Exotic Options with the Binomial Model

Zhang Zhuozhuo Calum Johnson

Waldemar Pietraszkiewicz

The binomial model is a very useful and popular technique for pricing an option.

The binomial option pricing formula is based on assumption that the stock price follows a multiplicative binomial process over discrete intervals.

Binomial Model

The rate of return of the stock over each period can have two possible values: u-1 with the probability q, or 1-d with the probability 1-q.

Binomial Model

The Cox-Ross-Rubenstein formula is the most popular formula for constructing binomial trees so it is what we use in our model.

where Δt is the time interval between observations of the price and σ the volatility.

Binomial Model

Binary options are options with discontinuous payoff.

The most common Binary options are:◦ Cash-or-nothing call◦ Cash-or-nothing put

◦ Asset-or-nothing call◦ Asset-or-nothing put

Binary options

Binary Call and Put Options pay either:

◦ a fixed cash settlement amount or,

◦ nothing at all, depending on the underlying price at expiration.

Binary options

Diagram of cash-or-nothing call and put options, with a payout 10 and strike price 50.

Call Put

Binary options

Cash-or-nothing options are valued by:

where Q is the amount paid at time T and r is the risk-free interest rate.

Cash-or-nothing options

Asset-or-nothing options are valued by:

where S is the initial stock price, q is the dividend rate.

Asset-or-nothing options

Power options are exotic options in which the payoff is multiplied by some power x of the stock.

For a call power option the payoff is

For a put power option the payoff is

where S – a stock with given price, K – strike price

Power options

The value of a power call option is given by:

while the value of a put is:

where and

Power options

Solution

Solution

We presented:◦ information about a binomial model which

valuates cash-or-nothing and asset-or-nothing options

◦ the theoretical research and presented information about the binomial model and binary options

◦ a program written on MS Excel which valuates the European cash-or-nothing and asset-or-nothing options.

Conclusion

The EndAny Questions?