Option valuation – Method 1.
by J Victor on October 15th, 2012So far, from our earlier posts we learned –
 The nature of option contracts
 Factors that affect the valuation of option contracts
 And, certain restrictions on upper and lower bound of options prices.
From this post onwards, we proceed to learn the methods of option pricing. The first point we would like to say is that, option valuation, however simple, requires a bit of mathematical calculations. The methods we would discuss in this series are:
 Risk neutral model
 Binomial model
 And, The Nobel prize winning BlackScholes model.
Each method is an attempt to suggest a price for an option contract, given the time to maturity, strike price, underlying stock price and the risk free interest rate. Among the above three methods, the Black –scholes method is the most widely used.
Two points to note here is that one, the main factor that influences the option price is the stock price which itself is again subject to valuation. second, Not all the factors that influence the option price are quantifiable. Demand and supply, bids and offers, unexpected events like war, earth quakes or scams can disturb the market equilibrium and drive the prices of underlying stocks to extreme levels. The effect of these events cannot be quantified or forecasted. Hence, what these pricing models suggest is a valuation based on the quantifiable factors. The value thus obtained may be used basically as an indicator to predict the reasonable value of options.
The first method: Risk neutral model
By ‘risk neutral’ we mean that the investor is indifferent to risk. In a risk neutral world, the expected return of the investors would be the same as the risk free rate of return or in other words, in a risk neutral environment, investors do not expect anything more than the risk free interest rate from their investments. Hence, it would be possible for us to calculate the future value of an option and discount it to the present value at the risk free rate.
For example: let’s assume that the stock of XY LTD is trading at Rs 200. You expect the price to move up 20% or fall by 10% in a year. The strike price is Rs 210 and the risk free rate is 10%. How would you calculate the value of option?
Step 1. We would value the calls both scenarios. So, if the stock rises 20% as expected, the call should be worth Rs 30 and if it falls 10%, naturally, the call is worthless since it’s out of the money. So the option values are Rs 30 ( if the stock price moves up 20%) and 0 in the other case.
Step2. Consider the upside and downside probability. We know that the stock can go up by 20% or drop down 10%. When the investors are indifferent to risk, the expected return on this stock must be the risk free rate of return ie, 10%. Therefore
 Expected rate of return = (probability of rise x 20%) + { (1probability of rise) x (10%)}
 Let’s assume that the probability of rise as ‘x’.
 Hence , 0.10 = .20 x + { (1x) x ( 0.10) }
 Therefore the probability of rise = 0.667 and the probability of fall is 0.333
Step 3. Find out the expected future value of the call which would be the weighted average of step 1 and step 2. So the expected value of option one year later is (0.667 x 30 ) + ( 0.333 x 0 ) = Rs 20 . Hence, if the investors are indifferent to risk, the value of option is Rs 20.
Step4. Now that we got the value of option 1 year hence, discount it at the risk free rate to get the present value of option. The present value of option would be –
 20/ 1.1 = 18.18
That’s the risk neutral method to value options. A deviation from the above rate would open up arbitrage opportunities.The risk neutral method is very simple. What we have done is – we have calculated the expected pay off from the option and discounted it to the present by applying the risk free rate.
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