Option Valuation

Binomial option pricing, Black-Scholes pricing model, implied volatility

1 Introduction

In previous lectures, we explored what options are, how they generate payoffs at expiration, and the basic strategies investors use to combine them. Now we turn to a fundamental question: how much should an option be worth before it expires? This is the challenge of option valuation—and it lies at the heart of modern derivatives theory.

Pricing an option correctly matters for several reasons. If you’re a trader, mispriced options represent arbitrage opportunities. If you’re a corporate finance professional, understanding option valuation helps you evaluate employee stock options, convertible bonds, and other securities with embedded optionality. If you’re a risk manager, option pricing models tell you how sensitive your portfolio is to changes in market conditions.

The intuition behind option pricing is both elegant and counterintuitive. We won’t price options by trying to forecast where the stock will go—that’s surprisingly irrelevant. Instead, we’ll construct portfolios that replicate an option’s payoffs using stocks and bonds. If two positions always produce identical outcomes, they must have the same price today. This replication principle is the foundation of everything that follows.

In this lecture, we’ll cover the key determinants of option value, develop the binomial model to build intuition for replication, introduce the celebrated Black-Scholes formula for continuous-time valuation, and explore implied volatility—the market’s “vote” on future uncertainty.

2 Intrinsic Value and Time Value

Before diving into pricing models, we need a framework for thinking about what an option is actually worth. Every option’s price can be decomposed into two components: intrinsic value and time value.

The intrinsic value of an option represents what it would be worth if you had to exercise it immediately. For a call option, this is the amount by which the stock price exceeds the strike price, or zero if it doesn’t. For a put option, it’s the amount by which the strike exceeds the stock price, or zero otherwise:

\[ \text{Intrinsic Value (Call)} = \max(S - X, 0) \]

\[ \text{Intrinsic Value (Put)} = \max(X - S, 0) \]

Here, \(S\) denotes the current stock price and \(X\) the strike price. When an option has positive intrinsic value, we say it is in the money. When intrinsic value is zero, the option is out of the money or at the money (if \(S = X\)).

The time value of an option is everything else—the premium investors pay above intrinsic value because the option still has time remaining until expiration. Time value reflects the optionality embedded in the contract: the chance that favorable price movements could make the option more valuable before it expires.

\[ \text{Option Price} = \text{Intrinsic Value} + \text{Time Value} \]

Consider a call option with a strike of $50 on a stock currently trading at $52. The intrinsic value is $2. If the option trades at $4.50, then the time value is $2.50. That extra $2.50 represents the market’s assessment of the option’s potential—the possibility that the stock could rise even further before expiration.

Time value is always non-negative for American options and typically positive whenever there’s time remaining. At expiration, time value vanishes entirely, and the option is worth exactly its intrinsic value. Understanding this decomposition helps investors think about whether an option is “expensive” relative to the immediate exercise value it provides.

3 Determinants of Option Values

What factors influence how much an option is worth? Understanding these determinants is crucial for both pricing and risk management. The table below summarizes how each factor affects call and put option values, holding everything else constant.

Factor	Effect on Call Value	Effect on Put Value	Intuition
Stock price (\(S\)) ↑	Increases	Decreases	Higher \(S\) makes calls more likely to finish in the money
Strike price (\(X\)) ↑	Decreases	Increases	Higher \(X\) means calls need more appreciation to pay off
Volatility (\(\sigma\)) ↑	Increases	Increases	More uncertainty means greater chance of large favorable moves
Time to expiration (\(T\)) ↑	Increases	Increases	More time for favorable price movements to occur
Risk-free rate (\(r\)) ↑	Increases	Decreases	Higher rates reduce present value of strike; favor calls
Dividend yield (\(\delta\)) ↑	Decreases	Increases	Dividends reduce stock price on ex-date; hurt call holders

Several of these relationships deserve further explanation. The effect of volatility is particularly important and perhaps counterintuitive. Because options have asymmetric payoffs—you benefit from favorable moves but your downside is limited to the premium paid—higher volatility always increases option value. A call holder benefits if the stock soars but loses nothing extra if it collapses (beyond the premium already paid). This asymmetry means option holders like uncertainty.

