CHAPTER 13
Binomial Trees
Practice Questions
Problem 13.1.
A stock price is currently $40. It is known that 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. What is the value of a one-month European call option with a strike price of $39?
Consider a portfolio consisting of
1-: Call option
+∆: Shares
If the stock price ris to $42, the portfolio is worth 423∆-. If the stock price falls to $38, it is worth 38∆. The are the same when
42338∆-=∆ or 075∆=.. The value of the portfolio in one month is 28.5 for both stock prices. Its value today must be the prent value of 28.5, or 0080083332852831e -.⨯..=.. This means that 402831f -+∆=.
where f is the call price. Becau 075∆=., the call price is 400752831$169⨯.-.=.. As an alternative approach, we can calculate the probability, p , of an up movement in a risk-neutral world. This must satisfy:
0080083334238(1)40p p e .⨯.+-=
so that
00800833344038p e .⨯.=-
or 05669p =.. The value of the option is then its expected payoff discounted at the risk-free rate:
008008333[305669004331]169e -.⨯.⨯.+⨯.=.
or $1.69. This agrees with the previous calculation.
Problem 13.2.
Explain the no-arbitrage and risk-neutral valuation approaches to valuing a European option using a one-step binomial tree.
In the no-arbitrage approach, we t up a riskless portfolio consisting of a position in the option and a position in the stock. By tting the return on the portfolio equal to the risk-free interest rate, we are able to value the option. When we u risk-neutral valuation, we first choo probabilities for the branches of the tree so that the expected return on the stock equals the risk-free interest rate. We then value the option by calculating its expected payoff and discounting this expected payoff at the risk-free interest rate.
Problem 13.3.
What is meant by the delta of a stock option?
The delta of a stock option measures the nsitivity of the option price to the price of the stock when small changes are considered. Specifically, it is the ratio of the change in the
price of the stock option to the change in the price of the underlying stock.
Problem 13.4.
机加
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. What is the value of a six-month European put option with a strike price of $50?
Consider a portfolio consisting of
1-: Put option
+∆: Shares
If the stock price ris to $55, this is worth 55∆. If the stock price falls to $45, the portfolio is worth 455∆-. The are the same when
45555∆-=∆
or 050∆=-.. The value of the portfolio in six months is 275-. for both stock prices. Its value today must be the prent value of 275-., or 010********e -.⨯.-.=-.. This means that 502616f -+∆=-.
where f is the put price. Becau 050∆=-., the put price is $1.16. As an alternative approach we can calculate the probability, p , of an up movement in a risk-neutral world. This must satisfy:
中国灵异故事01055545(1)50p p e .⨯.+-=
so that
010*******p e .⨯.=-
or 07564p =.. The value of the option is then its expected payoff discounted at the risk-free rate:
财务协同效应0105[007564502436]116e -.⨯.⨯.+⨯.=.
or $1.16. This agrees with the previous calculation.
Problem 13.5.
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. What is the value of a one-year European call option with a strike price of $100?
In this ca 110u =., 090d =., 05t ∆=., and 008r =., so that 0080509007041110090
e p .⨯.-.==..-. The tree for stock price movements is shown in Figure S13.1. We can work back from
the end o
f the tree to the beginning, as indicated in the diagram, to give the value of the option as $9.61. The option value can also be calculated directly from equation (13.10):
22200805[0704121207041029590029590]961e -⨯.⨯..⨯+⨯.⨯.⨯+.⨯=. or $9.61.
Figure S13.1: Tree for Problem
13.5
Problem 13.6.
殷当的灵异故事For the situation considered in Problem 13.5, what is the value of a one-year European put option with a strike price of $100? Verify that the European call and European put prices satisfy put –call parity.
Figure S13.2 shows how we can value the put option using the same tree as in Problem 13.5. The value of the option is $1.92. The option value can also be calculated directly from equation (13.10):
20080522[0704102070410295910295919]192e -⨯.⨯..⨯+⨯.⨯.⨯+.⨯=.
or $1.92. The stock price plus the put price is 10019210192$+.=.. The prent value of the strike price plus the call price is 008110096110192e $-.⨯+.=.. The are the same, verifying
that put –call parity holds.
Figure S13.2: Tree for Problem 13.6
Problem 13.7.
What are the formulas for u and d in terms of volatility?
跌宕的意思u e =and d e -=
余干辣椒炒肉
Problem 13.8.
Consider the situation in which 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 t up a position in the stock and the option that remains riskless for the whole of the life of the option.
The riskless portfolio consists of a short position in the option and a long position in ∆ shares. Becau ∆ changes during the life of the option, this riskless portfolio must also change.
Problem 13.9.
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 strikeprice of $49? U no-arbitrage arguments.
At the end of two months the value of the option will be either $4 (if the stock price is $53) or $0 (if the stock price is $48). Consider a portfolio consisting of:
国家投资公司shares 1option
+∆:-: The value of the portfolio is either 48∆ or 534∆- in two months. If
48534∆=∆- i.e.,
08∆=. the value of the portfolio is certain to be 38.4. For this value of ∆ the portfolio is therefore riskless. The current value of the portfolio is:
0850f .⨯-
where f is the value of the option. Since the portfolio must earn the risk-free rate of interest 010212(0850)384f e .⨯/.⨯-=.
<,
223f =. The value of the option is therefore $2.23.
This can also be calculated directly from equations (13.2) and (13.3). 106u =., 096d =. so that 01021209605681106096
e p .⨯/-.==..-. and
010212056814223f e -.⨯/=⨯.⨯=.
Problem 13.10.
A stock price is currently $80. It is known that at the end of four 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? U no-arbitrage arguments.
At the end of four months the value of the option will be either $5 (if the stock price is $75) or $0 (if the stock price is $85). Consider a portfolio consisting of:
shares 1option
-∆:+: (Note: The delta, ∆ of a put option is negative. We have constructed the portfolio so that it is +1 option and -∆ shares rather than 1- option and +∆ shares so that the initial investment is positive.)
The value of the portfolio is either 85-∆ or 755-∆+ in four months. If
85755-∆=-∆+
<,
05∆=-. the value of the portfolio is certain to be 42.5. For this value of ∆ the portfolio is therefore riskless. The current value of the portfolio is:
0580f .⨯+
逻辑分析法
where f is the value of the option. Since the portfolio is riskless
005412(0580)425f e .⨯/.⨯+=.
<,
180f =. The value of the option is therefore $1.80.
This can also be calculated directly from equations (13.2) and (13.3). 10625u =., 09375d =. so that 00541209375063451062509375
e p .⨯/-.==..-. 103655p -=. and
005412036555180f e -.⨯/=⨯.⨯=.
Problem 13.11.
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 rate of interest with quarterly compounding is 8% per annum. Calculate the value of a three-month European put option on the stock with an exerci price of $40. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.
At the end of three months the value of the option is either $5 (if the stock price is $35) or $0 (if the stock price is $45).
Consider a portfolio consisting of:
shares 1option
-∆:+: (Note: The delta, ∆, of a put option is negative. We have constructed the portfolio so that it is +1 option and -∆ shares rather than 1- option and +∆ shares so that the initial investment is positive.)
The value of the portfolio is either 355-∆+ or 45-∆. If:
