For parts (a), (b), and (c), consider the three-period binomial tree of stock pr

Question

For parts (a), (b), and (c), consider the three-period binomial tree of stock prices with the spot price 100, u = 1.25, and d = 0.8. Assume zero interest rates and dividends. (a) Find the time-zero value of an ATM European call option that expires at the end of three periods. (b) Suppose the stock price follows the following path: 100, 80, 100, 125. Describe the replicating portfolio and all the adjustments until the expiry. (c) Find the time-zero value of a lookback option that pays at expiration the maximum minus the initial stock price.

Explanation / Answer

Ans-

If the price goes to $110, our shares will be worth $110*d and we’ll lose $10 on short call payoff. The net value of our portfolio will be (110d – 10).

If the price goes down to $90, our shares will be worth $90*d, and option will expire worthless. The net value of our portfolio will be (90d).

If we want the value of our portfolio to remain the same, irrespective of wherever the underlying stock price goes, then our portfolio value should remain the same in either cases, i.e.:

=> (110d – 10) = 90d

=> d = ½

i.e. if we buy half a share (assuming fractional purchases are possible), we will manage to create a portfolio such that its value remains same in both possible states within the given timeframe of one year. (point 1)

This portfolio value, indicated by (90d) or (110d -10) = 45, is one year down the line. To calculate its present value, it can be discounted by risk free rate of return (assuming 5%).

=> 90d * exp(-5%*1 year) = 45* 0.9523 = 42.85 => Present value of the portfolio

Since at present, the portfolio comprises of ½ share of underlying stock (with market price $100) and 1 short call, it should be equal to the present value calculated above i.e.

=> 1/2*100 – 1*call price = 42.85

=> Call price = $7.14 i.e. the call price as of today.

Since this is based on the above assumption that portfolio value remains the same irrespective of which way the underlying price goes (point 1 above), the probability of up move or down move does not play any role here. The portfolio remains risk-free, irrespective of the underlying price moves.

In both cases (assumed to be up move to $110 and down move to $90), our portfolio is neutral to the risk and earns the risk free rate of return.

Hence both the traders, Peter and Paul, will be willing to pay the same $7.14 for this call option, irrespective of their own different perceptions of the probabilities of up moves (60% and 40%). Their individually perceived probabilities don’t play any role in option valuation, as seen from the above example.

If suppose that the individual probabilities matter, then there would have existed arbitrage opportunities. In real world, such arbitrage opportunities exist with minor price differentials and vanish in a short term.

But where is the much hyped volatility in all these calculations, which is an important (and most sensitive) factor affecting option pricing?

The volatility is already included by the nature of problem definition. Remember we are assuming two (and only two - and hence the name “binomial”) states of price levels ($110 and $90). Volatility is implicit in this assumption and hence automatically included – 10% either way (in this example).

Now let’s do a sanity check to see whether our approach is correct and coherent with the commonly used Black-Scholes pricing.

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