Valuation of Contingent Claims

CFA Level II · CFA · 7 min read

Welcome to the World of Contingent Claims!

Hello there! Today, we are diving into one of the most fascinating areas of the CFA Level II curriculum: Valuation of Contingent Claims. While "contingent claims" might sound like legal jargon, it’s just a fancy way of saying options. These are assets whose value "depends" (is contingent) on the value of something else, like a stock or a bond.

Don't worry if this seems a bit math-heavy at first. We are going to break it down into simple, logical steps. Think of this chapter as learning how to put a fair price tag on a "choice." By the end of these notes, you'll understand how we use tree diagrams and sophisticated formulas to figure out exactly what an option is worth. Let's get started!

1. The Binomial Option Pricing Model

The Binomial Model is our starting point. It assumes that in the next period, a stock price can only do two things: go up or go down. It’s like a "choose your own adventure" map for stock prices.

The One-Step Binomial Model

Imagine a stock is trading at $ S $. In one period, it can go up to $ S^u $ or down to $ S^d $. To value an option on this stock, we use the concept of No-Arbitrage. This means two portfolios with the same risk and same payoff must have the same price.

Step 1: Calculate the Hedge Ratio (Delta, $\Delta$)
Delta tells us how many shares of stock we need to buy to perfectly hedge one short call option. It is the change in the option price divided by the change in the stock price:
$ \Delta = \frac{c^u - c^d}{S^u - S^d} $
Where $ c^u $ is the option value if the stock goes up, and $ c^d $ is the value if it goes down.

Step 2: Risk-Neutral Probabilities
This is a "trick" that makes math easier. We pretend investors are risk-neutral (they don't care about risk, only expected returns). In this world, the expected return on a stock is the risk-free rate ($ r $).
The probability of an "up" move ($ \pi $) is:
$ \pi = \frac{1 + r - d}{u - d} $
Note: $ u $ is the "up factor" (e.g., 1.10 for a 10% increase) and $ d $ is the "down factor" (e.g., 0.90 for a 10% decrease).

Step 3: Calculate the Value
The value of the call option today ($ c_0 $) is the discounted expected value using these probabilities:
$ c_0 = \frac{\pi c^u + (1-\pi) c^d}{1 + r} $

Two-Step Binomial Model

A two-step model just repeats the process. You start at the end (the "leaves" of the tree), calculate the values at the middle nodes, and then work backward to today. It's like retracing your steps to find your keys.

Key Takeaway: The binomial model uses risk-neutral valuation. We don't need to know the actual probability of the stock going up; we only need the synthetic probability that makes the math work with the risk-free rate.

2. The Black-Scholes-Merton (BSM) Model

If the binomial model is a simple ladder, the BSM Model is a high-speed elevator. It assumes that the periods in our binomial tree become infinitely small, leading to a smooth, continuous curve of price movements.

BSM Assumptions (Must Know!)

The CFA exam loves to test these. To use the BSM model, we assume:
• The underlying stock price follows a lognormal distribution (prices can't be negative).
• The risk-free rate is constant and known.
• Volatility of the underlying asset is constant and known.
• Markets are continuous (no sudden jumps in price).
• There are no taxes or transaction costs.
• The options are European-style (can only be exercised at expiry).

The BSM Formula Components

You don't usually need to calculate the full formula by hand, but you must understand its parts:
$ c = S_0 N(d_1) - Ke^{-rt} N(d_2) $

• $ S_0 N(d_1) $: This is the benefit of owning the stock. $ N(d_1) $ is also the Delta of the option.
• $ Ke^{-rt} N(d_2) $: This is the "cost" of exercising the option (the strike price $ K $) discounted back to today. $ N(d_2) $ is the probability that the option will finish "in-the-money."

Quick Review: Think of the BSM formula as: (What you get) minus (What you pay), adjusted for probabilities and the time value of money.

3. The Greeks

The "Greeks" measure how sensitive an option's price is to different factors. Think of them as the "weather report" for your option.

1. Delta ($ \Delta $): Sensitivity to the price of the underlying asset.
• Calls have positive Delta (0 to 1).
• Puts have negative Delta (-1 to 0).
• Analogy: If Delta is 0.5, for every \$1 the stock rises, your call option rises by \$0.50.

2. Gamma ($ \Gamma $): Sensitivity of Delta to changes in the underlying price.
• It measures how "stable" your Delta is. High Gamma means Delta changes very quickly.
• Gamma is highest when an option is "at-the-money."

3. Theta ($ \theta $): Sensitivity to the passage of time.
• Options lose value as they get closer to expiration. This is called "time decay."
• Theta is almost always negative for option holders.

