Corporate Finance, Module 21: "Option Valuation"

(The attached PDF file has better formatting.)

Updated: July 7, 2005

{This posting has more information than is needed for the corporate finance on-line course. The past Course 2 exam problems on options pricing have been difficult, covering a wide range of topics. Most of the examples in this posting are for candidates taking the CAS transition exam, who may be confronted with problems like those shown here.}

Jacob: I can’t get the hang of these option pricing formulas.

Rachel: The first several times that you work with binomial tree valuation, you should draw the tree and work out the risk-free portfolio for both calls and puts. Focus on two items:

Question 21.1: Option Delta

Which of the following statements is true regarding option deltas?

Answer 21.1: A

The option delta is the partial derivative of the option price with respect to the stock price.

For binomial tree pricing, the formula for the option delta is the discrete analogue of the partial derivative. Let S₀ be the current stock price and f be the derivative security:

u is the upward movement in the stock price

d is the downward movement in the stock price

f_u is the value of the option if the stock price moves up (sometimes denoted f⁺)

f_d is the value of the option if the stock price moves down (sometimes denoted f^–)

The superscripts + and – are sometimes used for the stock, the option, the call, and the put:

S⁺, f⁺, c⁺, p⁺ and S^–, f^–, c^–, p^–

The option delta is

option Δ = (f_u – f_d)/(S₀ × u – S₀ × d)

Jacob: Are u and d factors, such as 1.150 and 0.850, or percentage changes, such as +15% and –15%?

Rachel: Either definition is fine. We use the difference between u and d. If we use factors, we get 1.150 – 0.850 = 0.300. If we use percentage changes, we get 15% – (–15%) = 30% = 0.300. When we use the risk-neutral probabilities, we must be consistent between stock price movements and interest rates. If u and d are factors, the interest rate is a factor, like 1.08; if u and d are percentage changes, the interest rate is a percentage, such as 8%.

Statement B is incorrect; it should say:

If we hold delta (Δ) shares of stock and we short one option, the portfolio is

ΔS – f.

You will sometimes see this risk-free portfolio as f – ΔS, since if a portfolio if risk-free, its negative is risk-free.

At the end of the period, there are two possibilities: the up state and the down state. If the up state occurs, S₀ becomes S₀ × u and f becomes f_u, so the value of the portfolio is

Δ × S₀ × u – f_u

We replace Δ by (f_u – f_d) / (S₀ × u – S₀ × d) to get

Δ × S₀ × u – f_u = (f_u – f_d) / (S₀ × u – S₀ × d) × S₀ × u – f_u

If the down state occurs, S₀ becomes S₀ × d and f becomes f_d, so the value of the portfolio is

Δ × S₀ × u – f_d = (f_u – f_d) / (S₀ × u – S₀ × d) × S₀ × d - f_d

A riskless portfolio has the same value in all states of the world, whether the stock price moves up or down, so if the portfolio is riskless, these two values must be the same:

(f_u – f_d) / (S₀ × u – S₀ × d) × S₀ × u – f_u = (f_u – f_d) / (S₀ × u – S₀ × d) × S₀ × d – f_d

We show this algebraically. We factor out the S₀’s to get

(f_u – f_d) / (u – d) × u – f_u = (f_u – f_d) / (u – d) × d – f_d

We add f_u to each side to get

(f_u – f_d) / (u – d) × u = (f_u – f_d) / (u – d) × d + (f_u – f_d)

We factor out the (f_u – f_d) to get

u / (u – d) = d / (u – d) + 1

(u – d) / (u – d) = 1

Statement C should be reversed:

Statement D should say that the call option delta equals the put option delta plus one. The put call parity relation says that c + PV(X) = p + S. We take the partial derivative of both sides with respect to the stock price S.

Statement E should say that an option is riskiest when its delta is near ½ or –½.

If the option has a delta of about ½, it may or may not be exercised, and it is risky.

Jacob: Suppose the exercise price is 50, and there is one day left to expiration.

The first scenario has a larger change in the call option price.

Rachel: You must also consider the present price of the option.

European call and put options are written on a share of stock, with a strike price of $80. As the stock price approaches infinity, which of the following are correct?

Answer 21.2: D

The intrinsic value of the call option is the stock price minus the strike price. The market price of the call option exceeds this, since the option also has a time-element value. As the stock price approaches infinity, the intrinsic value of the call option approaches infinity, so the full value of the call option approaches infinity.

As the stock price increases to infinity, the investor surely exercises the call option. A dollar rise in the stock price increases the value of the call option by one dollar, so the delta becomes unity.

The put option is exercised only if the stock price is less than the strike price at the expiration date. As the stock price approaches infinity, the probability that the put option will be exercised approaches zero, and its value approaches zero.

The delta of the put option is the change in price of the put option divided by the change in price of the underlying security. As the stock price approaches infinity, the value of the put option approaches zero. For a one dollar change in the stock price, the change in the value of the put option approaches 0 – 0 = 0.

Question 21.3: Risk-Neutral Valuation

In the binomial tree models that are used to value options, each node leads to two future nodes, an up path and a down path. All but which of the following statements is true?

Answer 21.3: A

This illustrative test question reviews the crux of the binomial tree pricing method. We explain each statement by examples.

Statement A above sounds reasonable. To see why it is not correct, we examine the two options here.

Stock A is trading at $20 a share. Option A is a European call option on Stock A with a strike price of $21. The stock price in three months will be either $22 or $18. Stock A has a 90% chance of moving up to $22 and a 10% chance of moving down to $18. This is the option with a high probability of an upward movement in the stock price.

Stock B is trading at $20 a share. Option B is a European call option on Stock B with a strike price of $21. The stock price in three months will be either $22 or $18. Stock B has a 75% chance of moving up to $22 and a 25% chance of moving down to $18. This is the option with a lower probability of an upward movement in the stock price.

Stocks A and B are unrelated. Stock A may move up when Stock B moves down, and Stock B may move up when Stock A moves down.

Solution 21.3:

Option pricing is tricky. In the dialogue between Jacob and Rachel, Jacob’s comments are not all correct, though they make sense at first.

Jacob: For Part A, we work out the expected values of Option A and Option B at the expiration date:

Option A has the higher expected value in three months.

Rachel: That’s correct.

Jacob: The difference between Stocks A and B is the probability of increasing in value. We know nothing else about the stocks or the options, so if Option A is worth more at the expiration date, it is worth more now.

For the risk of each option, we examine the standard deviation of their prices; higher standard deviation means higher risk.

Option A: The mean value of option A in three months is $0.90. The variance of the option value in three months is

90% × ($1.00 – $0.90)² + 10% × ($0.00 – $0.90)² = $0.09

The standard deviation in three months is 0.09^0.5 = $0.30.

Option B: The mean value of option B in three months is $0.75. The variance of the option value in three months is

75% × ($1.00 – $0.75)² + 25% × ($0.00 – $0.75)² = $0.19

The standard deviation in three months is 0.19^0.5 = $0.43

We infer that Option B is the riskier option.

Rachel: The arithmetic is correct. Option A has the higher expected value in three months. Option B has the higher variance and the higher standard deviation of its expected value.

But the two options have the same present value, and Option A is the riskier option.

Option A and Option B have the same present value because the pricing of the option does not depend on the actual probabilities of the stock movements. The actual probabilities are referred to as the subjective probabilities, actual probabilities, or real world probabilities, to distinguish them from risk-neutral probabilities.

To price these options with the binomial tree pricing method, we need the risk-free interest rate. We are not told this rate, but it is the same for both stocks. The risk-free interest rate does not depend on the particular option that we are pricing. Since everything else in the practice problem is the same for the two options, they must have the same price.

(Jacob is not yet persuaded; we come back to this topic further below.)

The standard deviation and the variance are not necessarily good proxies for risk. We use only systematic risk to price securities, not the total standard deviation or variance.

