One-Period Binomial Model
The binomial asset-pricing model serves as a crucial framework for understanding arbitrage pricing theory and probability This chapter focuses on introducing this model, while a deeper exploration of probability will occur in Chapter 2 We begin with the most basic binomial model, which operates over a single period, and will later expand to a more complex multiperiod binomial model for enhanced realism.
In the general one-period model illustrated in Figure 1.1.1, time is divided into two intervals: time zero marks the beginning of the period, while time one signifies its conclusion At time zero, the stock has a known price per share, denoted as S0 By the end of the period, at time one, the stock price will either be S1(H) or S1(T), corresponding to the outcomes of a coin toss, where H represents heads and T represents tails It is important to note that the coin used in this scenario is not necessarily fair, meaning the probability of landing on heads is not limited to one-half.
We assume only that the probability of hea.d, which we call p, is positive, and the probability of tail, which is q = 1 - p, is also positive
Fig 1.1.1 General one-period binomial model
2 1 The Binomial No-Arbitrage Pricing �lode!
At time zero, the outcome of a coin toss, which determines the stock price at time one, remains unknown We define any quantity that is uncertain at this initial moment as random, as it relies on the unpredictable result of the coin toss.
We introduce the two positive numbers d= S1(T)_ So ( 1 1 1 )
In our analysis, we assume that the down factor (d) is less than the up factor (u) If d were greater than u, we could simply relabel the sides of the coin to achieve the desired inequality When d equals u, the stock price at time one becomes predictable, rendering the model uninteresting We define u as the up factor and d as the down factor, where it is conceptually useful to view u as greater than one and d as less than one However, the mathematical framework we present does not necessitate these conditions.
We introduce an interest rate, denoted as r, where one dollar invested in the money market at time zero will yield 1 + r dollars at time one Similarly, borrowing one dollar from the money market at time zero results in a debt of 1 + r at time one Notably, the interest rate for borrowing matches the interest rate for investing It is generally true that r is greater than or equal to zero, but our mathematical framework only requires that r is greater than -1.
An efficient market operates under the principle that any trading strategy capable of generating profits from nothing also carries the risk of loss, preventing the existence of arbitrage opportunities Arbitrage is defined as a trading strategy that starts with no initial investment, has no chance of incurring losses, and offers a likelihood of profit Mathematical models that allow for arbitrage are unreliable for analysis, as they suggest that wealth can be created from nothing, leading to contradictory conclusions While real markets may occasionally present arbitrage opportunities, these are typically short-lived, as the discovery of such opportunities prompts trading actions that eliminate them.
In the one-period binomial model to rule out arbitrage we must assume
The inequality d > 0 is based on the positive nature of stock prices, which has already been established The other inequalities in (1.1.2) arise from the principle of no arbitrage If d were greater than 1 + r, an investor could start with zero wealth, borrow from the money market at time zero, and purchase stock In the worst-case scenario of a coin toss, the stock's value at time one would still be sufficient to cover the money market debt, with a positive chance of exceeding this amount, since u > d - 1 + r This situation would create an arbitrage opportunity.
If the condition 1t ≤ 1 + 1 holds true, investors can consider short-selling the stock and reallocating the proceeds into the money market Even in the most favorable scenario for the stock, this strategy could provide a cost-effective alternative for maximizing returns.
In the one-period binomial model, the cost of replacing a stock at time one may be less than or equal to the value of a money market investment Given that the down factor (d) is less than the up factor (u) and both are less than or equal to (1 + r), there exists a positive probability that the replacement cost of the stock will be strictly lower than the money market investment's value This scenario creates an opportunity for arbitrage.
In the previous discussion, we established that the absence of arbitrage opportunities in the stock and money market accounts necessitates the condition (1.1.2) Conversely, if condition (1.1.2) is satisfied, it confirms that arbitrage is not present For further insights, refer to Exercise 1.1.
Although it's common to set d equal to zero, the binomial asset-pricing model can be effectively applied with a more general assumption, as stated in equation 1.1.2, which is a fundamental requirement for the model's validity.
While stock price movements are more complex than those suggested by the binomial asset-pricing model, this model is valuable for several reasons Firstly, it clearly illustrates the relationship between arbitrage pricing and risk-neutral pricing Secondly, it is widely used in practice as it offers a computationally efficient approximation to continuous-time models when sufficient periods are considered Lastly, the binomial asset-pricing model enables the development of the theory of conditional expectations and martingales, which is fundamental to continuous-time models.
A European call option gives its holder the right, but not the obligation, to purchase one share of stock at a predetermined strike price K at a specified future time This type of option presents unique advantages and considerations that we will explore in detail.
