Derivatives Exam #3

•Exercising early is trading off the following:

-Capturing the dividend and earn interest on it -Remaining time value of the option

•The delta of a European put on the stock is...

-N(-d1)=N(d1)-1<0

Principal Protected Notes Variations on standard products:

-OTM strike price -Caps on investor return -Knock outs, averaging features, etc.

No Arbitrage Bounds for options

-Options never have negative value, c, p, C, P all greater than or equal to 0. -A call option is never worth more than the underlying and a put option is never worth more than the strike price: c(S,K,T)≤S and C(S,K,T)≤S, p(S,K,T)≤K and P(S,K,T)≤K •An European put option is never worth more than the present value of the strike price: p(S,K,T)≤K×e^(-rT)<K •An American option is worth at least as much as the corresponding European option C >= c P >= p •American options with more time to maturity are at least as valuable as the same options with less time to maturity. For T2 > T1: C(S,K,T_2 )≥C(S,K,T_1 ) P(S,K,T_2 )≥P(S,K,T_1 ) •An American option is worth at least its exercised value (the payoff you receive if you exercise today): C(S,K,T)≥max[0, S-K] P(S,K,T)≥max[0, K-S]

•The most popular underlying indices in the U.S. are:

-The S&P 100 Index (OEX and XEO) -The S&P 500 Index (SPX) -The Dow Jones Index times 0.01 (DJX) -The Nasdaq 100 Index (NDX) •Exchange-traded contracts are on 100 times index; they are settled in cash; OEX is American; the XEO and all others are European

The replicating portfolio properties

-The stock position is always positive. -The fraction of stock in the replicating portfolio is the hedge ratio or delta: ∆≤1

Risk of trading an option

-directional risk -volatility risk -time risk -IR risk

Principal Protected Notes Viability depends on:

-levels of dividends -level of IRs -Volatility of the port

N(0) = N(-infinity) = N(+infinity) =

0.5 0.0 1.0

We get the same probability distribution for the asset price at time T in each of the following cases:

1. The asset starts at price S0 and provides a yield = q 2. The asset starts at price S0e-qT and provides no income

Quick Summary: 1.Delta 2.Gamma 3.Theta 4.Vega 5.Rho

1. measures the sensitivity of option value to a change in und price 2. measures sensitivity of delta to und price 3.measures the sensitivity of option value to a change in time to expiration 4. measures the sensitivity of option value to a change in volatility 5. measures the sensitivity of option value to a change in interest rate

If your position is: 1. + Delta 2. - Delta 3. + Gamma 4. - Gamma 5. + Theta 6. - Theta 7. + Vega 8. - Vega 9. + Delta and - Gamma You wish for...

1. und price rises 2. und price falls 3.und big move or quick move 4.und to sit still or move slowly 5. the passage of time helps 6. the passage of time hurts 7. implied volatility to rise 8. implied volatility to fall 9. slow upward movement of und

Box Spread

A combination of Bull Call Spread and Bear Put spread •If all options are European a box spread is worth the present value of the difference between the strike prices •If they are American this is not necessarily so

Call, Puts, Und: 1. Delta 2. Gamma 3. theta 4. Vega 5. Rho

Delta: call >0, puts<0, und=1 Gamma: Call>0, puts>0, und=0 Theta: Call<0,Put<0, und=0 Vega: Call>0, puts>0, und=0 Rho: call >0, puts<0, und=0

•From European calls and puts with the same strike price and time to maturity •These formulas allow term structures of forward prices and dividend yields to be estimated

F_0=K+(c-p)e^rT q=-1/T ln〖(c-p+Ke^(-rT))/S_0 〗

Key point of Principal Protected Notes

If the underlying does not pay dividends at all, the principal note is NEVER possible!

stop-loss strategy cost of writing and hedging the option

Ignoring discounting, the cost of writing and hedging the option appears to be max(S0−K, 0).

Scenario Analysis

Involves testing the effect on the value of a portfolio of different assumptions concerning asset prices and their volatilities

Short Fwd

K×e^(-rT)-S

Bull Spread Using Calls

Long call w/ strike K1 (or X1) and Short call w/ strike K2 (or X2)

if you plug the implied volatility to the black sholes formula, the price will equal the...

Market price!

•In a short period of time of length T, the return on the stock is normally distributed: ln⁡( (S(T))/(S(0)))≈

N(μT,σ^2 T)

Currency options trade on...

NASDAQ OMX •There also exists a very active over-the-counter (OTC) market

Gamma for longing an option is...

Positive

Long Fwd

S-K×e^(-rT)

Cover call payoff equals the payoff of the...

Short Put b/c of the put-call parity

•For American options on non-dividend underlyings:

So- K≤C-P≤So- Ke^-rT

t/f: Hight interest makes early exercising more attractive.

TRUE

t/f: •The portfolio of a long position in the European call and a short position in the European put has the exact same future payoff as a long forward. Therefore, the two strategies have the same value today.

