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Binomial Trees

5 important questions on Binomial Trees

Real-world vs. Risk-neutral world

Note that the probabilities in both world are different! A position in a call option is riskier than a position in the stock. As a result, the discount rate to be applied to the payoff from a call option is greater than the one expected for the stock.

Risk-neutral valuation solves this as the discount rate and expected return are the same for all assets.

WHat about a two-step model?

Just start at the back and work your way upfront. Formula will be:
value of option (today) = e^(-2*r*t)*(p^2*value(uu) +p*(1-p)*value(ud) + (1-p)^2*value(dd)).

If we add more steps, the risk-neutral valuation principle continues to hold. THe option price is always equal to its expected payoff in a risk-neutral world discounted at the risk-free rate.

How about american option?

Do exactly the same but value early exercise at every step in the tree. Take the max of the value of a European option and immediate exercise.

WHat about volatility to calculate u and d?

The parameters u and d shoudl be chosen to match the volatility. When we generalize u = e^(sd*wortel(t)) and d=e^-(sd*wortel(t)) or d=1/u.

Match volatility in the risk-neutral world! For real-world formulas will be the same. This links to Girsanov's theorem: when we move from the risk-neutral world, the expected return from the stock price changes, but the volatility remains the same!! Switching risk-preference only changes the measure (p is real and q is neutral)

What if we increase the number of steps?

Although unrealistically simple, if we make the periods very small and the steps larger (30) we can get a good approximation of the true value. When the periods become smaller we move into the same assumptions as Black-Scholes-Merton model.

The question on the page originate from the summary of the following study material:

Part 4: Valuation and Risk Models

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