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University of Texas at Austin
Lecture 17
Option pricing in the one-period binomial model.
17.1. Introduction. Recall the one-period binomial tree which we used to depict the sim- plest non-deterministic model for the price of an underlying asset at a future time−h.
S
Sd
Su
Our next objective is to determine the no-arbitrage price of a European-style derivative security with the exercise date T coinciding with the length h of our single period. Consider such a derivative security whose payoff function is denoted by v. The payoff of this derivative security is, thus, a random variable
V (T ) = v(S(T )) = v(S(h)).
Per our stock-price model above, the random variable S(T ) can only attain values Su and Sd. So, the random variable V (T ) can only take the values Vu := v(Su) and Vd := v(Sd). We can depict the resulting derivative-security tree as follows:
V
Vd
Vu
Note that we constructed the stock-price tree by starting from the root node containing the initial observed stock price. Then, we used our model encapsulated in the pair (u, d) of multiplicative factors to “populate” the offspring nodes. In short, we moved from left to right. Now that we wish to figure out the option price dictated by our stock-price tree, we start from the only known quantities: the possible payoffs. Then, we move from right to left to calculate the price of the derivative security occupying the root node of the derivative-security tree.
17.2. Pricing by replication. The method by which we intend to accomplish the above goal is the following:
Step 1. Create the replicating portfolio for our derivative security consisting of an invest- ment in the underlying risky asset and a loan (given or taken) at the continuously compounded risk-free interest rate r. Step 2. Calculate the initial cost of the replicating portoflio. Step 3. Conclude that the no-arbitrage price V (0) of our derivative security must equal the initial cost of its replicating portfolio. Let us, again, focus on the underlying asset being a continuous-dividend-paying stock with the dividend yield δ. It is traditional to denote by ∆ the initial number of units of
in all states of the world. Formally, we obtain the following system of equations:
∆eδhSu + Berh^ = Vu ∆eδhSd + Berh^ = Vd
If the above two equalities hold, we can conclude that the initial cost of the replicating portoflio equals the price of the derivative security, i.e.,
V (0) = ∆S(0) + B (17.1)
The replicating portfolio will be completely determined once we solve for ∆ and B in the above system. We get
∆ = e−δh^
Vu − Vd Su − Sd
and B = e−rh^
uVd − dVu u − d
Problem 17.1. Solve for ∆ and B in the above system.
17.3. The pricing formula simplified. The above pricing formula is already straightfor- ward and simple. The procedure of finding ∆ and B also comes in handy when we need to explicitly determine the replicating portfolio (for instance, when an arbitrage opportunity presents itself due to mispricing). However, when we merely want to calculate the price of the derivative security of interest, we can make the calculation more streamlined. Moreover, we will have a pretty nifty interpretation of the resulting (simple) pricing formula. First, we can substitute the expressions for ∆ and B from (17.2) into the pricing formula (17.1). We obtain the
V (0) = ∆S(0) + B
= e−δh^
Vu − Vd Su − Sd
× S(0) + e−rh^
uVd − dVu u − d
= e−δh^
Vu − Vd S(0)(u − d)
× S(0) + e−rh^
uVd − dVu u − d
= e−rh
[
e(r−δ)h^ − d u − d
× Vu +
u − e(r−δ)h u − d
× Vd
]
If the numerators of the coefficients next to Vu and Vd look familiar, this is rightfully so. We have seen bits and pieces of those expressions in the no-arbitrage condition for the binomial asset-pricing model. In fact, we can conclude that both of the coefficients are non-negative and that they sum up to one. In other words, the weighted sum of the two possible payoffs is actually a convex combination of the two possible payoffs. In fact, the weights in the above convex combination can be interpreted as probabilities.
Definition 17.1. The risk-neutral probability of the asset price moving up in a single step in the binomial tree is defined as
p∗^ =
e(r−δ)h^ − d u − d
Remark 17.2. The probability measure P∗^ giving the probability p∗^ to the event of moving up in a single step and the probability 1 − p∗^ to the event of moving down in a single step in the binomial tree is called the risk-neutral probability measure. This probability measure and the rationale for its name will be discussed in M339W.
Combining Definition 17.1 with the result of calculations in (17.3), we get the following risk-neutral pricing formula:
V (0) = e−rT^ [p∗Vu + (1 − p∗)Vd] (17.4)
It is customary to interpret (and memorize) the above formula by noting that the initial value of the derivative security is equal to its discounted expected payoff under the risk-neutral probability measure. We even write
V (0) = e−rT^ E∗[V (T )]
where E∗^ denotes the expectation associated with the risk-neutral probability measure P∗.
Problem 17.2. MFE Exam, Spring 2007: Problem # For a one-year straddle on a non-dividend-paying stock, you are given:
- The straddle can only be exercised at the end of one year.
- The payoff of the straddle is the absolute value of the difference between the strike price and the stock price at expiration date.
- The stock currently sells for $60.00.
- The continuously compounded risk-free interest rate is 8%.
- In one year, the stock will either sell for $70.00 or $45.00.
- The option has a strike price of $50.00.
Calculate the current price of the straddle.
(A) $0. (B) $4. (C) $9. (D) $14. (E) $15.
Solution: Our intention is to use the risk-neutral pricing formula (17.4). The length of our one time-period is one year, so h = T = 1. The stock pays no dividends, so that δ = 0. With the remaining data explicitly provided in the problem statement, we get that the risk-neutral probability of the stock price going up equals
p∗^ =
e(r−δ)h^ − d u − d
S(0)e(r−δ)h^ − Sd Su − Sd
60 e^0.^08 − 45 70 − 45
The two possible payoffs of the straddle are
Vu = |Su − K| = | 70 − 50 | = 20 and Vd = | 45 − 50 | = 5.
Finally, we obtain
V (0) = e−^0.^08 [0. 8 × 20 + 0. 2 × 5] ≈ 15. 693.
We choose the offered choice E.
20 40 60 80 100
10
20
30
40
50
The replicating portfolio consists of:
- ∆ = 3/ 5 ⇒ 3 /5 purchased shares of stock;
- B = − 22 e−^0.^08 ⇒ borrowed $22e−^0.^08.
17.5. Arbitrage opportunities due to mispricing in the one-period binomial model. The source of an arbitrage opprotunity in this setting will be an observed option price which is inconsistent with the no-arbitrage price as dictated by the binomial asset-pricing model. To investigate how one can exploit such an arbitrage opportunity, let us look into a problem first.
Problem 17.4. MFE Exam, Spring 2009: Problem # You are given the following regarding stock of Widget World Wide (WWW):
- The stock is currently selling for $50.
- One year from now the stock will sell for either $40 or $55.
- The stock pays dividends continuously at a rate proportional to its price. The divi- dend yield is 10%.
- The continuously compounded risk-free interest rate is 5%. While reading the Financial Post, Michael notices that a one-year at-the-money European call written on stock WWW is selling for $1.90. Michael wonders whether this call is fairly priced. He uses the binomial option pricing model to determine if an arbitrage opportunity exists. What transactions should Michael enter into to exploit the arbitrage opportunity (if one exists)?
(A) No arbitrage opportunity exists. (B) Short shares of WWW, lend at the risk-free rate, and buy the call priced at $1.90. (C) Buy shares of WWW, borrow at the risk-free rate, and buy the call priced at $1.90. (D) Buy shares of WWW, borrow at the risk-free rate, and short the call priced at $1.90. (E) Short shares of WWW, borrow at the risk-free rate, and short the call priced at $1.90.
Qualitative analysis of offered answers.
(A) It would strike one as quite unlikely that the SoA would pose an arbitrage problem in which there is no arbitrage opportunity. So, one would be inclined to discard this option. (B) No obvious shortcomings in this answer!
(C) Simultanously longing both the shares of stock and the call option cannot eliminate risk. So, we discard this offered answer. (D) No obvious shortcomings in this answer! (E) Simultaneously short-selling the underlying and writing the call option cannot elim- inate risk. So, we discard this offered answer. The conclusion is that a cursory investigation of the offered answer allows one to increase the probability of guessing correctly if pressed for time! In the present problem, one would toss a mental coin to decide between (B) and (D). Solution: Although the offered answers are just sketches of potential arbitrage portfolio. This problem can serve as a template for all similar problems we may encounter in the future. So, let us solve it in a tad more detail than necessary. Diagnosis. Since we ultimately want to construct an arbitrage portfolio, it makes sense to immediately find the ∆ and B and use them for pricing. In general, we have
∆ = e−δh^
Vu − Vd Su − Sd
In this problem, the length of the period is one year so that h = 1. The two possible payoffs are Vu = 5 and Vd = 0. So, we get
∆ = e−^0.^1 ×
As for the risk-free investment, we have
B = e−^0.^05 ×
55 × 0 − 40 × 5
So, the no-arbitrage call price
V (0) = ∆S(0) + B = 0. 3016 × 50 − 12 .6831 = 2. 3969.
Since V (0) 6 = 1.9, we conclude that there is, indeed, an arbitrage opportunity. Construction. Since the no-arbitrage price exceeds the observed price, we conclude that the observed call option is underpriced. So, an arbitrage portfolio must include a purchase of the observed call option. At this point in the exam, you would choose B. and move on! For didactic purposes, let us completely construct the arbitrage portfolio to consist of the following components:
- one long observed call option,
- short-sale of ∆ shares of stock,
- a deposit of −B to earn the continuously compounded risk-free interest rate r.
The latter two components combined can be described a the short replicating portfolio of the call option. Verification. The inital cost of our proposed arbitrage portfolio equals
- 90 − V (0) < 0 ,
meaning that there is an initial inflow of funds. As for the payoff, note that the ∆ and B were chosen exactly so as to create the replicating portfolio, so the payoff of the total proposed arbitrage portfolio is by design equal to zero. We conclude that we have, indeed, created an arbitrage portfolio.
