91Ó°ÊÓ

Consider an \(N\)-period binomial model. An Asian option has a payoff based on the average stock price, i.e., 2 Probability Theory on Coin Toss Space $$ V_{N}=f\left(\frac{1}{N+1} \sum_{n=0}^{N} S_{n}\right) $$ where the function \(f\) is determined by the contractual details of the option. (i) Define \(Y_{n}=\sum_{k=0}^{n} S_{k}\) and use the Independence Lemme 2.5.3 to show that the two-dimensional process \(\left(S_{n}, Y_{n}\right), n=0,1, \ldots, N\) is Markov. (ii) According to Theorem 2.5.8, the price \(V_{n}\) of the Asian option at time \(n\) is some function \(v_{n}\) of \(S_{n}\) and \(Y_{n}\); i.e., $$ V_{n}=v_{n}\left(S_{n}, Y_{n}\right), n=0,1, \ldots, N $$ Give a formula for \(v_{N}(s, y)\), and provide an algorithm for computing \(v_{n}(s, y)\) in terms of \(v_{n+1}\). Exercise 2.14 (Asian option continued). Consider an \(N\)-period binomial model, and let \(M\) be a fixed number between 0 and \(N-1\). Consider an Asian option whose payoff at time \(N\) is $$ V_{N}=f\left(\frac{1}{N-M} \sum_{n=M+1}^{N} S_{n}\right) $$ where again the function \(f\) is determined by the contractual details of the option. (i) Define $$ Y_{n}= \begin{cases}0, & \text { if } 0 \leq n \leq M \\ \sum_{k=M+1}^{n} S_{k}, \text { if } M+1 \leq n \leq N\end{cases} $$ Show that the two-dimensional process \(\left(S_{n}, Y_{n}\right), n=0,1, \ldots, N\) is Markov (under the risk-neutral measure \(\tilde{\mathbf{P}}\) ). (ii) According to Theorem 2.5.8, the price \(V_{n}\) of the Asian option at time \(n\) is some function \(v_{n}\) of \(S_{n}\) and \(Y_{n}\), i.e., $$ V_{n}=v_{n}\left(S_{n}, Y_{n}\right), n=0,1, \ldots, N $$ Of course, when \(n \leq M, Y_{n}\) is not random and does not need to be included in this function. Thus, for such \(n\) we should seek a function \(v_{n}\) of \(S_{n}\) alone and have $$ V_{n}= \begin{cases}v_{n}\left(S_{n}\right), & \text { if } 0 \leq n \leq M \\\ v_{n}\left(S_{n}, Y_{n}\right), & \text { if } M+1 \leq n \leq N\end{cases} $$ Give a formula for \(v_{N}(s, y)\), and provide an algorithm for computing \(v_{n}\) in terms of \(v_{n+1}\). Note that the algorithm is different for \(nM\), and there is a separate transition formula for \(v_{M}(s)\) in terms of \(v_{M+1}(\cdot, \cdot)\).

Step by step solution

01

(i) Proving Markov property of \((S_n, Y_n)\)

First, we need to define \(Y_n = \sum_{k=0}^{n} S_k\) and prove that the two-dimensional stochastic process \((S_n, Y_n)\) is Markov. To show this, we need to use Independence Lemme 2.5.3, which states that given a financial market with two assets and IF the increments \(\Delta Y_n\) are independent in the risk-neutral measure, THEN the process is Markov.

02

(i) Markov property of \((S_n, Y_n)\) for exercise 2.14 part (i)

Similarly, we define \(Y_n\) as: $$ Y_{n}= \begin{cases}0, & \text { if } 0 \leq n \leq M \\\ \sum_{k=M+1}^{n} S_{k}, \text { if } M+1 \leq n \leq N\end{cases} $$ We need to prove again that the two-dimensional stochastic process \((S_n, Y_n)\) is Markov, under the risk-neutral measure \(\tilde{\mathbf{P}}\).

03

(ii) Finding \(v_N(s, y)\) formulas

Next, we need to find out the formula for \(v_N(s, y)\) for both exercises 2.13 and 2.14. These formulas are essential to price the options at time N.

04

(ii) Providing algorithms to compute \(v_n(s, y)\)

Lastly, we need to provide the algorithms for computing \(v_n(s, y)\) in terms of \(v_{n+1}\). This is important to efficiently price the options throughout the time. Now let's proceed with step by step solutions:

05

(i) Proving Markov property of \((S_n, Y_n)\)

Using Independence Lemme 2.5.3, we define the increments as \(\Delta Y_n = S_{n} - Y_{n-1}\). Since the \(S_n\) is a binomial process, the increments \(\Delta S_n\) are independent, which implies that \(\Delta Y_n\) are also independent. Therefore, the process \((S_n, Y_n)\) is Markov.

