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LATTICE MODELS

By ProbabilityofProfit.com

Lattice Models is referred to as a type of model, which includes binomial, trinomial, and multinomial models that create a lattice of potential future outcomes, determining value on a weighted average of calculated outcomes. 
Because of their capacity to predict a range of future possibilities, these models find greater relevance in valuing American-style options. This is because American options can be exercised at any point before their expiration date, resulting in a broader spectrum of potential outcomes to consider.

LATTICE MODEL VARIATIONS

BINOMIAL MODEL

Binomial Model often regarded as a popular method for its simple implementation to price options outcomes. The Binomial model was first introduced by John Carrington Cox, Stephen Ross, and Mark Edward Rubinstein in 1979. This model prices options each point in time up until the expiration term considering that the price of the underlying asset moves up or down each day.
  • Using the possible underlying prices that are generated from the binomial model, we can then calculate the price of the option. 
  • Unlike the Black-Scholes model, the Binomial model allows for a more flexible and intuitive approach to pricing options.

 

TRINOMIAL MODEL

Trinomial Model is an expanded version of the binomial model in which three potential movements are considered, either allowing for an increase, decrease or stable condition of the stock price. The Trinomial model was first introduced by Phelim Boyle in 1986
  • While the Black-Scholes model is more efficient in valuing an options price, the Trinomial model is more flexible in that it can accommodate more complex scenarios and variations in market conditions.

 

CALCULATIONS

BINOMIAL ASSUMPTIONS

Binomial Assumption #1: The continuous movement of the asset price consists of a discrete random walk.
  • Discrete Time Steps: The model assumes that the continuous movement of the asset price can be approximated by a discrete random walk. This means that the asset price only changes at specific, evenly spaced time intervals, denoted by Δt. For example, if Δt is one day, then the asset price changes only once per day.
  • Price Movements: At each time step (Δt), the asset price can either go up or down by a certain factor. If the asset price at time nΔt is Sn, then at time (n+1)Δt, it can either become u times Sn or d times Sn, where u is the factor by which the price increases, and d is the factor by which it decreases. These factors remain constant for all time steps.
  • Probabilities of Movements: The model assumes that the probabilities of the asset price moving up or down are known and remain constant over time. These probabilities are denoted by p for the probability of an up movement and (1 - p) for the probability of a down movement.
  • Dividing Time into Steps: The remaining life-time of the derivative security is divided into N time-steps, where each time-step has a size of Δt = T/N. T represents the total time until the expiry date of the derivative. So, if T is one year and N is 12, each time-step would represent one month.
  • Constructing the Tree: A binomial tree is created to represent all possible movements of the underlying asset price over time. The tree starts at the initial value of the underlying asset, denoted as S0. At each time-step, the asset price can move either up or down by certain factors (u and d) from its current value.
  • Possible Asset Prices at Each Step: At the first step (time 0), the tree starts with the initial asset price, S = S0. From there, there are only two possible asset prices at the next step: uS0 (if the price goes up) or dS0 (if the price goes down). This creates two branches in the tree representing the two possible price movements.
  • Continuing the Process: At each subsequent time-step, the process repeats. For example, at the second step, there are three possible asset prices: u2S0 (if the price goes up twice), udS0 (if the price goes up then down), and d2S0 (if the price goes down twice). This pattern continues until the expiry date T of the security is reached
Binomial Assumption #2: No arbitrage opportunity exists. Therefore leading to a determination of a riskless portfolio.
  • Value Equivalence in the Binomial Tree: In the binomial tree model, it's observed that the value of an asset remains the same when an up-jump is followed by a down-jump and vice versa (udS = duS). This implies that the binomial tree "reconnects" in such cases. Additionally, after n time steps, there are only n + 1 possible asset prices.
  • No Arbitrage Opportunity Assumption: The second assumption is that no arbitrage opportunities exist in the market. This assumption is crucial for ensuring fair pricing of financial derivatives.
  • Portfolio Construction: Consider a stock with a current price of S0 and an option on the stock with a current price of f. When the stock price moves up to uS0, the payoff from the option becomes fu, and when it moves down to dS0, the payoff becomes fd.
  • Determining Portfolio Value: Construct a portfolio consisting of a long position in Δ shares of the stock and a short position in one option. The value of this portfolio after the stock price moves up is uS0Δ - fu, and after it moves down is dS0Δ - fd.
  • Equating Portfolio Values: For the portfolio to be riskless, its value must remain the same regardless of the stock price movement. Equating the two values obtained above, we derive an equation for uS0Δ – fu = dS0 Δ – fd, the number of shares to hold in the portfolio.
  • Riskless Portfolio and Arbitrage: When the portfolio is riskless, it must earn the risk-free interest rate, implying that no arbitrage opportunities could exist. The value of Δ represents the ratio of the change in the option price to the change in the stock price Δ=(fu-fd)/(S0u-S0d).
  • Bridge to Continuous Models: The binomial tree model serves as a stepping stone towards continuous models, such as the Black-Scholes option pricing model. The Black-Scholes model builds upon these principles to provide more accurate pricing estimates for financial derivatives.

