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Risk-Neutral Valuation and the Binomial Model

Risk-Neutral Valuation and the Binomial Model. Chapter 11 Sorta. Objectives This segment introduces option pricing in a very simple setting. It has three key objectives: To describe the pricing of options by replication .

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Risk-Neutral Valuation and the Binomial Model

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  1. Risk-Neutral Valuation and the Binomial Model Chapter 11 Sorta

  2. Objectives This segment introduces option pricing in a very simple setting. It has three key objectives: To describe the pricing of options by replication. To introduce the concept of the option delta and to discuss the properties of the delta from an intuitive standpoint. To introduce the concept of risk-neutral pricing of derivatives. Later chapters build on this foundation and examine the pricing of options in more complex settings.

  3. Option Pricing Option pricing follows the same lines as forward pricing. The basic idea is replication: we look to create identical payoffs to the option's using Long /short positions in the underlying. Cash (default-free investment/borrowing). Once we have a portfolio that replicates the option, the cost of the option must be equal to the cost of replicating it. Thus, the challenge is in identifying the composition of the replicating portfolio.

  4. Risk-Neutral Valuation and the Binomial Model • We will develop below a simple model that is useful in understanding how options are priced. Although, it will seem unrealistic at first, it can be extended in a straightforward way to a form that is widely used by practitioners. • Assume that a stock currently sells at a price of $100, and at the end of the year will be worth either $200 or $50. A European call option is traded with one year to expiration and a strike price of $100. The risk-free rate is 10%

  5. Binomial Option pricing model (BOPM) • We begin with a single period. • Then, we stitch single periods together to form the Multi-Period Binomial Option Pricing Model. • The Multi-Period Binomial Option Pricing Model is extremely flexible, hence valuable; it can value American options (which can be exercised early), and most, if not all, exotic options.

  6. Assumptions of BOPM • There are two (and only two) possible prices for the underlying asset on the next date. The underlying price will either: • Increase by a factor of u% (an uptick) • Decrease by a factor of d% (a downtick) • The uncertainty is that we do not know which of the two prices will be realized. • No dividends and the one-period interest rate, r, is constant over the life of the option (r% per period). • Markets are perfect (no commissions, bid-ask spreads, taxes, price pressure, etc.)

  7. A Binomial Model The "binomial" assumption: Price moves from its current level S to one of two possible levels, uS or dS. u: "up" move. Occurs with probability p. d: "down" move. Occurs with probability 1 — p . Intuitively, binomial model volatilityul d. The larger is this ratio, the greater the variability of stock prices.

  8. Volatility and Black-Scholes Approximation Key input parameter in Black-Scholes model:  , the annualized volatility of the stock. To approximate the Black-Scholes model, set where h the length in years of one step of the tree. Thus, the annualized volatility  is related to u and d as As the number of steps in the binomial tree becomes large, the resulting tree approximates Black-Scholes arbitrarily closely.

  9. Stock Pricing “Process” Time T is the expiration day of a call option. Time T-1 is one period prior to expiration. ST,u = (1+u)ST‑1 ST‑1 ST,d = (1+d)ST‑1 Suppose that ST-1 = 40, u = 25% and d = -10%. What are ST,u and ST,d? ST,u = ______ 40 ST,d = ______

  10. Option pricing “Process” CT,u = max(0, ST,u‑K) = max(0,(1+u)ST‑1‑K) CT‑1 CT,d = max(0, ST,d‑K) = max(0,(1+d)ST‑1‑K) Suppose that K = 45. What are CT,u and CT,d? CT,u = ______ CT‑1 CT,d = ______

  11. The Equivalent Portfolio Buy  shares of stock and borrow $B. NB:  is not a “change” in S…. It defines the # of shares to buy. For a call, 0 <  < 1 (1+u)ST‑1 + (1+r)B = ST,u + (1+r)B  ST‑1+B (1+d)ST‑1 + (1+r)B = ST,d + (1+r)B Set the payoffs of the equivalent portfolio equal to CT,u and CT,d, respectively. (1+u)ST‑1 + (1+r)B = CT,u (1+d)ST‑1 + (1+r)B = CT,d These are two equations with two unknowns:  and B What are the two equations in the numerical example with ST-1 = 40, u = 25%, d = -10%, r = 5%, and K = 45?