TRUE

Greeks

The parameters which measure risk sensitivities of options -We want to know how the value of an option or a portfolio of options changes as market conditions change -We want to understand and manage the accumulated risk of our portfolio

Options on Assets with a Known Yield

We can value European options by reducing the asset price to S0e-qT and then behaving as though there is no income

Positions in an Option and the Underlying:

a) Long stock, Short Call (Covered Call) b) Short stock, Long call (Short Squeeze) c) Long stock, Long put (Protective Put) d) Short stock, Short Put (Opposite of Protective Put)

The value of a portfolio of derivatives dependent on an...

asset is a function of of the asset price S, its volatility s, and time t

•The prices of European put and call options on a underlying that pays a known dividend D must satisfy:

c - p = So- PV(D)- Ke^-rT

Arbitrage Free Bounds for Options: Put-Call Parity

c+Ke^(-rT)=p+S_0 e^(-qT) c+Ke^(-rT)=p+F_0 e^(-rT)

Arbitrage Free Bounds for Options: Lower Bound for Calls

c≥S_0 e^(-qT)-Ke^(-rT)

Risk of trading the underlying

directional risk

Theta large when option close to

expiration

Gamma (Γ)

is the rate of change of delta (Δ) with respect to the price of the underlying asset. Referred to as curvature. •The delta increases by the amount of the gamma as the underlying price rises; the delta declines by the amount of gamma as the price falls.

Theta (Θ) of a derivative (or portfolio of derivatives)

is the rate of change of the value with respect to the passage of time

Payoff of the protective put equals the payoff of the...

long call option b/c of the put-call parity

Gamma for shorting an option is...

negative

•The proper number of underlying contracts required to establish a _____________ or to maintain a delta neutral portfolio

neutral hedge (hedge ratio)

Arbitrage Free Bounds for Options: Lower Bound for Puts

p≥Ke^(-rT)-S_0 e^(-qT)

The value of the European call option equals the

replicating portfolio

Covered position: buy 100,000 shares today. •What are the risks associated with these strategies?

risk that stock price decreases and could suffer a large loss on those 100,00 shares bought today.

Naked Position: Take no action. •What are the risks associated with these strategies?

selling call could have a large unlimited risk of loss

•This involves: -Buying 100,000 shares as soon as price reaches K -Selling 100,000 shares as soon as price falls below K

stop-loss strategy

European options on dividend-paying stocks are valued by...

substituting the stock price less the present value of dividends into Black-Scholes •Only dividends with ex-dividend dates during life of option should be included •The "dividend" should be the expected reduction in the stock price expected

S_0 N(d_1) represents:

the amount spent on buying stocks.

K e^(-rT) N(d_2) represents:

the borrowing.

S_0 N(d_1)-K e^(-rT) N(d_2) represents:

the call premium

t/f: Adjust the stock position according to the underlying stock price movement

true

t/f: Low volatility makes early exercising more attractive

true

t/f: When we are valuing an option in terms of the price of the underlying asset, the probability of up and down movements in the real world are irrelevant

true

t/f: log stock price follows a normal distribution.

true

t/f: •American options require the dividend yield term structure

true

t/f: •Delta changes as stock price changes and time passes

true

t/f: •European call (put) options are always worth more than long (short) forward positions with same price K and maturity T.

true

t/f: •With dividend, sometimes it is optimal to exercise an American call. •For American options: S0 - PV(D)- K≤C-P≤S0 - Ke -rT

true

Delta hedging a written option involves a...

•"buy high, sell low" trading rule •When a call option is close to maturity, if S<K, Δ=0; if S>K, Δ=1.

Principal Protected Note

•Allows investor to take a risky position without risking any principal. -Not always reliable

Properties of Black-Scholes Formula

•As S0 becomes very large c tends to S0 - Ke-rT and p tends to zero •As S0 becomes very small c tends to zero and p tends to Ke-rT -S0 -As σ becomes very large, d1 will be very large and d2 will be very small and negative (call is almost like the stock itself) - As T becomes very large, Call option will finish ITM most likely, so almost like a Long Fwd; PV of the discount is almost the same value as the stock.

Options Strategies

•Bond plus option to create principal protected note •Stock plus option •Two or more options of the same type (a spread) •Two or more options of different types (a combination)

Managing Delta, Gamma, & Vega

•Delta can be changed by taking a position in the underlying asset •To adjust gamma and vega it is necessary to take a position in an option or other derivative

Delta vs Gamma

•Delta measures speed: rate of change in the option value (first derivative). •Gamma measures acceleration: rate of change in the delta value (second derivative)

Bounds for Non-dividend Paying Stock

•If the underlying does NOT pay dividends, then we have the following. -The price of a call option satisfies: c(S,K,T)≥max⁡[0, S-K×e^(-rT)] C(S,K,T)≥max⁡[0, S-K×e^(-rT)] -The price of a European put option satisfies: p(S,K,T)≥max⁡[0, K×e^(-rT)-S] P(S,K,T)≥max⁡[0, K×e^(-rT)-S]

Portfolio Insurance

•In October of 1987 many portfolio managers attempted to create a put option on a portfolio synthetically •This involves initially selling enough of the portfolio (or of index futures) to match the Δ of the put option •As the value of the portfolio increases, the Δ of the put becomes less negative and some of the original portfolio is repurchased •As the value of the portfolio decreases, the Δ of the put becomes more negative and more of the portfolio must be sold

Strangle Combination

•Long a call and a put with different strikes •A bet on volatility

A Straddle Combination

•Long a call and a put with the same maturity and strike •A bet on volatility - b/c both call and put have the same strike price, only 1 will be ITM. Both CANNOT be ITM at the same time. -Lose money in between the overall cost bounds left and right of K.

