Options are not only one of the most important topics in derivatives but also a fascinating one as well. CFA level I curriculum just touches the options and describes the basic kinds of options and the maximum and minimum value that an option can take, the types of options (call or put, American or European, out-of-money or in-the-money or at-the-money), and call-put parity along with two strategies (covered call and protective put). CFA level II curriculum goes deeper into options and covers the valuation of options (Binomial method and Black-Scholes-Merton model), properties of Greeks (delta, gamma, rho, vega, theta), and various kinds of hedging strategies (delta hedging). The questions expected in the level I curriculum are direct and less calculation intensive while in level II exam, there is a high probability that questions will be highly conceptual and it would be tough to score without getting the concepts correct. But you don’t have to worry. I will explain each and everything in this article and you can always refer it for solving every kind of questions.

**CFA level I**

A call option is an option which gives the holder a right to buy a stock at a particular predetermined price (known as strike price) on and before the option expiry date. The party which sells the call option has the obligation to deliver the stock at the strike price if the option holder decides to exercise the option. The option holder (long position) will exercise only when on exercising it will yield a profit. If on exercising the position is giving loss, then the holder will not exercise the position and the option will expire worthless. Similarly, a put option gives the option holder a right to sell the underlying security at the strike price. The increase in the underlying instrument will increase the value of call option and decrease the value of put option and vice versa.

The American option can be exercised anytime on and before the expiry while the European option can be exercised only at the time of expiry. Thus, the American option will always have a value greater than or equal to European option as it provides the additional facility of early exercise. In exercising a call option, the option holder will get the underlying while in exercising a put option; the option holder will be able to sell the stock at the strike price.

Suppose there is a cal option on a stock for a strike price of $25. If the stock is trading at $20, the option holder will not exercise the option to get the stock as he can buy from the market at $20. The intrinsic value of a call option is Max {0, (spot price-strike price)} which is zero in the given case. However, if the stock price would have been $30, then it would be having an intrinsic value of $5. The intrinsic value of a put option is Max {0, (strike price-spot price)}. The time value of an option is the total value of the option minus its intrinsic value.

A call option for which strike price is greater than spot price is out-of-money and if the strike price is lesser than the spot price, then the option is in-the-money. The opposite is true for the put option. However, if the strike price is equal to the spot price, then both call and put options are at-the-money.

The European and American call option can have a maximum value equal to the spot price because of the simple reason that if the value exceeds the spot price, then the option holder can simply buy the stock from the market rather than buying an option for that. The American put option can have a maximum value equal to the strike price as the minimum value of the underlying is zero. However, the minimum value of European put is equal to the present value of the strike price from expiry till today. In the case of a European option, we can exercise at the expiry only. So, we can get the strike price at the expiry only and thus the minimum value will be the present value of that strike price. That explains the positive value of theta for the deep in the money European put options which we will discuss later.

The put-call parity equation is c + PV(X) = p + S, where c and p are the European call prices and European put prices with a strike price as X, S is the stock price and PV(X) means the present value of X with a discount rate as the risk-free rate. It can be proved by taking 2 portfolios i.e. one as RHS and the other as LHS and comparing their payoffs with all the possible scenarios at expiry (spot price>strike price, spot price=strike price, and spot price<strike price). The payoffs of both portfolios will be same in all the cases.

From the put-call parity equation, the value of put option = c + PV(X) – S. But the value of European put option cannot be negative. So, c + PV(X) – S ≥ 0 => c ≥ S – PV(X). So, the minimum value of European call option is Max {0, (S – PV(X))}. Similarly, the minimum value of European put option is Max {0, (PV(X)-S)}. The minimum value of American put option is Max{0, (X-S)} because if the option value is lesser than that then the option holder can exercise it and get risk-free profit. Similarly, the minimum value of American call option is Max {0, (S-X). But as explained above that the value of American option cannot be less than the value of European option. So, the minimum value of American call option is Max {0, (S-PV(X)}.

Options can be on interest rates as well. The most common options are interest rate cap and interest rate floor. Interest rate floor means that if the interest rate goes below the floor rate, the option holder (the issuer of the loan) will receive the floor rate as coupon and the option will have some positive value equal to the present value of the differential of the interest multiplied by the notional principal. The payoff of interest rate floor is Max {0, (floor rate-floating rate)*notional principal}/ (1+floating rate) for one particular period. Similarly, the interest rate cap is beneficial for the borrower as if the interest rate rises above the cap rate, he has to pay only the cap rate and the payoff of the option is Max{0, (floating rate-cap rate)*notional principal}/(1+floating rate) for a particular period. A collar is the combination of floor and cap. The issuer of loan buys floor and sells a cap to make a collar while it is opposite for the borrower who sells floor and buys a cap.

The amount of money paid to get an option is called as option premium. Option premium for a call option is inversely proportional to the strike price (exercise price) because higher is the exercise price, the higher is the amount of which the buyer will be able to buy the underlying and thus lower will be the premium required. The option premium for put option is directly proportional to the strike price or exercise price. Higher is the volatility, higher is the value of both call and put options as both options can provide a higher profit on and before the option expiry date.