The effect of time to expiration works similarly. More time means more opportunities for the stock to make large moves. It also means the present value of the strike price (which call holders pay upon exercise) is lower. Both effects increase call value.

Interest rates affect options through the present value of the strike. For a call, you pay \(X\) at expiration if you exercise. Higher rates reduce the present value of that future payment, effectively making calls cheaper to exercise in present-value terms. Dividends work in the opposite direction: when a stock pays dividends, its price drops on the ex-dividend date, which hurts call holders who don’t receive the dividend.

4 Binomial Option Pricing

Now we arrive at the central insight of option pricing: replication. The idea is simple but powerful: if we can construct a portfolio of stocks and bonds that produces exactly the same payoffs as an option in every possible future state, then that portfolio and the option must have the same price today. Any other price would create an arbitrage opportunity.

4.1 The Replicating Portfolio Approach

Consider a world where a stock can only move to one of two prices next period—either up to \(S_u\) or down to \(S_d\). This “binomial” assumption is restrictive, but it allows us to develop the key intuitions that carry through to more realistic models.

Our goal is to replicate a call option using a portfolio consisting of \(\Delta\) shares of stock and \(B\) dollars invested in risk-free bonds. At the end of the period, this portfolio must match the option’s payoff in both the up state and the down state:

\[ \text{Portfolio Value} = \Delta \cdot S + \frac{B}{1 + r_f} \]

where \(r_f\) is the risk-free rate for the period.

If the stock goes up, the option pays \(C_u = \max(S_u - X, 0)\). If it goes down, the option pays \(C_d = \max(S_d - X, 0)\). For replication, we need:

\[ \Delta \cdot S_u + B = C_u \]

\[ \Delta \cdot S_d + B = C_d \]

These are two equations in two unknowns. Subtracting the second from the first gives us the hedge ratio (or delta):

\[ \Delta = \frac{C_u - C_d}{S_u - S_d} \]

This formula has an intuitive interpretation: delta measures how much the option’s value changes relative to changes in the stock price. A delta of 0.5 means the option moves about 50 cents for every dollar move in the stock.

Once we have delta, we can solve for the bond position:

\[ B = C_u - \Delta \cdot S_u \]

Finally, the option’s fair value today equals the cost of establishing the replicating portfolio:

\[ V_0 = \Delta \cdot S + \frac{B}{1 + r_f} \]

Notice something remarkable: we never asked about the probability that the stock goes up versus down. The option price depends only on the possible stock prices, not their probabilities. This is because we’re pricing by replication, not by expected value. As long as we can replicate the option perfectly, probability doesn’t enter the picture.

Example 1: One-Period Binomial Call Pricing

Consider a stock currently trading at $100. After one period, the stock can rise to $120 or fall to $90. A call option has a strike price of $110, and the risk-free rate is 10% for the period.

Solution

Step 1: Calculate option payoffs in each state

If the stock rises to $120: \(C_u = \max(120 - 110, 0) = \$10\)

If the stock falls to $90: \(C_d = \max(90 - 110, 0) = \$0\)

Step 2: Calculate the hedge ratio

\[ \Delta = \frac{C_u - C_d}{S_u - S_d} = \frac{10 - 0}{120 - 90} = \frac{10}{30} = \frac{1}{3} \]

Step 3: Calculate the bond position

\[ B = C_u - \Delta \cdot S_u = 10 - \frac{1}{3} \times 120 = 10 - 40 = -30 \]

The negative value indicates we borrow $30 (shorting the bond).

Step 4: Calculate the option value

\[ V_0 = \Delta \cdot S + \frac{B}{1 + r_f} = \frac{1}{3} \times 100 + \frac{-30}{1.10} = 33.33 - 27.27 = \$6.06 \]

The fair value of this call option is $6.06.

Verification: Let’s confirm replication works in both states.

• Up state: \(\frac{1}{3} \times 120 + (-30) = 40 - 30 = \$10\) ✓
• Down state: \(\frac{1}{3} \times 90 + (-30) = 30 - 30 = \$0\) ✓

The replicating portfolio matches the option payoffs exactly.