81 (T) < K < 81 (H) If we get a tail on the toss, the option expires worthless
In the event of a head on the coin toss, the option can be exercised, resulting in a profit of 81 (H) minus K We represent this scenario by stating that the option's value at time one is (81 - K)+, where the notation (x)+ signifies the maximum of the expression in parentheses or zero This approach aligns with standard probability practices, allowing us to omit the random variable's argument.
81 The fundamental question of option pricing is how much the option is worth at time zero before we know whether the coin toss results in head or tail
The arbitrage pricing theory addresses the option-pricing problem by replicating options through strategic trading in both stock and money markets This concept is exemplified through a specific case study, after which we will revisit the broader context of the one-period binomial model.
Multiperiod Binomial Model
In this section, we expand upon the concepts introduced earlier by analyzing a multi-period model where a coin is flipped repeatedly Each time the coin lands on heads, the stock price increases by a factor of u, while tails result in a decrease by a factor of d Alongside this stock, we also consider a money market asset that offers a constant interest rate r Our analysis is grounded in the no-arbitrage condition, ensuring that these parameters maintain a consistent relationship throughout the model.
Fig 1.2.1 General three-period model
The initial stock price is represented as S0, which is a positive value Depending on the outcome of the first coin toss, the stock price at time one will either be S1(H) = uS0 if it lands on heads or S1(T) = dS0 if it lands on tails Following the second toss, the stock price will take on one of these potential values.
S2(TH) = uS1(T) = udSo, S2(TT) = dS1(T) =�So
After three tosses, there are eight possible coin sequences, although not all of them result in different stock prices at time 3 See Figure 1.2.1
Example 1.2.1 Consider the particular three-period model with So= 4, u 2, and d = 4 We have the binomial "tree" of possible stock prices shown in
Fig 1.2.2 A particular three-period model
In the general three-period binomial model, we examine a European call option that allows the holder to purchase one share of stock for K dollars at time two Following this analysis, we will broaden our discussion to encompass any European derivative security that expires at time N-2.
At expiration, the payoff of a call option with strike price K is expressed as V2 = (S2 - K)+, where V2 and S2 are influenced by the outcomes of two coin tosses To establish the no-arbitrage price of this option at time zero, an agent sells the option for an amount Vil dollars, while the price Vo remains to be determined The agent then purchases Ll0 shares of stock and invests the difference, Vo - Ll0S0, in the money market, which will ultimately result in a negative amount, indicating that the agent is borrowing LloSo - Vo dollars By time one, the agent's portfolio, excluding the short position in the option, will have a certain valuation.
10 1 The Binomial No-Arbitrage Pricing J'vlodel
Although we do not indicate it in the notation, S1 and therefore X1 depend on the outcome of the first coin toss Thus, there are really two equations implicit in ( 1 2 1 ) :
XI (H) = d0St(H) + ( 1 + r)(Vo- doSo), Xt(T) = doSt (T) + (1 + r)(Vo- doSo)
After the first coin toss, the agent has a portfolio valued at X1 dollars and can readjust her hedge Suppose she decides now to hold 6 1 shares of stock, where
At time one, the agent's decision to invest 6.1 is influenced by the outcome of the initial coin toss, which she is aware of when making her choice She allocates the remaining portion of her wealth, calculated as X - 6.1, into the money market In the subsequent period, her goal is to achieve a wealth level denoted as V2, prompting her to strategize accordingly.
S2 and V2 are influenced by the results of the initial two coin tosses By analyzing all four potential outcomes, we can express this relationship through four distinct equations.
We now have six equations, the two represented by ( 1 2 1 ) and the four rep resented by ( 1 2.4), in the six unknowns Vo, 110, 111 (H), L1I (T), X1 (H) and
To determine the no-arbitrage price V0 of the option and the replicating portfolio Llo, we start by analyzing the last two equations, (1.2.7) and (1.2.8) By subtracting equation (1.2.8) from (1.2.7) and solving for d1(T), we derive the delta-hedging formula.
( 1 2.9) and substituting this into either ( 1 2.7) or ( 1 2.8), we can solve for
( 1 2 10) where p and ij are the risk-neutral probabilities given by ( 1 1.8) We can also obtain ( 1 2 10) by multiplying ( 1 2.7) by p and ( 1 2.8) by ij and adding t hem together Since www.pdfgrip.com
The multiperiod binomial model simplifies the valuation of options by eliminating terms related to L11(T) According to Equation (1.2.10), if the stock price decreases between time zero and one, the value of the replicating portfolio at time one is established This value represents the option price at time one, contingent upon the outcome of the first coin toss resulting in tails.