  12. Interest Rates The default risk-free interest rate is denoted r. Important change from earlier notation (for binomial models only): r is expressed in simple terms as the rate of interest applicable to the time-period represented by each step of the binomial tree. r is the gross rate of interest per time step. That is, $1 invested at the beginning of the period will grow to $r at the end of the period.

  13. Summary of Notation Summing up, a binomial model is described by: u and d : the up and down moves. p : Probability of the up move. r : risk-free interest rate per time step. n : number of steps in the tree. S : initial price of the asset. Important: Note that we must have d < R< u (why?) R denotes (1+r).

  14. Key Point • If two assets offer the same payoffs at time T, then they must be priced the same at time T-1. (Law of One Price) • Here, we have set the problem up so that the equivalent portfolio offers the same payoffs as the call. • Hence the call’s value at time T-1 must equal the $ amount invested in the equivalent portfolio. • CT-1 = ST-1 + B

  15. Pricing Examples We use the following parameter values in the example: u = 1.10, d = 0.90, p = 0.75. r = 1.02. S = 100. In this model, we will price: A one-period call option with K = 100. A one-period put option with K = 100.

  16. Pricing the Call Option There are two possible prices of the security after one period: uS = 110. dS = 90. In the state uS, the call with a strike of 100 is worth Cu = 10. In the state dS, the call expires worthless: Cd = 0. Question: What is the initial price C of the call?

  17. The Call Pricing Problem Stock Cash Call

  18. Replicating the Call To replicate the call, consider the following portfolio: ∆c units of stock. B units of cash. Note that ∆c and B can be positive or negative. ∆c > 0: we are buying the stock. ∆c< 0: we are selling the stock. B > 0: we are investing. B < 0: we are borrowing. We need to identify ∆c and B so that the replicating portfolio mimics the call.

  19. The Replicating Portfolio For the portfolio to replicate the call, we must have: Δc • (110) + B • (1.02) = 10 Δc • (90) + B • (1.02) = 0 Solving, we obtain: ΔC= 0.50 B = –44.12. In words, the following portfolio will perfectly replicate the call option: A long position in 0.50 units of the stock. A borrowing of 44.12.

  20. The Arbitrage-Free Call Price Cost of the replicating portfolio: (0.50)•(100) — (44.12)(1) = 5.88. Since the portfolio perfectly replicates the payoffs from the call, we must have C = 5.88. Any other price leads to arbitrage! If C > 5.88, we can sell the call and buy the replicating portfolio. If C < 5.88, we can buy the call and sell the replicating portfolio. Thus, replication yields us a unique arbitrage-free price for the call.

  21. Arbitraging an Undervalued Call For example, suppose C = 5.50. Then the call is undervalued relative to the replicating portfolio by $0.38. So: buy the call and sell the replicating portfolio. Specifically: Buy one call with a strike of 100. Short 0.50 units of the stock. Invest 44.12 for one period at the rate r = 1.02.

  22. Cash Flows at Inception Cash flows at inception:

  23. Cash Flows at Maturity No net cash flows at maturity because we have bought a call and sold the synthetic call.

  24. Arbitraging an Overvalued Call Now, suppose C = 6.25. Then the call is overvalued relative to the replicating portfolio by $0.37. So: sell the call and buy the replicating portfolio. Specifically: Sell one call with a strike of 100. Buy 0.50 units of the stock. Borrow 44.12 for one period at the rate r = 1.02.

  25. Cash Flows at Inception Cash flows at inception:

  26. Cash Flows at Maturity

  27. Pricing the Put Option The arguments here are essentially the same. There are two possible prices of the security after one period: uS = 110. dS= 90. In the state uS, the put with a strike of 100 expires worthless: Pu = 0. In the state dS, the put is worth Pd = 10. Question: What is the initial price P of the put?

  28. The Put Pricing Problem 110 90 Stock Cash Put 0 10

  29. Replicating the Put To replicate the put, consider the following portfolio: Δp units of stock. B units of cash. Again, it is to be stressed that Δp and B can be positive or negative. Δc > 0: we are buying the stock. Δc < 0: we are selling the stock. B > 0: we are investing. B < 0: we are borrowing.

