(t), i.e., the forward price coincides with the spot price at time t.

2.4 Market assumptions

We consider a “perfect” market with the following properties:

Frictionless market:

• No transaction costs
• No taxes
• Securities are infinitely divisible (we can trade arbitrary fractions)
• Short selling is allowed (we can sell securities we don’t own)

Competitive market:

• Market agents are profit maximizers (they prefer more to less)
• Market agents are price takers (they cannot influence the market with their activities)

Additionally, we assume:

• Absence of riskless arbitrage opportunities: It cannot happen that an agent obtains a positive cash flow without risk.
• No insolvencies: Both parties fulfill their obligations regardless of the price evolution. 5

2.5 Payoff of a forward contract

For the buyer (long position) of a forward contract, denote:

• L(t): price at time t of the forward contract
• L(T ): payoff at time T of the forward contract

We have: L(t) = 0 (3)

This is because neither party pays to enter the contract at time t. Meanwhile:

−L(T ) = S(T ) F (t, T ) = ϕ(S(T )) (4)

−Where the contract function ϕ(x) = x F (t, T ) is a linear function known at time t.

For the seller (short position), the payoff at time T is simply:

−L(T −) = F (t, T ) S(T ) (5)

2.6 Determining the forward price

The hypothesis of absence of riskless arbitrage opportunities (and its conse-·]quences: the linearity of V [t; and the law of one price) allows us to determine a constraint for the forward price F (t, T ):

L(t) = V [t; L(T )] = V [t; S(T )−F (t, T )] = V [t; S(T )]−V [t; F (t, T )] = V [t; S(T )]−v(t, T )F (t, T ) = 0(6)

Where v(t, T ) is the value at time t of a deterministic Zero-Coupon Bond (ZCB) paying 1 at time T.

Hence, we deduce: V [t; S(T )]F (t, T ) = (7)v(t, T )

Where V [t; S(T )] is the value at time t of a stochastic ZCB paying S(T ) at time T.