• Understand and calculate the Greek Letters:
  • Delta ($\mathrm{\Delta }$) • Gamma ($\mathrm{\Gamma }$)
  • Vega ($\mathcal{V}$) • Rho ($\rho$) • Theta ($\mathrm{\Theta }$)
• Make a portfolio of options:
  • Delta neutral • Gamma neutral • Vega neutral
• Discuss the hedging of option positions:
  • Naked positions • Covered positions
  • Stop-loss strategies • Dynamic delta hedging
• Discuss the creation of synthetic options & static vs dynamic hedges

• Definition: Delta is the rate of change of the option price $f$ with respect to the underlying asset price $S$: $\mathrm{\Delta }=\frac{\partial f}{\partial S}$.
• European Calls and Puts on Non-Dividend Paying Assets:
  $${\mathrm{\Delta }}_{\text{call}}=N(d_1)>0,{\mathrm{\Delta }}_{\text{put}}=N(d_1)-1<0$$
• Delta Ranges:
  • Call Delta lies between $0$ and $+1$ ($\text{ATM}\approx 0.5$, deep ITM $\to 1$, deep OTM $\to 0$).
  • Put Delta lies between $-1$ and $0$ ($\text{ATM}\approx -0.5$, deep ITM $\to -1$, deep OTM $\to 0$).

• Slope Interpretation: Delta is the slope of the curve plotting option price against the underlying asset price $S$, where $\mathrm{\Delta }S$ and $\mathrm{\Delta }f$ are the changes in the asset price and in the option price $f$ (here $\mathrm{\Delta }$ in front of a symbol means "change in", while $\mathrm{\Delta }$ alone is the option's delta):
  $$\text{Slope of tangent line}=\mathrm{\Delta }=\underset{\mathrm{\Delta }S\to 0}{lim}\frac{\mathrm{\Delta }f}{\mathrm{\Delta }S}$$
  • Relate this to the previous graphs.
• Relation between Delta and Time to Maturity:
  • As time to maturity $T\to 0$, the delta curve steepens around the strike $K$:
    • For an ITM call ($S>K$): $\mathrm{\Delta }\to 1.0$
    • For an OTM call ($S<K$): $\mathrm{\Delta }\to 0.0$
    • For an ATM call ($S=K$): $\mathrm{\Delta }\approx 0.5$, transitioning like a step function.

• Delta measures the sensitivity of option value to small changes in the underlying asset price:
  • If the underlying price $S$ changes by $\mathrm{\Delta }S$, then:
    • Call value changes by approximately ${\mathrm{\Delta }}_c\times \mathrm{\Delta }S$
    • Put value changes by approximately ${\mathrm{\Delta }}_p\times \mathrm{\Delta }S=-|{\mathrm{\Delta }}_p|\times \mathrm{\Delta }S$
• Numerical Example:
  The stock price rises from $100 to $102 ($\mathrm{\Delta }S=+\mathrm{\$}2$).
  A call option on this stock has a delta of 0.5.
  The call value is expected to increase by: $$\mathrm{\Delta }\times \mathrm{\Delta }S=0.5\times \mathrm{\$}2=\mathrm{\$}1$$
• For deep ITM options: ${\mathrm{\Delta }}_{\text{call}}\approx 1,{\mathrm{\Delta }}_{\text{put}}\approx -1$.
• For deep OTM options: ${\mathrm{\Delta }}_{\text{call}}\approx 0,{\mathrm{\Delta }}_{\text{put}}\approx 0$.

• The delta of the spot contract (underlying asset itself) is $1$.
• The delta of a forward contract with maturity $T$ on an asset paying continuous dividend yield $q$:
  $${\mathrm{\Delta }}_{\text{forward}}=e^{-qT}(\text{if }q=0⟹{\mathrm{\Delta }}_{\text{forward}}=1)$$
• The delta of a futures contract with maturity $T$ on an asset paying continuous dividend yield $q$:
  $${\mathrm{\Delta }}_{\text{futures}}=e^{(r-q)T}$$
• Examples:
  • Index Futures: $\mathrm{\Delta }=e^{(r-q)T}$
  • Currency Futures: $\mathrm{\Delta }=e^{(r-r_f)T}$ (where $r_f$ is foreign risk-free rate)
  • Commodity (Gold/Silver) Futures: $\mathrm{\Delta }=e^{rT}$

• The delta of an option with maturity $T$ on an asset paying continuous dividend yield $q$:
  $${\mathrm{\Delta }}_{\text{call}}=e^{-qT}N(d_1),{\mathrm{\Delta }}_{\text{put}}=e^{-qT}[N(d_1)-1]$$
• Examples:
  • Index Options: ${\mathrm{\Delta }}_c=e^{-qT}N(d_1),{\mathrm{\Delta }}_p=e^{-qT}[N(d_1)-1]$
  • Currency Options: ${\mathrm{\Delta }}_c=e^{-r_{f}T}N(d_1),{\mathrm{\Delta }}_p=e^{-r_{f}T}[N(d_1)-1]$
  • Futures Options: ${\mathrm{\Delta }}_c=e^{-rT}N(d_1),{\mathrm{\Delta }}_p=e^{-rT}[N(d_1)-1]$
• where by choice of $q$ (or $r_f$ or $r$), the appropriate $d_1$ is used: $$d_1=\frac{\ln (S_0/K)+(r-q+\frac{1}{2}{\sigma }^2)T}{\sigma \sqrt{T}}$$

• Definition: Theta is the rate of change of the option price $f$ with respect to the passage of time $t$: $\mathrm{\Theta }=\frac{\partial f}{\partial t}$.
• Theta of a European call:
  $${\mathrm{\Theta }}_c=-\frac{S_0{N}^{\prime }(d_1)\sigma }{2\sqrt{T}}-rK{e}^{-rT}N(d_2)$$
  or, for an asset paying a continuous yield $q$:
  $${\mathrm{\Theta }}_c=-\frac{S_0{N}^{\prime }(d_1)\sigma e^{-qT}}{2\sqrt{T}}-rK{e}^{-rT}N(d_2)+q{S}_{0}N(d_1)e^{-qT}$$
  where $N^{\prime }(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$.
• Theta of a long option position is usually negative – as time passes, the option continually loses time value.
• Example: $S_0=100,K=100,r=0,\sigma =0.20,T=1$ (ATM call, $d_1=0.1$, $N^{\prime }(0.1)\approx 0.397$): $$\mathrm{\Theta }=-\frac{100\times 0.397\times 0.20}{2}=-3.97\text{ per year}$$ Equivalently $-3.97/252=-0.0158$ per trading day – about $0.016 of value lost per trading day.

• Relation between Theta and the underlying asset price:
  • Parameters: $K=100,r=0,\sigma =0.20,T=1$.
  • Theta is most negative (time value decays fastest) when the option is at-the-money ($\text{ATM},S\approx K=100$).
  • For deep ITM or deep OTM options, Theta approaches 0 (because the option price is dominated by intrinsic value, or the time value itself is tiny).
• Relation between Theta and time to maturity:
  • For ATM options, time decay accelerates sharply as expiration approaches ($T\to 0$).
  • Traders holding near-expiry ATM long options must watch out for severe daily time-value erosion.

• Definition: Gamma is the rate of change of Delta ($\mathrm{\Delta }$) with respect to the underlying asset price $S$ (second derivative of the option price with respect to $S$):
  $$\mathrm{\Gamma }=\frac{\partial \mathrm{\Delta }}{\partial S}=\frac{{\partial }^{2}f}{\partial S^2}$$
• Formula for European calls and puts (identical for both!):
  $$\mathrm{\Gamma }=\frac{N^{\prime }(d_1)}{S_0\sigma \sqrt{T}}\text{with yield }q:\mathrm{\Gamma }=\frac{N^{\prime }(d_1)e^{-qT}}{S_0\sigma \sqrt{T}}$$
  where $N^{\prime }(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$.
• If the underlying price moves by $\mathrm{\Delta }S$, Delta changes by approximately $\mathrm{\Gamma }\times \mathrm{\Delta }S$.
• Example: $S_0=100,K=100,r=0,\sigma =0.20,T=1$, giving $\mathrm{\Gamma }\approx 0.02$, $\mathrm{\Delta }\approx 0.54$.
  If the stock price rises from $100 to $110 ($\mathrm{\Delta }S=+\mathrm{\$}10$): $${\mathrm{\Delta }}_{\text{new}}\approx 0.54+(0.02\times 10)=0.54+0.20=0.74$$

• Moneyness Effect on Gamma:
  • Deep OTM: Delta $\approx 0$, changes very slowly with $S$ → $\mathrm{\Gamma }\approx 0$.
  • At-the-Money (ATM): Delta $\approx 0.5$, highly sensitive to small moves in $S$ → $\mathrm{\Gamma }$ reaches its maximum!
  • Deep ITM: Delta $\approx 1.0$, changes very slowly with $S$ → $\mathrm{\Gamma }\approx 0$.
• Relation between Gamma and Asset Price:

  The curve of Gamma against asset price forms a sharp bell-shaped peak close to the strike price (slightly below $K$).

• Hedgers must be especially attentive to portfolios with ATM options where Gamma risk is concentrated.
• Left chart: the same $2 move in the stock price changes the call's delta a lot only near the strike $K$; Gamma (right chart) is the slope of this delta curve.

• Relation between Gamma and time to maturity:
  • As $T\to 0$, ATM Gamma tends to infinity, concentrating extreme sensitivity at $S=K$.
• Gamma accounts for curvature in delta hedging:
  • If $\mathrm{\Gamma }$ is small: Delta changes slowly → infrequent rebalancing suffices.
  • If $\mathrm{\Gamma }$ is large: Delta is highly sensitive → leaving the hedge unadjusted causes severe tracking error!
• Curvature Gain Illustration:
  At $S=100$, the ATM call is $7.97 ($\mathrm{\Delta }=0.54$, $\mathrm{\Gamma }=0.02$). Stock rises to $110 ($\mathrm{\Delta }S=+10$):
  Linear Delta prediction: $7.97+0.54\times 10=\mathrm{\$}13.37$
  With Gamma correction: $13.37+\frac{1}{2}\times 0.02\times {10}^2=13.37+1.00=\mathrm{\$}14.37$
  Actual call price with curvature (BSM): $\mathrm{\$}14.29$
  The ~+$1 correction ($\frac{1}{2}\mathrm{\Gamma }\mathrm{\Delta }{S}^2$) explains most of the gap between $13.37$ and $14.29$!

In one line: delta hedging removes only the risk of small moves; what is left is a bet on big moves versus no moves.

Position	Delta	$\mathrm{\Gamma }$	$\mathrm{\Theta }$
Long Call	+	+	−
Long Put	−	+	−
Short Call	−	−	+
Short Put	+	−	+

$\mathrm{\Gamma }$ and $\mathrm{\Theta }$ always have opposite signs: protection against big moves ($\mathrm{\Gamma }$) is paid for by daily time decay ($\mathrm{\Theta }$).

Taylor expansion for a delta-neutral portfolio (portfolio Delta $=0$):

$\mathrm{\Pi }$ is the portfolio value (not a Greek). The $\mathrm{\Delta }$ in $\mathrm{\Delta }S$, $\mathrm{\Delta }t$, $\mathrm{\Delta }\mathrm{\Pi }$ means “change in”, not the Greek Delta. The portfolio is delta-neutral, so the first term is 0.

• Buy options and hedge ($\mathrm{\Gamma }>0$, $\mathrm{\Theta }<0$) ≈ buying insurance: no move ($\mathrm{\Delta }S=0$): pay a little $\mathrm{\Theta }$ each day; a big move up or down: the $\mathrm{\Gamma }$ gain exceeds that cost.
• Sell options and hedge ($\mathrm{\Gamma }<0$, $\mathrm{\Theta }>0$) ≈ selling insurance: quiet days: earn $\mathrm{\Theta }$; a big move: a loss, larger the bigger the move.