Example 2: One-Period Binomial Put Pricing

Stock A is trading at $60. Assume it can only go up to $75 or down to $45 in the next month. A put option on stock A has a strike of $65 and expires in a month. The risk-free rate is 4%.

Use the one-period binomial model to compute the fair value of the put option.

Solution

Step 1: Calculate put option payoffs in each state

If the stock rises to $75: \(P_u = \max(65 - 75, 0) = \$0\)

If the stock falls to $45: \(P_d = \max(65 - 45, 0) = \$20\)

Step 2: Calculate the hedge ratio

\[ \Delta = \frac{P_u - P_d}{S_u - S_d} = \frac{0 - 20}{75 - 45} = \frac{-20}{30} = -\frac{2}{3} \]

The negative delta indicates we short shares to replicate a put option. This makes sense—puts gain value when stocks fall.

Step 3: Calculate the bond position

\[ B = P_u - \Delta \cdot S_u = 0 - \left(-\frac{2}{3}\right) \times 75 = 0 + 50 = 50 \]

We invest $50 in risk-free bonds.

Step 4: Calculate the put option value

\[ V_0 = \Delta \cdot S + \frac{B}{1 + r_f} = -\frac{2}{3} \times 60 + \frac{50}{1.04} = -40 + 48.08 = \$8.08 \]

The fair value of this put option is $8.08.

Verification:

• Up state: \(-\frac{2}{3} \times 75 + 50 = -50 + 50 = \$0\) ✓
• Down state: \(-\frac{2}{3} \times 45 + 50 = -30 + 50 = \$20\) ✓

4.2 Generalizing to Multiple Periods

The assumption that stocks can only move to two possible prices is obviously unrealistic. However, we can make the model more realistic by dividing time into many short periods, each with its own up and down possibilities. By repeatedly applying the one-period valuation procedure and working backward from expiration, we can price options over arbitrarily complex price paths.

In practice, the binomial model with many periods converges to the continuous-time Black-Scholes model that we’ll explore next. The binomial approach remains valuable for building intuition and for pricing American options, where the possibility of early exercise complicates matters.

5 Black-Scholes Option Valuation

In 1973, Fischer Black and Myron Scholes (with contributions from Robert Merton) published a formula that revolutionized derivatives pricing. The Black-Scholes model takes the replication logic to its continuous-time limit, assuming that stock prices follow a smooth random process and that investors can trade continuously to maintain their replicating portfolios.

5.1 The Black-Scholes Formula

For a European call option on a non-dividend-paying stock, the Black-Scholes formula gives:

\[ C = S \cdot N(d_1) - X \cdot e^{-rT} \cdot N(d_2) \]

For a European put option:

\[ P = X \cdot e^{-rT} \cdot N(-d_2) - S \cdot N(-d_1) \]

where:

\[ d_1 = \frac{\ln(S/X) + (r + \sigma^2/2)T}{\sigma\sqrt{T}} \]

\[ d_2 = d_1 - \sigma\sqrt{T} \]

The variables are defined as follows:

Symbol	Description
\(S\)	Current stock price
\(X\)	Strike (exercise) price
\(r\)	Risk-free interest rate (annualized, continuously compounded)
\(T\)	Time to expiration (in years)
\(\sigma\)	Volatility of stock returns (annualized standard deviation)
\(N(\cdot)\)	Cumulative standard normal distribution function
\(C\)	Call option price
\(P\)	Put option price

The formula might look intimidating, but it has an elegant interpretation. The term \(S \cdot N(d_1)\) represents the expected stock value you receive if the option finishes in the money, adjusted for the probability of that outcome. The term \(X \cdot e^{-rT} \cdot N(d_2)\) is the present value of the strike price, weighted by the probability that you’ll have to pay it. The call price is the difference between what you expect to receive and what you expect to pay.

5.2 Understanding \(N(d_1)\) and \(N(d_2)\)

The \(N(\cdot)\) function is the cumulative distribution function of the standard normal distribution—it gives the probability that a standard normal random variable is less than or equal to the argument. For example, \(N(0) = 0.50\) (50% chance of being below zero), and \(N(1.96) \approx 0.975\).

