5. Option Greeks and Risk Management

Delta: definition, Call and Put formulas, call and put relationship, signs and bounds

(1) Def: change in the option's value corresponding to an increase in the stock price or δV/δS (2) Call: Δ_c = e^(-δT) N(d_1) Put: Δ_p = -e^(-δT) N(-d_1) (3) Δ_c - Δ_p = e^(-δT) (4) ↑ stock price -> ↑ call payoff -> + Δ_c ↑ stock price -> ↓ put payoff -> - Δ_p (5) 0 <= Δ_c <= 1 because of e^(-δ) bounds -1 <= Δ_p <= 0 because of e^(-δ) bounds Note: the delta are derived from taking the derivative of the Value of the option with respect to stock price for the BS and replicating portfolio method. Thus the delta's are connected. Also delta refers to the delta of a long position. For a short position, its delta is -delta.

Gamma: definition, Call and Put formulas, call and put relationship, signs an bounds

(1) Def: measures the change in the option's delta corresponding to an increase in the stock price or δ²V/δ²S (2) Γ_c = Γ_p = e^(-δT) N'(d_1) * (1/S*σ√T) Note that: δ_d2/δS = δ_d1/δS = 1 / [S*σ√T] (3) Γ_c & Γ_p are both positive

Risk premium (stocks and options) (3)

(1) Definition: the excess of the expected return over the risk-free rate of return (2) For stock = α - r For option = γ - r (3) γ - r = Ω (α - r) [holds for portfolios too] Note: The expected return (γ or α) can be referred to as the instantaneous expected rate of return on the stock.

Inflation Indexing

(1) Feature which adjusts for inflation in defined benefit plans. (2) Full Indexing occurs when the purchasing power of each payment is never less than the first payment (Note that the basis is first payment, not previous payment). (3) Partial indexing means that the loss of purchasing power will not be more than some specified amount. (4) For a pension plan with full indexing benefit, let: I_t denote level of the consumer price index P_0 be the first pension payment P_t be the pension payment at time T P_t = max [ P_0 * {I_t/I_0} , P_(t_1) ] (5) Two guarantees are present (a) the payment will never be less than the purchasing power of the first (b) The size of payments will never decline (5) The result is a look back put option (picture attached)

Dynamic hedging strategy

(1) Involves the frequent buying and selling of assets and or derivatives with the goal of matching changes in the value of the guarantee. (2) May appear simple but are very complex and costly in general.

Guaranteed replacement cost coverage

(1) Is an optional benefit that can be added ti a property insurance policy to cover the cost of replacing a physical piece of property when the cost is more than the amount of insurance in the base policy. (2) Valuable when cost of replacement rises above the initial insurance amount. (3) Let I denote the amount of coverage in base policy C be the cost of replacement The coverage provided by the guaranteed replacement benefit is a CALL OPTION with payoff max(C-I,0) i.e. you receive the excess of the replacement cost over coverage in the event that C > I.

GMBD with a Return of Premium guarantee:

(1) It is a guarantee which returns the greater of the account value and the original amount invested K (2) max( S(T) , K) = S(T) + max( K-S(T) , 0) Thus this has two pieces the account value and amount needed to get the befit to K if S(T) is below K (3) Similar to a floor? (4) The put embedded put option is called called a LIFE-CONTINGENT put option since the exact time of expiration is the policy holder's time of death. (5) Let : T_x be the future lifetime continuous random variable for a policy holder aged x. P(T_x) be the value of the European put option for the policyholder's time of death f_Tx_(t) be T_x's density function The probability weighted values of the European put options is: E[P(T_x)] = Int_0_inf { P(t) * f_Tx_(t) } dt (6) If P(t) satisfies the conditions of the BS model, then numerical techniques can be used to calculate value. Where K is the original amount invested and T is the time when the policyholder dies.

Benefits of mortgage insurance as a product

(1) Less qualified borrowers can take out mortgages (2) Lenders can manage their credit risk (3) Credit risk is spread out so that liquidity is increased

Which four exotic options are useful for hedging variable annuity guarantees?

(1) Lookback Options (2) Shout Options (3) Rainbow Options (4) Forward Start Options

Vega: definition, Call and Put formulas, signs

(1) Measures the change in the option's value corresponding to an increase in volatility or δV/δσ (2) Vega_c = Vega_p = S * e^(-δT) * N'(d_1) * √T (derivation not easy) (3) Only option greek without a greek letter (4) Vega is positive for both calls and puts because option values increase when volatility increases Note: memory aid, both vega and volatility start with v.

Theta: definition, Call and Put formulas, call and put relationship, signs an bounds

(1) Option's value with respect to the passage of time or δV/δt (2) Θ_c and Θ_p in picture (derivation not easy) (3) Signs: usually negative for both call and put thetas. However, it may be positive in some special cases including: (a) A deep in-the-money European put. (b) A deep in-the-money European call option on stock with a high dividend yield. Note: memory aid is that both theta and time start with t. Also, since as t increases, T-t decreases, theta also actually measures the change in the option value as time to expiration decreases. Add more after response to question from coaches

Earnings-Enhanced Death Benefit

(1) Pays the beneficiary an amount based on the increase in the account value over the original amount invested (2) For example it could be 40% * max(S(T)-K,0) (3) The embedded call option's value is: E[C(T_x)] = Int_0_inf { C(t) * f_Tx_(t) } dt

3 other cases in which the guaranteed amount can vary by besides the fixed strike assumption

(1) Performance of underlying assets (2) Promised rate of return (3) Age of policyholder

Hedging of Catastrophic Risk

(1) Property insurance assumes independent losses e.g. among 50 houses. (2) However catastrophes present big risks. (3) Reinsurance and weather derivatives or cat bonds can be used. (4) Cat bond def: a bond issued to investors where repayments and principal payments are contingent upon not there being a catastrophe which causes large losses for the insurer. In general cat bond holders typically receive higher interest as compensation for risk.

Psi: definition, Call and Put formulas, call and put relationship, signs an bounds

(1) Psi measures the change in the option's value corresponding to an increase in the dividend yield or δV/δ-delta (2) Ψ_c = -T * S * e^(-δT) * N(d_1) Ψ_p = T * S * e^(-δT) * N(-d_1) (3) psi is negative for calls and positive for puts. This is because for increasing delta, Se^(-δT) will reduce call value and increase put value. Derived from BS formula

Mortgage Guaranty Insurance / Mortgage Loan as Put

(1) Purchased by mortgage lenders as protection from borrower defaults (Secured by physical property such as a home). (2) Mortgage guaranty insurance typically provides coverage for a variety of settlement costs (does not include the outstanding Loan balance) (3) For an uninsured position, Let: (a) B be the outstanding loan balance at default (b) C* be the lender's total settlement cost (c) R be the amount recovered on the sale of property (4) The mortgage loan (LOSS TO LENDER) represents a PUT with payoff max [ B + C* - R , 0 ] with strike B+C and R representing the underlying. (5) Note: B declines overtime and C depend on property conditions. R also depends on market conditions too. The lender has SHORT position since this is in the form of a loss to them. (6) If the value of the home (R) falls below the value of the mortgage, the borrower can relinquish the property and go into default (There are negative consequences though).

Static Hedging

(1) Purchasing options and holding them to expiration for hedging purposes. (2) Also known as hedge-to-forget strategies. (3) Even if review is done (i.e. sale/purchase of options in the event of lapse/more than assumed guarantees remaining in place), it is still considered to be static if original intention was to hold till expiration. (4) Exotic options usually used because of the variance and complexity of the guarantees

Rho: definition, Call and Put formulas

(1) Rho measures the change in the option's value corresponding to an increase in the risk-free interest rate or δV/δr (2) ρ_call = TK*e^(-rT) * N(d_2) ρ_put = -TK*e^(-rT) * N(-d_2) (3) ρ is + for calls and - for puts (same signs as delta) This is because as r increase, present value of strike decreases, increase and decreasing payoffs for calls and puts respectively. Derived from B.S. Formula Note: memory aid, both rho and risk-free rate start with r.

Sharpe ratio (definition for option and stock & relationship)

(1) Sharpe ratio of an asset is the ration of its risk premium to its volatility (2) Φ_stock = [α - r] / σ_stock Φ_option = [γ - r] / σ_stock (3) Φ_option = [Ω/|Ω|] * Φ_stock (4) since Ω is + for call and - for puts Φ_call = Φ_stock Φ_put = -Φ_stock

GMAB with a Return of Premium Guarantee

(1) Similar to GMDB with return of premium guarantee (2) However, contingent upon policy holder surviving to the end of the guarantee period (3) Payoff of the embedded put P(m) is max(0, K - Sm), that is either the beneficiary dies/Sm > K, otherwise K- Sm is paid out if Sm<K. (4) Expected cost of the guarantee is: = Pr(T*_x >= m) * P(m) where : (a) T*x is the future lifetime of the policy, different from T_x (future lifetime of policyholder) (b) m is the time to expiration

Elasticity and Sharpe ratio of a portfolio

(1) The elasticity of the portfolio can be calculated as the weighted average of the elasticity of the instruments in it. Ω_port = sum all i ___ w_i * Ω_i (2) Risk premium of a portfolio is also: γ_port - r = Ω_port * (α - r)

3 ways derivatives are used by companies with policy guarantees

(1) Transferring risk (2) Modifying the characteristics of an asset portfolio (3) Protecting or controlling a firm's surplus

Key assumption under delta-hedging and related problems for Exam IFM

(1) We assume that the initial cash outflow is zero. (2) Thus the market-maker either borrows or lends depending on the initial investment (paid or received)

Elasticity (4)

(1) Ω = %change in option price / %change in stock price (2) Relationship between and Ω and Δ Ω = Δ * (S/V) (3) Call bound: Ω_c >= 1 (from relationship above) (4) Put bound: Ω_p <= 0 (from relationship above) Note: that S and V are initial stock and option prices.

Approach 2 to calculating the profit (overnight) on a delta-hedged portfolio

1) Calculate gain on option ignoring interest 2) Calculate gain on stocks ignoring interest 3) Interest on borrowed/lent money Why does this work? this is because the a zero initial investment thus the profit is simply the gain on options & stocks + the interest on the money lent or borrowed

3 components of a delta-hedged portfolio

1. Buy or selling options 2. Buying or selling shares of stock 3. Borrowing or lending (selling/buying a risk-free zero-coupon bond)

list the four options (and 2 other) embedded in insurance products covered on Exam IFM

1. GMDB - A guaranteed minimum death benefit. 2. GMAB - A guaranteed minimum accumulation benefit. a.k.a guaranteed minimum maturity benefit 3. GMWB - A guaranteed minimum withdrawal benefit. 4. GMIB - A guaranteed minimum income benefit. 5. Guaranteed replacement cost coverage on property. 6. Inflation Indexing of Pension benefits.

What is the focus of the SOA study note, the "Actuarial Applications of Options and Other Financial Derivatives" ?

1. The focus of the study note is to describe how financial derivatives are used to manage insurance and investment products. 2. Actuaries must recognize when insurance and savings products contain embedded options or guarantees. 3. If present, the actuary should also be able to know how to use derivatives to manage the risk.

What does it mean for a position to be delta-hedged?

A delta hedged position has a delta of zero!

What is the difference between a traditional annuity and a variable annuity?

A traditional annuity provides the beneficiary, called the annuitant, payments over a specified period of time. However, a variable annuity is a variable savings product with a VARIABLE ACCUMULATION AMOUNT. Thus: 1) Variable annuity is not an annuity, It is instead, a variable savings product with the option to convert the accumulated amount to an annuity. 2) Since the accumulation amount is not known, payment amounts are not know in advance.

Note about short position signs

For short positions, just reverse the signs

Greeks for a portfolio of options

Greeks are additive That is for a portfolio of N options, where the quantity of each option is n_i, we have: Greek port = Sum_i_N { n_i * greek_i)

Delta-Gamma_Theta approximation

Let ε = S(t + h) - S(t) (1) The delta-gamma approximation is V(t+h) ≈ V(t) + {Δ* ε} + {0.5*Γ*ε²} (2)The delta-gamma-theta approximation is V(t+h) ≈ V(t) + {Δ* ε} + {0.5*Γ*ε²} + {Θ*h) Note: the approximation is based on the Taylor series f(x) = f(x_0) + f'(x_0)*(x-x_0) + 0.5*f''(x_0)*(x-x0)² Also make sure thet the units if Θ and h are the same!!!

Hedging Multiple Greeks

Note that all other stock Greeks beside delta are zero. Thus by necessity, it is required for the market maker to Simply set the sum of the Greeks you are hedging to zero. However, you must start will other Greeks and delta hedge last.

Rebalancing

Purchasing or selling additional share to get Δ_portfolio = 0 Come to down the following Δ_portolio = 0 = Δ_stock*(x+y) + Δ_option * (z) Where: x is already purchased or sold shares. y the shares needed to be bought or sold z the number of options already purchased or sold Signs are in cost perspective. You may also use cashflow perspective too.

Assuming the BS framework, given current stock price S(t), the two prices after period h for which the market-maker would breakeven are what?

S(t) +/- S*σ√h

Approach 1 to calculating the profit (overnight) on a delta-hedged portfolio

Simply calculate the profit on each of the thee components and add them up. 1) Profit on the options 2) Profit on the stock 3) Profit on the bond

Important note about where Greeks apply

Under BS Framework

What are the six Greeks and their descriptions:

Δ - Change in option price per increase in the stock price Γ - Change in delta per increase in the stock price Θ - change in option price per increase in the passage of time N/A - change in option price per increase in the volatility ρ - change in option price per increase in the risk-free rate Ψ - change in option price per increase in the dividend yield

Delta of a stock & All Other greeks of a stock

Δ_stock = 1 Greek_All Other = 0 This is a simple but very important fact to remember.

Relationship between volatility of stock and option

σ_option = |Ω| * σ_stock