We determine the relative risk of the two options by comparing their capitalization rates. The option with the higher capitalization rate has the greater risk.

For any problem comparing the risk of securities, security #1 must have the higher variance but security #2 may have the higher systematic risk. In fact, security #1 may have a very high variance and standard deviation and security #2 may be risk free, but if security #1 is inversely correlated with the overall market return, security #2 is riskier.

Jacob: How can a security have less risk than a security that is risk-free?

Rachel: A security which has negative risk is less risky. A security which has a negative beta has negative risk.

We don’t know the risk of either option. We know only that they have the same present value and that Option A has the higher expected value in three months. This implies that Option A has the higher capitalization rate and is the riskier security.

Exercise 21.4: Two Stocks

Jacob: I recognize the phrases you used; they come from the Brealey and Myers text. The text discusses total risk and systematic risk, along with total variance (or total standard deviation) vs the CAPM beta. But I don’t understand how these options can have the same value. You admit that the options have different expected values in three months. That is all we know about these two options; you admit that we don’t know the risk of the options. If all we know about two securities is that one has a greater expected value in three months, shouldn’t we assume it have the greater current value as well?

Rachel: Let us revise the illustration to speak about the stocks, not about call options.

Stock A is trading at $20 a share. Option A is a European call option on Stock A with a strike price of $21. The stock price in three months will be either $22 or $18. Stock A has a 90% chance of moving up to $22 and a 10% chance of moving down to $18.

Stock B is trading at $20 a share. Option B is a European call option on Stock B with a strike price of $21. The stock price in three months will be either $22 or $18. Stock B has a 75% chance of moving up to $22 and a 25% chance of moving down to $18.

Solution 21.4:

We work out the expected values of Stock A and Stock B.

Option A has the higher expected value in three months.

Part B: The present value of the stocks is $20 for each stock (as mentioned in the problem). The present value of a stock is its market value, which is $20 for both stocks.

Part C: Whichever stock has the higher capitalization rate is the riskier stock.

Stock A is the riskier stock.

Jacob: How can stock A be the riskier stock? The only difference between Stock A and Stock B is that Stock A has a higher probability of increasing in value. Stock B has the higher probability of decreasing in value. Shouldn’t this make Stock B the riskier stock?

Rachel: Stocks A and B are not related. Perhaps stock A is a high beta stock, which increases in value when the overall stock market increases in value. Stock B might be a low beta (defensive) stock, which increases in value when the overall stock market declines and decreases in value when the overall stock market increases.

For the pricing of options, we are not concerned with the type of risks accompanying the underlying securities. For whatever reason, Stock A is the riskier stock, as shown by the market valuation of the stock. The option contract has the same risk as the underlying security, but the risk is leveraged (magnified). Option A has the higher expected value at the expiration date, but the same present value, so it is the riskier security.

Jacob: We have two scenarios, a favorable scenario and an adverse scenario, or a favorable state of the world and an adverse state of the world.

How can you say that Stock A is not worth more than Stock B, if it is always in at least as good a state of the world and sometimes in a better state of the world?

Rachel: You are mis-using the terms. A state of the world is just what it says: all securities have the same probability of being in that state, since the state has nothing to do with the security. If there were just two states of the world, and Stock A had a 90% probability of being in one of these states in three months time, Stock B would have the same probability of being in that state, since the entire world is in that state.

In the problem with Stocks A and B, there are at least thee states of the world, and there may be many more. The two stocks don’t move up or down together.

Let us show the three possible states:

Let us return to the original illustrative test question.

Rachel: It is natural to assume that as the probability of an upward movement in the stock price increases, the value of a call option on the stock increases and the value of a put option on the stock decreases. This is not the case.

Jacob: if there is a greater chance for the stock price to increase, then the option must be worth more.

Rachel: We do not value the option in absolute terms. We calculate its value in terms of the price of the underlying stock. The probabilities of future up or down movements are already incorporated into the price of the stock. We do not need to take them into account again when valuing the option in terms of the stock price.

Jacob: When we price an option, we get a dollar value, not a factor that is applied to the stock price. What do you mean by in terms of the price of the underlying stock?

Rachel: The value of the option depends on the value of the underlying security, not just on the future cash flows of the derivative security.

We summarize the conclusion regarding stocks A and B as follows:

The Black-Scholes formula prices an option in terms of five input values: the stock price, the exercise price, the expected, the risk-free rate, and the stock price volatility. Let us denote the exercise price as a multiple of the stock price, such as 90% of the stock price, or 115% of the stock price. If we double the stock price, the call option value and the put option value both double. If we multiply the stock price by a constant k, the values of the call option and the put option are multiplied by k.

Jacob: The up and down probabilities in the binomial tree pricing model are not related to the real-world probabilities of up and down movements in the stock price. Can we use any up and down movements that we wish?

Rachel: No. Two constraints limit the choice. The no-arbitrage constraint requires that the risk-free interest rate lie between the up and down movements.

Question 21.5:

The risk-free interest rate is 10% per annum. We are pricing a one year option with a four period binomial tree. Which of the following are possible values for the up and down movements of the binomial tree?

Answer 21.5: B

The up and down movements must straddle the risk-free rate: the up movement must be greater and the down movement must be lower. This question uses a four period binomial tree, so each period is three months, or one quarter of a year. The risk-free rate for three months is 1.10^¼ – 1 = 2.41% if we use annual compounding and 10% × ¼ = 2.5% if we use continuous compounding.

Jacob: Why do we often use continuous compounding for options pricing? Does this make a difference?

Rachel: No. Continuously compounded rates are easier for the Black-Scholes model and for pricing with various different terms to maturity. 10% per annum with continuous compounding is 2.5% per quarter with continuous compounding. We could use annual effective yields and take the fourth root.

Jacob: What should we use for the final exam?

Rachel: The final exam problems use annual effective yields. If a final exam problem uses continuous compounding, it explicitly says that the effective interest rate is e^r.

In this problem, the risk-free interest rate for a three month period is 2.4% with annual compounding and +2.5% with continuous compounding. All statements except for statement B lead to arbitrage profits.

Jacob: What does it mean to short the bond?

Rachel: To short the bond means to borrow at 10% per annum.

The arbitrage constraint is that u > r > d. r must be positive, so u must also be positive. d can be positive or negative. For a high volatility stock, u is very high and d is usually negative. For a low volatility stock, u is slightly above the risk-free rate and d is slightly below the risk-free rate.

Jacob: Why must r be positive?

Rachel: If r were negative we would have an arbitrage opportunity, by shorting the risk-free bond and placing the cash under the mattress.

Question 21.6:

The risk-free interest rate is 10% per annum with continuous compounding. We are pricing a one year option with a four period binomial tree. The volatility parameter σ of the stock is 30% per annum. Which of the following are possible values for the up and down movements of the binomial tree? All the movements below use continuous compounding.

Answer 21.6: Volatility Constraint

E (only Statement 3 is correct)

The most common method of selecting the up and down movements is to set

i = e^{σ × %t} and d = e^{–σ × %t}

In this illustrative test question, t = ¼ and σ = 30%, so e^{σ × %t} = e^15% and e^{–σ × %t} = e^–15%. The textbook mentions this method. It is not the only method, and it does not always give an arbitrage free scenario, but it is the easiest method to use (when it works).

Jacob: If we used the movements in statement 1 or statement 2, would the computed option price be too high or too low?

Rachel: The movements in statement 1 imply an annual volatility of 20%. Since the actual volatility is 30% per annum, the option price computed from the binomial tree pricing method would be too low. This is true for both a call option and a put option.

The movements in statement 3 imply an annual volatility of 40%. Since the actual volatility is 30% per annum, the option price computed from the binomial tree pricing method would be too high. This is true for both a call option and a put option.

Jacob: Are there other possibilities for the up and down movements?

