Tutorial Questions
Question 1 () [path-dependent payoff; up-and-in barrier put option] In Lecture 11, I demonstrated a series of barrier options (down-and-in, up-and-in, down-and-out, up-and-out). However, they were all barrier call options. Of course, we could also study barrier put options. The barrier operates in exactly the same manner as we saw in Lecture 11. A knock-in option starts off “dead” and gets “activated” if the underlying asset price hits the barrier. A knock-out option starts off alive but gets killed off if the underlying asset price hits the barrier. The defining feature of this question is that we will value a barrier put option (a right to sell at the strike) rather than a barrier call option (a right to buy at the strike). Assume that the stock on which the barrier is written is currently priced at $30 and the standard deviation of its return (σ) is 40% pa. The riskfree rate of interest (r) is 5% per annum. The up-and-in barrier put option has a strike price (X) of $32 and a barrier (H) of $35. It expires in one year (T=1). Required: Estimate the value of this up-and-in barrier put option using a 5-step Binomial tree and the risk-neutral approach BFF3751 Derivatives 1 Philip Gray 2020 3 Question 2 () [path-dependent payoff: fixed lookback call option]
A vanilla call option has a fixed strike price (X) and a payoff that only depends on the expiry-date share price (ST). To be specific, the vanilla call option payoff is max(0, ST – X).
In contrast, a fixed lookback call option also has a fixed strike price (X). However, path dependency is introduced because the payoff is a function of the maximum share price achieved over the life of the option. To be specific, the payoff is max(0, Smax – X).
Consider a stock with a current price of $60. The volatility (σ) of the share returns is 50% pa and the riskfree interest rate is 6% pa. A fixed lookback call option is written on this stock. The option as a strike price of $45 and one year to expiry.
Required :
Use a four-step Binomial tree and risk-neutral valuation to calculate an approximate value for this fixed lookback call option.
note: on each path, when assessing the maximum share price along that path, we will not consider the starting share price of $60. In other words, only consider the four share price that come after the $60.
BFF3751 Derivatives 1 Philip Gray 2020 4
Question 3 (*) [Asian derivative with path-dependent payoff]
Assume that the evolution of a stock price is well-described by a four-step Binomial model with:
S0 = $18
u = 1.20 per month
d = 0.90 per month
r = 10% per annum continuously compounded.
Let S1, S2, S3 and S4 denote the stock price at the end of the first, second, third and fourth months respectively.
An ‘average rate’ Asian derivative, which matures in four months’ time, is written on this stock with a payoff function tht takes the average stock price at the end of the second and fourth months. Specifically:
()().15,0maxPayoff4221−+=SS
Required:
Use a four-step Binomial tree and the risk-neutral approach to estimate the value of this average-rate derivative.
Note: usually u and d are calculated such that d = 1/u. Clearly, this is not the case when u = 1.20 and d = 0.90. Nevertheless, it does not affect the way we value this derivative. We can still draw the Binomial tree with whatever u and d are given and proceed as normal with the risk-neutral approach.
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