V1 (T) We have just shown that
The risk-neutral pricing formula, represented as (1.2.1 1), serves as a delayed version of formula (1.1.10) This relationship is reflected in the first two equations, (1.2.5) and (1.2.6), which similarly lead to corresponding pricing formulas.
(1.2.12) and X1 (H) = V1 (H), where V1 (H) is the price of the option at time one if the first toss results in head, defined by
The process mirrors the previously established formula (1.1.10), adjusted by one period By substituting the values XI(H) = VI(H) and X1(T) = V1(T) into the two equations inherent in (1.2.1), we find that the solutions for L10 and V0 align with those of equations (1.1.3) and (1.1.4), ultimately leading us back to results (1.1.9) and (1.1.10).
To recap, we have three stochastic processes, (L10, L11 ) , (X0, X� , X2), and
A stochastic process refers to a sequence of random variables indexed by time, where these variables are influenced by the outcomes of coin tosses, indicated by their subscripts Starting with an initial wealth \(X_0\) and defining values for \(L_{10}\), \(L_{li}(H)\), and \(L_{li}(T)\), one can calculate the portfolio's value that corresponds to the specified number of stock shares, while managing financing through borrowing or money market investments as needed The portfolio's value is recursively determined from \(X_0\) using the wealth equation.
This equation serves as a contingent framework for defining random variables, where the actual values remain uncertain until the results of the coin toss are disclosed However, even at the initial moment, this equation allows us to calculate the potential value of the portfolio at each future point in time, considering all possible outcomes of the coin toss.
A derivative security expiring at time two is defined by the random variable V2, which is contractually linked to the outcome of a coin toss For instance, if the coin toss yields a result of WIW2, the stock price at time two will be S2 (wiw2).
The Binomial No-Arbitrage Pricing Model establishes that for a European call option, the value can be represented as V2(w1w2) = (S2(w1w2) - K)+ To ensure that the recursive application of the formula (1.2.14) yields X2 such that X2(w1w2) = V2(w1w2) for any values of w1 and w2, we need to determine appropriate values for X0, as well as V1(H) and V1(T) The specified formulas guide us in achieving this objective, with V0 representing the value of X0 that facilitates the desired outcome.
In this article, we denote the values of X1(H) and X1(T) as specified in equation (1.2.14), based on the initial wealth X0 and the chosen Llo Generally, we use the symbols Ll, and Xn to represent the number of shares held in the portfolio and the corresponding portfolio values, irrespective of the initial wealth and Lln selections When X0 and Lln are specifically chosen to replicate a derivative security, we refer to the portfolio value as Vn instead of Xn.
(no-arbitrage) price of the derivative security at time n
Computational Considerations
If an agent sells the lookback option at time zero for 1.376, she can hedge her short position in the option by buying
St (H)-St (T) 8 - 2 shares of stock are priced at $0.6933, leaving an available investment of $0.6827 in the money market at a 25% interest rate By the end of the first time period, this investment will grow to $0.8533 in the money market If the stock price increases, the potential for returns will also rise significantly.
At time one, the stock value is 1.3867, resulting in a total portfolio value of 2.24, designated as V1 (H) If the stock price drops to 2, the stock value at time one becomes 0.3467, leading to a total portfolio value of 1.20, referred to as V1 (T) By consistently applying this method, the agent can confidently ensure a portfolio value of V3 at time three, regardless of the outcomes from the coin tosses.
The computational demands of a basic implementation of the derivative security pricing algorithm, as outlined in Theorem 1.2.2, increase exponentially with the number of periods In practical applications, the binomial models employed often face similar challenges.
100 or more periods, and there are 2100 � 1030 possible outcomes for a se quence of 100 coin tosses An algorithm that begins by tabulating 2100 values for V100 is not computationally practical
Fortunately, the algorithm given in Theorem 1.2.2 can usually be organized in a computationally efficient manner We illustrate this with two examples
In the given model with initial stock price So = 4, up factor u = 2, down factor d = 4, and risk-free rate r = 0, we analyze the pricing of a European put option with a strike price K = 5, set to expire at time three The risk-neutral probabilities are p = 4 and q = 0 The stock price movement is illustrated in Figure 1.2.2 The option's payoff at expiration is determined by V3 = (5 - S3)+, which can be organized into a payoff table.
The table contains 23 entries, but it can be simplified to 8 We define v3(s) as the option's payoff at time three, where s represents the stock price at that time Unlike V3, which relies on the sequence of three coin tosses, v3 focuses solely on the stock price At time three, there are only four potential stock prices, allowing us to create a concise table of relevant v3 values.