Rachel: There are many possibilities. We could use +20% and –10%, and we could use +10% and –20%. Both of these give a quarterly volatility of about +15%, which is the same as an annual volatility of 30%.

Jacob: On the final exam, what should we use?

Answer: If you are given the stock price volatility but not a pair of price movements, use

u = e^{σ × %t} and d = e^{–σ × %t}.

To value an option, we use a binomial tree where the risk-free interest rate, r, the up movement, u, the down movement, d, and the time step Δt are kept constant. Which of the following statements are correct?

Answer 21.7: "B: 1 and 3 only."

The binomial tree pricing method is a discrete method. To keep a risk-free portfolio, we re-balance the portfolio at every step of the binomial tree model. The Black-Scholes model is a continuous model. To keep a risk-free portfolio, we re-balance the portfolio continuously in the Black-Scholes pricing model.

Jacob: Does an investor using Black-Scholes have to hold a risk-free portfolio?

Rachel: No. The risk-free portfolio is used for the pricing (arbitrage) argument. As long as some investors seek out mispriced securities and form risk-free portfolios to earn arbitrage profits, the derivative securities will not remain mispriced in an efficient market.

Jacob: Are there actually any arbitragers who seek out mispriced derivative securities?

Rachel: Yes; some hedge funds seek mispriced derivative securities to gain arbitrage profits. The hedge fund searches for relative mispricing. It can’t say that option A is overpriced or option B is underpriced, since the price depends on the volatility of the underlying security, which is hard to estimate. But the hedge fund can say that a certain call option on a stock is overpriced relative to a certain put option on the same stock. The hedge fund would buy the put option and simultaneously sell the call option.

Jacob: Is it easy to find such pairs of relatively mispriced derivative securities?

Rachel: It is hard to find them in efficient markets, such as the markets for derivative securities on stock indices. It is easier to find them in thinly traded markets for unusual options, such as the markets for bonds with embedded options.

Jacob: Rebalancing incurs transaction costs. Do traders form risk-free portfolios and rebalance them?

Rachel: Yes. Some financial institutions who sell options rebalance their portfolios every day.

Statement 2 is correct. We assume that the risk-free rate is not stochastic, and that the values of the up and down parameters remain constant.

The risk-neutral probability p satisfies p × u + (1 – p) × d = r. That’s what risk-neutral valuation means: if investors are risk-neutral, they are indifferent between the uncertain return with the up and down parameters or the certain return at the risk-free rate.

The delta changes at the each node of the tree. The delta depends on the price of the stock (among other variables). The option delta is the ratio of the price change of the call option to the price change of the stock itself.

If the price of the stock increases to infinity, delta becomes unity.

At each node, the stock price of the stock moves up or down and the option delta changes.

Jacob: Is this the only reason for the change in the option delta? If the stock price does not change, would the option delta change?

Rachel: Yes. The option delta changes with the passage of time. Suppose we have a one year call option on stock ABC with strike price $85 and stock price $80. Because the duration of the option is long (one year), the call option may be worth now $10. If the stock price increases from $80 to $81, the call option value may increase from $10 to $10.50, for a 50% delta.

Suppose the stock price is $80 one day before the expiration of the option. The value of the option is a few cents. If the stock price increases by $1 during the morning, the value of the option may increase by five cents, for a 5% delta.

Jacob: What if the stock were far in the money instead of far out of the money? If the stock were $90 one day before the expiration of the option, what would the delta be?

Rachel: At a stock price of $90, the option may be worth $5 plus four cents. If the stock price rises to $91, the option may be worth $6 plus two cents. The option delta is 98%.

The delta represents the amount of shares that one must sell short to hedge one long option, or the amount of shares that one must hold long to hedge against one short option. If the delta changes, then the hedging portfolio must be re-balanced.

A stock price is currently $40. At the end of one month it will be either $42 or $38. The risk-free interest rate is 8% per annum with continuous compounding. A one-month European call option is trading with a strike price of $39?

Solution 21.8:

Two mathematically equivalent methods of solving these problems are the no-arbitrage method (the option delta method) and the risk-neutral valuation method.

The no-arbitrage method lets you see what is happening; the risk-neutral valuation method is faster for solving problems. In some problems, it may not be clear how to use the risk-neutral valuation method. The no-arbitrage method shows the intuition; once you have solved the problem, you can see how to use the risk-neutral valuation method to solve it more quickly.

Part A: The no-arbitrage argument runs as follows. We set up a portfolio consisting of the call option c minus delta (Δ) shares of stock such that this portfolio is risk free over the time interval in the problem: c – ΔS

Part B: At the end of the month, we know the value of the option in two scenarios. The strike price is $39.

If the portfolio is risk-free, its value in these two scenarios must be the same:

$3 – Δ × $42 = $0 – Δ × $38, or

Δ = ¾.

Part C: We wrote out the intuition to make the procedure clear. More simply, the delta is the ratio of the change in the option price to the change in the stock price:

Δ = ($3 – $0) / ($42 – $38) = ¾.

Part D: If the portfolio is risk-free, then its return must be the risk-free rate. This is true by a no-arbitrage argument.

The risk-free rate is 8% with continuous compounding. The risk-free return for one month is e^8%/12 = e^0.00667 = 1.0067.

Jacob: For the final exam, will any problems use continuous compounding?

Rachel: Most problems use effective annual yields. If a problem uses another compounding interval, it will explicitly tell you.

Part E: The one month return on the risk-free portfolio must be +0.67% in the two scenarios. Let’s use the up scenario to determine the option price.

In the up scenario, the stock begins at $40 and ends at $42. The option price begins at c and ends at $3. The equation is

(c – ¾ × $40) × 1.0067 = ($3 – ¾ × $42), or

c = $1.69

We do the same with the down scenario where the stock begins at $40 and ends at $38 and the option price begins at c and ends at $0:

(c – ¾ × $40) × 1.0067 = ($0 – ¾ × $38), or

c = $1.69

Part F: The risk-neutral valuation method using the binomial tree puts this algebraic procedure into formulas. First we find the probability of the up movement in a risk-neutral world. If the risk-free rate is 8% per annum with continuous compounding, the equation becomes

p × u + (1 – p) × d = e^rT

p = (e^rT – d) / (u – d)

In our problem, u = 42/40 = 1.05 and d = 38/40 = 0.95, so we have

p × 42/40 + (1 – p) × 38/40 = e^8%/12

p = (1.0067 – 0.95) / (1.05 – 0.95) = 0.567

Part G: We now value the option using risk-neutral valuation. The values of the option at the end of the month are $3 in the "up" state and $0 in the "down" state. The expected value of the option at the expiration date in the risk-neutral world is

Part H: The value of the option at the beginning of the month is

c = 1/1.0067 × [0.567 × $3 + (1 – 0.567) × $0] = 1.689

This is the same answer as we got in the no-arbitrage method.

We have emphasized the intuition; that’s what you need to understand the text. We also provide a seven-step cookbook procedure for exam problems.

Jacob: This means that d = 1/u. But some examples don’t have this relation.

Rachel: There are many ways to choose the up and down movements. The movements must provide a volatility of σ. The expressions above do this; other choices do this as well.

Jacob: Don’t we also have to take into consideration the stock’s expected return F? After all, if the stock has an expected return of +20% and we choose up and down movements of +15% and –5%, the binomial tree won’t agree with the actual movements of the stock.

Rachel: The binomial tree pricing method assumes a risk-neutral world. In a risk-neutral world, the stock’s expected return is the risk-free interest rate. We need the relation that

down movement < risk-free rate < up movement.

We don’t need to do anything with the actual drift of the stock (F).

Jacob: Is the binomial tree always recombining?

Rachel: If we use this definition of the up movement and the down movement the binomial tree is recombining. If is also recombining if we have a proportional stock dividend.

Jacob: When is the tree not recombining?

