AUD/USD

Sell at 1.070  Target: Buy at 1.055

The Elasticity and Risk Premium of an Option Portfolio

The elasticity of a portfolio of call options can be expressed as

Ω_portfolio = _i=1ⁿΣω_iΩ _i where Ω _i is the elasticity of the ith call option and ω_i is the percentage of the portfolio comprised of the ith call option.

The risk premium on the portfolio - where all call options are based on the same underlying asset - is

γ - r = Ω_portfolio(α - r)

Meaning of variables:

γ = expected annual continuously compounded return on the option.

α = expected annual continuously compounded return on the underlying asset (most often a stock).

Ω = option elasticity.

r = annual continuously compounded risk-free interest rate.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 12,

p. 395 

The delta-gamma approximation and A delta-hedged

A market-maker sells assets or contracts to buyers and buys them from sellers. He is an intermediary between the buyers and sellers. A market-maker's function is in contrast toproprietary trading, which is "trading to express an investment strategy" (McDonald, p. 414).

A delta-hedged position is a position designed to earn the risk-free rate of interest and is used to offset the risk of an option position.

The delta-gamma approximation is used to estimate option price movements if the underlying stock price changes.

The delta-gamma approximation for call options can be expressed via the following formula:

C(S_t+h) = C(S_t) + є∆(S_t) + (1/2)є²Γ(S_t)

For a put option, the same formula holds, but delta is now negative - so the put price will decrease if the stock price increases.

Meaning of variables:

S_t = stock price at time t.

S_t+h = stock price at time t+h.

C = call option price.

є = stock price change from time t to time t + h.

∆ = option delta.

Γ = option gamma.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 13, pp. 413-425.

Put-call parity for European options

Put-call parity for European options with the same strike price and time to expiration is

Call - put = present value of (forward price - strike price)

Equation for put-call parity:
C(K, T) - P(K, T) = PV_0,T(F_0,T - K) = e^-rT(F_0,T - K)

Meaning of variables:

K = strike price of the options

T = time to expiration of the options

C(K, T) = price of a European call with strike price K and time to expiration T.

P(K, T) = price of a European put with strike price K and time to expiration T.

F_0,T = forward price for the underlying asset.

PV_0,T = the present value over the life of the options.

e^-rT*F_0,T = prepaid forward price for the asset.

e^-rT*K= prepaid forward price for the strike.

r = the continuously compounded interest rate.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 9, p. 282.

MFE question from SOA's Sample question

 Company A is a U.S. international company, and Company B is a Japanese local
company.  Company A is negotiating with Company B to sell its operation in
Tokyo to Company B.  The deal will be settled in Japanese yen.  To avoid a loss at
the time when the deal is closed due to a sudden devaluation of yen relative to
dollar, Company A has decided to buy at-the-money dollar-denominated yen put of
the European type to hedge this risk.
 You are given the following information:
(i) The deal will be closed 3 months from now.
(ii) The sale price of the Tokyo operation has been settled at 120 billion Japanese
yen.
(iii) The continuously compounded risk-free interest rate in the U.S. is 3.5%.
(iv) The continuously compounded risk-free interest rate in Japan is 1.5%.
(v) The current exchange rate is 1 U.S. dollar = 120 Japanese yen.
(vi) The natural logarithm of the yen per dollar exchange rate is an arithmetic
Brownian motion with daily volatility 0.261712%.
(vii)  1 year = 365 days; 3 months = ¼ year.
 Calculate Company A’s option cost.