Revise | Dr. Phil's Quant Lab

What are Derivatives?

Value is derived from an **underlying asset** or rate

Hedging vs Speculation

\sigma_{\text{hedged}}^2 = \sigma_S^2 + h^2 \sigma_F^2 - 2 h \rho \sigma_S \sigma_F

**Hedging** reduces variance of an existing exposure

Exchange-traded vs OTC

**Exchange**: standardised, listed, CCP-cleared, transparent

Counterparty Risk

EAD_t = \max(V_t, 0)

**Risk the other side defaults** before final settlement

Clearing House

\text{Exposure}_i = V_i^{\text{CCP}}

**Central counterparty** (CCP) to every cleared trade

Novation

(A \leftrightarrow B) \;\longrightarrow\; (A \leftrightarrow \text{CCP}) \;\text{and}\; (\text{CCP} \leftrightarrow B)

**Extinguishes** the original contract, creates two new ones

Futures Contract Basics

F_0 = S_0 e^{(r - q) T}

**Standardised** exchange-traded contracts with central clearing

Long vs Short Futures

\Pi_{\text{long}} = S_T - F_0

**Long** profits when $S_T > F_0$, payoff $= S_T - F_0$

Futures Payoff

\Pi_{\text{long}} = S_T - F_0

Linear payoff with **no premium** and no floor or cap

Cost of Carry

F_0 = S_0 e^{(r - q) T}

Net carry $= r$ + storage $-$ income $-$ convenience yield

Continuous Compounding Futures Pricing

F_0 = S_0 e^{rT}

$F_0 = S_0 e^{rT}$ for a non-dividend asset

Discrete Compounding Futures Pricing

F_0 = S_0 (1 + r)^T

$F_0 = S_0 (1 + r)^T$ for annual compounding

Arbitrage in Futures

F_0 = S_0 e^{(r - q) T}

**Cash-and-carry** sells overpriced futures and buys spot

Convergence of Futures to Spot

F_T = S_T

$F_t \to S_t$ as $t \to T$, enforced by arbitrage

Effective Price

P_{\text{eff}} = S_2 + (F_1 - F_2) = F_1 + (S_2 - F_2) = F_1 + b_2

$P_{\text{eff}} = F_1 + b_2$ for short hedge

Initial Margin

L = \frac{\text{Notional}}{\text{Initial Margin}}

Refundable **performance bond** posted with the clearing house

Maintenance Margin

\text{Account Balance} < \text{Maintenance Margin}

Minimum balance before a **margin call** is triggered

Variation Margin

VM_t = (F_t - F_{t-1}) \times \text{multiplier} \times N

Daily cash flow equal to change in settlement price times multiplier times contracts

Marking to Market

\text{P\&L}_t = (F_t - F_{t-1}) \times \text{multiplier} \times N

Daily revaluation of positions at the official **settlement price**

Margin Call

\text{Account Balance} < \text{Maintenance Margin}

Triggered when balance falls below **maintenance margin**

Closing Out a Position

N_{\text{net}} = N_{\text{long}} - N_{\text{short}} = 0

Exit by entering an equal and **opposite** trade in the same contract

Open Position at Close

\Delta OI = \text{new opening positions} - \text{closing positions}

**Open interest** counts all contracts still outstanding

Tick Value

\text{Tick Value} = \text{Tick Size} \times \text{Contract Multiplier}

Tick size is the **minimum price increment** for a contract

Interest Rate Futures

P = 100 - Y

**Exchange-traded** rate contracts

Bank Bill Futures

P = 100 - Y

Underlying, **90-day Prime Bank bill**

Bond Futures

P = 100 - Y_{\text{bond}}

**Notional bond**, A$100k face, 6% coupon

100 minus Yield Quoting

Q = 100 - Y

$Q = 100 - Y$

Forward Rate Agreement (FRA)