In the Black-Scholes context, \(N(d_2)\) can be interpreted (under risk-neutral pricing) as the probability that the call option will be exercised. The term \(N(d_1)\) is related but also accounts for the delta of the option—how much stock you’d need to hold to replicate it.

5.3 Put-Call Parity

There’s a fundamental relationship between European call and put prices on the same underlying with the same strike and expiration:

\[ P = C + X \cdot e^{-rT} - S \]

This relationship, called put-call parity, allows us to price puts from calls or vice versa. It arises from a no-arbitrage argument: a portfolio of a call plus bonds (with face value \(X\)) produces the same payoffs as a portfolio of a put plus the stock. If the equation didn’t hold, traders could earn risk-free profits.

5.4 Excel Implementation

To compute Black-Scholes prices in Excel, use the following formulas. Assume the stock price is in cell A1, strike in A2, risk-free rate in A3, time to expiration in A4, and volatility in A5:

d1 = (LN(A1/A2) + (A3 + 0.5*A5^2)*A4) / (A5*SQRT(A4))
d2 = d1 - A5*SQRT(A4)
Call = A1*NORM.S.DIST(d1, TRUE) - A2*EXP(-A3*A4)*NORM.S.DIST(d2, TRUE)
Put = A2*EXP(-A3*A4)*NORM.S.DIST(-d2, TRUE) - A1*NORM.S.DIST(-d1, TRUE)

The NORM.S.DIST(x, TRUE) function returns \(N(x)\), the cumulative standard normal distribution evaluated at \(x\).

Example 3: Black-Scholes Call Pricing

Stock A is trading at $50 per share and has a volatility of 25%. A call option on A with strike of $52 expires in 6 months. The risk-free rate is 3%. Use the Black-Scholes model to estimate the fair value of this option.

Solution

Given:

• \(S = 50\)
• \(X = 52\)
• \(\sigma = 0.25\) (25%)
• \(T = 0.5\) (6 months = 0.5 years)
• \(r = 0.03\) (3%)

Step 1: Calculate \(d_1\)

\[ d_1 = \frac{\ln(50/52) + (0.03 + 0.25^2/2) \times 0.5}{0.25 \times \sqrt{0.5}} \]

\[ d_1 = \frac{\ln(0.9615) + (0.03 + 0.03125) \times 0.5}{0.25 \times 0.7071} \]

\[ d_1 = \frac{-0.0392 + 0.0306}{0.1768} = \frac{-0.0086}{0.1768} = -0.0487 \]

Step 2: Calculate \(d_2\)

\[ d_2 = d_1 - \sigma\sqrt{T} = -0.0487 - 0.25 \times 0.7071 = -0.0487 - 0.1768 = -0.2255 \]

Step 3: Find \(N(d_1)\) and \(N(d_2)\)

Using standard normal tables or Excel’s NORM.S.DIST function:

• \(N(d_1) = N(-0.0487) = 0.4806\)
• \(N(d_2) = N(-0.2255) = 0.4108\)

Step 4: Calculate the call price

\[ C = S \cdot N(d_1) - X \cdot e^{-rT} \cdot N(d_2) \]

\[ C = 50 \times 0.4806 - 52 \times e^{-0.03 \times 0.5} \times 0.4108 \]

\[ C = 24.03 - 52 \times 0.9851 \times 0.4108 \]

\[ C = 24.03 - 21.05 = \$2.98 \]

The fair value of this call option is approximately $2.98.

Example 4: Black-Scholes Put Pricing

A put option has a strike of $100 and expires in 2 years. The underlying is currently trading at $105 and has a volatility of 18%. The risk-free rate is 6%. Use the Black-Scholes formula to calculate the fair value of this put option.