Rachel: The binomial tree is not recombining when there is a constant dollar dividend. {Note: The final exam problems do not have dividend paying stocks.}

p = (e^rT – d) / (u – d)

Some authors call this the equivalent Martingale probability; this term is not in the syllabus readings, but you will see it on the Course 6 readings (Panjer’s textbook).

Jacob: Are these values related to the risk-neutral probabilities in the previous step?

Rachel: No. These values have not included the probability of arriving at any of the nodes in the final column. You can reverse these two steps; they are not related.

This is not the expected value of the derivative at the expiration date. If we obtained the expected value of the derivative at the expiration date, we would not be able to go on, since we don’t know the proper capitalization rate to discount back to the present time. Rather, this is the expected value in a risk-neutral world.

Explain the no-arbitrage (option delta) and risk-neutral valuation approaches to valuing a European option using a one-step binomial tree.

The no-arbitrage argument says that if we have only 2 paths leading from each node, then we can set up a risk-free portfolio consisting of one option and the short sale of delta shares of the underlying asset. We know that we can set up such a risk-free portfolio since we have one variable (delta) and one equation (the value of the risk-free portfolio must be equal in the two states stemming from this node).

Once we set up the risk-free portfolio, we know that the return on the risk-free portfolio must be equal to the risk-free rate. We set up an equation saying that the value of the portfolio at the beginning of the period times the risk-free interest rate equals the value of the portfolio (at either node, since they are equal) at the end of the period. This equation solves for the value of the option.

The risk-neutral valuation approach stems from the no-arbitrage approach. The risk-neutral valuation approach says that our previous argument was not dependent upon the risk aversion of investors.

Jacob: We never discuss the risk aversion of investors in financial theory, do we?

Rachel: We do this all the time, though it’s not obvious. We discuss the returns of various securities, such as the returns on stocks, or corporate bonds, or mortgages, or real estate. These securities have different returns because they have different types of risk and because investors are risk averse. Any time that we use the actual return of a security in an argument, we are using the risk aversion of investors.

In the no-arbitrage argument, we never discuss the actual return on the underlying security, so we never use the risk aversion of investors for our argument. If so, the same argument is true in any type of world, whether a risk-neutral world or a world where investors are risk averse. It’s simplest to solve for the value of the option in a risk-neutral world. We solve for the value of the option in that type of world, but this value is the same in any type of world.

A stock price is currently $50. It is known that at the end of six months it will be either $45 or $55. The risk-free interest rate is 10% per annum with continuous compounding. A six-month European put option has a strike price of $50.

Part A: To determine the risk-neutral probability, we must know the values of the up and down movements and the risk-free interest rate. We do not have to know the current stock price or the exercise price, since the risk-neutral probability doesn’t change as long as the up and down movements and the risk-free interest rate don’t change.

Jacob: What if we are given the stock values at the nodes, not the up and down movements?

Rachel: Call the stock values S₀, S⁺, and S^–. The up movement is S⁺ / S₀. The down movement is S^– / S₀.

In our problem, e^rT = e^10%/2 = 1.0513, u = 55/50 = 1.10, and d = 45/50 = 0.90.

Jacob: Is it true that u > 1 and d < 1?

Rachel: No. All we need is that u > e^rt and d < e^rt. If the risk-free interest rate is 8% for the time period, then u = 6% and d = –5% doesn’t work, but u = +12% and d = +1% works fine.

Part B: If the stock price moves up to 55, the put option expires worthless (= $0). If the stock price moves down to 45, the put option is worth 50 – 45 = 5.

p = (1.0513 – 0.90) / (1.10 – 0.90) = 0.756

Part D: The expected value of the put option at expiration in the risk-neutral world is

[0.756 × $0 + (1 – 0.756) × $5] = $1.22

Part E: The present value of the put option is

1/1.0503 × [0.756 × $0 + (1 – 0.756) × $5] = 1.1588.

A stock price is currently $100. Over each of the next two six-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 8% per annum with continuous compounding. A one-year European call option has a strike price of $100.

The first several times that you do this, draw the binomial tree. Almost all of the examples in the textbook have recombining trees. This means that the price of the underlying security is the same whether an up movement is followed by a down movement or a down movement is followed by an up movement. For options on non-dividend paying stocks, where the up and down movements don’t change over time, this means that the up and down movements are multiplicative factors.

Practice drawing the binomial tree for several problems. The formats for both one period trees and two period trees are shown in the textbook.

Nodes D, E, and F are at the exercise date. Fill in the values of the options at the exercise date based on the exercise price, the price of the underlying asset, and the type of option (call or put). That’s as far as we can get without any options pricing technique. For nodes A, B, and C, we must use either the no-arbitrage method or the risk-neutral valuation method.

Using the figures in this exercise, we first fill in the stock prices. We start at $100, and we move up or down by 10% in each period. We calculate uu as +21%, ud as –1%, and dd as –19%, so the ending stock prices are $121, $99, and $81 in nodes D, E, and F.

This is a European call option, so the option values at exercise are $21, $0, and $0 at nodes D, E, and F, respectively.

Jacob: How does the option pricing technique differ for European call vs put options?

Rachel: Only one piece differs:

Part B: The risk-neutral probability does not vary over the life of the option, since the up and down movements do not change and the risk-free interest rate does not change. We have

p = (e^rT – d) / (u – d)

e^rT = e^8%/2 = 1.0408, u = 1.10, and d = 0.90, so

p = (1.0408 – 0.90) / (1.10 – 0.90) = 0.704

Part C: T is the length of the period, which is six months or ½ of a year in this problem. We use the risk-neutral valuation procedure to solve for the value of the option at each node, using e^-rT = 1/1.0408 = 0.9608. At node B, we have

value of call at node B = 0.9608 × [0.704 × $21 + (1 – 0.704) × $0] = $14.2054.

Part D: For node C, the arithmetic is easy. Since the option value is $0 at both node E and node F, the option value is $0 at node C as well. Conceptually, this says that if the option will be worth $0 at the next time period, whether an "up" or "down" movement occurs, then it is worth $0 now. This is a characteristic of the binomial tree model. In real life, it is almost always possible that the option will end up worth something next period, either by a large rise in the stock price (for a call option) or by a large decline in the stock price (for a put option).

Part E: For node A, the value of the call option is

0.9608 × [0.704 × $14.2054 + (1 – 0.704) × $0] = $9.6092.

Part F: The one-line formula for the value of the option is

f = e^–2rΔt [p² f_uu + 2p(1–p) f_ud + (1–p)² f_dd]

f = e^{–2(0.08)(0.5)}[(0.704²)($21) + 2(0.704)(1–0.704)($0) + (1–0.704)²($0)] = $9.6092

Part A: This question helps you see the differences between call option pricing and put option pricing. The only difference is the values of the option at each node in the final column (i.e., at the exercise date). For the put option, the values of the option are $0 at node D (up-up), $1 at node E (up-down), and $19 at node F (down-down). We use these numbers to get

f = e^{–2(0.08)(0.5)}[(0.704²)($0) + 2(0.704)(1–0.704)($1) + (1–0.704)²($19)] = $1.9208.

Part B: The put-call parity relationship helps verify the answer. The put-call parity relationship says that "call plus cash = put plus stock." Cash is the amount of cash that would accumulate to the strike price if invested at the risk-free interest rate. Similarly, the other values in the put call parity relationship are the values at the initial date.

In this problem, cash = e^{–2(0.08)(0.5)}($100) = $923.12. The put-call parity relationship says that $9.6092 + $92.31 = $1.9208 + $100. This is true.

Exercise 21.13: Consider the situation where stock price movements during the life of a European option are governed by a two-step binomial tree. Explain why it is not possible to set up a position in the stock and the option that remains riskless for the whole of the life of the option.

The volatility of the stock price doesn’t change over the life of the option in the binomial tree model. However, the price of the underlying asset changes and the time remaining until the exercise date changes, and both of these changes affect the option delta.

