The price of any asset in the world is the present value of future cashflows. If we talk about call options, future cashflows are the pay-off from the call option at maturity, which is the difference of stock price (St) at expiry date and exercise price (E) but could not be negative i.e., max(St-E, 0). Stock price at maturity is uncertain and we need to model it to see how it evolves over time. There are two models for modelling St over time:
- Binomial Model
- Black-Scholes Model
Assumptions of Binomial Model:
- Binomial model says there are only two possible values that stock price can have after one period (say one period = one year). One is the upside possibility and other is the downside possibility. This possibility is calculated on the basis of underlying stock’s annual volatility. Hence, we get upward movement and downward movement factor in this way.
- Investors are in the risk neutral world and hence we calculate risk neutral probability.
Binomial process is not realistic though. In fact, we can have binomial process at each moment such that stock price can take infinite possibilities on a continuous scale. This is achieved by Black-Scholes model. So, Binomial model is a discrete process and Black-Scholes is the continuous version of Binomial model.
Methods of Binomial Model:
Within binomial process, option price could be found out using any of the 4 methods.
- Risk Neutral Valuation Approach: In this approach, pay-off does not change and price of the option is the present value of expected pay-off.
- Delta Hedging Approach: In this approach, we kill directional exposure by buying or selling delta number of shares and make the pay-off certain.
- Bankruptcy Free Portfolio Approach: In this approach, we buy or sell one stock and borrow something to replicate the option pay-off (leveraged stock buying). In this approach, pay-off is matched.
- Replicating Portfolio Approach: In this approach, we replicate the option pay-off by buying or selling delta number of shares and borrowing say X amount. In this approach, once again pay-off is matched.
In this blog, we are focusing on Risk Neutral Valuation Approach. However, all the above-mentioned approaches will yield same results and we call it valuing a derivative instrument on no-arbitrage principle.
Risk Neutral Valuation Approach:
In risk neutral valuation approach, we assume that investor is in risk neutral world. So, we use risk free rate of return to discount expected future cashflows. Reasons for assuming risk neutrality world are:
- Finding out the risk adjusted rate of return is very difficult and highly debatable regarding which model to use for calculation of risk premium.
- Financial studies have proved that even if we take the risk averse world, the answer would have been same as yielding by risk neutrality assumption. Since both the numerator and denominator would have been adjusted upward and hence cancelling the effect.
One of the most important input in risk neutrality valuation approach is the risk neutral probability. In risk neutral world, the probability of upward movement and downward movement is adjusted in such a way that the expected value of stock yields risk free rate of return. Formula for calculating upward movement risk neutral probability is given below:
Downward Movement risk neutral probability = 1 – q
q = Upward movement risk neutral probability
ert = Risk Free rate factor
d = Downward movement factor
u = Upward movement factor
Upward movement factor and downward movement factor is calculated from underlying’s annualized volatility.
We have the complete binomial set-up along with each of the inputs. Now, we can value options. For example, we will discuss about how to value call option under risk neutrality valuation approach.
Value of call option is calculated as the present value of expected future cashflows where we use risk neutral probability to calculated future cashflows discounted at risk free rate of return. Formula for calculating value of call option given below:
Value of call option = (q*Su) + ((1-q)*Sd)/ert
Su = Pay-off from call option in upward movement of underlying
Sd = Pay-off from call option in downward movement of underlying
Pitfalls of Binomial Model:
The main limitation of the binomial model is its relatively slow speed because as we increase the number of periods, complexity increases. Other limitation is that it does not capture stochastic behavior of stock.
The Bottom Line:
The binomial model allows multi-period views of the underlying asset price and the price of the option for multiple periods as well as the range of possible results for each period, offering a more detailed view. While both the Black-Scholes model and the Binomial model can be used to value options, the Binomial model simply has a broader range of applications and it is very useful in valuing interest rate options, bond options, callable bond, puttable bonds, etc., It is also very intuitive and easier to use.