Solution

Given:

• \(S = 105\)
• \(X = 100\)
• \(\sigma = 0.18\) (18%)
• \(T = 2\) years
• \(r = 0.06\) (6%)

Step 1: Calculate \(d_1\)

\[ d_1 = \frac{\ln(105/100) + (0.06 + 0.18^2/2) \times 2}{0.18 \times \sqrt{2}} \]

\[ d_1 = \frac{\ln(1.05) + (0.06 + 0.0162) \times 2}{0.18 \times 1.4142} \]

\[ d_1 = \frac{0.0488 + 0.1524}{0.2546} = \frac{0.2012}{0.2546} = 0.7903 \]

Step 2: Calculate \(d_2\)

\[ d_2 = d_1 - \sigma\sqrt{T} = 0.7903 - 0.18 \times 1.4142 = 0.7903 - 0.2546 = 0.5357 \]

Step 3: Find \(N(-d_1)\) and \(N(-d_2)\)

For put options, we need \(N(-d_1)\) and \(N(-d_2)\):

• \(N(-d_1) = N(-0.7903) = 0.2147\)
• \(N(-d_2) = N(-0.5357) = 0.2961\)

Step 4: Calculate the put price

\[ P = X \cdot e^{-rT} \cdot N(-d_2) - S \cdot N(-d_1) \]

\[ P = 100 \times e^{-0.06 \times 2} \times 0.2961 - 105 \times 0.2147 \]

\[ P = 100 \times 0.8869 \times 0.2961 - 22.54 \]

\[ P = 26.26 - 22.54 = \$3.72 \]

The fair value of this put option is approximately $3.72.

6 Assumptions Behind Black-Scholes

The Black-Scholes model is elegant and widely used, but it rests on several assumptions that may not hold in practice. Understanding these limitations is crucial for applying the model appropriately.

The model assumes that stock prices follow geometric Brownian motion, meaning returns are normally distributed and price changes are continuous (no sudden jumps). In reality, markets do experience sudden discontinuous moves, and return distributions have “fat tails”—extreme events occur more frequently than the normal distribution predicts.

Volatility is assumed constant over the life of the option. In practice, volatility fluctuates over time and can spike during market crises. Moreover, implied volatility varies across strikes (the “volatility smile” or “skew”), contradicting the model’s assumptions.

Interest rates are assumed constant and known, which is approximately true for short-dated options but less so for longer maturities. The basic model assumes no dividends, though extensions exist to handle dividend-paying stocks.

The model assumes frictionless markets: no transaction costs, no taxes, and the ability to trade any quantity (including fractional shares) at any time. Real markets have bid-ask spreads, commissions, and other frictions.

Finally, the model assumes no arbitrage opportunities and that investors can borrow and lend freely at the risk-free rate. While these assumptions are reasonable in liquid markets, they can break down during periods of market stress.

Despite these limitations, Black-Scholes remains the industry standard because it provides a common language for discussing option prices. Traders typically use the model as a framework while adjusting inputs (particularly volatility) to account for its shortcomings.

7 Implied Volatility

Of all the inputs to the Black-Scholes formula, volatility is the most difficult to estimate. Historical volatility—calculated from past returns—is backward-looking and may not reflect future uncertainty. This is where implied volatility becomes valuable.

Implied volatility is the value of \(\sigma\) that, when plugged into the Black-Scholes formula, produces the option’s current market price. It’s what the market is implying about future volatility, given the price at which the option trades.

To find implied volatility, we solve the Black-Scholes equation in reverse. Given the market price \(C_{market}\), find \(\sigma\) such that:

\[ C(\sigma) = C_{market} \]

There is no closed-form solution for this equation—we must solve it numerically. This is typically done using iterative methods like Newton-Raphson or, in Excel, using Goal Seek or Solver.

7.1 Computing Implied Volatility in Excel

Follow these steps to find implied volatility using Excel’s Goal Seek:

1. Set up cells for \(S\), \(X\), \(r\), \(T\), and an initial guess for \(\sigma\) (try 20% or 0.20)
2. Calculate \(d_1\), \(d_2\), and the Black-Scholes price using the formulas above
3. Go to Data → What-If Analysis → Goal Seek
4. Set the Black-Scholes price cell to equal the market price by changing the volatility cell
5. Excel will iterate to find the implied volatility

Implied volatility has become a crucial market indicator in its own right. The VIX index, often called the “fear gauge,” measures implied volatility on S&P 500 index options. When investors are nervous, they bid up option prices, driving implied volatility higher.

Example 5: Calculating Implied Volatility

Stock A is trading at $50 per share. A call option on A with strike of $52 expires in 6 months. The risk-free rate is 3%. The call option is trading at $2. What is the implied volatility of stock A?

