Question

In: Finance

The option writer in Example 1.6.6 sells a digital option to a speculator. This amounts to...

The option writer in Example 1.6.6 sells a digital option to a speculator. This amounts to a bet that the asset price will go up. The payoff is a fixed amount of cash if the exchange rate goes to $165 per £100, and nothing if it goes down. If the speculator pays $10 for this bet, what cash payout should the option writer be willing to write into the option? You may assume that interest rates are zero.

(Example 1.6.6 -Suppose that in the US dollar markets the current Sterling
exchange rate is 1.5 (so that £100 costs $150). Consider a European call option that
offers the holder the right to buy £100 for $150 at time T . The riskless borrowing
rate in the UK is u and that in the US is r . Assuming a single period binary model in
which the exchange rate at the expiry time is either 1.65 or 1.45, find the fair price
of this option.)

Solutions

Expert Solution

For the given question, we would try to construct a portfolio that consists of the asset and the option, so that the risk is cancelled and the portfolio value is the same in both states. This portfolio becomes riskless; therefore, it must have the same value to begin with as the final payoff

In the current scenario we have a call option with strike of $165

Current Asset Price is $150 and there are two possible states at time T:

  • State A – When the asset price is at $165
  • State B – When the asset price is at $145

Current Payoff of the call:

  • State A – $10
  • State B – $0

We need to find out the cash pay-out in which the option writer would be willing to write into the option

Suppose we sell one call and need to hedge. We would buy some options for eg x assets

Total value of the portfolio at T:

If A is reached = 165x-10

If B is reached = 145x

If x = 1/2 , the risk is eliminated as the portfolio value will be $72.5 in both states. So

value of the portfolio must be $72.5 to begin with,

  • 110x-C =72.5
  • 55-C = 72.5
  • C = $17.5

So the cash pay-out in which the option writer would be willing to write the option would be $17.5


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