\text{Settlement} = \frac{(R_{\text{market}} - R_{\text{FRA}}) \times \text{NP} \times \frac{d}{B}}{1 + R_{\text{market}} \times \frac{d}{B}}

**OTC**, cash-settled rate contract

FRA vs Futures

R_{\text{fwd}} \approx R_{\text{fut}} - \tfrac{1}{2}\sigma^2 T_1 T_2

**FRA**, OTC, custom, settled at start

Options Basics

\text{Payoff}_{\text{call}} = \max(S_T - K, 0)

**Right not obligation** for the buyer of an option

Call Option

\text{Payoff} = \max(S_T - K, 0)

**Right to buy** at strike $K$

Put Option

\text{Payoff} = \max(K - S_T, 0)

**Right to sell** at strike $K$

Premium

\text{Premium} = \text{Intrinsic Value} + \text{Time Value}

Premium $=$ **intrinsic value** $+$ **time value**

American vs European Options

C_{\text{Am}} \geq C_{\text{Eu}} \quad \text{and} \quad P_{\text{Am}} \geq P_{\text{Eu}}

**American**: exercise anytime up to $T$

ITM / ATM / OTM / DITM

\text{ITM: } S > K, \quad \text{ATM: } S \approx K, \quad \text{OTM: } S < K

Call **ITM** when $S > K$. Put **ITM** when $S < K$

Intrinsic Value

\text{IV}_{\text{call}} = \max(S - K, 0)

Call: $\max(S - K, 0)$

Time Value

\text{Time Value} = \text{Premium} - \text{Intrinsic Value}

Time value $=$ premium $-$ intrinsic value

Break-even

S_T^* = K + C

**Call BE** $= K + C$

Long Call

\text{Payoff} = \max(S_T - K, 0)

**Buy** a call to pay $C$ for the right to buy at $K$

Short Call

\text{Profit} = C - \max(S_T - K, 0)

**Sell** a call to collect premium $C$

Long Put

\text{Payoff} = \max(K - S_T, 0)

**Buy** a put to pay $P$ for the right to sell at $K$

Short Put

\text{Profit} = P - \max(K - S_T, 0)

**Sell** a put to collect premium $P$

Long Straddle

\text{Payoff} = \max(S_T - K, 0) + \max(K - S_T, 0) = |S_T - K|

Buy call $+$ put, same $K$ and same $T$

Short Straddle

\text{Profit} = (C + P) - |S_T - K|

Sell call $+$ put, same $K$ and same $T$

Option Combinations and Spreads

\text{Payoff} = \max(S_T - K_2, 0) + \max(K_1 - S_T, 0), \quad K_1 < K_2

Payoff of any position $=$ sum of its **leg payoffs**

4 pitfallsquiz

Synthetic Stock Position

C - P \equiv S_0 - K e^{-rT}

**Long call + short put** at same $K$, $T$ replicates stock

Synthetic Call

S_0 + P \equiv C + K e^{-rT}

**Long stock + long put** at strike $K$ = synthetic long call

Put-Call Parity

Put-Call Parity Equation

C + K e^{-rT} = P + S_0

$C + K e^{-rT} = P + S_0$ (European only)

Put-Call Parity

Arbitrage when Parity is Violated

C + K e^{-rT} < P + S_0 \Rightarrow \text{Buy call, lend } K e^{-rT}, \text{ sell put, short stock}

Parity violation $\Rightarrow$ **riskless arbitrage profit**

One-step Binomial Tree

p = \frac{e^{rT} - d}{u - d}

Stock moves to $S_0 u$ or $S_0 d$ at expiry

Two-step Binomial Tree

f_0 = e^{-2r\Delta t}\bigl[p^2 f_{uu} + 2p(1-p) f_{ud} + (1-p)^2 f_{dd}\bigr]

**Three terminal prices** $S_0 u^2$, $S_0 ud$, $S_0 d^2$

Risk-neutral Probability

p = \frac{e^{r\Delta t} - d}{u - d}

**Risk-neutral probability** $p = (e^{r\Delta t} - d)/(u - d)$

Backward Induction

f_i = e^{-r\Delta t}\bigl[p\, f_{i,u} + (1-p)\, f_{i,d}\bigr]

