Option Pricing Models

Introduction

An option is a contingent claim whose payoff depends on the value of an underlying asset. A call option gives its holder the right, but not the obligation, to buy the underlying at a specified strike price; a put gives the right to sell. The central pricing problem is to determine the present value of these uncertain future payoffs without relying on subjective forecasts of the underlying asset’s expected return.

Modern option pricing rests on two complementary ideas. First, a portfolio can sometimes be constructed to replicate the option payoff. Second, if two portfolios generate the same future cash flows, the no-arbitrage principle requires them to have the same current value. The binomial model makes replication transparent in discrete time, while the Black–Scholes–Merton model extends the logic to continuous time.

These models are not merely formulas. They are frameworks for understanding hedging, risk-neutral valuation, implied volatility, dividend effects, model risk, and the limits of equilibrium assumptions in real markets.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Explain no-arbitrage pricing, replication, risk-neutral probabilities, and martingale valuation.
2. Price European and American options using a one- or multi-period binomial model.
3. State and interpret the Black–Scholes–Merton assumptions and formulas.
4. Adjust option valuation for continuous dividends and understand early exercise.
5. Calculate and interpret implied volatility and option sensitivities.
6. Evaluate the empirical limitations and governance implications of pricing models.

1. Foundations of No-Arbitrage Valuation

Let S₀ denote the current asset price, S_T its value at maturity T, and r the continuously compounded risk-free interest rate. A derivative payoff at maturity is written H_T. Under deterministic interest rates, its discounted value is e^-rTH_T when valuation is performed under an appropriate risk-neutral probability measure.

The no-arbitrage principle states that a zero-investment strategy cannot produce a non-negative payoff in every state and a strictly positive payoff in at least one state. In a complete market, every attainable derivative payoff has a unique replicating portfolio. In an incomplete market, several prices may be consistent with no arbitrage, and additional assumptions or preferences are required.

Replication

Suppose a portfolio contains ∆ units of the underlying and B invested in a risk-free account. Its value is

where B_t is the account value at time t. If this portfolio produces exactly the same payoff as an option in every relevant state, the option price must equal the portfolio value. Otherwise, buying the cheaper and selling the more expensive position creates an arbitrage opportunity.

Risk-neutral valuation

A risk-neutral probability is a pricing device, not a claim that investors are actually risk-neutral. Under a risk-neutral measure Q, traded assets, after reinvested income is included, have expected growth at the risk-free rate. For a claim with payoff H_T,

This equation applies when the market is arbitrage-free and the claim is priced using a valid equivalent martingale measure. The expected return under the real-world probability measure is generally not used directly in the formula.

A discounted asset price is a martingale under Q if its conditional expected future discounted value equals its current discounted value:

for a non-dividend-paying asset, where F_t represents information available at time t. With a continuous dividend yield q, the reinvested asset process is the martingale, and the corresponding relation is

2. The Binomial Option-Pricing Model

The one-period economy

Assume that over one period the asset can move from S₀ to either S_u = uS₀ or S_d = dS₀, where u > d. Let R be the gross risk-free return over the period, usually R=e^r∆t or R=1+r∆t under simple compounding.

For a European claim, let the up-state and down-state payoffs be V_u and V_d. A replicating portfolio satisfies

Subtracting the equations gives the hedge ratio:

The initial option value is

Solving through risk-neutral probabilities produces the equivalent expression

where

The risk-neutral probability is valid as a probability when d < R < u. This condition is the one-period no-arbitrage condition. Notice that p^* depends on the risk-free return and the up and down factors, not on the asset’s actual expected return.

Worked example: a one-period call

Suppose S₀=100, u=1.20, d=0.90, the one-period gross risk-free return is R=1.05, and the strike is K=105. The terminal asset prices are 120 and 90. The call payoffs are therefore 15 and 0.

The risk-neutral probability is

Thus,

The replicating hedge is

The bond position satisfies 0.5(120)+1.05B=15, giving B=-42.8571. The initial portfolio value is 0.5(100)-42.8571=7.1429, confirming the risk-neutral calculation.

