L10 期权定价:二叉树



二叉树模型

无套利理论

二叉树模型

无套利理论

价值 ST=$18S_T=\$18 ST=$22S_T=\$22
股票的头寸 18Δ18\Delta 22Δ22\Delta
期权的头寸 00 1-1
组合 18Δ18\Delta 22Δ122\Delta-1

一般化

问题:pp是概率吗?


风险中性定价

风险中性定价(Risk-neutral valuation)

价值 ST=S0uS_T=S_0u ST=S0dS_T=S_0d 期望
风险中性概率 pp 1p1-p 不适用
股票 S0uS_0u S0dS_0d pS0u+(1p)S0dpS_0u+(1-p)S_0d
期权 fuf_u fdf_d pfu+(1p)fdpf_u+(1-p)f_d

多期(步)二叉树

欧式期权

欧式看涨期权

欧式看跌期权


美式期权

美式看跌期权


二叉树模型的实现与应用

二叉树模型的实现

Delta

确定uudd

其他的标的资产


DerivaGem

DerivaGem


课后阅读与练习

课后阅读与练习

课堂练习

  1. Explain the no-arhitrage and risk-neutral valuation approaches to valuing a European option using a one-step binomial tree.


  2. What is meant by the delta of a stock option?


  3. A stock price is currently $40. It is known that at the end of one month it will he either $42 or $38. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a 1-month European call option with a strike price of $39?


  4. A stock price is currently $100. Over each of the next two 6-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 1-year European call option with a strike price of $100?


  5. For the situation considered in the above problem, what is the value of a 1-year European put option with a strike price of $100? Verify that the European call and European put prices satisfy put-call parity.


  6. A stock price is currently $50. It is known that at the end of 6 months it will be either $60 or $42. The risk-free rate of interest with continuous compounding is 12% per annum. Calculate the value of a 6-month European call option on the stock with an exercise price of $48. Verify that no-arhitrage arguments and risk-neutral valuation arguments give the same answers.


  7. A stock price is currently $40. Over each of the next two 3-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.

    • (a) What is the value of a 6-month European put option with a strike price of $42?
    • (b) What is the value of a 6-month American put option with a strike price of $42?