Tuckman's option pricing

[shanlane]

In the chapter, it always assumes that the future spot rate is going to go up. All of the RN probs of going up are higher than the ones going down (positive drift). This, in turn makes the option less valuable than its expected value. If, however, the spot rates were favored to go down (negative drift), wouldnt this make the price of the option greater than the expected value (because we would be discounting at a lower rate than the average of the two nodes)?

Thanks!

Shannon

 

[David Harper CFA FRM]

Hi Shannon,

Yes, exactly! To illustrate, the Tuckman base case (as you suggest) is a slightly upward sloping term structure with spot(0.5) = 5.00% and s(1.0) = 5.15% such that Tuckman's Chapter 9 risk-neutral option price = 0.58 versus it's expected discount value of $1.46.

If I plug in a lower S(1.0) of 4.9%, the risk-neutral option price increases to $2.04 (> 1.46). See below. The the new risk-neutral probabilities reflect a change from p (up) = 80% to p (up) = only 30.11%.

How to interpret? By applying the implication of Tuckman:

"The true option price is less than this value because investors dislike the risk of the call option and, as a result, will not pay as much as its expected discounted value. Put another way, the risk penalty implicit in the call option price is inherited from the risk penalty of the one-year zero, that is, from the property that the price of the one-year zero is less than its expected discounted value."

The inverted spot curve here would signify an (un-natural) actual risk-seeking preference. It would imply we actually want to pay-up for the uncertainty itself. The upward sloping zero curve reflects natural risk-aversion; e.g., if you are risk-averse, you would pay less than $0.50 for payoff = [50% of $1.00, 50% of zero] which has an average value of $0.50; why give up a sure $0.50 for the same expected $0.50 but with added dispersion? The inverted scenario reflects un-natural risk-seeking: you would pay more than a certain $0.50 for an expected payoff of $0.50 with dispersion. Not the same as this example, just simplifying.

I hope that helps, hard idea,