Multi-period trees

A multi-period binomial tree applies the one-period logic recursively. At each node, the option value equals the discounted risk-neutral expected value of the two next-node values:

For a European call at maturity,

and for a European put,

The tree is recombining when an up move followed by a down move produces the same price as a down move followed by an up move. This reduces computational cost. In a model with n periods, terminal states are then n+1 rather than 2ⁿ.

American options and early exercise

An American option may be exercised at any time up to maturity. At each node, its value is the greater of its immediate exercise value and its continuation value:

For an American call on a non-dividend-paying stock, early exercise is generally suboptimal because exercising destroys time value and forfeits the interest that could be earned on the strike. An American put may optimally be exercised early, particularly when it is deep in the money and interest rates are positive. Dividends can make early exercise of a call rational immediately before an ex-dividend date.

Choosing tree parameters

A common parameterization uses volatility σ and step length ∆t:

With continuous dividends, a frequently used risk-neutral probability is

where q is the continuously compounded dividend yield. The parameters must satisfy d < e^(r-q)∆t < u to avoid one-step arbitrage. Alternative trees, including the Cox–Ross–Rubinstein and Jarrow–Rudd trees, differ in how they choose u, d, and probabilities. As the number of steps increases, a properly constructed tree can converge toward the Black–Scholes–Merton value for European options.

3. The Black–Scholes–Merton Model

Assumptions

The standard model assumes:

• The underlying follows continuous geometric Brownian motion.
• Volatility and the risk-free interest rate are constant over the option’s life.
• Trading is continuous, with no transaction costs, taxes, or liquidity constraints.
• The asset is divisible and short selling is permitted.
• The risk-free rate is known and borrowing and lending occur at that rate.
• The option is European unless an additional treatment is provided.
• Markets are arbitrage-free and sufficiently complete.

These assumptions create a tractable benchmark. They are not descriptions of every real market.

Under the real-world measure, the asset process is often represented as

where µ is the real-world expected instantaneous return, σ is instantaneous volatility, and W_t is a standard Brownian motion. Under the risk-neutral measure, the drift becomes r for a non-dividend-paying asset, or r-q with continuous dividends:

European option formulas

For a European call and put on a non-dividend-paying asset,

where

Here, N(·) is the cumulative distribution function of a standard normal variable. S₀ and K are measured in currency units, r and σ are annualized rates, and T is measured in years. The products σ√T and rT are dimensionless, as required by the logarithm and exponential functions.

With a continuous dividend yield q, the formulas become

with

and d₂=d₁-σ√T. The dividend yield reduces the present value of the asset benefit received by the holder and generally lowers call values while increasing put values.

Conceptual derivation

The Black–Scholes–Merton result follows from dynamic hedging. Let the option value be V(S,t). Applying Itô’s lemma gives

A portfolio containing one option and -∆ units of the asset eliminates the stochastic term when ∆=V_S. Because the resulting portfolio is locally riskless, its return must equal the risk-free rate. For a non-dividend-paying asset, this yields the Black–Scholes partial differential equation:

The terminal condition for a call is V(S,T)=max(S-K,0), and for a put it is V(S,T)=max(K-S,0). The remarkable feature is that µ disappears. The option price is determined by hedgeability and no arbitrage, not by the asset’s risk premium.

4. Worked Black–Scholes Example

Consider a six-month European call with S₀=100, K=100, r=5%, q=0, and σ=20%. Thus T=0.5.

First,

Using N(0.2475)≈0.5978 and N(0.1061)≈0.5423,

The estimated fair value is approximately 6.89 currency units per share, subject to the model assumptions and the accuracy of the input data. The corresponding put value can be obtained directly or from put–call parity.

5. Put–Call Parity and Bounds

For European options with the same strike and maturity on a non-dividend-paying asset,

With a continuous dividend yield,

Put–call parity is a no-arbitrage relationship, not a statistical estimate. It is useful for detecting inconsistent market quotes and synthesizing one option from the other plus positions in the underlying and bond.

