Option Valuation and Risk Sensitivities

Introduction

Option valuation connects uncertain future payoffs with current market prices. Unlike a forward contract, an option has a nonlinear payoff: its value depends not only on the expected direction of the underlying asset but also on volatility, time to expiry, interest rates and dividends. These characteristics make options useful for hedging, speculation and structured finance, but they also create significant model and risk-management challenges.

This chapter develops two foundational valuation methods: binomial trees and the Black–Scholes–Merton (BSM) model. It then examines implied volatility and the principal option risk sensitivities—delta, gamma, theta, vega and rho. Throughout the chapter, interest rates and dividend yields are continuously compounded unless stated otherwise.

Learning Objectives

After studying this chapter, students should be able to:

• construct and apply a risk-neutral binomial option-pricing tree;
• calculate European call and put prices using the BSM model;
• explain implied volatility, volatility smiles and volatility skews;
• calculate and interpret delta, gamma, theta, vega and rho;
• use Greeks to estimate option and portfolio value changes;
• distinguish European from American exercise features; and
• critically evaluate option-pricing assumptions, limitations and governance requirements.

1. Foundations of Option Value

A call option gives its holder the right, but not the obligation, to buy an underlying asset at an exercise price. A put gives the right to sell it. Let:

• S = current underlying price, in currency units;
• K = exercise or strike price;
• T = time to expiry, in years;
• S_T = underlying price at expiry;
• C and P = current call and put values.

The European expiry payoffs are:

An option’s premium consists of intrinsic value and time value. For example, a call’s intrinsic value is max(S-K,0). Time value reflects the possibility that market movements before expiry will increase the payoff.

1.1 Moneyness

1.2 European Put–Call Parity

For European options with the same strike and expiry on an asset paying a continuous dividend yield q, no-arbitrage requires:

Here r is the continuously compounded risk-free rate. The relationship assumes frictionless trading and identical contractual terms. It does not generally apply unchanged to American options because early exercise rights have value.

2. Option Valuation by Binomial Trees

A binomial tree models the underlying price as moving to one of two possible values over each short interval. With sufficiently many intervals, a suitably calibrated tree can approximate a continuous price process.

2.1 One-Period Model

Suppose the current asset price is S. Over a period of length ∆t, it rises to Su or falls to Sd, where u>d. For a derivative with corresponding values V_u and V_d, the risk-neutral probability of an upward movement is:

The current derivative value is:

The model requires:

This no-arbitrage condition ensures 0<p<1. The risk-neutral probability is a pricing device, not necessarily the real-world probability that the asset will rise.

The replicating portfolio contains:

units of the underlying. The remaining investment is held in or borrowed through the risk-free asset.

2.2 Two-Period Worked Example

Consider a non-dividend-paying stock with:

• S=100;
• K=100;
• maturity of one year divided into two periods;
• u=1.20 and d=0.80 per period;
• r=5% continuously compounded;
• ∆t=0.5.

The risk-neutral up probability is:

The terminal stock prices are 144, 96 and 64. A European call with strike 100 therefore has terminal payoffs of 44, 0 and 0.

Figure: Backward valuation of a two-period European call.

At the upper first-period node:

At the lower node, both subsequent payoffs are zero, so C_d=0. The initial value is:

The initial hedge ratio is:

Thus, approximately 0.6043 shares are required to replicate one call over the first period.

2.3 American Options

An American option can be exercised at any time up to expiry. At every tree node its value is:

This makes binomial trees particularly useful for American puts and dividend-paying American calls. Under standard assumptions, early exercise of an American call on a non-dividend-paying stock is not optimal, so its value equals that of the corresponding European call. That result does not generally hold when dividends, funding constraints or market frictions are material.

3. The Black–Scholes–Merton Model

The BSM model values European options by constructing a continuously rebalanced risk-free hedge. Under its assumptions, the option satisfies the partial differential equation:

The equation follows because a portfolio containing the option and an offsetting amount of the underlying can eliminate instantaneous price risk. A locally riskless portfolio must earn the risk-free rate to avoid arbitrage.

The principal assumptions are:

• the underlying follows a continuous lognormal diffusion with constant volatility σ;
• trading is continuous and assets are divisible;
• there are no transaction costs, taxes or binding short-sale restrictions;
• the risk-free rate and continuous dividend yield are known and constant;
• no arbitrage opportunities exist; and
• the option is European.

3.1 European Call and Put Formulas

where:

Here N(x) is the cumulative standard normal distribution and σ is annualized volatility expressed as a decimal.