Solution

Given:

• \(S = 50\)
• \(X = 52\)
• \(r = 0.03\) (3%)
• \(T = 0.5\) years
• Market call price = $2.00

Step 1: Set up the problem

We need to find \(\sigma\) such that the Black-Scholes call price equals $2.00.

Step 2: Use iterative approach

Let’s try \(\sigma = 0.20\) (20%):

\[ d_1 = \frac{\ln(50/52) + (0.03 + 0.20^2/2) \times 0.5}{0.20 \times \sqrt{0.5}} = \frac{-0.0392 + 0.025}{0.1414} = -0.1004 \]

\[ d_2 = -0.1004 - 0.1414 = -0.2418 \]

\[ C = 50 \times N(-0.1004) - 52 \times e^{-0.015} \times N(-0.2418) \] \[ C = 50 \times 0.4600 - 51.23 \times 0.4045 = 23.00 - 20.72 = \$2.28 \]

This is too high ($2.28 > $2.00), so we need lower volatility.

Step 3: Iterate

Try \(\sigma = 0.15\) (15%):

\[ d_1 = \frac{-0.0392 + 0.01875}{0.1061} = -0.1929 \]

\[ d_2 = -0.1929 - 0.1061 = -0.2990 \]

\[ C = 50 \times 0.4235 - 51.23 \times 0.3824 = 21.18 - 19.59 = \$1.59 \]

This is too low. The true volatility is between 15% and 20%.

Step 4: Narrow down

Continuing the iteration (or using Excel’s Goal Seek), we find:

Implied volatility ≈ 18.0%

At \(\sigma = 0.18\), the Black-Scholes formula produces a call price of approximately $2.00.

8 Key Takeaways

Option valuation represents one of the most elegant applications of financial theory, combining the principle of no-arbitrage with sophisticated mathematics to answer a practical question: what should an option cost? The journey from intrinsic value to implied volatility reveals how deeply interconnected these concepts are.

We began with the fundamental decomposition of option prices into intrinsic value and time value. Intrinsic value tells us what an option would be worth if exercised immediately—it’s the “sure thing” component. Time value captures everything else: the possibility that favorable price movements will increase the option’s worth before expiration. This decomposition explains why options trade above their intrinsic value and why that premium erodes as expiration approaches.

The factors that determine option values follow logically from the nature of optionality. Stock price, strike price, volatility, time to expiration, interest rates, and dividends all affect option prices, but the direction of these effects makes intuitive sense once you understand what options really are. Most notably, volatility helps option holders due to the asymmetric nature of option payoffs—you capture the upside while your downside is limited.

The binomial model introduced the revolutionary concept of replication. By constructing a portfolio of stocks and bonds that exactly matches an option’s payoffs in all states, we can determine the option’s fair price without knowing the probability of each outcome. The hedge ratio (delta) tells us precisely how many shares we need, and the model extends naturally to multiple periods by working backward through time.

Black-Scholes takes replication to its continuous-time limit, producing a closed-form formula that has become the lingua franca of options markets. While the formula’s assumptions don’t perfectly match reality—markets aren’t frictionless, volatility isn’t constant, and returns aren’t perfectly lognormal—Black-Scholes provides a crucial benchmark for pricing and risk management. Put-call parity connects puts and calls through a simple no-arbitrage relationship.

Finally, implied volatility turns the pricing problem on its head: given market prices, what volatility is the market expecting? This “reverse engineering” has made implied volatility one of the most important indicators of market sentiment, embodied in measures like the VIX index. Understanding implied volatility helps traders identify whether options are cheap or expensive relative to their expectations for future volatility.