Basic no-arbitrage bounds include C₀≥max(0,S₀e^-qT-Ke^-rT) for a European call and P₀≥max(0,Ke^-rT-S₀e^-qT) for a European put. Violations may indicate mispricing, but apparent arbitrage can disappear after transaction costs, bid–ask spreads, funding differences, taxes, or settlement constraints are considered.

6. Volatility, Greeks, and Implied Volatility

Volatility is the most important and difficult input in the standard model. Historical volatility estimates past variation; realized volatility measures subsequent variation; implied volatility is the value of σ that makes a model price equal to an observed market price:

There is usually no closed-form solution for σ_imp, so numerical methods such as bisection or Newton–Raphson are used. For a standard European call, the model price is increasing in volatility, which supports a unique implied volatility under normal conditions. Implied volatility varies by strike and maturity, producing a volatility smile or skew rather than the constant-volatility surface assumed by Black–Scholes.

Important sensitivities, called Greeks, include:

For a non-dividend-paying European call,

and its gamma is

where φ(·) is the standard normal density. Greeks are local approximations; they can become unreliable during large price moves, near maturity, or when volatility and correlations change abruptly.

7. Limitations and Real-Market Practice

Nonconstant volatility

Observed implied-volatility smiles and skews contradict the single-volatility assumption. Local-volatility models allow volatility to depend on price and time. Stochastic-volatility models allow volatility itself to evolve randomly. These models may fit market prices better but introduce additional parameters, estimation risk, and hedging complexity.

Jumps and extreme events

Geometric Brownian motion has continuous paths and thin-tailed log returns. Markets can jump because of earnings announcements, policy decisions, defaults, or geopolitical events. Jump-diffusion and regime-switching models address some of these effects, although no model eliminates model risk.

Interest rates, dividends, and funding

Rates may be stochastic, dividends may be discrete and uncertain, and derivatives may be collateralized under different contractual arrangements. Practitioners therefore use yield curves, discount curves, dividend forecasts, repo rates, and collateral conventions rather than a single universal risk-free rate.

Incomplete markets and liquidity

Perfect replication is often impossible for options with volatility risk, jump risk, credit risk, or illiquid underlyings. Bid–ask spreads and market impact affect both execution and hedging. A model price should therefore be interpreted as a benchmark or mid-market estimate, not automatically as a tradable price.

American and exotic options

Closed-form Black–Scholes formulas generally do not value American options exactly. Binomial trees, finite-difference methods, Monte Carlo methods with early-exercise techniques, and specialized numerical methods are used. Path-dependent and barrier options require careful treatment of monitoring frequency and boundary behavior.

8. Applied Case: Hedging an Equity Call Book

This is an illustrative case based on a hypothetical investment bank.

Background

A bank has sold 10,000 European calls on a dividend-paying stock. Each option covers one share. The stock is priced at 80, the strike is 82, maturity is three months, the continuously compounded dividend yield is 1%, the risk-free rate is 4%, and implied volatility is 25%.

Problem

The bank is exposed to increases in the stock price. It must estimate the option value, calculate a hedge, and assess risks that a static Black–Scholes calculation may conceal.

Analysis and approach

The desk inputs the market yield curve and dividend forecast into the dividend-adjusted Black–Scholes formula. Suppose the resulting call delta is 0.47. A short call position has delta approximately -0.47 per option. For 10,000 short calls, the aggregate delta is -4,700. The desk therefore buys approximately 4,700 shares to create a delta-neutral position.

Delta neutrality is only local. If the stock moves, gamma changes the delta, requiring rebalancing. The position also has vega exposure: a rise in implied volatility increases the value of the calls and harms the short position. It has theta exposure, interest-rate exposure, dividend risk, and potential gap risk between rebalancing times.

Outcome

The hedge reduces first-order exposure to small stock-price changes but does not eliminate risk. The desk monitors delta, gamma, vega, liquidity, stress scenarios, and model-versus-market discrepancies. If the volatility surface steepens, a constant-volatility model can misstate both the option value and the hedge ratio.

Learning points

1. A model value and a hedge are inseparable: the hedge ratio depends on the model’s sensitivities.
2. Delta hedging does not remove volatility, jump, liquidity, or discrete-rebalancing risk.
3. Implied volatility should be taken from comparable market instruments and interpolated consistently.
4. Governance requires independent price verification, documented assumptions, limits, and stress testing.

9. Governance, Ethics, and Contemporary Practice

Financial institutions should document model purpose, inputs, calibration procedures, limitations, and validation results. Independent model validation should test mathematical correctness, implementation accuracy, stability, sensitivity, and performance under stressed conditions. Valuation adjustments may be required for credit, funding, collateral, liquidity, and uncertainty.

Ethically, a model must not be presented as an objective truth when its assumptions are known to be unrealistic. Selective use of volatility inputs to improve reported profits, failure to recognize illiquidity, or concealment of model uncertainty can mislead clients, management, regulators, and investors. Governance frameworks should assign responsibility for model approval, change control, data quality, and escalation of breaches.

Modern computational practice uses high-performance numerical methods, automatic differentiation for Greeks, and machine-learning tools for volatility-surface interpolation or calibration. AI can improve pattern recognition and computation, but it does not remove the need for economic constraints, explainability, out-of-sample testing, and controls against unstable extrapolation. A machine-learning price that violates monotonicity, convexity, or put–call parity is not acceptable merely because it fits historical data.

Chapter Summary

Option pricing is founded on no arbitrage and replication. In a one-period binomial model, the hedge ratio and bond position replicate the option, while risk-neutral probabilities provide an equivalent valuation method. Multi-period trees extend this logic and can handle American exercise by comparing immediate exercise with continuation value.

The Black–Scholes–Merton model uses dynamic hedging and a continuous-time limit to produce closed-form European option prices. Its central formulas depend on the underlying price, strike, maturity, interest rate, dividends, and volatility. The real-world expected return disappears because pricing is performed under a risk-neutral measure. Martingale reasoning formalizes the idea that discounted traded values are fairly priced under that measure.

Implied volatility is extracted from market prices and is often more useful for quoting and comparison than a single historical estimate. Greeks describe local sensitivities and support hedging, but they do not eliminate model, liquidity, jump, or parameter risk. Volatility smiles, stochastic rates, discrete dividends, early exercise, and incomplete markets require extensions or numerical methods. Sound professional practice combines models with stress testing, independent validation, transparent assumptions, and ethical governance.

Key Terms

No-arbitrage: The principle that a zero-cost strategy cannot guarantee a positive payoff.

Replication: Constructing a portfolio with the same payoff as a derivative.

Risk-neutral probability: A probability used for arbitrage-free pricing under which discounted traded prices are martingales.

Martingale: A process whose conditional expected future value, after appropriate discounting, equals its current value.

Complete market: A market in which every relevant contingent claim can be replicated.

Binomial model: A discrete-time model in which the underlying takes an up or down movement at each step.

Risk-neutral probability p^*: The probability used in a binomial pricing recursion.

Hedge ratio: The quantity of underlying required in a replicating portfolio; commonly represented by ∆.

Black–Scholes–Merton model: A continuous-time no-arbitrage model for European options.

Geometric Brownian motion: A diffusion process commonly used to model continuously compounded asset prices.

Volatility: The standard deviation of continuously compounded returns, usually annualized.

Implied volatility: The volatility input that makes a pricing model match an observed market price.

Put–call parity: A no-arbitrage relationship linking European calls, puts, the underlying, and the bond.

Delta: Sensitivity of option value to the underlying price.

Gamma: Sensitivity of delta to the underlying price.

Vega: Sensitivity of option value to volatility.

Theta: Sensitivity of option value to the passage of time.

