# options stuff #FRM

**Strategy**

collar – insurance for long position in stock

straddle – betting for high volatility(with ATM call and put, higher cost)

strangle – betting for higher volatility too but with a lower cost

**Binomial Pricing**

U = size of going up

D = size of going down

Probability of going up ( risk neutral)

discount back

**Option Greeks**

delta

to hedge one long call, short delta shares stock

to hedge one long stock, short 1/delta shares call

dela = N(d1)

* probability of default = N(d2)

* expected loss = adjusted exposure * P(d) * loss given degault

Vega

call Vega = put Vega , due to put call parity

Gamma

highest when at the money

Theta

As time more and more approach to maturity, the loss of one day has more and more negative impact on option value

**Early exercise of american option on dividend paying stock**

American call

benefit: dividend received, PV(dividend)

loss: prepayment of K ,interest loss, K – PV(K)

American put

benefit: interest received, K – PV(K)

loss: dividend loss, PV(dividend)

**Black&Scholes**

call = S*N(d1) -PV(K)*N(d2)

expected return = mu

mean return = mu – 0.5* SD^2

flaws:

price lognormally distributed, return normally distributed

continuous prices, no jumps

volatility and interest rate is known and stay constant