**CFA level II**

CFA level I covers the basics of the options and the level II curriculum enters into the valuation of the options. There are two methods to calculate the value of options. One is based on the discrete-time option pricing (Binomial model) and other is based on the continuous-time option pricing (Black-Scholes-Merton model).

The discrete time option pricing means that the option is priced such that the time moves in distinct increments. When these increments become infinitely small, this will lead to continuous option pricing. More are the number of intervals; more accurate would be the valuation. However, in the case of BSM model, we cannot value American options. BSM model is suitable only for European options. To value American options, one has to use Binomial model. So, Binomial model is also very important to study. The Binomial model can be of any period i.e. one-period, two-period and so on. CFA level II curriculum deals with one-period and two-period binomial models.

In a simple one-period binomial model, we will be provided with the size of up and down move. It is based on the assumption that after one-period price can either go up or come down. If u is the size of up move and d is the size of down move, then after 1-period price can be either Su or Sd where S is the price of underlying today. Now there will be some risk neutral probabilities which we need to calculate to value the option. The risk neutral probability for the up-move will be π_{U} = (1+r-d)/ (u-d) where r is the risk-free rate. The probability of the down-move will be π_{D} = 1 – π_{U}. The price of the call option will be (π_{U}*c^{+} + π_{D}*c^{–})/ (1+r). Let us try to understand it with an example. Suppose a share of Konvexity is trading at $20. It is expected to either go up by 20% or fall down by 15% in one year. Let us calculate the price of the 1-year call and put options with a strike price of $20. The price after one year will be either $24 (=20*1.2) or $17 (=20*0.85). The risk neutral up and down probabilities will be 0.6 and 0.4 assuming a risk-free rate of 6% per annum. The call option price will be {(4*0.6 + 0*0.4)}/1.06 = $2.264 and the put option price will be {(0*0.6 + 3*0.4)}/1.06 = $1.132. The two-period binomial method is also very similar to one-period binomial method. In this we need to discount the option value at the node 2 to the node 1 and then again discount back the values at node 1 to get the value of option at node 0. However, in case of valuing an American option, we should be careful at node 1 and node 0. If the value at node 1 or node 0 are less than the intrinsic value of the option at that node, then we need to keep the intrinsic value as American option can be exercised any time and the value of American option can’t be less than the intrinsic value. Can you think of a European option having option value less than the intrinsic value? (Hint: European deep-in-the-money put option)

The binomial method can be applied to value call and put options on bonds as well. The procedure of valuation is quite simple. We will be given the interest rates at each particular node. We need to calculate the price of bond at each node at the maturity of the option and calculate the value of call or put option. Then we need to discount both bond price and option price to one earlier node with the discount rate as the earlier node rate. In case of American options, we need to check the intrinsic value of the option as well. If the intrinsic value is more than the calculate value, we need to keep the intrinsic value at that particular node. By this way, we can get the value of option by backward induction method. Let us try to understand it with an example.

Suppose we want to value an American put option with 2 years to expiration on a bond having face value as $100 and coupon rate as 6% per annum. The strike price is $102.

Let us calculate the price of the bond at the node on which the option will mature. Using financial calculator, for the upper node: FV=100, N=1, I/Y=7.5, PMT=6, CPT->PV=98.60. Similarly, the price at middle and lower nodes will be $99.62 and $100.28 respectively. The value of put option will be equal to its intrinsic value at the maturity i.e. $1.4, $0.38 and $0 at upper, middle and lower nodes respectively.

The bond value at the node with 6.1% as the interest will be {(98.6+6)*0.5 + (99.62+6)*0.5}/1.61 = $99.07 and the put option value will be {(1.4*0.5) + (0.38*0.5)}/1.061 = $0.84. But since this is an American option which can be exercised any time on or before the maturity date, the option value cannot be less than the intrinsic value which is 0.93. So, the value of put option at that node will be $0.93. Similarly, we can calculate all the values and find out the value of the put option as $0.53. The value of put option would have been $0.49 if it would have been a European option.

Let us now discuss the Black-Scholes-Merton model, also known as BSM model of option valuation. The formula to calculate the option price is out of the scope of the CFA level II curriculum. The candidate should be aware of the implications, assumptions, and limitations of the model. This model is based on the continuous period i.e. the stock price keeps on changing on a continuous basis rather than on discrete basis as is the case with the Binomial model. The BSM model assumes that the continuous risk-free rate and volatility of the underlying asset are constant and known. Markets are frictionless i.e. there are no taxes, transaction costs and no restrictions on short sales. The options are European in nature and the price of the underlying asset follows a lognormal distribution. The main limitation of the model is that it cannot be used for valuing American options. It is also not useful in pricing options on bond prices and interest rates as interest rate volatility is the main factor in determining the value of those options.