American option: An option exercisable at any time up to expiration.

Model risk: The risk of loss caused by incorrect model structure, inputs, implementation, or use.

Volatility smile or skew: The variation of implied volatility across strikes or maturities.

Review Questions

5 Short Answer Questions

1. Why can risk-neutral probabilities be used even when investors are not risk-neutral?
2. State the one-period binomial formula for the hedge ratio.
3. What is the main difference between a European and an American option?
4. Why does the expected real-world return µ disappear from the Black–Scholes partial differential equation?
5. Define implied volatility.

5 Long Answer Questions

1. Explain replication and derive the one-period binomial risk-neutral pricing formula.
2. Describe the assumptions of the Black–Scholes–Merton model and evaluate their realism.
3. Explain how dividends affect European call and put values.
4. Compare binomial trees with Black–Scholes–Merton valuation for European and American options.
5. Discuss why volatility smiles create a challenge for traditional option pricing.

3 Analytical / Application Questions

1. A one-period asset has current price 50, an up factor of 1.30, a down factor of 0.80, and a gross risk-free return of 1.05. Calculate the risk-neutral probability and price a call with strike 50.
2. A bank is short a portfolio of calls with aggregate delta −8,000 and gamma 250. Explain how a stock-price increase affects the hedge and why delta rebalancing may be required.
3. An observed option price is materially above the Black–Scholes value calculated using historical volatility. Identify at least three possible explanations and describe how a practitioner should investigate them.

Multiple Choice Questions with Answers

Question 1

In a one-period binomial model, which expression gives the risk-neutral probability of an up move?

A. (u-d)/(R-d)

B. (R-d)/(u-d)

C. (R-u)/(d-u)

D. (u-R)/(u-d)

Correct Answer: B. (R-d)/(u-d)

Question 2

What is the central purpose of a replicating portfolio?

A. To maximize the investor's expected return

B. To forecast the underlying asset's real-world drift

C. To produce the same payoff as the derivative

D. To eliminate all forms of market risk

Correct Answer: C. To produce the same payoff as the derivative

Question 3

For a non-dividend-paying asset, the Black–Scholes European call delta is:

A. N(d₁)

B. N(d₂)

C. -N(d₁)

D. e^-rTN(d₁)

Correct Answer: A. N(d₁)

Question 4

Which condition prevents one-period arbitrage in the basic binomial model?

A. u<d<R

B. R<u<d

C. d<R<u

D. d<u<R

Correct Answer: C. d<R<u

Question 5

Under risk-neutral valuation, the expected growth rate of a non-dividend-paying traded asset is:

A. The historical average return

B. The risk-free rate

C. The volatility rate

D. The dividend yield

Correct Answer: B. The risk-free rate

Question 6

Which option is generally more likely to be exercised early when interest rates are positive?

A. A non-dividend-paying American call

B. An American put

C. A European call

D. A European put

Correct Answer: B. An American put

Question 7

Implied volatility is best defined as:

A. The maximum historical return of the asset

B. The volatility that makes a model price equal the market price

C. The risk-free rate adjusted for inflation

D. The standard deviation of option payoffs at maturity

Correct Answer: B. The volatility that makes a model price equal the market price

Question 8

For a long plain-vanilla option, gamma is typically:

A. Negative

B. Zero at all times

C. Positive

D. Equal to the risk-free rate

Correct Answer: C. Positive

Question 9

With a positive continuous dividend yield, the standard Black–Scholes call formula uses which present-value term for the underlying?

A. S₀e^qT

B. S₀e^-qT

C. S₀e^rT

D. S₀(1+qT)

Correct Answer: B. S₀e^-qT

Question 10

A volatility smile primarily indicates that:

A. Implied volatility is identical for every strike

B. The underlying has no price risk

C. Market prices are inconsistent with a single constant-volatility assumption

D. Risk-neutral probabilities cannot be calculated

Correct Answer: C. Market prices are inconsistent with a single constant-volatility assumption