BINOMIAL CALCULATION STEPS

Once you've constructed the binomial tree representing all possible movements of the underlying asset price over time, the next steps typically involve:
  • Step 1 - Calculating Option Prices: Using the binomial tree, you can calculate the option prices at each node of the tree. This involves working backward from the final nodes (i.e., at expiration) to the initial node (current time). At each node, you compute the option price based on the expected future payoff discounted back to the present time using the risk-neutral probability. 
    • If ST - K > 0 (where ST is the asset price at expiration and K is the strike price), the call option has intrinsic value because the asset price is higher than the strike price. In this case, the payoff is ST - K
    • If ST - K ≤ 0, the call option has no intrinsic value because exercising the option would result in a negative payoff. In this case, the payoff is 0
    • If K - ST > 0, the put option has intrinsic value because the strike price is higher than the asset price at expiration. In this case, the payoff is K - ST.
    • If K - ST ≤ 0, the put option has no intrinsic value because exercising the option would result in a negative payoff. In this case, the payoff is 0.
       
  • Step 2 - Discount Future Payoff: Once you've calculated the option payoff at each possible future price, you discount it back to the present time using the risk-free interest rate r. Where Δt is the time step and e is the base of the natural logarithm.
    This is done to bring future cash flows to their present value. The formula for discounting is:
  • Step 3 - Calculate Expected Payoff: To compute the expected payoff, you consider both possible future prices weighted by their respective probabilities. The risk-neutral probability p is used for the up movement, and (1-p) is used for the down movement. The expected payoff formula is:
  • Repeat for Each Node: Repeat steps 1-3 for each node of the binomial tree, starting from the layer just before expiration and working backward towards the initial node. 
  • Value of the Option: Once you've calculated the option prices at each node, the value of the option at the initial node (current time) represents its fair market price under the assumptions of the model.

BINOMIAL EXAMPLE

Imagine you're an investor interested in purchasing a call option on a stock. The current price of the stock (S0) is $50, and the option has a strike price (K) of $55. The expiration date of the option is one year from now, and you've divided this time period into monthly intervals (N = 12). 

You've also determined that there are two possible movements for the stock price at each time step: either it can go up by a factor of u = 1.1 or down by a factor of d = 0.9. Additionally, you've calculated the probabilities of an up movement (p) and a down movement (1 - p) based on market data.

Here are the input values:
  • At time 0 (initial node), the stock price is $50 (S0).
  • At time 1 (first step), the stock price can either go up to $50 * 1.1 = $55 (uS0) or down to $50 * 0.9 = $45 (dS0).
  • At time 2 (second step), if the price went up in the first step, it can go up again to $55 * 1.1 = $60.50 (u2S0), stay the same at $55, or go down to $55 * 0.9 = $49.50 (udS0). If the price went down in the first step, it can go down again to $45 * 0.9 = $40.50 (d2S0), stay the same at $45, or go up to $45 * 1.1 = $49.50 (udS0).

 

  • This process continues until we reach the expiration date with all possible stock prices represented in the binomial tree. To keep this example straight and to the point, we will leave this at the second time step.
     