3.2 Worked BSM Valuation

Consider a three-month European option with:

• S=100 and K=100;
• T=0.25 years;
• r=0.04;
• q=0;
• σ=0.20.

Then:

Using N(0.15)=0.55962, N(0.05)=0.51994 and e^−0.01=0.99005:

Put–call parity confirms the result:

and S-Ke^-rT=100-99.005≈1.00.

4. Implied Volatility and the Volatility Surface

Historical volatility is estimated from past returns. Implied volatility is the value of σ that makes a pricing model reproduce an observed market option price:

Because the BSM equation cannot generally be inverted algebraically for volatility, implied volatility is obtained numerically, commonly through Newton–Raphson iteration, bisection or robust hybrid algorithms.

If C(σ) denotes the model call value, Newton–Raphson updates volatility as:

The method can become unstable when vega is very small, such as for deep in- or out-of-the-money options near expiry. Bisection is slower but more robust when valid price bounds are available.

4.1 Volatility Smile and Skew

If BSM assumptions held exactly, options on the same asset with the same expiry would have one common implied volatility. In practice, implied volatility varies with strike and maturity.

• A volatility smile has higher implied volatility for both low and high strikes than for at-the-money strikes.
• A volatility skew or smirk displays an asymmetric pattern. Equity-index markets commonly exhibit higher implied volatility for low-strike puts, reflecting crash risk and demand for downside protection.
• A term structure describes variation in implied volatility across maturities.
• A volatility surface combines strike and maturity dimensions.

The surface reflects market expectations, supply and demand, jump risk, stochastic volatility, leverage effects and liquidity conditions. It is therefore misleading to treat one implied volatility as a complete description of an option market.

5. Option Risk Sensitivities: The Greeks

Greeks measure local changes in option value with respect to model inputs. For small movements:

This is a local approximation. It becomes less accurate for large simultaneous changes or when Greeks themselves vary materially.

5.1 Delta

Delta measures sensitivity to the underlying price:

For BSM European options:

Using the BSM example, where q=0:

A one-currency-unit increase in the stock price therefore increases the call value by approximately 0.5596 and decreases the put value by approximately 0.4404, for a sufficiently small movement.

5.2 Gamma

Gamma measures the rate at which delta changes as the underlying price changes:

The function φ(x) is the standard normal probability density. Gamma is the same for otherwise identical European calls and puts.

For the example, φ(0.15)=0.39448:

The unit is change in delta per one-unit change in the underlying price. If the stock rises by 2, the call delta is estimated to increase by:

A long option position normally has positive gamma; a short option position has negative gamma. Delta hedging does not remove gamma risk because the hedge ratio changes as the underlying moves.

5.3 Theta

Theta measures the change in option value as calendar time passes, holding other inputs constant. The following formulas report annualized calendar-time theta:

For the non-dividend-paying call:

On a 365-day convention:

Thus, with all else unchanged, the call loses approximately 0.0273 in value over one day. Theta is often negative for long vanilla options, but it is not universally negative for every option structure or parameter combination.

5.4 Vega

Vega measures sensitivity to annualized volatility:

For the example:

This is the value change for a 1.00 change in volatility—for example, from 20% to 120%. Market practice usually quotes vega per one percentage-point change:

If volatility rises from 20% to 22%, then ∆σ=0.02 and:

Calls and puts with identical terms have the same BSM vega. Long vanilla options have positive vega.

5.5 Rho

Rho measures sensitivity to the continuously compounded risk-free rate:

For European options:

For the call example:

This is per 1.00 change in the annual rate. For a one-percentage-point increase, ∆r=0.01:

Higher rates generally increase European call values and reduce European put values, other factors held constant.

5.6 Summary of Major Greeks

6. Illustrative Case: Hedging an Option-Writing Desk

Background

This is an illustrative case. Meridian Securities has written 10,000 three-month calls with the same parameters as the BSM example. Each option relates to one share. The desk must assess its short-term risk and establish an initial hedge.

Problem

The short call position is exposed to rising share prices, changing volatility and nonlinear losses. Management wants to know whether a delta hedge provides adequate protection.

Analysis and Approach

The call delta is 0.5596. Because the desk is short the calls, its aggregate delta is:

The desk buys approximately 5,596 shares to become initially delta neutral.

The option liability has aggregate gamma:

Suppose the share price falls by 2. Ignoring volatility and time initially, the delta–gamma change in the hedged portfolio is approximately:

The first-order option delta effect is substantially offset by the shares, but the negative gamma loss remains.