Butterfly Spread Using Calls

•Long a call with strike K1, long a call with strike K3 •Short 2 calls with strike K2 •c(K1)-2c(K2)+c(K3)=[c(K1)-c(K2)]+[c(K3)-c(K2)] •A combination of bullish and bearish call spreads at different strike prices. •A bet on volatility

Butterfly Spread Using Puts

•Long a put with strike K1, long a put with strike K3 •Short 2 puts with strike K2

Calendar Spread Using Calls

•Long call with maturity T2 Short call with maturity T1

Bear Spread Using Calls

•Long call with strike K2 •Short call with strike K1

Calendar Spread Using Puts

•Long put with maturity T2 •Short put with maturity T1

Bull Spread Using Puts

•Long put with strike K1 •Short put with strike K2

Bear Spread Using Puts

•Long put with strike K2 •Short put with strike K1

The delta of a European call on a non-dividend paying stock is...

•N(d1)>0

Why Not Exercising a Call Early?

•No income is sacrificed •You delay paying the strike price •Holding the call provides insurance against stock price falling below strike price •If you would like to exercise and sell the stock, it is better to directly sell the call. •Formally: C(S,K,T)≥max⁡[0, S-K×e^(-rT) ]>S-K

Properties of Gamma

•Non-negative (by convexity of option prices). •Gamma is greatest for options that are close to the money •Gamma of call and put of the same strike/maturity is equal (why?). •When stock price goes to 0 or infinity, Gamma goes to 0. •For an option close to expiration and still at-the-money, Gamma goes to infinity (why?).

Risk-Neutral Valuation on Forwards

•Payoff is ST - K •Expected payoff in a risk-neutral world is S0erT - K •Present value of expected payoff is e-rT[S0erT - K]=S0 - Ke-rT

Multi-period Binomial Trees

•Project price of the stock forward until maturity of the option. •Compute payoffs of the option at maturity •Work your way back one step a time (on the option tree) till the tree root. •C is the price of the call today

Strip & Strap

•Strip: Long one call and two puts with the same maturity and strike. A straddle plus one put. •Strap: Long two calls and one put with the same maturity and strike. A straddle plus one call.

Index Options for Portfolio Insurance

•Suppose the value of the index is S and the strike price is K •If a portfolio has a β of 1.0, the portfolio insurance is obtained by buying 1 put option contract on the index for each 100S dollars held •If the β is not 1.0, the portfolio manager buys β put options for each 100S dollars held •In both cases, K is chosen to give the appropriate insurance level

Risk-Neutral Valuation

•The expected return does not appear in the Black-Scholes model •The equation is independent of all variables affected by risk preference •The solution is therefore the same in a risk-free world as it is in the real world •This leads to the principle of risk-neutral valuation

Implied Volatility

•The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price •There is a one-to-one correspondence between prices and implied volatilities •Traders and brokers often quote implied volatilities rather than dollar prices

Binomial Tree

•To price an option we have to make assumptions about the behavior of the underlying security's prices. •A binomial tree is a diagram that represents all the possible future values of a security. •Future values are represented by node of a tree. Each node is a possible outcome that can be reached after a period of time. •If we are using a tree to price an option the terminal nodes of the tree represent the option maturity.

Other Payoff Patterns

•When the strike prices are close together a butterfly spread provides a payoff consisting of a small "spike" •If options with all strike prices were available any payoff pattern could (at least approximately) be created by combining the spikes obtained from different butterfly spreads

Currency options are used by corporations to...

•buy insurance when they have an FX exposure

Put-Call Parity

•c + Ke^-rT = p + So c - p = So - Ke^-rT

Binomial trees illustrate the...

•general result that to value a derivative we can assume that the expected return on the underlying asset is the risk-free rate and discount at the risk-free rate. •This is known as using risk-neutral valuation, p risk-neutral probability.

The N(x) Function

•is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x

Rho

•is the rate of change of the value of a derivative with respect to the interest rate •Hedging a call involves borrowing, so Rho for a call is positive. For the same reason, Rho for a put is negative.

Vega (v)

•is the rate of change of the value of a derivatives portfolio with respect to volatility. •Vega always positive. •When stock price very low (high), Vega goes to 0.

Delta (Δ)

•is the ratio of the change in the price of a stock option to the change in the price of the underlying stock The value of Δ varies from node to node

The theta of a call or put

•is usually negative. This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of a long call or put option declines.

Delta (Δ) is the...

•rate of change of the option price with respect to the underlying

The value of the option depends only upon the...

•risk free-rate, the value of the stock, and the allocations in the portfolio (how much of the stock I have to hold along with the option)

OTC European options are typically valued...

•using the forward prices (Estimates of q are not then required)