Now, let's calculate: 
  • Once the binomial tree is established and all possible stock prices are represented we calculate the option prices based on the payoff at expiration and then discount them back to the present using the risk-free interest rate for each node of the tree.Lastly, we consider the expected payoff based on the probabilities of future movements for each node. 
  • Initial Node: $50 (S0) 
    • Option Payoff at Expiration:
      • Intrinsic value = $50 - $50 = $0 (since $50 = $50). 
  • Node 1: $55 (uS0) 
    • Option Payoff at Expiration:
      • Intrinsic value = $55 - $50 = $5 (since $55 > $50).  
    • Discount Future Payoff to Present Time: 
      • Discount factor = e-0.05*1/12 ≈ 0.995
      • Option Price = $5 * 0.995 ≈ $4.975 
    • Calculate Expected Payoff: 
      • Expected Payoff = $4.975 (since there's no chance of the price going down).
  • Node 2: $45 (dS0) 
    • Option Payoff at Expiration:
      • Intrinsic value = $0 (since $45 ≤ $50).
    • Discount Future Payoff to Present Time: 
      • Option Price = $0 (since there's no intrinsic value).
    • Calculate Expected Payoff: 
      • Expected Payoff = $0 (since there's no chance of the price going up).
  • Node 3: $60.50 (u2S0)
    • Option Payoff at Expiration:
      • Intrinsic value = $60.50 - $50 = $10.50 (since $60.50 > $50).
      Discount Future Payoff to Present Time: 
      • Discount factor = e-0.05*1/12 ≈ 0.995 
      • Option Price = $10.50 * 0.995 ≈ $10.4475.
    • Calculate Expected Payoff: 
      • Expected Payoff = $0.6 * $10.4475 ≈ $6.2685 (since there's a 60% chance of the price going up).
  • Node 4: $49.50 (udS0)
    • Option Payoff at Expiration:
      • Intrinsic value = $0 (since $49.50 ≤ $50).
      Discount Future Payoff to Present Time: 
      • Option Price = $0 (since there's no intrinsic value).
    • Calculate Expected Payoff: 
      • Expected Payoff = $0 (since there's no chance of the price going up).
  • Node 5: $40.50 (d2S0)
    • Option Payoff at Expiration:
      • Intrinsic value = $0 (since $40.50 ≤ $50).
      Discount Future Payoff to Present Time: 
      • Option Price = $0 (since there's no intrinsic value).
    • Calculate Expected Payoff: 
      • Expected Payoff = $0 (since there's no chance of the price going up).
  • Node 6: $49.50 (duS0)
    • Option Payoff at Expiration:
      • Intrinsic value = $0 (since $49.50 ≤ $50).
      Discount Future Payoff to Present Time: 
      • Option Price = $0 (since there's no intrinsic value).
    • Calculate Expected Payoff: 
      • Expected Payoff = $0 (since there's no chance of the price going up).

BREAKDOWN

  • Option Payoff Dynamics: 
    • Nodes where the stock price is higher than the strike price result in positive option payoffs, indicating the option is "in the money" and has intrinsic value.
    • Conversely, nodes where the stock price is equal to or lower than the strike price yield zero option payoffs, indicating the option is "out of the money" and has no intrinsic value. 
  • Impact of Price Movements: 
    • The option payoffs reflect the impact of price movements on the option's value. For example, when the stock price increases (as in Node 1 and Node 3), the option payoff also increases, leading to higher option prices and expected payoffs.
    • Conversely, when the stock price decreases (as in Node 2, Node 4, Node 5, and Node 6), the option payoffs are zero, resulting in zero option prices and expected payoffs. 
  • Discounted Option Prices:
    •  The discounted option prices provide insights into the present value of future option payoffs. Nodes with positive option payoffs result in discounted option prices greater than zero, while nodes with zero option payoffs lead to zero discounted option prices. 
  • Expected Payoffs: 
    • ​​​​​​​The expected payoffs at each node represent the weighted average of possible future option payoffs, considering the probabilities of different price movements.
    • Nodes with higher probabilities of favorable price movements have higher expected payoffs, reflecting the impact of probabilities on option pricing.

EXAMPLE CONCLUSION

This example highlights the complexity of stock option pricing and explains how investors can use a Binomial model to assess their fair market value of options now and in the future. Understanding the dynamics of option payoffs, discounted option prices, and expected payoffs is crucial for pricing options accurately and identifying potential market inaccuracies and opportunity for market arbitrage.

PROS & CONS

PROS

CONS

Flexibility: Lattice models, including binomial, trinomial, and multinomial models, offer flexibility in modeling various scenarios and market conditions.

Computational Intensity: Lattice models can be computationally intensive, especially as the number of time steps or potential outcomes increases.

Ability to Value American Options: Lattice models are particularly useful for valuing American options due to their capacity to consider a range of potential future outcomes.

Model Calibration: Lattice models require calibration to market data, which can introduce additional complexity and uncertainty. Ensuring that the model accurately reflects market conditions and price movements may require ongoing adjustments and refinements.

Simple Implementation: Binomial models, in particular, are known for their relatively simple implementation compared to more complex models like Black-Scholes.

Assumption dependency: Like any other financial model, the lattice model relies on assumptions about the behavior of underlying assets and market conditions.

CONCLUSION

Lattice models, including binomial, trinomial, and multinomial models, provide flexible and intuitive methods for option pricing, particularly American-style options that can be exercised at any time before expiration. By modeling a range of potential future outcomes, these models accommodate various scenarios and market conditions, offering significant advantages over more rigid approaches like the Black-Scholes model.

 

Ultimately, while the Black-Scholes model remains a cornerstone in option pricing due to its efficiency and systematic approach, lattice models offer a practical alternative for valuing American options and understanding dynamic market scenarios. As you engage in options trading, a thorough grasp of these models and their underlying assumptions will enhance your ability to navigate the market effectively. Thank you for exploring this guide, and best of luck with your trading endeavors!

REFERENCES

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