The short position’s aggregate vega is −197,240 per unit volatility, or approximately −1,972.4 per percentage point. A two-percentage-point increase in implied volatility therefore produces:

The short calls gain approximately 272.6 from one day of time decay because the long-call daily theta is about −0.0273:

Combining the estimated effects gives a one-day loss of approximately:

Outcome

The delta hedge controls only first-order price exposure. It does not eliminate losses caused by gamma and vega. Meridian should establish limits for aggregate gamma and vega, perform scenario tests, and consider purchasing options to reduce nonlinear exposure. Frequent rebalancing may reduce delta drift but introduces transaction costs and gap risk.

Learning Points

• Delta neutrality is local and temporary.
• Short-option portfolios are commonly short gamma and vega.
• A large volatility movement may dominate ordinary daily time decay.
• Scenario analysis complements Greek-based approximations.
• Hedge design must consider liquidity, transaction costs and governance limits.

7. Model Limitations and Professional Practice

7.1 Limitations of BSM

Actual markets depart from BSM assumptions in several important ways:

1. Volatility is not constant. It varies over time and across strikes and maturities.
2. Prices may jump. Earnings announcements, defaults and geopolitical events can produce discontinuous movements that cannot be continuously hedged.
3. Trading is costly and discrete. Bid–ask spreads, brokerage charges and market impact make continuous rebalancing impossible.
4. Interest rates and dividends are uncertain. This is particularly relevant for long-dated options.
5. Liquidity differs across contracts. Observed prices may contain liquidity premiums or stale quotations.
6. Returns are not perfectly lognormal. Empirical distributions commonly exhibit skewness and heavy tails.

Consequently, BSM is better viewed as a benchmark and quoting framework than as a complete description of market dynamics.

7.2 Alternative Models

Local-volatility models fit an observed volatility surface by making volatility a function of price and time. Stochastic-volatility models allow volatility itself to follow a random process. Jump-diffusion models introduce discontinuous price changes. Monte Carlo simulation is useful for path-dependent and high-dimensional derivatives, while finite-difference methods solve valuation equations numerically. Model choice should reflect product features, available data and the risk being managed.

7.3 Governance, Ethics and Regulation

Option valuation affects financial reporting, collateral, capital, customer disclosures and trader compensation. Weak controls can encourage deliberate model selection or input manipulation. Sound governance includes:

• independent model validation;
• approved market-data sources and volatility-surface controls;
• documented assumptions and model limitations;
• valuation adjustments for liquidity, funding and counterparty risk where applicable;
• separation of trading, valuation control and risk oversight;
• back-testing and profit-and-loss attribution; and
• stress testing beyond historically observed market movements.

Professionals must communicate uncertainty honestly. Reporting a model estimate as though it were an objectively certain value is misleading, particularly for illiquid or complex derivatives.

7.4 Technology and Contemporary Developments

Modern option systems calibrate volatility surfaces in real time, calculate portfolio Greeks and run large scenario sets. Automatic differentiation and adjoint methods accelerate sensitivity calculations for complex portfolios. Machine-learning methods can assist with pricing approximations, anomaly detection and surface interpolation, but they do not remove the need for no-arbitrage constraints, explainability, validation and human oversight.

Best practice is to combine model prices, observable market quotations, Greek limits and severe but plausible stress scenarios. A model that matches current prices may still provide inadequate hedges under future market dynamics.

Chapter Summary

Option valuation is founded on replication and no-arbitrage. Binomial trees value derivatives by applying risk-neutral probabilities and backward induction, while also accommodating American exercise decisions. The Black–Scholes–Merton model provides closed-form values for European calls and puts under restrictive assumptions.

Implied volatility is the volatility input consistent with a market price. Its variation across strike and maturity creates volatility smiles, skews and surfaces, revealing departures from constant-volatility assumptions.

Delta measures first-order price exposure, gamma measures delta curvature, theta measures time decay, vega measures volatility exposure and rho measures interest-rate exposure. These sensitivities support hedging and risk reporting, but they are local approximations. Effective risk management therefore combines Greeks with rebalancing policies, stress testing, liquidity assessment and robust model governance.

Key Terms

American option: An option exercisable at any time up to and including expiry.

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

Binomial tree: A discrete model in which the underlying can move to one of two values per period.

Delta: The first derivative of option value with respect to the underlying price.

European option: An option exercisable only at expiry.

Gamma: The rate of change of delta with respect to the underlying price.

Hedge ratio: The quantity of the underlying used to offset derivative price exposure.