The relative amounts of stock shares versus options in the risk-free portfolio at any time depend on the option delta at that moment. Thus, the relative amounts change over the life of the option. To keep the portfolio risk-free, the portfolio must be rebalanced at each node of the binomial tree.

Exercise 21.14: A stock price is currently $50. It is known that at the end of two months it will be either $53 or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a two-month European call option with a strike price of $49?

The textbook often asks you to use no-arbitrage arguments in the solution. We used no-arbitrage arguments in our solution to Question 9.1. These arguments take a long time to write down, and they are conceptually the same for all the problems. The model solutions here simply show the answers from the risk-neutral valuation method. In your mind, make sure that you can formulate the no-arbitrage arguments for each situation. On the exam, you may be asked to evaluate the option delta at various nodes.

The risk-free portfolio changes at each node. For a two-step binomial tree, we must form as many as 3 risk-free portfolios for the no-arbitrage argument.

This problem is a simple one step binomial tree. The "up" movement is 53/50 = 1.06. The down movement is 48/50 = 0.96. For the risk-neutral probability we have

p = (e^rT – d) / (u – d)

Note that the "up" movement does not have to be the reciprocal of the "down" movement. In fact, the examples in Hull’s text generally do not use reciprocals; that is, u … 1/d. However, Hull notes that certain options pricing formulas do use reciprocal movements, where u = 1/d. On the exam, you might be expected to assume that u = 1/d.

In this problem, e^rT = e^10%/6 = 1.0168, so

p = (1.0168 – 0.96) / (1.06 – 0.96) = 0.568

We take the expected value of the "up" and "down" values of the call option at the risk-neutral probability and we discount by the risk-free interest rate to get

value of call = 1/1.0168 × [0.568 × $4 + (1 – 0.568) × $0] = $2.2347

Remember that the value of the call option at the exercise date is Max (0,S_T–X). Review the model solution to Question 9.1 and fill in the no-arbitrage argument.

Exercise 21.15: A stock price is currently $80. It is known that at the end of two months it will be either $75 or $85. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a four-month European put option with a strike price of $80?

This question is just like the previous one, except that the call option is replaced by a put option and the parameters are changed. Since the type of option is relevant only for the boundary conditions, the changes to the solution are trivial.

This problem is a one step binomial tree. The "up" movement is 85/80 = 1.0625. The down movement is 75/80 = 0.9375. We convert these to "up" and "down" movements.

For the risk-neutral probability we have

p = (e^rT – d) / (u – d)

In this problem, e^rT = e^5%/3 = 1.0168, so

p = (1.0168 – 0.9375) / (1.0625 – 0.9375) = 0.63445

We take the expected value of the "up" and "down" values of the put option at the risk-neutral probability and we discount by the risk-free interest rate to get

value of put = 1/1.0168 × [0.63445 × $0 + (1 – 0.63445) × $5] = $1.7975

Remember that the value of the put option at the exercise date is Max (0,X-S_T). Review the model solution to Question 9.1 and fill in the no-arbitrage argument.

Exercise 21.16: A stock price is currently $40. It is known that at the end of three months it will be either $45 or $35. The risk-free interest rate with continuous compounding is 8% per annum. Calculate the value of a three-month European call option and of a European put option on the stock with an exercise price of $40. Solve the problem is the risk-free interest rate is 8% per annum with quarterly compounding.

We use the risk neutral valuation approach, which is fastest for exam problems.

The up movement is 45/40 = 1.125; the down movement is 35/40 = 0.875.

The risk-free interest rate is 8% per annum or 2% per three months.

The risk-neutral probability of an up movement is

The value of the European call option at the final two nodes is $5 if the stock rises to $45 and $0 if the stock falls to $35.

The expected value of the call option at the expiration date in a risk-neutral world is

The present value of the call option is e^–4% × $2.904 = $2.85.

The expected value of the put option at the expiration date in a risk-neutral world is

The present value of the put option is e^–4% × $2.096 = $2.05.

If we use quarterly compounding, the three month risk-free interest rate is 1.02 instead of e^2%. Everything else in the problem is the same. The risk-neutral probability of an up movement is

The expected value of the call option at the expiration date in a risk-neutral world is

The present value of the call option is e^–4% × $2.9 = $2.85.

The expected value of the put option at the expiration date in a risk-neutral world is

The present value of the put option is e^–4% × $2.1 = $2.06.

Exercise 21.17: A stock price is currently $50. Over each of the next three-month periods it is expected to go up by 6% or down by 5%. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a six-month European call option on with a strike price of $51?

Exercise 21.18: For the situation considered in problem 9.11, what is the value of the six-month European put option with a strike price of $51? Verify that the European call option and the European put option satisfy put-call parity. If the put option were American, would if ever be optimal to exercise it early at any of the nodes on the tree?

We discuss the last sentence in this problem. First, not ask the same question about American options in the previous problem?

Answer: American call options on non-dividend paying stocks should never be exercised early. If we exercise the American call option early, we lose in two ways. (1) We lose the interest on the cash that we pay early to obtain the underlying asset, and (2) We lose the chance to defer our decision until the last possible date. This latter part, the chance to defer our decision, is particularly important if the current asset price is close to the strike price.

For the put option, if we exercise early, the first of these two pieces is reversed. Instead of losing interest on the cash, we get interest on the cash that we receive early for the sale of the underlying asset. The second part remains the same: we still lose the chance to defer our decision whether or not to exercise the option until the last possible date.

When is it preferable to exercise an American put option at an early date? Answer: When the value of the extra interest exceeds the value of the chance to defer the exercise decision. The value of the interest is easy to determine, since we know the strike price and the risk-free interest rate. The value of deferring the exercise decision depends on various factors and it is hard to estimate. The general rule is as follows: If the put option is deep in the money, and the volatility of the stock price movement is low, and the interest rate is high, then it pays to exercise early. If the put option is near the money, or the volatility of the stock price movement is high, or the risk-free interest rate is low, then it does not pay to exercise early.

This is not a rule to use in solving options pricing problems. This is an intuitive rule to help you understand what is happening in these problems.

The up movement is +6% and the down movement is –5%. The risk-free interest rate is 5% per annum or 5% × ¼ for a three month period. The value of the risk-neutral probability of an up movement is

We determine the values of the stock and the put option at the three nodes in the final column. At the highest node, the value is $50 × 1.06 × 1.06 = $56.18; the value of the put option at this node is $0. At the middle node, the value of the stock is $50 × 1.06 × 0.95 = $50.35, and the value of the put option is $0.65. At the lowest node, the value of the stock is $50 × 0.95 × 0.95 = $45.125, and the value of the put option is $5.875.

The risk-neutral probability of getting to the highest node in the final column is 0.5689². The risk-neutral probability of getting to the middle node in the final column is 2 × 0.5689 × 0.4311. The coefficient "2" is included because there are two paths leading to the middle node. The risk-neutral probability of getting to the lowest node in the final column is 0.4311².

The present value of the European put option is

e^{–5% × 0.5} × (0.5689² × $0 + 2 × 0.5689 × 0.4311 × $0.65 + 0.4311² × $5.875) = 1.376

What about the American put option? There are two ways to view this question. One way is to calculate the value of the European put option at the two nodes in the middle column (i.e., at the end of the first period).

Question: Is there a way to see that we should exercise early at the lower node in the middle column (after 1 period)?

Answer: Yes, the intuition is simple. If we have a put option, and we will exercise the option at both nodes in the next period (both the upward node and the downward node), then we should exercise now and not wait.

Question: What is the intuition for this?

Answer: If we wait, we get the cash from the strike price one period earlier and we can invest this cash at the risk-free interest rate.

Exercise 21.19: A stock price is currently $25. It is known that at the end of two months it will be either $23 or $27. The risk-free interest rate is 10% per annum with continuous compounding. Suppose that S_T is the stock price at the end of two months. What is the value of a derivative that pays of S_T² at this time?

There is nothing special about the square of the stock price. We use the general approach outlined above.

Most of the arithmetic is already done for us in this problem. The up movement is 27/25 = 1.08; the down movement is 23/25 = 0.92 (plus and minus 8%, respectively).

The risk-free interest rate is 10% per annum or 0.1/6 for two months.

The risk-neutral probability of an up movement is (e^0.1/6 – 0.92) / (1.08 – 0.92) = 0.60504.

The value at the two final nodes (at expiration) are 27² = 729 and 23² = 529.

The expected value of the derivative at the expiration date in a risk-neutral world is

0.60504 × 729 + (1 – 0.60504) × 529 = 650.01

The present value of the derivative is e^–0.1/6 × 650.01 = $639.26.

When there is a single period in the binomial tree, the option delta approach is simple.

We purchase one derivative and we short Δ (delta) shares of stock for a net outlay of "f(S) – Δ × S." We choose Δ so that this portfolio is risk-free. This implies that the portfolio has the same value in the "up" state as in the "down" state, or

We now solve for the present value of the derivative. Since the portfolio is risk-free, it earns a risk-free rate of return over the two months, or

At time 0, S = $25. Solving for f(25) gives $639.26.

Exercise 21.20: A stock price is current $40. Over each of the next two three-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding.

Exercise 21.21: Estimate how high the strike price has to be in Problem 9.15 for it to be optimal to exercise the option immediately.

Exercise 21.22: The risk free rate of interest is 5% per annum, the common stock of the ABC Company has a market beta of 1.00, and the market risk premium is 8 points. A May 2001 European call option on ABC stock with an exercise price of $50 has a value of $10 per option on March 1, 2001. If all investors suddenly became risk-neutral, which of the following statements is true?

In truth, there is no such risk-neutral world; this is only a construct that helps us conceive of option pricing.

Actually a third thing happens as well: The stock price changes at the moment that we turn to a risk-neutral world. That is because the stock is not a real asset, such as a loaf of bread, that has a consumption value to the owner. The stock is a financial security that gives the owner the future cash flows.

Hull is assuming that we change to a risk-neutral world and that the stock price doesn’t change. Some readers ask themselves: "Does this makes sense?" Well, Hull is not claiming that it makes sense. In truth, there is no risk-neutral world; this is all just a mathematical construct to help our understanding of option pricing. Instead, just assume that we move to a risk-neutral world and nothing else changes. All current prices stay exactly where they are for the primary securities.

Exercise 21.23: A stock trades for $120. Each year, it will rise by 10% or fall by 20%. The risk-free interest rate is 6% per annum. A call option on this stock has an exercise price of $130.

(A) The problems doesn’t say if the risk-free interest rate is a continuously compounded rate or an effective annual rate. Texts on option pricing generally assume that the risk-free interest rate is continuously compounded. Observed risk-free interest rate are the rates on Treasury securities. These are semiannually compounded if they are rates on Treasury notes or bonds, and they are discount rates if they are rates on Treasury bills.

Question: Does the compounding frequency have a large effect on the price of options?

Answer: Generally the effect is not material. To illustrate this, let us first assume that the rate given in this practice problem is an effective annual rate. Let r = 6% per annum. The risk-neutral probability of a rise in the stock price is

We could also express this as

Intuition: The return from the stock equals the risk-free rate in a risk-neutral world. Let Z be the risk-neutral probability:

We have used annual compounding only to show the method needed if annual compounding is specified in the exam problem. On the exam itself, use continuous compounding unless specifically told to use another compounding frequency; see below.

(B) The delta of the option is the change in the option price divided by the change in the price of the underlying security. When the stock increases by 10%, it is worth $120 × 1.10 = $132 and the call option is worth $2. When the stock decreases by 20%, it is worth $120 × 0.80 = $96 and the call option is worth $0. The change in the option price divided by the change in the price of the underlying security is [$2 – $0] / [$132 – $96] = $2/$36 = 1/18 = 5.56%.

Question: With such a small delta, the stock is probably well out-of-the-money; is that correct?

Answer: Yes; the call option is not worth much. If you calculate a high delta and a low call option price or a low delta and a high call option price, check your work.

(C) To hold a risk-free portfolio, we can purchase 18 options and sell short one share of stock. Alternatively, we can purchase one share of stock and sell short 18 call options.

(D) The value of the call option in the up state is $2, and the risk-neutral probability of moving up is 86.67%. The value of the call option in the down state is $0, so it is not relevant.

The expected value of the call option at the expiration date in the risk-neutral world is 86.67% × $2 = $1.7333. The expected value of the call option at inception in the risk-neutral world is $1.7333 / 1.06 = $1.63522 . $1.64. Since the value of the call option does not depend on the risk aversion of the investor(s), this is also the value of the call option in the real world.

(E) If the option expires in two years and all other input parameters remain the same, only a few items change in the pricing method.

There are three possible states at the expiration date: up-up, up-down (= down-up), and down-down. The values of the stock and of the call option in these three states are

The risk-neutral probability of two successive up movements is 0.8667 × 0.8667 = 0.7512. This is the only probability that is needed to solve the problem.

The expected value of the call option at the expiration date in the risk-neutral world is 0.7512 × $15.20 = $11.41824. The expected value of the call option at the inception date in the risk-neutral world is 0.7512 × $15.20 = $11.41824 / (1.06)² = $10.16. This is also the value in the real world.

(F) If the strike price is $135 and there is one year to expiration, the value of the call option is $0, since the stock price does not exceed the strike price even in the up state.

Question: Is the call option actually worth $0?

Answer: No. A call option is always worth something (as long as the stock price is not worth $0), since the stock price might always increase. A one-period binomial tree is a crude approximation, and it gives a price of $0 for any strike price exceeding $132.

Question: Doesn’t the price determined from the binomial tree pricing method converge to the true option price?

Answer: It does converge, but the convergence is slow. It might take several hundred periods before the calculated option price was exact (to the penny).

Question: Along the path to convergence, does the binomial tree pricing method overstate or understate the true price?

Answer: In most cases, the binomial tree pricing method alternates. Suppose the true price of the option is $10. The binomial tree pricing method with one valuation period may show a price of $9; with two periods, it may give a price of $11; with three periods, it may give a price of $9.50; and so forth.

Question: If the interest rate is continuously compounded, how does the solution change?

Answer: The changes are minor.

The value of the call option at the inception date is 0.87279 × $2 / 1.061837 = $1.6439 . $1.64. The value of the call option does not change much for a slight change in the risk-free interest rate.

Question: The value of the call option increased when the risk-free interest rate increased. Is this always true?

Answer: Yes, this is always true. The call option allows the investor to hold onto the cash until the expiration date, instead of using the cash to purchase the stock at the inception date. The greater the risk-free interest rate, the more valuable is the ability to hold onto the cash, since the investment income on the cash is greater.

Question: It seems like there is a second reason as well. If the risk-free interest rate is greater, then the expected value of the stock at the expiration date in the risk-neutral world is greater.

Answer: Not quite. If the risk-free interest rate is greater, then the discount rate in the risk-neutral world from the expiration date to the inception date is also greater. This exactly offsets the expected increase in the value of the stock in the risk-neutral world.

Question: Can one see the effect of the higher interest rate in the binomial tree pricing method?

Answer: Yes. Let R be the original risk-free interest rate and let R’ be the higher risk-free interest rate. The original risk-neutral probability is (R – D)/(U – D), and the revised risk-neutral probability is (R’ – D)/(U – D). To go from the value of the call option at the expiration date in the risk-neutral world using the original interest rate of R to the revised interest rate of R’, we multiply by (R’ – D)/(R – D).

In addition, we discount by R’ instead of by R. The multiplicative factor to go from using the original interest rate of R to the revised interest rate of R’ is R/R’.

To go from the value of the call option at the inception date in the risk-neutral world using the original interest rate of R to the revised interest rate of R’, we multiply by the product of these two factors, or (R’ – D)/(R – D) × R/R’. As long as D > 0, R > D, and R’ > D, this product is greater than unity.

Exercise 21.24: A stock has a current price of $80 and a market beta of 1.2 and a volatility of 32%. The risk-free interest rate is 7% per annum and the market risk premium is 8%. All rates use continuous compounding.

Part (a): The CAPM formula uses effective annual rates of return. The expected return on this stock using annual effective rates of return is

The continuously compounded rate of return on this stock = ln(1.172453) = 15.9% per annum. The expected value of the stock in six months $80 × e^{0.159 * ½} = $86.62.

Part (b): The expected rate of return that is realized over this half year period depends on the probability distribution of the stock price. We assume that stock prices have a lognormal distribution, so the rates of return have a normal distribution. The expected (annual) rate of return that is realized over any period is F – σ²/2 = 10.8%.

Question: Should we divide 10.8% in half for a 5.4% continuously compounded rate of return?

Answer: This depends on the examiner. Some examiners want the annual rate of return; others want the rate of return over a half year period.

Question: Why does the CAPM use annual effect rates of return whereas option pricing texts use continuously compounded rates of return?

Answer: The simple answer is that the derivation of the CAPM is based on annual effective rates of return. Investment theorists differ on the proper rates for return factor models.

Exercise 21.25: A stock’s current price is $110. The expected rate of return is 15% per annum, the volatility is 25% per annum, and the risk-free interest rate is 11% per annum. A European call option and a European put option on this stock have 43 days to expiration.

Part (a): We use the Black-Scholes model to solve for the prices of European options. The input data to the Black-Scholes formula are

We begin by calculating values for d₁ and d₂. The formulas are:

The values of d₁ and d₂ in this problem are

We use a cumulative normal distribution table or a built-in spreadsheet function to calculate

The values of the put and call options in the Black-Scholes formula are

The value of the European call option is

Part (b): The put option is computed in a similar fashion. The only input figure which changes is the strike price, which is now $140. For the put option, the values of d₁ and d₂ are

We use a cumulative normal distribution table or a built-in spreadsheet function to calculate

The value of the European put option is

Exercise 21.26: A stock trading for $75 has an expected return of 16% per annum and a volatility of 35% per annum. The risk-free rate is 9% per annum. All rates use continuous compounding.

Part (a): If there are no dividends, the computation is as follows:

The input data to the Black-Scholes formula are

We calculate values for d₁ and d₂.

The values of d₁ and d₂ are

We use a cumulative normal distribution table or a built-in spreadsheet function to calculate

Similarly,

The values of the put and call options in the Black-Scholes formula are

The value of the European call option is

The put option is computed in a similar fashion. We compute the values of N(–d₁) and N(–d₂):

The value of the European put option is

Part (b): The continuous dividend is interpreted as an adjustment to the stock price. Instead of buying one share of stock for S₀ at time 0, we purchase an amount of stock equal to S₀ × e^–rθ at time 0 and we reinvest the continuous dividends in additional shares of stock. At time

T, the maturity date of the option, we have one share of stock S.

The revised stock price at time 0 is $75 × e^{–0.04 × 150/365}. The Black-Scholes expressions for d₁ and d₂ for the call option are

We can rewrite the equation for d₁ as

This is the more common method of valuing d₁ when there is a continuous dividend payment.

We evaluate the cumulative normal distribution values as

and

The value of the European call option is

The value of the European put option is

Exercise 21.27: We are using a binomial lattice to price an American call option and an American put option on a stock that pays no dividends. The current stock price is $120 and the exercise price for both the put and the call is $110. The volatility of the stock returns is 40% per annum, and the risk-free interest rate is 10% per annum. The options expire in 120 days.

The options expire in 120 days. The four period binomial tree has 30 days in each period.

The volatility of the stock price is 40% per annum. The upward movement in the binomial tree is e^σ%Δt = e^{0.40 × %(30/365)}.

The downward movement is the reciprocal of the upward movement, or 1/1.1215 = 0.8917.

The risk-neutral probability of an upward movement equals

The risk-neutral probability of a downward movement is the complement of the risk-neutral probability of an upward movement, or 1 – 0.5073 = 0.4927.

The risk-free discount rate for each period is e^{–r × Δ t} = 0.9918.

We draw the binomial tree for the stock price, with an upward movement of 1.1215 and a downward movement of 0.8917. Since the stock does not pay dividends, the American call option has the same value as the European call option. However, the American put option may be more valuable than the European call option. We show the valuation procedure for both types of option, since the subsequent problems deal with dividend paying stocks.

We draw the binomial tree for the call option, using the strike price of $110 and the binomial tree for the stock price. The figure below shows two prices for each cell. The upper price is the price of the corresponding European call option, based on the binomial tree pricing method. The lower value is the intrinsic value of the option. If the intrinsic value exceeds the value of the European call option, we should exercise the American call option early.

We determine the value of the option in the last column from the stock price and the strike price. We use a backwards recursive method to determine the value of the option in each preceding column.

Illustration A: The value in the uppermost node in the right-most column of 79.84 equals the stock price minus the strike price, or 189.843 – 110.000 = 79.84 (rounded).

Illustration B: The topmost cell in the preceding column has two values: 60.17 and 59.27. The upper value is the value of the European call option in that cell. The lower value is the intrinsic value of the call option in that cell, which is the value that the long position receives if the call option is exercised early.

In every cell, the European call option price exceeds the intrinsic value, so the American call option has the same value as the European call option. This is always true for options on non-dividend paying stocks.

We draw the binomial tree for the put option, using the strike price of $110 and the binomial tree for the stock price. Each cell has two prices. The upper figure is the value of the corresponding European call option, based on the binomial tree pricing method. The lower figure is the intrinsic value of the option. If the intrinsic value exceeds the value of the European put option, we should exercise the American put option early.

When the intrinsic value of the put option exceeds the value of the corresponding European put option, the value of the corresponding European put option is replaced by the intrinsic value and the figures are in boldface. The recursive valuation procedure uses the higher of the intrinsic value and the European put option value to determine the value of the American put option in the preceding cells.

Question: How can we tell whether it would be optimal to exercise an American put option early? Is there any simple rule-of-thumb?

Answer: For the binomial tree pricing method, there is a rule-of-thumb that is sufficient, though it is not necessary. If the cell in the binomial tree satisfies this rule-of-thumb, it is optimal to exercise the American put option early. However, it may be optimal to exercise early even when the rule-of-thumb is not satisfied.

There are offsetting forces for the early exercise of an American put option:

strike price × (e^rt – 1) × (e^–rt) = strike price × (1 – e^–rt)

Consider the lowest cell in the column labeled "time=4" in the binomial tree above. The intrinsic value in this cell is $24.93. Both paths from this cell lead to cells in the fifth column which have positive payoffs for the put option. The intrinsic values in these two cells in the fifth column (bottom two nodes) are $14.59 and $34.15.

Since the investor in the bottom cell of the fourth column will exercise the American put option in the fifth column no matter which path is taken, the volatility value of waiting is zero. The value of the investment income is positive, so it is advantageous to exercise early.

Question: Can you generalize these comments?

Answer: The general rule is as follows. When the option is at the money (that is, when the stock price equals the strike price), the value of volatility is greatest. (To be exact, we should define "at the money" to mean that the stock price equals the present value of the strike price.)

When the option is deep in the money or deep out of the money, the value of volatility is low. When the investment income exceeds the value of volatility, it pays to exercise the option early.

When the option is deep out of the money, it has no intrinsic value, so we do not exercise early. When the option is deep enough in the money that the value of volatility is less than the expected investment income from early exercise, it pays to exercise early,

The values in the bottom node in column four are shown in bold. In this cell, it is advantageous to exercise the American put option early. The lower value of $24.93 (the intrinsic value) replaces the European put option that was originally shown as the upper value. The original value was

To check the rule-of-thumb, we examine the present value of the investment income on the strike price. The strike price is $110 and the value of (1 – e^-rt) for one period is 0.0082. The product of $110 and 0.0082 is $0.90, which is the difference between $24.93 and $24.03. The additional value from early exercise is the value of the additional investment income.

