CFA: Option Markets and Contracts (64)
here we go….I don’t think they actually mentioned Credit Derivatives outright before. i could be wrong. i always remember it as an aside in this section and not actually part of the title of the section. looking ahead at section 67, its all about credit derivatives, mainly CDS. sweet!
the holidays bogged me down a bit but i’m back and getting my study on….enjoy.
LOS 64a: calculate and interpret the prices of a synthetic call option, synthetic put option, synthetic bond, and a synthetic underlying stock, and infer why an investor would want to create such investments.
calculate the price:
the price of any of these instruments is basically the utilization of this formula:
C + X/(1+r)expT = P + S, where
C = call
X = strike price
P = put
S = underlying stock
solving for any of these variables will create a synthetic “fill-in-the-blank” instrument. for example, a synthetic call will equal:
C = P + S - X/(1+r)expT, or
long a put, long the stock and short a discount bond equal to the amt of the strike price
An investor would want to create one of these instruments in order to take advantage of arbitrage opportunities.
LOS 64b: calculate and interpret prices of interest rate options and options on assets using one- and two-period binomial models.
i think this is one of the LOS that killed me in the last exam. i got twisted with the formulas and a little cocky on options and was smacked with an entire question about this. a question on
check this little spreadsheet, hope this helps
basically, the steps are:
1. find the share price at each node assuming the percent that they will move by (this will be given in the problem as well as the risk-free rate)
2. find the call price at each node assuming c = Max[0, S - X]
3. Plug into the formula to solve for pi. pi = (1 + r - d)/(u - d)
4. Then plug pi into the formula to solve for the call price. c = [pi(c+) + (1 - pi)(c-)]/(1 + r)
5. If its a 2 period tree, then you have to solve for the call price twice
LOS 64c: explain the assumptions underlying the Black-Scholes-Merton model and their limitations
Assumptions/Limitations:
- underlying price follows a geometric lognormal diffusion process: easier than it sounds. this is where the price volatility is found using lognormal price changes or continuously compounded returns.
- risk-free rate is known and constant: assumed that it does not change
- volatility of the underlying asset is known and constant
- no taxes or transactions costs
- no cash flows on the underlying
- option are european
The limitations here are obvious. The risk-free rate is not constant and neither is the volatility of the underlying. Everything is constantly in flux. This also goes for transaction costs and taxes. Those are a reality and do impact the real-world analysis. Also, this means that you cannot use this model to value American options or underlyings that have cash flows. The binomial model is used to price American options.
LOS 64d: explain how an option price, as represented by the Black-Scholes-Merton model, is affected by each of the input values (the option Greeks)
delta: this is probably one of the greeks that has the most impact. this is the most straightforward too. its basically ratio of the change of the option price to the change in the underlying price and measures the sensitivity of the option price to the change in the price of the underlying.
rho: measures the sensitivity to the risk-free rate, more specifically, the continuously-compounded risk-free rate that matches the maturity of the option contract. the option price is not very sensitive to this input value
theta: measures the sensitivity to the time to maturity or time value decay. this is the exact same concept as with bonds you are slowly accreting towards par as the bond reaches maturity. instead of a bond, we are accreting to the payoff value of the option at maturity. if this number is negative, that means that the option price decreases as time moves forward and vice-versa if the number is positive. generally, the longer the maturity, the higher the option price. also, most options (puts or calls) will have a negative theta value except for certain cases with Euro puts. [remember that when talking about the greeks, we can include American and Euro options]
vega: measures the sensitivity to volatility. this is always one the biggest driving inputs into the option price. as volatility increases, so the option price. vega is also larger the closer the option is to being at-the-money.
LOS 64e: explain the delta of an option, and demonstrate how it is used in dynamic hedging
delta is the ratio of the change of the option price and the change of the underlying (change in option price/change in underlying). delta is something that is constantly changing. this measure can tell us how large of a position is needed in order to hedge an option position.
the formula is:
delta = (call price T - call price T-1)/(share price T - share price T-1)
for calls, use this delta formula. for puts, the delta is call option delta - 1. so if the call option delta is .3, then the put delta is -.7.
re-arrange the forumla:
(call price T - call price T-1) = change in call price = delta * ( share price T - share price T-1)
or
change in call price = delta * change in share price
if you are selling options (as the text describes in the example), you need to buy shares in order to hedge the position. to find out how many shares to buy we would apply the formula and assume a $1 price movement.