Implied volatility: The volatility input that equates a model value to an observed market price.

Intrinsic value: The immediate exercise value of an option.

Moneyness: The relationship between the underlying price and the option strike.

No-arbitrage: The condition that equivalent future cash flows must have equivalent current values.

Put–call parity: The price relationship between European calls, puts, the underlying and a risk-free bond.

Risk-neutral probability: A pricing probability under which discounted tradable asset values are martingales.

Rho: Sensitivity of option value to the risk-free interest rate.

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

Time value: The part of an option premium exceeding intrinsic value.

Vega: Sensitivity of option value to volatility.

Volatility surface: Implied volatility represented across strikes and maturities.

Review Questions

Short Answer Questions

1. Distinguish an option’s intrinsic value from its time value.
2. What does a risk-neutral probability represent?
3. Explain why delta neutrality is only temporary.
4. Define implied volatility.
5. Why is gamma normally positive for a long vanilla option?

Long Answer Questions

1. Explain how replication and no-arbitrage lead to risk-neutral option valuation.
2. Describe the construction and backward valuation of a multi-period binomial tree.
3. Derive and explain the economic meaning of European put–call parity.
4. Critically evaluate the principal assumptions of the BSM model.
5. Explain volatility smiles, skews and term structures and discuss what they reveal about market expectations.

Analytical / Application Questions

1. A stock is priced at 80 and may rise to 96 or fall to 68 over six months. The continuously compounded risk-free rate is 4% and the stock pays no dividends. Calculate the risk-neutral probability and value a six-month European call with a strike of 82.
2. Using the BSM model, value a one-year European put where S=50, K=55, r=3%, q=1% and σ=25%. Verify the result using put–call parity.
3. A dealer is short a portfolio with delta −2,500, gamma −180 and vega −900 per volatility percentage point. Design an initial hedge and evaluate the remaining exposure to a three-unit price movement and a two-percentage-point volatility increase.

Multiple Choice Questions with Answers

Question 1

In the two-period binomial example, what is the risk-neutral upward probability when u=1.20, d=0.80, r=5% and ∆t=0.5?

A. 0.4367

B. 0.5000

C. 0.5633

D. 0.6250

Correct Answer: C. 0.5633

Question 2

Using S=K=100, T=0.25, r=4%, q=0 and σ=20%, what is the approximate BSM European call value?

A. 3.49

B. 4.49

C. 5.60

D. 9.90

Correct Answer: B. 4.49

Question 3

For the same inputs as the BSM illustration, what is the approximate European put value?

A. 2.50

B. 3.49

C. 4.49

D. 5.48

Correct Answer: B. 3.49

Question 4

A call has delta 0.56. For a sufficiently small one-unit increase in the underlying price, with other inputs unchanged, what is the estimated call-value change?

A. An increase of 0.56

B. A decrease of 0.56

C. An increase of 1.00

D. No change

Correct Answer: A. An increase of 0.56

Question 5

An option has gamma 0.03945. If the underlying rises by 2, what is the approximate change in delta?

A. 0.0197

B. 0.0395

C. 0.0789

D. 0.1578

Correct Answer: C. 0.0789

Question 6

A call has annualized theta of −9.95. Using a 365-day convention, what is its approximate one-day value change if all other inputs remain constant?

A. −0.0273

B. −0.9950

C. 0.0273

D. 9.9500

Correct Answer: A. −0.0273

Question 7

An option's vega is 0.18 per unit change in volatility. If volatility rises from 20% to 22%, what is the first-order estimated change in option value?

A. 0.0036

B. 0.0360

C. 0.1800

D. 0.3600

Correct Answer: A. 0.0036

Question 8

A European call has rho of 12.87 per unit interest rate. What is its approximate value change if the annual rate rises by one percentage point?

A. −0.1287

B. 0.0129

C. 0.1287

D. 12.8700

Correct Answer: C. 0.1287

Question 9

What does a typical downward equity volatility skew indicate?

A. Low-strike options have lower implied volatility than high-strike options

B. All strikes have identical implied volatility

C. Low-strike puts tend to have higher implied volatility than at-the-money options

D. Historical volatility must equal implied volatility

Correct Answer: C. Low-strike puts tend to have higher implied volatility than at-the-money options

Question 10

Under standard frictionless assumptions, which statement is correct for an American call on a non-dividend-paying stock?

A. Immediate exercise is always optimal

B. Its value equals that of the corresponding European call

C. It must be worth less than the corresponding European call

D. Its delta must always equal one

Correct Answer: B. Its value equals that of the corresponding European call