**Start at terminal payoffs**, walk back to root

American Early Exercise Logic

f_i = \max\bigl(\text{intrinsic}_i,\; e^{-r\Delta t}[p f_{i,u} + (1-p) f_{i,d}]\bigr)

**Compare intrinsic** against **continuation value** at every node

Delta

\Delta = \frac{f_u - f_d}{S_u - S_d}

**Binomial delta** $\Delta = (f_u - f_d)/(S_u - S_d)$

Delta Hedging

\Pi = -1 \text{ option} + \Delta \cdot \text{shares}

Hold **$\Delta$ shares per option** to be delta-neutral

Dividend-adjusted Binomial Tree

p = \frac{e^{(r-q)\Delta t} - d}{u - d}

**Continuous yield** $q$: use $r - q$ in $p$ formula

The Greeks Dashboard

\Gamma = \frac{\varphi(d_1)}{S\,\sigma\,\sqrt{T}}

**Five Greeks**: $\Delta$ (spot), $\Gamma$ (delta's change), $\mathcal{V}$ (vol), $\Theta$ (time), $\rho$ (rate)

Black-Scholes-Merton Intro

C = S_0 N(d_1) - K e^{-rT} N(d_2)

**Closed-form price** for European options

Normal Distribution Intuition

N(x) = P(Z \leq x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-z^2/2}\, dz

**Standard normal** has mean 0 and variance 1

Z-table and $N(x)$

N(z) \text{ from table where row} = z \text{ to first decimal, column} = z \text{ to second decimal}

**Pre-computed** values of $N(x)$

$d_1$ and $d_2$

d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}

$d_1 = [\ln(S_0/K) + (r + \sigma^2/2)T] / (\sigma\sqrt{T})$

European Call Price

C = S_0 N(d_1) - K e^{-rT} N(d_2)

$C = S_0 N(d_1) - K e^{-rT} N(d_2)$

European Put Price

P = K e^{-rT} N(-d_2) - S_0 N(-d_1)

$P = K e^{-rT} N(-d_2) - S_0 N(-d_1)$

Dividend-adjusted BSM

C = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2)

Replace $S_0$ with $S_0 e^{-qT}$ for **continuous yield**

Implied Volatility

C_{\text{market}} = \text{BSM}(S_0,\, K,\, r,\, T,\, \sigma_{\text{imp}})

$\sigma$ that makes **BSM = market price**

Money Market

P = \frac{FV}{1 + y \times d/365}

**Short-term debt** with maturity of one year or less

T-bills and Bank Bills

P = \frac{FV}{1 + y \times d/365}

Sold at a **discount**, redeemed at face value

Commercial Paper

P = FV \left(1 - r \times \frac{d}{360}\right)

**Unsecured** corporate short-term IOU

Certificates of Deposit

P = \frac{FV + I}{1 + y \times d/365}

**Bank-issued time deposit** with fixed term and rate

Money Market Fund

WAM = \sum_{i=1}^{n} w_i \times t_i

**Pooled** short-term debt investment

Bank Bill Pricing Formula

P = \frac{FV}{1 + y \times d/365}

$P = FV/(1 + y \times d/365)$ for AUD bills

Yield Curve Types

(1 + r_2)^2 = (1 + r_1)(1 + f_{1,2})