4. Vega ($ \nu $): Sensitivity to volatility.
• If the market gets "crazier" (higher volatility), options become more valuable because there is a higher chance they will end up deep in-the-money.
• Common Mistake: Don't confuse Vega with Delta. Vega is about the "speed" of the market, not the "direction."

5. Rho ($ \rho $): Sensitivity to interest rates.
• Call options generally increase in value when interest rates rise.

Key Takeaway: Delta is the most important for hedging. If you are "Delta-neutral," you have neutralized your risk to small price changes in the stock.

4. Valuing Options on Other Underlyings

We can use the BSM model for more than just non-dividend-paying stocks. We just need to adjust the formula for "carry costs" or "leakages."

Options on Dividend-Paying Stocks

Dividends are like "leaks" in the stock price. When a dividend is paid, the stock price drops. We adjust the BSM by replacing the stock price $ S_0 $ with $ S_0 e^{-qt} $, where $ q $ is the dividend yield.

Options on Currencies (The Garman-Kohlhagen Model)

Currencies are unique because both currencies have interest rates. The "foreign" interest rate ($ r_f $) acts just like a dividend yield. We replace $ q $ in the formula with $ r_f $.

Options on Futures (The Black Model)

Futures are different because they cost nothing to enter today. The Black model uses the Futures Price ($ F $) instead of the spot price and assumes the "cost of carry" is zero because the futures price already incorporates interest rates.

Did you know? The Black model is widely used for valuing options on bonds and interest rates, not just commodity futures!

5. Implied Volatility

Usually, we plug volatility into the BSM formula to get the price. But in the real world, we see the price in the market. Implied Volatility (IV) is the volatility number that, when plugged into the BSM, makes the formula equal the market price.

• If IV is high, the option is "expensive."
• If IV is low, the option is "cheap."
• The Volatility Smile: Real-world IV often differs across strike prices. This suggests that the BSM assumption of constant volatility isn't always true in the real world!

Final Encouragement

You've made it through the core concepts of Valuation of Contingent Claims! Remember, the CFA exam doesn't just want you to memorize formulas; it wants you to understand the relationships. If volatility goes up, what happens to the call price? (It goes up!). If time passes, what happens to the put price? (Usually, it goes down!).

Keep practicing those binomial trees and Greek definitions. You've got this!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is a contingent claim in financial markets?

A contingent claim is a derivative asset, such as an option, whose value depends on the value of another underlying asset.

How is the Hedge Ratio (Delta) calculated in the Binomial Model?

$ \Delta = \frac{c^u - c^d}{S^u - S^d} $

What is the formula for the risk-neutral probability of an upward move ($ \pi $)?

$ \pi = \frac{1 + r - d}{u - d} $

How does a Two-Step Binomial Model determine the option value today?

The model uses backward induction, starting at the final nodes and working backward through middle nodes to find the initial value.

What are the key assumptions regarding stock prices and volatility in the Black-Scholes-Merton (BSM) model?

The BSM model assumes stock prices follow a lognormal distribution and that volatility is constant and known.

In the BSM formula, what is the significance of $ N(d_1) $?

$ N(d_1) $ represents the Delta of the option, measuring its sensitivity to the underlying stock price.

In the BSM formula, what does the term $ N(d_2) $ represent?

$ N(d_2) $ is the risk-neutral probability that the option will expire in-the-money (exercise probability).

What is the Delta ($ \Delta $) of a call option versus a put option?

Call options have positive Delta (0 to 1), while put options have negative Delta (-1 to 0).

What does Gamma ($ \Gamma $) measure in the context of option Greeks?

Gamma measures the sensitivity of the option's Delta to changes in the underlying asset's price.

What is Theta ($ \theta $) and how does it usually affect an option holder?

Theta measures time decay; it is usually negative, meaning the option loses value as it approaches expiration.

What does Vega ($ \nu $) represent in option valuation?

Vega measures the sensitivity of the option's price to changes in the volatility of the underlying asset.

How is the BSM model adjusted for a stock that pays a continuous dividend yield ($ q $)?

The current stock price $ S_0 $ is replaced with $ S_0 e^{-qt} $.

How are currency options valued using the BSM framework (Garman-Kohlhagen model)?

The foreign interest rate ($ r_f $) is treated as a continuous dividend yield ($ q $) in the formula.

What is the Black Model and when is it used?

The Black Model is a variation of BSM used for valuing options on futures, bonds, and interest rates.

What is Implied Volatility (IV)?

Implied Volatility is the volatility value that makes the BSM model price equal the current market price of the option.

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