9 Key Formulas Summary

Concept	Formula	When to Use
Intrinsic Value (Call)	\(\max(S - X, 0)\)	Finding immediate exercise value of a call
Intrinsic Value (Put)	\(\max(X - S, 0)\)	Finding immediate exercise value of a put
Option Price Decomposition	\(\text{Price} = \text{Intrinsic Value} + \text{Time Value}\)	Analyzing option premium components
Hedge Ratio (Delta)	\(\Delta = \frac{C_u - C_d}{S_u - S_d}\)	Binomial model: shares needed to replicate option
Bond Position	\(B = C_u - \Delta \cdot S_u\)	Binomial model: bond investment for replication
Binomial Option Value	\(V_0 = \Delta S + \frac{B}{1 + r_f}\)	Binomial model: current option value
Black-Scholes \(d_1\)	\(d_1 = \frac{\ln(S/X) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}\)	Input for Black-Scholes call/put formulas
Black-Scholes \(d_2\)	\(d_2 = d_1 - \sigma\sqrt{T}\)	Input for Black-Scholes call/put formulas
Black-Scholes Call	\(C = S \cdot N(d_1) - X \cdot e^{-rT} \cdot N(d_2)\)	Pricing European calls on non-dividend stocks
Black-Scholes Put	\(P = X \cdot e^{-rT} \cdot N(-d_2) - S \cdot N(-d_1)\)	Pricing European puts on non-dividend stocks
Put-Call Parity	\(P = C + X \cdot e^{-rT} - S\)	Converting between call and put prices

10 Practice Problems

Practice Problem 1: Binomial Put Option

Stock B is currently trading at $80. Over the next period, it can rise to $100 or fall to $65. A put option on Stock B has a strike of $85 and expires in one period. The risk-free rate is 5%.

Use the one-period binomial model to compute the fair value of this put option.

Solution

Step 1: Calculate put payoffs in each state

If stock rises to $100: \(P_u = \max(85 - 100, 0) = \$0\)

If stock falls to $65: \(P_d = \max(85 - 65, 0) = \$20\)

Step 2: Calculate the hedge ratio

\[ \Delta = \frac{P_u - P_d}{S_u - S_d} = \frac{0 - 20}{100 - 65} = \frac{-20}{35} = -0.5714 \]

Step 3: Calculate the bond position

\[ B = P_u - \Delta \cdot S_u = 0 - (-0.5714) \times 100 = 57.14 \]

Step 4: Calculate the put value

\[ V_0 = \Delta \cdot S + \frac{B}{1 + r_f} = (-0.5714) \times 80 + \frac{57.14}{1.05} \]

\[ V_0 = -45.71 + 54.42 = \$8.71 \]

The fair value of this put option is $8.71.

Verification:

• Up state: \((-0.5714) \times 100 + 57.14 = -57.14 + 57.14 = \$0\) ✓
• Down state: \((-0.5714) \times 65 + 57.14 = -37.14 + 57.14 = \$20\) ✓

Practice Problem 2: Black-Scholes Call Option

Company XYZ stock trades at $75 with a volatility of 30%. A call option with a $70 strike expires in 3 months. The risk-free rate is 4%. Calculate the fair value of this call using Black-Scholes.

Solution

Given:

• \(S = 75\)
• \(X = 70\)
• \(\sigma = 0.30\) (30%)
• \(T = 0.25\) (3 months)
• \(r = 0.04\) (4%)

Step 1: Calculate \(d_1\)

\[ d_1 = \frac{\ln(75/70) + (0.04 + 0.30^2/2) \times 0.25}{0.30 \times \sqrt{0.25}} \]

\[ d_1 = \frac{\ln(1.0714) + (0.04 + 0.045) \times 0.25}{0.30 \times 0.5} \]

\[ d_1 = \frac{0.0690 + 0.02125}{0.15} = \frac{0.0903}{0.15} = 0.6017 \]

Step 2: Calculate \(d_2\)

\[ d_2 = 0.6017 - 0.30 \times 0.5 = 0.6017 - 0.15 = 0.4517 \]

Step 3: Find \(N(d_1)\) and \(N(d_2)\)

• \(N(d_1) = N(0.6017) = 0.7263\)
• \(N(d_2) = N(0.4517) = 0.6743\)

Step 4: Calculate the call price

\[ C = 75 \times 0.7263 - 70 \times e^{-0.04 \times 0.25} \times 0.6743 \]

\[ C = 54.47 - 70 \times 0.9900 \times 0.6743 \]

\[ C = 54.47 - 46.73 = \$7.74 \]

The fair value of this call option is approximately $7.74.

Practice Problem 3: Black-Scholes Put Option

A put option has a strike of $45 and expires in 9 months. The underlying stock trades at $42 and has volatility of 22%. The risk-free rate is 5%. Calculate the put’s fair value.