  30. The Replicating Portfolio For the portfolio to replicate the put, we must have:Δp• (110) + B • (1.02) = 0 Δp• (90) + B • (1.02) = 10 Solving, we obtain: Δp= −0.50 B = 53.92. In words, the following portfolio will perfectly replicate the put option: A short position in 0.50 units of the stock. Investment of 53.92.

  31. Pricing the Put Cost of this portfolio: (-0.50) • (100) + (53.92)(1) = 3.92. Since the portfolio perfectly replicates the put, we must have P = 3.92. Any other price leads to arbitrage! If P > 3.92, we can sell the put and buy the replicating portfolio. If P < 3.92, we can buy the put and sell the replicating portfolio. Thus, replication yields us a unique price for the put that is "arbitrage free."

  32. Comments What happened to the probability p in the pricing exercise? That p does not matter explicitly for the pricing seems surprising. But since the replication is done state-by-state, it does not matter what the probability of the state is. Nonetheless, does p enter implicitly through u and d ? (What happens to u and d if p = 1? If p = 0?

  33. Risk-Neutral Pricing An alternative approach used to price options in practice is called risk-neutral pricing. Risk-neutral pricing uses on a very intuitive approach: the value of the option equals the present value of the payoffs received at maturity. Since these payoffs are uncertain, present-valuation requires taking expectations of the payoffs and discounting them back to the current time. In the risk-neutral pricing approach, these expectations are taken under what is called the risk-neutral (i.e., "risk-adjusted") probabilities. By construction, since the probabilities are risk-adjusted, no further adjustment needs to be made for risk, and the expectations can be discounted back to the present at the risk-free rate. Of course, this approach results in the same price as we obtain from replication.

  34. Advantages of Risk-Neutral Pricing Many computational advantages of risk-neutral pricing: It only involves "simple" operations like taking expectations and discounting. All expectations are taken with respect to a particular fixed probability measure called the risk-neutral measure (or the martingale measure). In particular, the risk-neutral measure does not depend on which particular option or derivative security we wish to value.

  35. Steps in Risk-Neutral Pricing Risk-neutral pricing involves a three-step procedure: We compute the "probabilities" of the states u and d that makes the expected return on the risky asset equal to the risk-free rate r. These are the model's risk-neutral probabilities. Using these probabilities, we compute the expected payoff from the option at maturity. We discount these expected payoffs back to the current period using the risk-free rate r. The result of Step 3 will be precisely the arbitrage-free price of the option. This same procedure is valid in any model, not just the binomial, though, course, Step 1 (identifying the risk-neutral measure), becomes more complicated.

  36. Examples We have u = 1.10, d = 0.90, and r = 1.02. We first compute the risk-neutral probability. If q denotes the risk-neutral probability of the state u, we must have q • u +( 1—q) • d = r, or q • (1.10) + (1 — q) • (0.90) = 1.02, This gives us q = 0.60.

  37. Example 1 Consider pricing a call with K = 100. The call pays 10 in state u and 0 in state d. Therefore, its expected payoff under q is (0.60) • 10 + (0.40) • 0 = 6. Discounting this expected payoff at the risk-free rate, we obtain This is the same as the call price we derived earlier!

  38. Two State Binomial • The fact that option prices do not depend on investors’ risk preferences is important because it allows us to assume risk-neutrality (as a convenience) in computing option prices • Consider the earlier example. In a risk-neutral world, the expected return on the stock must be 10%. This allows us to calculate the risk-neutral probability for the stock price to go up (p) as follows: • Solving for the risk-neutral probability (p*), p* = .4 • Using this risk-neutral (artificial) probability we can now calculate the expected value and price for the call option p*200 + (1 - p*) 50 = 100 (1 + .10)

  39. Two State Binomail • Recall that 100 is the value of the call if the stock goes up, and 0 is its value if the stock goes down 0.4 * 100 + 0.6 * 0 = 40 • Thus ($40) is the expected price of the call option at the end of the year (using the risk-neutral probabilities). The price today must be equal to the present value of ($40), or: 40/(1 + .10) = 36.36