16 1 The Binomial No-Arbitrage Pricing Model
If the put expired after 100 periods, the argument of Vwo would range over the
2100 possible outcomes of the coin tosses whereas the argument of v100 would range over the 101 possible stock prices at time 100 This is a tremendous reduction in computational complexity
According to Theorem 1 2.2, we compute V2 by the formula
Equation (1.3.1) consists of four equations corresponding to each possible combination of w1 and w2 We define v2(8) as the price of the put option at time two when the stock price is 8 By utilizing this function, we can simplify (1.3.1) to represent three equations, each reflecting a distinct value of the stock price at time two Consequently, we can compute the necessary results based on these parameters.
In the given calculations, the value of the put option at time one, denoted as v1(8), is derived from the formula v2(16) = 0, v2(4) = 1.20, and v2(1) = 3 Additionally, v3(32) and v3(8) contribute to the evaluation of v2(16), while v2(4) and v2(1) are used to calculate v1(2) Ultimately, the price of the put option at time zero is determined through these interconnected values.
At each time n = 0, 1 2, if the stock price is 8, the number of shares of stock that should be held by the replicating portfolio is £ ( ) _ Vn+1 (28) - Vn+I ( �8)
This is the analogue o f formula ( 1 2.17) D www.pdfgrip.com
In Example 1.3.1, the option's price at any time n is determined solely by the stock price Sn, independent of any coin toss outcomes This allows for the formulation of functions Vn in relation to the random variables Vn, expressed as v; = vn(Sn) A similar simplification can frequently be applied when the option price is influenced by the stock price trajectory instead of just the current stock price, which we will demonstrate in a subsequent example.
In the lookback option discussed in Example 1.2.4, the option's value at each time n is determined by the stock price Sn and the maximum stock price Afn, which is defined as the highest stock price up to that point By time three, there are six potential pairs of values for (S3, A3).
We define VJ(S, m ) to be the payoff of the option at time three if s3 = s and AJ3 = m We have
In general, let vn(s, m ) denote the value of the option at t.ime n if Sn = s and Ain = m The algorithm of Theorem 1.2.2 can be rewritten in terms of the functions Vn a.'l
Vn(tl, m ) = H vn+l (28, m V (2s)) + Vn+l ( � s , m ) ] , where m V (2s) denotes the maximum of m and 2s Using this algorithm, we compute then compute v2(16, 16) = � [v3(32, 32) + v2(8, 16) ) = 3.20, v2(4, 8) = � [v3(8, 8) + v3(2, 8)] = 2.40, v2(4, 4)= �[v3(8, 8) + v3(2, 4)] = 0.80, v2(1, 4)= �[v3(2, 4) + v3(.50, 4)] = 2.20,
Vt(8, 8) = � [ v2(16, 16) + v2(4, 8) ] = 2.24, vl (2, 4) = �[vi (4, 4) + v1(1, 4)] = 1.20, and finally obtain the time-zero price vo(4, 4) = �[v1(8, 8) + vi(2, 4)] 1.376
18 I The Dinomial No-Arbitrage Pricing l\lodel
At each time n = 0, 1, 2, the stock price is denoted as s, while the maximum stock price to date is represented by m The optimal number of shares to hold in the replicating portfolio is calculated using the formula r = IJ11+t(28 m V (2s)) - 11n H(�s, m) u.(s, m) = 2s - 2s1 This approach ensures that the portfolio is effectively managed based on current and historical stock performance.
This is the analogue o f formula ( 1.2.17).
Summary
This chapter explores a multiperiod binomial model in which a coin toss at each period dictates the stock price movement, either increasing by a factor of u or decreasing by a factor of d, where 0 < d < 1 Alongside the stock, there exists a money market account that offers a consistent per-period interest rate r, applicable for both investments and loans.
Arbitrage is a trading strategy that starts with no capital and involves trading in stock and money markets to generate profit with a positive probability while eliminating the risk of loss The multiperiod binomial model confirms that arbitrage opportunities do not exist if and only if specific conditions are met.
We shall always impose this condition
A derivative security's payoff is contingent upon the outcomes of coin tosses over a specified period, N The arbitrage pricing theory provides two approaches to determine the price of a derivative security before expiration First, it establishes a no-arbitrage condition that prevents profit through trading the derivative, the underlying stock, and the money market, thereby uniquely defining the derivative's price at all times Second, prior to expiration, one can sell the derivative for a certain price and use the proceeds to create a portfolio that involves dynamic trading of the stock and money market assets until the expiration time, N.
This portfolio effectively hedges the short position in the derivative security by aligning its value at time N with the derivative's payoff, irrespective of the results of the coin toss between times n and N To establish this hedge, the derivative security must be sold at time n for the same no-arbitrage price determined by the initial pricing method.