The formula for valuing the call option according to the BSM model is C = S*N(d_{1}) – X*e^{-RFR*T}*N(d_{2}) where RFR=continuously compounded risk-free rate, S=asset price, C=call option price, X=exercise price, N(x)=cumulative normal probability of x. d_{1}=[ln(S/X) + {RFR+(0.5*σ^{2})}*T]/σ*√T and d_{2}=d_{1}– σ*√T where σ is the volatility of continuously compounded returns on the stock. Don’t need to worry. You don’t have to remember this formula. But yes, if you can remember, then it is always good. The put option price can be calculated using put-call parity equation.

The most asked area from the topic of options in level II is from the Greeks. There are mainly five inputs in the BSM model. Those are asset price, exercise price, volatility of asset price, time to expiration and the risk-free rate. The Greeks captures the relationship between each of these inputs (except exercise price) with the option price. There are five Greeks covered in level II curriculum and those are delta, gamma, vega, theta, and rho.

Delta is the change in option price per unit change in the option price. The call value increases with increase in option price and opposite is true for the put option. So, the value of call delta is positive and the value of put delta is negative. The call delta ranges from 0 to 1 and the put delta ranges from -1 to 0. The delta is the slope the curve between the option price and the underlying price. The maximum value of slope can be 1 when the option is deep-in-the-money for call option. Similarly the slope value goes to -1 for the put option when the option is deep-in-the-money. The value is near to zero for deep-out-of-money options for both call and put. With an increase in the price of the underlying, the call delta moves from 0 to 1 and the put delta moves from -1 to 0. We can do delta hedging as well. Delta hedging is a combination of options and the underlying so that the entire portfolio is hedged. No. of shares + Δ*No. of options = 0. The delta is positive for call, so the call option will have opposite position with respect to the underlying for the hedging and the put option will have the same position as of the underlying as its delta is negative. The hedging would be such that the change in underlying would be compensated by the equally change in the option price. However, delta also keeps on changing with the underlying and for dynamic hedging; we need to continuously rebalance our portfolio which could lead to lots of transaction costs. The change in delta with respect to the change in the underlying is called as gamma. Gamma is the maximum for at-the-money option and decreases when the option goes deep-in-the-money and deep-out-of-money. So, we can conclude that the dynamic hedging when done with the relatively out-of-money or in-the-money options will be more cost effective that doing it with the at-the-money options because there will be less portfolio rebalancing required. The changes in call and put options’ prices with a change in the underlying can be calculated from BSM as:

ΔC ≈ N (d_{1})*ΔS

ΔP ≈ [N (d_{1}) -1]*ΔS

Vega, theta and rho are easy to remember. Vega is the change in option price with respect to the change in the volatility. Theta is with respect to the time to expiration and rho is with respect to the risk-free rate. We can see that the first syllables are the same: (vega-volatility, theta-time, rho-risk free rate). The options prices increase with increase in volatility (both call and put options) because the higher volatility makes options more valuable as there can be higher price appreciation or depreciation. The increase in risk-free rate leads to an increase in call option price and a decrease in put option price. It can be easily remembered by looking at the put-call parity equation. c + PV(X) = p + S. The PV(X) decreases with an increase in risk-free rate. Hence, the decrease in put option price and an increase in call option price. The option price decrease as the time to expiration decrease as the time value of the option decreases and there are less chances that the option will valuable as volatility would be applied for the few days only. Suppose that the volatility is 5% per month. Then, according to normal distribution, 95% of the times the underlying price will remain within the range of current price ± 2*standard deviation. With an decrease in the time to expiration, the standard deviation will go down by 1/n^{0.5} i.e. if the monthly volatility is 5%, then the daily volatility will be 5%*(1/30^{0.5}) and thus the change in the underlying will be less and the option price will decrease. However, for deep-in-the-money European put options, the value of option increases with a decrease in time to expiration. Let us take an example. Suppose a share of a company was trading at $80 and we bought the European put option for the strike price of $80. Now, the company goes bankrupt. Now, the value of put option should be $80. But since it is European, we can exercise it as the expiry only and its value will be the present value of $80 and will increase with a decrease in time to expiration.

The effect of cash flows on the underlying is straight forward. If we are getting cash flows, then the price of underlying will go down and thus it will increase the value of put option and decrease the value of call option. The put-call parity equation will become: c + PV(X) = p +{S – PV (CF)}. The BSM model can also be modified for the cash flows.

The options can be on forwards and futures as well. The put-call parity equation will be modified to: c + PV(X) = p + PV (F_{T}). The American options will be more valuable than the European option on futures because we can exercise anytime and because of mark-to-market feature of the futures, we can get the funds and can earn an extra yield on that. While in the case of options on forwards, we can only get the benefit at the contract expiry as mark-to-market feature is not there in forwards. Thus, the price of European and American options on forwards do not differ. The Black model can be used to price options on futures and forwards. It is just a little modification of BSM model. We don’t need to learn the formula as it won’t be asked in the examination. We just need to substitute e^{-Rfc*T}*F_{T} for S_{0} in the BSM equation to get the equation.

I hope that you guys have understood everything. I have covered everything which can be asked in the exam.

Tagged: CFA level I, CFA level II

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