Solution

Given:

• \(S = 42\)
• \(X = 45\)
• \(\sigma = 0.22\) (22%)
• \(T = 0.75\) (9 months)
• \(r = 0.05\) (5%)

Step 1: Calculate \(d_1\)

\[ d_1 = \frac{\ln(42/45) + (0.05 + 0.22^2/2) \times 0.75}{0.22 \times \sqrt{0.75}} \]

\[ d_1 = \frac{\ln(0.9333) + (0.05 + 0.0242) \times 0.75}{0.22 \times 0.866} \]

\[ d_1 = \frac{-0.0690 + 0.0557}{0.1905} = \frac{-0.0133}{0.1905} = -0.0698 \]

Step 2: Calculate \(d_2\)

\[ d_2 = -0.0698 - 0.22 \times 0.866 = -0.0698 - 0.1905 = -0.2603 \]

Step 3: Find \(N(-d_1)\) and \(N(-d_2)\)

• \(N(-d_1) = N(0.0698) = 0.5278\)
• \(N(-d_2) = N(0.2603) = 0.6027\)

Step 4: Calculate the put price

\[ P = X \cdot e^{-rT} \cdot N(-d_2) - S \cdot N(-d_1) \]

\[ P = 45 \times e^{-0.05 \times 0.75} \times 0.6027 - 42 \times 0.5278 \]

\[ P = 45 \times 0.9632 \times 0.6027 - 22.17 \]

\[ P = 26.12 - 22.17 = \$3.95 \]

The fair value of this put option is approximately $3.95.

Practice Problem 4: Implied Volatility

Stock Z trades at $65. A call option with a $60 strike expires in 4 months. The risk-free rate is 3.5%. The call currently trades at $8.50. What is the implied volatility?

Solution

Given:

• \(S = 65\)
• \(X = 60\)
• \(r = 0.035\) (3.5%)
• \(T = 0.333\) (4 months)
• Market price = $8.50

Approach: Iterate to find \(\sigma\) that produces $8.50

Try \(\sigma = 0.25\) (25%):

\[ d_1 = \frac{\ln(65/60) + (0.035 + 0.25^2/2) \times 0.333}{0.25 \times 0.577} = \frac{0.0800 + 0.0221}{0.1443} = 0.7075 \]

\[ d_2 = 0.7075 - 0.1443 = 0.5632 \]

\[ C = 65 \times N(0.7075) - 60 \times e^{-0.0117} \times N(0.5632) \] \[ C = 65 \times 0.7604 - 59.30 \times 0.7133 = 49.43 - 42.30 = \$7.13 \]

Too low—need higher volatility.

Try \(\sigma = 0.35\) (35%):

\[ d_1 = \frac{0.0800 + (0.035 + 0.0612) \times 0.333}{0.35 \times 0.577} = \frac{0.0800 + 0.0321}{0.202} = 0.5549 \]

\[ d_2 = 0.5549 - 0.202 = 0.3529 \]

\[ C = 65 \times 0.7105 - 59.30 \times 0.6379 = 46.18 - 37.83 = \$8.35 \]

Close but slightly low.

After further iteration: Implied volatility ≈ 36.1%

At \(\sigma \approx 0.361\), the Black-Scholes formula produces approximately $8.50.

11 Ask an LLM

Here are three questions you might ask an AI assistant to deepen your understanding of these concepts:

• The Black-Scholes model assumes constant volatility, but in real markets we observe the “volatility smile” where implied volatility varies across strike prices. What causes this phenomenon, and how do practitioners adjust for it when pricing options?
• Can you walk me through an intuitive explanation of why the probability of stock price movements doesn’t appear in the binomial option pricing formula? This seems counterintuitive—shouldn’t expected returns matter?
• How would I modify the Black-Scholes formula to price options on a stock that pays a known dividend yield? What changes in the formula and why?
• The VIX index is often called the market’s “fear gauge.” Can you explain how it’s calculated from S&P 500 option prices and why it spikes during market crises?
• What are the main differences between pricing American options versus European options, and why is the binomial model still widely used for American options even though Black-Scholes is available?