The no-arbitrage price of the de riv a t iv e security that pays VN at time N can he computed recursively, backward in time, by the formula www.pdfgrip.com
The number of shares of the stock that should be held by a portfolio hedging a short position in the derivative security is given by
The numbers p and ij appearing in ( 1 2.16) are the risk-neutral probabilities given by l + r - d p = - ,- u-d ' u - 1 - r ij = d u - ( 1 2.15)
These risk-neutral probabilities are positive because of (1.1.2) and sum to
1 They have the property that, at any time, the price of the stock is the discounted risk-neutral average of its two possible prices at the next time:
-[fiSn+1 (w1 WnH) + ii.Sn+l (w1 WnT)]
Under risk-neutral probabilities, the average return for stocks aligns with the money market rate, denoted as r If these probabilities dictated coin tossing, a trader would face two investment options yielding the same mean return, ensuring that their overall portfolio return remains r Specifically, if a trader aims for a portfolio value of VN (w1 WN) at time N, the portfolio value at time N - 1 must reflect this target.
-[fJVN(WI WN-1/l) + ijVN(WI WN 1T)] ã
This is the right-hand side of ( 1.2 16) with n = N -1, and repeated application of this argument yields (1.2.16) for all values of n
In the context of hedging a short position in a derivative security, it is essential that the hedge aligns with the derivative's payoff across all potential stock price paths, regardless of the underlying randomness, such as coin tossing The validity of this approach stems from the fact that if a path has a positive probability, the hedge should be effective along that path, regardless of the actual probability values To identify these hedges, we solve a system of equations that does not incorporate probabilities, as seen in equations (1.2.2)-(1.2.3), (1.2.5), and (1.2.8) By introducing risk-neutral probabilities, we can establish that the average return for any investment strategy remains consistent at rate r, a critical aspect that is not achievable with other probability frameworks.
The Dinornial No-Arbitrage Pricing Model utilizes probabilities to simplify the resolution of complex equations Unlike actual probabilities, which vary based on the portfolio and complicate the process, the model offers a more straightforward approach by eliminating uncertainty about which portfolio to select when solving these equations.
An alternative explanation of Theorem 1.2.16 can be provided without discussing probability, similar to the method used in the proof of Theorem 1.2.2 In that proof, the numbers p and ij were utilized but were not considered probabilities; instead, they were defined solely by the formula in (1.2.15).
Notes
No-arbitrage pricing, initially implicit in the work of Black and Scholes, was explicitly developed by Merton, who utilized the no-arbitrage axiom to derive numerous conclusions The concept was further refined in continuous-time models by Harrison and Kreps, as well as Harrison and Pliska, who introduced martingales and risk-neutral pricing The binomial model, created by Cox, Ross, and Rubinstein, serves as a valuable tool on its own and can also be used to rederive the Black-Scholes-Merton formula as a limit of the binomial model, demonstrating the log-normality of stock prices.
Exercises
Exercise 1 1 Assume in the one-period binomial market of Section 1 1 that both H and T have positive probability of occurring Show that condition
Arbitrage is precluded when X0 equals zero, indicating that if X1 is strictly positive with a positive probability, then X must also be strictly negative with a positive probability This relationship holds true regardless of the chosen number, demonstrating the interconnected nature of these variables in financial contexts.
In Exercise 1.2, we analyze a scenario where an agent starts with zero wealth and purchases Ll0 shares of stock and F0 options at a price of 1.20 The values of Ll0 and F0 can be positive, negative, or zero, resulting in a cash position of -4Ll0 - 1.20F0 If this cash position is positive, the agent invests the surplus in the money market; if negative, it indicates a borrowing situation from the money market.
At time one, the value of the agent 's portfolio of stock, option, and money market assets is www.pdfgrip.com l 6 Exercises 2 1
In a scenario where both heads (H) and tails (T) have a positive probability of occurring, it can be demonstrated that if there is a positive probability of the outcome being positive (X1 > 0), then there must also be a positive probability of the outcome being negative (X1 < 0) This indicates that an arbitrage opportunity cannot exist when the initial price of the option is set at 1.20.
Exercise 1.3 In the one-period binomial model of Section 1 1 , suppose we want to determine the price at time zero of the derivative security V1 = 81
The derivative security, which pays off based on the stock price, can be viewed as a European call option with a strike price of K = 0 To determine the time-zero price V0, we can apply the risk-neutral pricing formula outlined in equation (1.1.10).
Exercise 1 4 In the proof of Theorem 1 2 2, show under the induction hy pothesis that
In Example 1.2.4, we analyzed an agent who sold a look-back option for a value of V0 = 1.376 and purchased Ll0 = 0.1733 shares of stock at the initial time If the stock price increases at time one, her portfolio will be valued at VI(H).
At time two, if the stock price increases, the portfolio value will be V2(HH) = 3.20; conversely, if the stock price decreases, it will be V2(HT) = 2.40 Assuming the stock rises in the first period and falls in the second, the agent holds a position of Llz(HT) At time three, if the stock goes up again, the portfolio value will be V3(HTH) = 0, while if it goes down, the value will rise to V3(HTT) = 6 This indicates that the agent has effectively hedged her short position in the option.
In Exercise 1.6, we analyze a bank's long position in a European call option on a stock, which expires in one period with a strike price of K = 5 The initial price of the call option is determined to be V0 = 1.20 At time zero, the bank holds this option, effectively tying up capital of V0 = 1.20 The bank aims to earn a 25% interest rate on this capital until expiration, seeking to maximize returns without additional investment, regardless of the outcomes of the underlying stock's price movements.
To achieve a payoff of 1.50 at time one, the bank's trader should strategically allocate funds between the stock and money markets This involves investing in stocks that are expected to appreciate in value, while simultaneously placing a portion of the investment in low-risk money market instruments to ensure liquidity and stability By balancing these investments, the trader can optimize returns and secure the desired payoff effectively.
22 The Binomial No-Arbitrage Pricing Model
In Exercise 1.7, we examine a bank with a long position in a lookback option, as outlined in Example 1.2.4 The bank plans to hold this option until expiration to secure a payoff of V3 At the outset, the bank's capital amounts to V0 = 1.316, which is invested in the option The bank aims to earn an interest rate of 25% on this capital until time three, without any additional investment and irrespective of the outcomes from the coin tosses.
To achieve the payoff from the lookback option at time three, the bank's trader should strategically allocate investments between the stock market and a money market account This involves determining the appropriate proportion of funds to invest in stocks while simultaneously managing the balance in the money market account to ensure optimal returns By carefully analyzing market conditions and potential outcomes, the trader can effectively position the portfolio to meet the financial goals associated with the lookback option.
Exercise 1.8 ( Asian option) Consider the three - period model of Example
In this analysis, we define the initial stock price \( S_0 = 4 \), with a constant interest rate \( r = 1 \) and a price increment \( d \) The variable \( p \) is set to \( ij = \frac{1}{2} \) For \( n = 0, 1, 2, 3 \), we denote \( S_n \) as the cumulative sum of stock prices from time zero to \( n \) We focus on an Asian call option that matures at time three, featuring a strike price \( K = 4 \), which determines its payoff at expiration.
The option described functions similarly to a European call option, but its payoff is determined by the average stock price instead of the final stock price Denoted as Vn(s, y), the price of this option at time n is contingent on the stock price Sn = s and the variable Yn = y Specifically, the value at time three is represented as v3(s, y) = (h - 4) +.
(i) Develop an algorithm for computing v11 recursively In particular, write a formula for Vn in terms of v,+ 1
(ii) Apply the al gorithm developed in (i) to compute v0(4 4) , the price of the Asian option at time zero
(iii) Provide a formula for On (s y), the number of shares of stock that should be held by the replicating portfolio at t ime n if Sn = s and Yn = y
In a binomial pricing model that incorporates stochastic volatility and random interest rates, the up factor \( U_n \), down factor \( d_n \), and interest rate \( r_n \) are influenced by the time step \( n \) and the outcomes of the first \( n \) coin tosses \( w_1, w_2, \ldots, w_n \) Unlike these variables, the initial conditions such as the up factor \( u_0 \), down factor \( d_0 \), and interest rate \( r_0 \) remain constant and non-random Consequently, the stock price at time one can be determined based on these initial values and the stochastic factors that evolve with each time step.
S ( l �, , ) { = u0S0 doSo if if w1 = w1 = T H and, for rz ;:::: 1 , the stock price at time 11 + 1 is given by
S ( ) { 11n (WJW2 wn )S, (WJW2 Wn) if Wn+l = H, n+I WtW2 WilWn+ l = d ( n WJW2 ã ã ã '"-'n )S ( n WtWz ã ã ã Wn ) I '"-'"f n+ l = T ã www.pdfgrip.com
Investing or borrowing one dollar in the money market at the initial time grows to an amount of 1 + r0 at the next time period For subsequent periods, a dollar invested or borrowed at time n increases to 1 + r n (w1 w2 Wn) at time n + 1 It is assumed that the no-arbitrage condition is satisfied for each n and all corresponding values of w1, w2, , Wn, with the constraints that 0 < d0 < 1 + r0 < u0.
To determine the price at time zero for a derivative security that pays off a random amount \( V_N \) based on the outcomes of the first \( N \) coin tosses, we can develop an algorithm This algorithm will evaluate the expected payoff at time \( N \) by analyzing the probabilities of each possible outcome from the coin tosses By discounting the expected payoff back to time zero using an appropriate discount rate, we can derive the present value of the derivative security This method ensures an accurate pricing strategy for the security based on the stochastic nature of the coin tosses.
(ii) Provide a formula for the number of shares of stock that should be held at each time n (0 ::::; n :S N - 1 ) by a portfolio that replicates the derivative security V N
Finite Probability Spaces
A finite probability space models scenarios involving random experiments with a limited number of outcomes For instance, in the binomial model, if we flip a coin three times, the complete set of possible outcomes is represented as {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.
In each coin toss, the probability of landing on heads is denoted as p, while the probability of tails is represented as q, where q equals 1 minus p Assuming that each toss is independent, we can determine the probabilities of various sequences of three tosses, represented as w = w1w2w3, within the larger framework of outcomes.
'P(THH) = p2q, 'P(THT) = pq2, 'P(TTH) = pq2, 'P(TTT) = q3 (2.1.2)
The subsets of fl are called events, and these can often be described in words as well as in symbols For example, the event
"The first toss is a head" = {w E fl;w1 = H}
The event represented by {HHH, HHT, HTH, HTT} can be described using both words and symbols To calculate the probability of this event, we sum the probabilities of its individual elements.
'P(First toss is a head) = 'P(H H H) + 'P(H HT) + IP'(HT H) + IP'(HTT)
26 2 Probability Theory on Coin Toss Space
= p Thus the mathematics agrees with our intuition
Relying solely on intuition when using mathematical models can be misleading, as it may result in discrepancies between our understanding and the model's outcomes To avoid potential issues, it's crucial to ensure that our intuition aligns with the mathematical results If there is a mismatch, we must investigate and reconcile the differences before moving forward with our analysis.
In our study, we aimed to develop a model where the probability of obtaining heads on each coin toss is denoted as p To achieve this, we defined the probabilities of the elements of the sample space and established that the probability of an event, which is a subset of the sample space, is the sum of the probabilities of its constituent elements This framework necessitates the computation outlined in section 2.1.3, which we must perform to ensure that our model yields the expected results If the outcomes do not align with our expectations, we will need to reevaluate our mathematical model for coin tossing.
We expand upon the previously discussed scenario by permitting \( f? \) to represent any finite set and by allowing certain elements within \( f? \) to possess a probability of zero These adjustments culminate in the following definition.
A finite probability space is defined as a sample space, denoted as \( \Omega \), along with a probability measure, \( P \) The sample space \( \Omega \) consists of a nonempty finite set, while the probability measure \( P \) is a function that assigns a value between 0 and 1 to each element \( \omega \) in \( \Omega \).
An event is a subset of f?, and we define the probability of an event A to be
This article discusses a model for a random experiment, where the set \( f? \) represents all possible outcomes The notation \( IP'(w) \) denotes the probability of a specific outcome \( u \) occurring, while \( IP'(A) \) indicates the probability that the outcome falls within the set \( A \) If \( IP'(A) = 0 \), it guarantees that the outcome will not be in \( A \), whereas \( IP'(A) = 1 \) ensures that the outcome will definitely be in \( A \) This leads to a fundamental equation based on the principles outlined.
The probability function IP'(f?) = 1 indicates that the outcome will certainly be within the set Jl However, since IP'(w) can equal zero for certain values of w, we can exclude specific outcomes from the experiment that are guaranteed not to happen Moreover, as noted in equation (2.1.5), if A and B are disjoint subsets of f?, then their probabilities can be analyzed independently.
IP' ( A U B ) = IP'(A) + IP'( B ) (2 1 7) www.pdfgrip.com
Random Variables, Distributions, and Expectations
A random experiment generally generates numerical data This gives rise to the concept of a random variable
Definition 2 2 1 Let (!7, IP) be a finite probability space A random variable is a real-valued function defined on !7 (We sometimes al.so permit a random variable to take the values +oo and -oo.)
In the context of stock prices influenced by three independent coin tosses, we define the pricing outcomes based on the results of the tosses Specifically, if all three tosses result in heads (H), a certain price is assigned; if two tosses are heads and one is tails (T), a different price applies; conversely, if all tosses yield tails, another distinct price is set This framework illustrates the relationship between coin toss outcomes and stock pricing dynamics.
In this article, we discuss the arguments of So, SI 's2, and s3 as WIW2WJ, highlighting that some random variables are not influenced by all coin tosses Notably, S0 is a degenerate random variable, consistently taking the value 4 regardless of the outcomes of the coin tosses.