$1 = delta * $1
so if the delta is .65, then we would expect that for a $1 price move in the option price, we would expect a $.65 price move in the underlying. so to hedge that, we would need to buy 65 shares. and also, you don’t ever buy 1 option. you buy them in lots of 100, or put another way, an option to buy 100 shares.
Schweser explains this with a formula:
nbr of options needed to delta hedge = nbr of shares to hedge/delta of call option
applying that to our example, if we had 100 options, or one option contract….
100 = x/.65
x = 65 shares
so if the delta changed to .7, then you would have to buy 5 more shares to properly hedge the position. delta is constantly changing so dynamically hedging the position is the process of buying/selling shares to keep a risk-neutral position so that price changes in both the options and the shares offset each other. this is never perfect in practice and delta is a rough way of doing it, especially for large moves. the text compares duration on bonds to delta.
LOS 64f: explain the gamma effect on an option’s price and delta and how gamma can affect a delta hedge.
gamma is the amount of the change in delta. its the second derivative of the underlying’s price move to the option’s price move, or the amount of the amount of the price moves between the call and share price. gamma is an indicator of how effective delta hedging can be. Gamma is generally larger when there is more uncertainty about whether the option will expire in- or out-of-the-money. so the closer that the share price is to the strike price, or the closer the option is to being at-the-money, the higher the gamma. it will also be larger the closer to expiration the option contract is. when the gamma is large, delta hedging works poorly.
LOS 64g discuss the effect of the underlying asset’s cash flows on the price of an option.
cash flows on the underlying affect an option’s boundary conditions (in a binomial model) and put-call parity by lowering the price of the underlying by the present value of the cash flows over the life of the option.
all else equal, it will decrease the value of a call option and increase the value of a put option. the reason being is that call options typically increase in value when the underlying price increases and vice-versa for put options. so if the underlying price is decreasing by the present value of the future cash flows, then call options on this underlying will also decrease by a similar amount.
LOS 64h: demonstrate the methods for estimating the future volatility of the underlying asset (i.e.: the historical volatility and the implied volatility methods)
i thought we had discussed this previously, but basically historical volatility is the lognormal distribution or the standard deviation of the continuously compounded returns from a sample of recent data on the underlying. the implied volatility is done by using the BSM model’s formula and backing into the volatility number by making the option price equal to the market price and solving for volatility. this assumes how the market is pricing the option and the volatility used in that pricing.
LOS 64i: illustrate how put-call parity for options on forwards (or futures) is established.
This is very similar to the original put-call parity formula:
C = P + S - X/(1+r)expT
but its a little different. i would recommend just memorizing the formula. the big difference is that there is no underlying. the value of the forward contract is basically including the underlying.
P = C + [X - F]/(1+r)expT, or
C = P - [X - F]/(1+r)expT, or
F = (C - P)(1+r)expT + X
where X is the strike, and F is the forward/futures price.
This implies that a forward contract equals a long call, short put and a zero-coupon bond valued at the strike price minus the forward price.
LOS 64j: compare and contrast American options on forwards and futures to European options on forwards and futures, and identify the appropriate pricing model for European options.
American options on forwards are meaningless and unjustified because there are no underlying cash flows. Since forwards don’t payoff until maturity, exercising an option early does nothing because the forward still has not matured and therefore will not payout. This implies that options on forwards are basically European options.
Because futures are marked-to-market everyday, there are cashflows and therefore interest rate implications. If the option on a futures position is deep-in-the-money, an investor would exercise that option in order to realize interest in the margin account. Until the option is exercised, there would be no cash exchanged. So if there are high interest rates, its in the best interest of the investor to exercise prior to expiration in order to capture those profits.
Due to this phenomenon, American options on futures are different from European options on futures and are therefore priced higher than Euro options.