**Normal** curve slopes upward (most common)

Zero Rates

P = FV \times e^{-r_T T}

Yield on a **single future payment**, no coupons

Forward Interest Rates

f_{T_1, T_2} = \frac{r_2 T_2 - r_1 T_1}{T_2 - T_1}

**Implied** rate for a future period from today's zero curve

Bond Pricing

P = \sum_{t=1}^{n} \frac{C}{(1+y)^t} + \frac{FV}{(1+y)^n}

Price equals **present value of all future cash flows**

Duration

D = \frac{1}{P}\sum_{t=1}^{n} t \cdot \frac{CF_t}{(1+y)^t}

**Macaulay duration** is the PV-weighted average time to cash flows, in years

Convexity

C = \frac{1}{P}\sum_{t=1}^{n} \frac{t(t+1)\,CF_t}{(1+y)^{t+2}}

**Convexity** is the curvature of the price-yield curve (second order)

Coupon, Yield, and Accrued Interest

AI = C \times \frac{\text{days since last coupon}}{\text{days in coupon period}}

**Coupon rate** is annual coupon as $\%$ of face value

Settlement Date and Accrued Interest

\text{Amount} = \text{Face} \times \frac{P_{\text{clean}} + AI}{100}

Trade date $\to$ settlement date (T+1 for AGS and UST, T+2 for most corporates)

Hedgers vs Speculators

**Hedgers** reduce existing risk in the cash market

Short Hedge

P_{\text{eff}} = S_2 + (F_1 - F_2) = F_1 + b_2

**Sell futures** today, lift them at the close date

Long Hedge

P_{\text{eff}} = S_2 - (F_2 - F_1) = F_1 + b_2

**Buy futures** today, sell them at the close date

Portfolio Beta Hedging

N^* = \beta \times \frac{V_P}{V_F}

Use **index futures** to hedge market exposure

Changing Portfolio Beta with Futures

N^* = (\beta^* - \beta) \times \frac{V_P}{V_F}

**Long futures** raise beta

Basis

b = S - F

**Basis** $= S - F$ in the Hull convention

Basis Risk

P_{\text{eff}} = F_1 + b_2

Risk that **basis $b_2$** at close differs from forecast

Cross Hedging

h^* = \rho \times \frac{\sigma_S}{\sigma_F}

Hedge with a **correlated but different asset**

Minimum Variance Hedge Ratio

h^* = \rho_{SF} \times \frac{\sigma_{\Delta S}}{\sigma_{\Delta F}}

**$h^* = \rho\,\sigma_S / \sigma_F$** minimises hedged variance

Tailing the Hedge

N_{\text{naive}} = h^* \times \frac{Q_A}{Q_F}

Adjusts for **daily mark-to-market** cash flows

Fixed vs Floating Rate

C = N \times (\text{BBSW} + s) \times \frac{d}{365}

**Fixed**: constant coupons, certainty on cash flow

LIBOR / BBSW Concept

C = N \times (r_{\text{benchmark}} + s) \times \frac{d}{365}

**LIBOR retired**: USD LIBOR ended 30 June 2023

Interest Rate Swap Mechanism

\text{Net} = N \times (r_{\text{fix}} - r_{\text{float}}) \times \frac{d}{365}

**Exchange fixed for floating** payments on a notional

Comparative Advantage in Swaps

\text{QSD} = |(r^A_{\text{fix}} - r^B_{\text{fix}}) - (r^A_{\text{float}} - r^B_{\text{float}})|

**Total gain** = |fixed gap − floating gap| = QSD

Direct vs Intermediated Swap

M = N \times (r^A_{\text{fix-paid}} - r^B_{\text{fix-received}}) \times \frac{d}{365}

**Direct**: bilateral, full QSD split between parties

Bank Fee and Each Leg of the Swap

\text{Fee} = \text{QSD} - \text{Saving}_A - \text{Saving}_B

**Fixed leg** and **floating leg** make up the two sides of the swap

Revision

Common Formulas

F_0 = S_0 e^{rT}

**Forward price**: $F_0 = S_0 e^{rT}$ with carry adjustments

Revision

Common Payoff Shapes

\pi_{\text{LC}} = \max(S_T - K, 0) - C

**Futures**: straight 45-degree lines through $F_0$

Revision

Exam Hotspots

P_{\text{eff}} = F_1 + b_2

**Parity arbitrage**: identify which side is cheap, list the four trade legs