Derivatives Flashcards
(52 cards)
How are equity forward prices calculated (considering dividends)?
Ft = FV(St) − FV(Dividends) = (St−PV(Dividends))×(1+Rf)^T−t
Price is the future value of the spot price minus the future value of dividends paid during the contract life.
How is the value of an existing equity forward contract calculated?
Vt(long)=St−PV(Dividends)−PV(F0)=(St−PV(Dividends))−(1+Rf)T−tF0.
Value is the current spot (adjusted for PV of dividends) minus the present value of the original forward price. Alternatively, Vt=PV(Ft−F0).
How are bond forward/futures prices calculated (considering coupons)?
Price F0=FV[B0+AI0−PV(CI)], where B0 is quoted clean price, AI0 is accrued interest at initiation, PV(CI) is present value of coupons during contract life. The price represents the future value of the full bond price less the future value of coupon income. For futures, adjust quoted price by conversion factor (CF) for cheapest-to-deliver bond: QFP=(F0−AIT)/CF.
How is the value of an existing bond forward contract calculated?
Vt(long)=(Bt+AIt)−PV(CI)−PV(F0). Value is the current full price adjusted for PV of remaining coupons minus the PV of the original forward price.
What is a Forward Rate Agreement (FRA) and how is it priced?
An agreement to lock in an interest rate (like MRR) for a future period. The FRA fixed rate (F0) is the implied forward rate derived from the current yield curve (spot MRR rates) for the relevant periods. F0=[(1+Short Rate×360Short Days)(1+Long Rate×360Long Days)−1]×Loan Days360.
How is an FRA settled and valued?
Settled in cash at FRA expiry (loan start date) based on the difference between the market MRR at expiry (ST) and the FRA rate (F0). Payoff =[( ST - F0) * (Days / 360) * NP] / [1 + ST * (Days / 360)] Value before expiry Vt=PV(Payoff using new FRA rate).
How is the fixed rate on a plain vanilla interest rate swap determined?
The fixed rate is set such that the PV of the fixed leg equals the PV of the floating leg at initiation (making swap value zero).
It’s calculated like a par bond coupon: C=∑DFs1−Final DF×Periodicity.
Discount factors (DFs) are derived from the appropriate MRR curve.
Valuation of pay-fixed receive floating interest rate swap
NA * (FS0 - FRt) * ∑DFs
Important drill in brain
How is an equity swap priced and valued?
Fixed rate is determined same way as interest rate swap. Value for Fixed Payer = Vequity−Vfix. Vfix calculated using original fixed rate and current DFs. Vequity resets to (NP * Index value at t / Index value at t-1) at each reset date.
How is a currency swap priced and valued?
Two separate bonds in different currencies. Fixed rate for each currency leg is determined using the respective currency’s yield curve (like pricing a par bond). Notional principal is exchanged at start and end.
Value = PV(Foreign Bond) * Spot Rate - PV(Domestic Bond).
Value of currency swap formula
𝑉𝐶𝑆=
𝑁𝐴𝑎 [𝐴𝑃×𝑟𝐹𝑖𝑥,𝑎∑𝑛𝑖=1𝑃𝑉𝑖(1)+𝑃𝑉𝑛(1)]
−𝑆𝑡 𝑁𝐴𝑏 [𝐴𝑃×𝑟𝐹𝑖𝑥,𝑏∑𝑛𝑖=1𝑃𝑉𝑖(1)+𝑃𝑉𝑛(1)]
Key steps, find notional amounts expressed in both currencies
Accrual period * Fixed rate * Sum discounts + Final discount
Explain the one-period binomial option valuation model.
Assumes stock price can move to one of two possible values (up or down) in one period. Value derived by constructing a replicating portfolio (long stock + short bond or vice versa) that has the same payoffs as the option in both states. Option value is the cost of this portfolio. Uses risk-neutral probabilities (π). C0=PV[πCu+(1−π)Cd].
What are the assumptions of the Black-Scholes-Merton (BSM) model?
Underlying price follows geometric Brownian motion (lognormal returns), risk-free rate & volatility are constant & known, no dividends (or constant yield), no transaction costs/taxes, continuous trading possible, European options.
What are the components of the BSM formula for a call option: C0=S0N(d1)−e−rTXN(d2)?
S0N(d1) is PV of expected stock value if option exercised (leveraged stock position). e−rTXN(d2) is PV of exercise price payment if option exercised (PV of loan repayment). N(d1) relates to delta; N(d2) is risk-neutral probability of option finishing in-the-money.
How is the BSM model adapted for dividends, currencies, and futures (Black Model)?
Dividends (Continuous Yield γ): Replace S0 with S0e−γT. Use (r−γ) in d1. Currencies (Foreign Rate rf): Replace S0 with S0e−rfT. Use (r−rf) in d1. Futures (Black Model): Replace S0 with PV(F0)=F0e−rT. No adjustment needed in d1 (cost of carry is in futures price). Formula: C0=e−rT[F0N(d1)−XN(d2)].
Define the main Option Greeks (Delta, Gamma, Vega, Theta, Rho).
Delta (Δ): Change in option price per 1changeinunderlyingprice[cite:660].∗∗Gamma(\Gamma$):** Change in Delta per 1changeinunderlyingprice(curvature)[cite:663].∗∗Vega:∗∗Changeinoptionpriceper1\Theta$):** Change in option price per 1 day decrease in time to expiry (time decay). Rho (ρ): Change in option price per 1% change in risk-free rate.
Delta
Delta_call = e–δTN(d1), where δ=dividend
Delta_put = –e–δTN(–d1)
Static term, by its nature always between 0-1 for calls, -1-0 for puts.
Delta * Hedging_unit = portfolio delta
Delta approximation
cˆ= c + Delta_c * (Sˆ−S)
Gamma
n(d1) * e^(−δT) / S𝜎√T = gamma_call = gamma_put
Always non-negative, highest value when option is near the money
To get gamma neutral portfolio, get desired option portfolio. Then we adjust the exposure to the underlying (stock) to get the right delta, because this has no gamma but delta equal to one
Gamma approximation
cˆ= c + Delta_c * (Sˆ−S) + Gamma_c/2 * (Sˆ−S)^2
Vega
Vega positive, equal for calls and puts
Highest when option is near the money
Theta
Theta typically negative for options
Becomes more negative as we go closer towards 0
Rho
Positive for calls, negative for puts
Intuition is that this is the money that someone who owns a call can receive by not having to finance the underlying
How is a delta hedge constructed and maintained?
Offset the delta of an option position by taking an opposite position in the underlying asset. Number of shares = Option Delta * Number of options. Hedge ratio needs to be dynamically adjusted (rebalanced) as delta changes with underlying price and time, due to Gamma risk.