In probability theory, it is common to denote the argument of random variables as w, even when w represents a sequence like w = w1w2w3 We will use both notations interchangeably More frequently, random variables are expressed without arguments; for instance, we will write S3 instead of S3(w1w2w3) or S3(w).
A random variable is defined as a function that maps a sample space to real numbers, while its distribution specifies the probabilities associated with the variable's potential values It is crucial to understand that a random variable and its distribution are distinct concepts This distinction becomes particularly important when transitioning between the actual probability measure, estimated from historical data, and the risk-neutral probability measure, as this change in measure will affect the outcomes.
28 2 Probability Theory on Coin Toss Space distributions of random variables but not the random variables themselves
We make this distinction clear with the following example
ExarnplP 2.2.3 Toss a coin three times, so the set of possible outcomes is
!? = {H H H, H HT, HTH, HTT TH H THT, TTH, TTT}
X = Total number of heads, Y = Total number of tails
Y(TTT) = 3, Y(TTH) = Y(THT) = Y(HTT) = 2 Y(THH) = Y(HTH) = l'(HHT) = l ,
To define random variables, we do not need to know the probabilities of different outcomes However, once we establish a probability measure, we can ascertain the distributions of the random variables X and Y For instance, by specifying a probability measure where the likelihood of heads on each coin toss is p, and the probability of each outcome in the sample space is q, we can effectively determine the distributions involved.
IP{w E J2; X(w) = 1 } = IP{HTT THT, TTH} = 8'
IP{w E D; X(w) = 2 } = IP{H HT, HTH, TH H} = 8'
We simplify the notation P{w E !?; X(w) = j} to P{X = j}, but it’s important to note that P{X = j} represents the probability of a subset of !?, specifically the elements w for which X(w) equals j According to section iii, the probabilities for X taking the values 0, 1, 2, and 3 are calculated accordingly.
IP{X = 2} = B' IP{X = 3} = 8 www.pdfgrip.com r �
2.2 Random Variables, Distributions, and Expectations 29
This table of probabilities where X takes its various values records the distri bution of X under PI
The random variable Y is different from X because it counts tails rather than heads However, under P, the distribution of Y is the same as the distri bution of X:
A random variable is a function defined on a sample space, while its distribution represents a table of probabilities for the different values that the random variable can assume It is important to note that a random variable and its distribution are distinct concepts.
Suppose, moreover, that we choose a probability measure lP' for fl that corresponds to a � probability of head on each toss and a ! probability of tail Then
The random variable X, representing the total number of heads, exhibits a different distribution under the probability measure lP' compared to P Despite this variation in distribution, X remains the same random variable This concept becomes particularly relevant when analyzing asset prices under both actual and risk-neutral probability measures.
Incidentally, although they have the same distribution under JPi, the random variables X and Y have different distributions under IP' Indeed, lP'{Y = 0} = 27 ' 8 12 lP' { Y = 1 } = 27 '
Definition 2.2.4 Let X be a random variable defined on a finite probability space ( fl, lP') The expectation (or expected value) of X is defined to be lEX = L X (w)IP'(w) wEfl
_!Vhen we compute the expectation using the risk-neutral probability measure lP', we use the notation
30 2 Probability Theory on Coin Toss Space
It is clear from its definition that expectation is linear: if X and Y are random variables and ct and c2 are constants, then
In particular, if f(x) = n.r + h is a linear function of a dummy variable x (a and b are constants) , then IE[f(X )] = f(IEX ) When dealing with convex functions, we have the following inequality
Theorem 2.2.5 (Jensen's inequality) Let X be a random variable on a finite pmbability space and let 0, contradicting Corollary 2.4.6.
The First Fundamental Theorem of Asset Pricing states that if a risk-neutral measure exists within a model—one that aligns with the actual probability measure regarding price paths with zero probability and under which discounted prices of all primary assets are martingales—then arbitrage opportunities are absent in the model The proof highlights that under a risk-neutral measure, the discounted wealth process maintains a constant expectation, implying that it cannot start at zero and later become strictly positive with a positive probability unless it also has a chance of being strictly negative This theorem is crucial for eliminating arbitrage in term structure models and contributes to the Heath-Jarrow-Morton no arbitrage condition on forward rates.
Theorem 2.4.5 leads to an important version of the risk-neutral pricing formula, which states that a random variable VN, representing a derivative security that pays off at time N based on the outcomes of the first N coin tosses, can be evaluated using an initial wealth X0 and a replicating portfolio process L0, , LN-1 This process ensures that the resulting wealth XN equals VN, regardless of the outcomes of the coin tosses Additionally, since the sequence (1 - p)^n, for n = 0, 1, , N, forms a martingale, the multistep ahead property highlighted in Remark 2.4.2 further supports this relationship.