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LM10 Valuing a Derivative Using a One-Period Binomial Model 2025 Level I Notes © IFT. All rights reserved 1 LM10 Valuing a Derivative Using a One-Period Binomial Model 1. Introduction ........................................................................................................................................................ 2 2. Binomial Valuation ........................................................................................................................................... 2 3. The Binomial Model ......................................................................................................................................... 2 4. Pricing a European Call Option ................................................................................................................... 3 5. Risk Neutrality ................................................................................................................................................... 5 Summary ................................................................................................................................................................... 7 Required disclaimer: IFT is a CFA Institute Prep Provider. Only CFA Institute Prep Providers are permitted to make use of CFA Institute copyrighted materials which are the building blocks of the exam. We are also required to create / use updated materials every year and this is validated by CFA Institute. Our products and services substantially cover the relevant curriculum and exam and this is validated by CFA Institute. In our advertising, any statement about the numbers of questions in our products and services relates to unique, original, proprietary questions. CFA Institute Prep Providers are forbidden from including CFA Institute official mock exam questions or any questions other than the end of reading questions within their products and services. CFA Institute does not endorse, promote, review or warrant the accuracy or quality of the product and services offered by IFT. CFA Institute®, CFA® and “Chartered Financial Analyst®” are trademarks owned by CFA Institute. © Copyright CFA Institute Version 1.0
LM10 Valuing a Derivative Using a One-Period Binomial Model 2025 Level I Notes © IFT. All rights reserved 2 1. Introduction This learning module covers:  How to value a derivative using a one-period binomial model  The concept of risk-neutrality in derivatives pricing 2. Binomial Valuation According to the law of one price, if the payoffs from any two assets at a given future date are identical, then the value of these two assets must also be identical today. Forward commitments can be priced without making assumptions about the future price of the underlying asset. However, due to their asymmetric payoffs, the pricing of options requires a model to factor in the random price behavior of the underlying asset. A commonly used model to determine the no-arbitrage value of an option is the binomial model. 3. The Binomial Model The binomial model assumes that over a given time period, there are only two possible outcomes - the asset’s price will either go up to S1 u or down to S1 d . Let q denote the probability of an upward price movement and 1 – q denote the probability of a downward price movement. With only two possible outcomes, the sum of these two probabilities must equal to 1. To value options, knowing q is not required, only the values S1 u and S1 d are needed. The difference between S1 u and S1 d represents the spread of possible future price outcomes or the volatility of the underlying asset. We can also define the upward and downward moves in terms of returns R u = S1 u /S0 > 1 R d = S1 d /S0 < 1 Exhibit 1 from the curriculum, illustrates the binomial model.
LM10 Valuing a Derivative Using a One-Period Binomial Model 2025 Level I Notes © IFT. All rights reserved 3 4. Pricing a European Call Option Consider a one-year European call option with an exercise price (X) of 100. The underlying spot price (S0) is 80. The two possible stock prices after 1 year are: S1 d = 60 and S1 u=110 We can compute the up and down returns as: Ru = 110/80 = 1.375 Rd = 60/80 = 0.75 At t = 0, the call option value is c0. At t = 1, the call option value is either c1 u (if the price goes up) or c1 d (if the price goes down). We can compute the values c1 u and c1 d as: Up move: The call option ends in-the-money c1 u = Max (0, ST − X) = Max(0,110 − 100) = 10 Down move: The call option ends out-of-the-money c1 d = Max (0, ST − X) = Max(0,60 − 100) = 0 To compute c0 we can use replication and no-arbitrage pricing. Assume that at t = 0 we sell the call option at a price of c0 and purchase h units of the underlying asset. V0 represents the initial investment in the portfolio, and V1 u and V1 d represent the portfolio value if the underlying price moves up or down, respectively. V0 = hS0 − c0 V1 u = hS1 u − c1 u V1 d = hS1 d − c1 d
LM10 Valuing a Derivative Using a One-Period Binomial Model 2025 Level I Notes © IFT. All rights reserved 4 We solve for h, such that the portfolio payoff in both the up and down scenarios are equal, i.e.: V1 u = V1 d hS1 u − c1 u = hS1 d − c1 d Solving for h, we get: h = c1 u − c1 d S1 u − S1 d This equation gives us the hedge ratio, or the proportion of the underlying that will offset the risk associated with an option. The hedge ratio for our example is: h = c1 u − c1 d S1 u − S1 d = 10 − 0 110 − 60 = 0.20 i.e. for each call option unit sold, we must buy 0.2 units of the underlying asset. The portfolio values at t=1 for the up and down scenarios are: V1 u = hS1 u − c1 u = 0.20 × 110 − 10 = 12 V1 d = hS1 d − c1 d = 0.20 × 60 − 0 = 12 As expected, the portfolio values are the same, which has two implications: 1. We can use either portfolio to value the derivative, and 2. The return V1 u /V0 = V1 d /V0 must equal one plus the risk-free rate. To prevent arbitrage, the portfolio value at t =1 should be discounted at the risk-free rate so that: V0 = V1 1 + r For our example, V1 =12. Assume that the annual risk-free rate is 5%. Therefore, V0 = 12 1.05 = 11.42 Replacing V0 by the hedged portfolio we get: hS0 − c0 = 11.42 c0 = hS0 − 11.42 = 0.2 × 80 − 11.42 = 4.58 The value of the call option at t = 0 is 4.58. Example: