CHAPTER 12
Introduction to Binomial Trees
Practice Questions
Problem 12.8。
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 科学探究 changes during the life of the option， this riskless portfolio must also change.
Problem 12.9。
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：
The value of the portfolio is either or in two months。 If
i。e.,
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：
where is the value of the option。 Since the portfolio must earn the risk—free rate of interest
朗诵素材
i.e。,
The value of the option is therefore $2。23。
This can also be calculated directly from equations (12。2） and (12.3)。 ， so that
and
Problem 12。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 strikeprice of ＄80？ U no—arbitrage arguments.
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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:
(Note： The delta, of a put option is negative。 We have constructed the portfolio so that it is +1 option and shares rather than option and shares so that the initial investment is positive。）
The value of the portfolio is either or in four months。 If
i.e.，
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：
where is the value of the option. Since the portfolio is riskless
i.e.,
The value of the option is therefore ＄1。80。
This can also be calculated directly from equations （12。2) and (12.3)。 , so that
and
Problem 12。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.
宝宝积食的症状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:
（Note： The delta， , of a put option is negative。 We have constructed the portfolio so that it is +1 option and shares rather than option and shares so that the initial investment is positive。）
The value of the portfolio is either or . If:
i。e.，
the value of the portfolio is certain to be 22.5。 For this value of 福剪纸 the portfolio is therefore riskless。 The current value of the portfolio is
where f家庭用英语怎么说 is the value of the option. Since the portfolio must earn the risk—free rate of interest
Hence
i。e., the value of the option is $2。06.
This can also be calculated using risk-neutral valuation。 Suppo that is the probability of an upward stock price movement in a risk—neutral world. We must have
i.e。，
or：
The expected value of the option in a risk—neutral world is：
This has a prent value of
This is consistent with the no-arbitrage answer.
Problem 12.12.
A stock price is currently ＄50。 Over each of the next two 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 with a strike price of $51？
A tree describing the behavior of the stock price is shown in Figure S12。1. The risk-neutral probability of an up move， p， is given by
自动档档位There is a payoff from the option of for the highest final node (which corresponds to two up moves) zero in all other cas. The value of the option is therefore
This can also be calculated by working back through the tree as indicated in Figure S12。1。 The value of the call option is the lower number at each node in the figure.