  40. Risk-Neutral Valuation • The steps for calculating an option price in the binomial model using the risk-neutral valuation approach are therefore as follows: • Solve for the risk-neutral probability (p*): • Find the expected value of the call option, using the risk-neutral probabilities: • Discount the expected payoff: p* Su + (1- p*) Sd= S0(e-rt) Where p* = ((e-rt)S0- Sd)/ (Su - Sd) p* Cu + (1- p*) Cd= E[CT] C0 = E[CT] /(e-rt)

  41. Why the Method "Works“ For an intuitive explanation, consider a thought experiment: Imagine two worlds in which all securities have the same current price and the same set of future prices. Only the probabilities of these prices differ. One world is risk-neutral, so the probabilities are such that all expected returns are the same. The other world has risk-averse investors (like our own), so the probabilities reflect this. Consider a call option in this setting that can be priced by replication.

  42. Why the Method "Works" As we have seen, the replicating portfolio does not depend on the probabilities of the future states. Therefore, the replicating portfolio is the same in the two worlds, so the price of the option must be the same too. But in the risk-neutral world, the price of the option must also be just its expected payoff discounted back to the present. This means that the replication-based price of the option must coincide with its expected payoff in the risk-neutral world discounted back to the present. But this is precisely the statement of risk-neutral pricing!

  43. Volatility and Option Prices • What happens to the value of an option when the volatility of the underlying stock increases? • We assume a world in which the stock price moves during the year from $100 to one of two new values at the end of the year when the option matures • Assume risk neutrality

  44. Volatility and Option Prices, P0 = $100, Strike = $100 Stock Price Call Payoff Put Payoff Low Volatility Case Rise 120 20 0 Fall 80 0 20 Expectation 100 10 10 High Volatility Case Rise 140 40 0 Fall 60 0 40 Expectation 100 20 20 Vols and Option Pricing

  45. Effect of Volatility • The stock volatility in the second scenario is higher, and the expected payoffs for both the put and the call are also higher • This is the result of truncation (lower bound), and holds in all empirically reasonable cases • Conclusion: Volatility increases all option prices • TWO STATE, RISK NEUTRAL BINOMAL EXAMPLE…

  46. Binomial Model • Although the single period binomial model is extremely useful in understanding the intuition of option pricing, it is, of course, unrealistic since the end-of-period stock price can take on many values • The binomial model can be easily extended however to allow for many possible ending stock prices. Consider for example a 2-period binomial setup

  47. u2S uS S0 udS = duS dS d2S (1 + r ) (e-rt)2 B0 Cuu Cu Cud = Cdu C0 Cd Cdd Binomial Model To get three possible end-of-period values for the stock price, we add an additional period

  48. Binomial Model • How do we calculate the price of the call in this extended setup? We determine it by working backwards from the last period, using the valuation technique we developed for the single period model. • At the end of the period, depending on the value of the stock, the values of the call option are: Cuu = max(Suu - X, 0) Cud = Cdu = max(Sdu - X, 0) Cdd = max(Sdd - X, 0)

  49. Binomial Model • As in the one-period model, the call prices in the intermediate period are: Cu= (p*Cuu+ (1- p*) Cdu)/(e-rt) Cd= (p*Cdu+ (1- p*) Cdd)/(e-rt) where p* = ((e-rt)S0- Sd)/ (Su - Sd). • Therefore the call price today is given as follows: C = (p* Cu + (1- p*) Cd)/(e-rt) = (p*2Cuu+ 2p*(1- p*) Cud + (1- p*)2Cdd)/(e-rt)2 • Note that the first equality above indicates that the expected call price is the expected payoff in one period discounted back to today. The second equality indicates that we can also calculate the call price by discounting back to today the payoffs in two periods.

  50. Binomial Model In Class Exercise: Assume the current stock price is 100, u = 11.8%, d = 10.56% , rf = 7%, and the number of periods is equal to 2. What is the current value of a call option with X = 50? Comparing this answer with the answer for the one period example a few pages back, what is the effect of increasing the time to expiration on the value of an option?

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