06

(i) Markov property of \((S_n, Y_n)\) for exercise 2.14 part (i)

Similarly, to prove the Markov property of \((S_n, Y_n)\) for this case, we find the increments as: $$ \Delta Y_n = Y_{n} - Y_{n-1} = \begin{cases}0, & \text { if } 0 \leq n \leq M \\\ S_{n} - Y_{n-1}, & \text { if } M+1 \leq n \leq N\end{cases} $$ Under the risk-neutral measure \(\tilde{\mathbf{P}}\), we have \(\Delta Y_n\) independent, which implies the process \((S_n, Y_n)\) is Markov.

07

(ii) Finding \(v_N(s, y)\) formulas

For exercise 2.13, the formula for \(v_N(s, y)\) can be found as: $$ v_N(s, y) = f\left(\frac{y}{N+1}\right) $$ For exercise 2.14, the formula for \(v_N(s, y)\) can be found as: $$ v_N(s, y) = f\left(\frac{y}{N-M}\right) $$

08

(ii) Providing algorithms to compute \(v_n(s, y)\)

For exercise 2.13, the algorithm for computing \(v_n(s, y)\) can be represented as: 1. Initialize \(v_N(s, y)\) using the formula found in step (ii). 2. For \(n=N-1, N-2, \dots, 0\): - Calculate \(v_{n}(s, y)\) using the risk-neutral measure and the values of \(v_{n+1}\). For exercise 2.14, the algorithm for computing \(v_n(s, y)\) can be represented as: 1. Initialize \(v_N(s, y)\) using the formula found in step (ii). 2. For \(n=N-1, N-2, \dots, M+1\): - Calculate \(v_{n}(s, y)\) using the risk-neutral measure and the values of \(v_{n+1}\) 3. For \(n=M, M-1, \dots, 0\): - Calculate \(v_{n}(s)\) using the risk-neutral measure and the values of \(v_{n+1}\), considering the transition from \(v_M(s)\) to \(v_{M+1}(\cdot, \cdot)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Binomial Asset Pricing Model

The binomial asset pricing model is a fundamental tool used to value derivatives, such as options, in finance. It relies on a simple yet powerful premise: over a small time period, an asset can either increase or decrease in value by specific factors with given probabilities. To visualize this, imagine a tree with branches splitting at each time interval, showing the up and down movements possible for the asset price.

This model is particularly convenient for understanding Asian option pricing, where the option's payoff depends on the average price of the underlying asset. To value such options, it's essential to accurately represent the asset's price dynamics over time, and the binomial model does this effectively. By dividing the option's lifespan into multiple periods, the model allows us to assess the possible asset prices and the corresponding option values at each node on the tree.

Algorithmic Execution in Asian Option Pricing

Applying the binomial asset pricing model to price an Asian option means that we need to recursively calculate option values from the terminal nodes (at expiration) backward to the present. By doing so, we respect the time value of money and uncertainty at each step, using a risk-neutral measure to obtain present values which can be complicated, but the model's step-by-step approach simplifies it.

Exploring the Markov Property in Pricing

When analyzing stochastic processes, such as stock price movements, the Markov property is a critical concept. A process has the Markov property if the future is independent of the past given the present. In other words, only the current state influences what happens next, not how we got there.

This property is important in pricing options because it allows us to simplify the complexity of future predictions. For the Asian option problem, proving that the two-dimensional process \(S_n, Y_n\) is Markov means that the future option price depends only on the current stock price and the accumulated sum of prices till that point, which is a requirement for the risk-neutral valuation we employ in the binomial model.

By understanding this property, students can grasp why certain steps are taken to compute option prices: each computation only needs to consider the immediate next step, just like only the current position on a chessboard, not the moves that led there, matters in determining your next best move.

The Role of Risk-Neutral Measure in Option Pricing

The concept of a risk-neutral measure is at the heart of modern financial theory and especially crucial in derivatives pricing. It is a probabilistic framework under which all investors are deemed to be risk-neutral, meaning they do not demand extra expected return for taking on more risk.

Under this measure, the expected growth rate of the asset's price equals the risk-free interest rate, which simplifies pricing considerably. Any expected excess return is adjusted for by the market, so an investor is indifferent to risk when evaluating financial securities.

The risk-neutral measure is essential when using the binomial model to calculate the price of an Asian option through the algorithm described earlier. It ensures that the recursion works correctly and that the present value of the expected payoffs from the option is consistent with the market expectations. Essentially, it's like calibrating a telescope correctly to see the stars—the calibration won't change what's in the sky, but it will help us see it more clearly.