Exercise 21.28: A stock trading for $120 has a volatility of 40% per annum. The risk-free rate is 10% per annum. American put and call options expire in 120 days with an exercise price of $110.

Exercise 21.29: A stock currently trades for $80 with an expected return of 15% per annum and a volatility of 30% per annum. The stock is expected to pay a $5 dividend in 70 days. The risk-free interest rate is 10% per annum. Options written on this stock have an exercise price of $80 and expire in 120 days. Calculate the price of these options using a four period tree.

We show just the preliminaries; the binomial tree pricing method is the same as in the previous practice problem.

The options expire in 120 days. The four period binomial tree has 30 days in each period.

The volatility of the stock price is 30% per annum. The "up" movement in the binomial tree is e^σ%Δt = e^{0.30 × %(30/365)}.

The "down" movement is the reciprocal of the up movement, or 1/1.089814 = 0.9176.

The risk-neutral probability of an up movement equals

The risk-neutral probability of a down movement is the complement of the risk-neutral probability of an up movement, or 1 – 0.5264 = 0.4736.

The risk-free discount rate for each period is e^{–r × Δ t} = 0.9918.

The manner of drawing the binomial tree for the stock price differs when the stock pays dividends. For a fixed dollar dividend, we subtract the present value of the dividend from the initial stock price, draw the tree, and then add back in the appropriate dividend value to the cells preceding the ex-dividend date.

The present value of the dividend at time 0 is $5.00 × e^{0.01 × (70.365)} = $4.9050. The adjusted stock price at time 0 is $80.0000 – $4.9050 = $75.0950.

CAS Spring 2003, Exam 8, Question 36 (2.5 points)

A stock price is currently $40. It is known that at the end of one month the stock’s price will be either $42 or $38. The risk-free interest rate is 8% per annum with continuous compounding.

Show all work.

Part A: The upward and downward movements in the stock price are ± $2/$40 = ± 5%, or 1.05 and 0.95. The risk-neutral probability of an upward movement in the stock price is

(e^{0.08 / 12} – 0.95) / (1.05 – 0.95) = 56.69%.

If the stock price moves up, the option payoff is $3; if the stock price moves down, the option payoff is zero. The expected option payoff in a risk-neutral world is 56.69% × $3 = $1.70. The present value of this expected payoff at the risk-free interest rate is $1.70 × e^{–0.08 / 12} = $1.69.

Part B: In the real world, the probability of the stock price moving up is

(e^{0.10 / 12} – 0.95) / (1.05 – 0.95) = 58.37%.

The expected option payoff is 58.37% × $3 = $1.75. The capitalization rate for the call option is ($1.75 / $1.69 – 1) × 12 = 42.60%. Since we used only two decimal places for the dollar amounts, the answer is not exact; for the exam, use fractions of a penny for the dollar amounts so that all the digits in the answer are significant.

CAS Spring 2003, Exam 8, Question 39 (3 points)

Show all work.

Solution:

Step 1: The upward movement is 1.06 and the downward movement is 0.95. The risk-free interest rate for a six month period e^0.015.

Step 2: The risk-neutral probability of an upward movement is

(e^0.015 – 0.95) / (1.06 – 0.95) = 59.19%.

Step 3: The stock price in one year’s time may take three values, for which the call option values and the risk-neutral probabilities are as shown below:

Top node: stock = $50 × 1.06² = $56.18; option = $4.18; probability = 59.19%² = 35.03%

Middle node: stock = $50 × 1.06 × 0.95 = $50.35; option = $0; probability = 2 × 59.19% × (1 – 59.19%) = 48.31%

Bottom node: stock = $50 × 0.95² = $45.13; option = $0; probability = (1 – 59.19%)² = 16.65%

The expected value of the option at the final column of nodes (in one year) is

$4.18 × 35.03% = $1.46.

The present value at time 0 is $1.46 × e^–0.03 = $1.42.

So, I didn't read all of these 71 pages, but I did work through some of them. I thought I'd post the typos I found, as well as my answers to a couple of questions for which the posting does not give an exact solution. Here goes:

In Answer 21.6, on the first displayed line, I think that "i" should be changed to "u." Four paragraphs below that line, I think that the first sentence in the paragraph is incorrect, when it says, "The movements in statement 3 imply an annual volatility of 40%." I'm not sure what NEAS intended to say here, but since statement 3 is the only correct answer, that would seem to imply that statement 3 implies an annual volatility of the given 30%, not 40%.

In Answer 21.7, under the heading "Risk-Neutral Probability," I believe that the first line should begin, "Statement 2 is not correct. We assume that the risk-free rate..."

In Solution 21.8, Part G, the last part of the solution is missing. I believe that The last sentence of this solution should read, "The expected value of the option at the expiration date in the risk-neutral world is 0.567 x $3 + (1 - 0.567) x $0 = 1.701.

Just after Solution 21.8, Part H, under the heading "Cookbook," in the second sentence, change "seven" to "six." The sentence should read, "We also provide a six-step cookbook procedure for exam problems."

In Exercise 21.15, the title above the exercise and the given information in the question do not match the solution. I think that the title should be changed from "Two Period Put" to "European Put Option." In the second sentence of the exercise, change "two" to "four" so the sentence reads, "It is known that at the end of four months it will be either $75 or $85."

In Exercise 21.16, just above the heading "Quarterly Compounding," there are two lines which begin with the words, "The present value..." In both of these lines, the exponent of e should be -2%, not -4%. So, "The present value of the call option is e^-2% x $2.904 = $2.85." and "The present value of the put option is e^-2% x $2.096 = $2.05." Then, a few lines below the heading "Quarterly Compounding," there are two more lines which also begin with the words, "The present value..." In the first of these two lines, the answer should be $2.84 (not $2.85) and in both of these lines, we should change e^-4% to 1.02^-1. So, "The present value of the call option is 1.02^-1 x $2.9 = $2.84." and "The present value of the put option is 1.02^-1 x $2.1 = $2.06."

In Exercise 21.18, in the first sentence, change "9.11" to "21.17." Under the heading "American Options," in the second sentence of the first paragraph, change "not" to "why not."

In Exercise 21.20, NEAS does not give a solution to these problems. I worked them and got an answer of $2.12 for Part A and $2.54 for Part B.

In Exercise 21.21, in the first sentence, change "9.15" to "21.20."

In Exercise 21.22, NEAS does not explicitly state the correct answer to the problem. I believe that the correct answer is B, given my interpretation of the NEAS explanation below this problem.

In the solution to Exercise 21.29, under the heading "Stock Price Binomial Tree," in the second paragraph, the exponent of e is incorrect. This sentence should be corrected to read, "The present value of the dividend at time 0 is $5.00 x e^{-0.10 x (70 / 365)} = $4.9050."

In 21.3 E The statement is false as written.  Instead of "true probability" it should be "risk-neutral probability".

You cannot directly get true probabilities from risk-free returns: true probabilities go with actual expected returns and risk-neutral probabilities go with risk-free returns.

In 21.3 A I must respectfully disagree with NEAS: there are two ways to value an option - both perfectly valid:

(1) with risk-neutral probabilities and risk free rates

(2) with true probabilities and actual rates of return

If NEAS says the option price stays the same when the true probability changes, they must be assuming that the actual rate of return is changing as well (in the opposite direction).  If the actual rate stays the same, the option price does change. 

A better way to ask this would be "holding the risk-neutral probabilities and risk free rates constant, if the true probabilities change, the option price does not change."  In that case, we would know that the actual rate of return is changing in such a way as to keep the option price the same.