One trading day ($\mathrm{\Delta }t=1/252$), from the previous slide, where $\mathrm{1,000}\times \mathrm{\Theta }\mathrm{\Delta }t=-15.75$ and $\mathrm{1,000}\times \frac{1}{2}\mathrm{\Gamma }=9.92$:

• No move: the insurance buyer loses 15.75 (a trading day of $\mathrm{\Theta }$); the seller earns 15.75.
• Break-even: $-15.75+9.92(\mathrm{\Delta }S{)}^2=0\Rightarrow \mathrm{\Delta }S\approx \pm 1.26=\sigma S\sqrt{\mathrm{\Delta }t}$, one standard deviation of a trading day's move.

• Definition: Vega is the rate of change of option price $f$ with respect to the volatility $\sigma$ of the underlying asset: $\mathcal{V}=\frac{\partial f}{\partial \sigma }$.
• Formula for European Call and Put (identical for both):
  $$\mathcal{V}=S_0\sqrt{T}{N}^{\prime }(d_1)\text{or with dividend }q:\mathcal{V}=S_0\sqrt{T}{N}^{\prime }(d_1)e^{-qT}$$
  where $N^{\prime }(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$.
• Vega is strictly positive for both long calls and long puts.
• Interpretation: If volatility $\sigma$ increases by 1% (0.01), the option price increases by approximately $0.01\times \mathcal{V}$.
• Relation between Vega and asset price (chart below): Vega is highest for at-the-money options and drops off for deep ITM/OTM options.

• Definition: Rho is the rate of change of option price $f$ with respect to the risk-free interest rate $r$:
  $$\rho =\frac{\partial f}{\partial r}$$
• Formulas for European Options:
  $${\rho }_{\text{call}}=KT{e}^{-rT}N(d_2)>0,{\rho }_{\text{put}}=-KT{e}^{-rT}N(-d_2)<0$$
• Interpretation: If interest rate $r$ increases by 1% (0.01), option price changes by approximately $0.01\times \rho$.
• Currency Options: There are two rho parameters:
  • Domestic Rho (${\rho }_{\text{domestic}}$) with respect to domestic rate $r$.
  • Foreign Rho (${\rho }_{\text{foreign}}$) with respect to foreign rate $r_f$.

• The Greek letters of a portfolio of options are weighted sums of the individual Greeks (the portfolio has $n$ options; $w_i$ = position in the $i$-th option):
  $${\mathrm{\Delta }}_P=\sum _{i=1}^n{w}_i{\mathrm{\Delta }}_i,{\mathrm{\Gamma }}_P=\sum _{i=1}^n{w}_i{\mathrm{\Gamma }}_i,{\mathcal{V}}_P=\sum _{i=1}^n{w}_i{\mathcal{V}}_i$$
• Delta Neutrality (${\mathrm{\Delta }}_P=0$): Can be achieved using positions in the underlying spot asset or in futures contracts (${\mathrm{\Gamma }}_{\text{spot}}=0,{\mathcal{V}}_{\text{spot}}=0$).
  • If $w_F$ is futures position and $w_S$ is spot position: $w_F=e^{-(r-q)T^{\ast }}{w}_S$, where $T^{\ast }$ is the maturity of the futures contract.
• Gamma & Vega Neutrality (${\mathrm{\Gamma }}_P=0,{\mathcal{V}}_P=0$):
  • The underlying asset and futures have $\mathrm{\Gamma }=0$ and $\mathcal{V}=0$.
  • Therefore, to alter Gamma or Vega, traded options must be added to the portfolio!

An OTC option portfolio has $\mathrm{\Delta }=-500$, $\mathrm{\Gamma }=-100$. A traded option has ${\mathrm{\Delta }}_T=0.5,{\mathrm{\Gamma }}_T=0.02$. Find positions in the option ($w$) and in units of spot ($w_S$) for joint Gamma and Delta neutrality.

• Neutrality conditions (spot has $\mathrm{\Gamma }=0$):
  $$\mathrm{\Gamma }:-100+w\times 0.02+w_S\times 0=0$$ $$\mathrm{\Delta }:-500+w\times 0.5+w_S\times 1=0$$
• Gamma gives $w=\frac{100}{0.02}=5,000$ options; Delta then gives $w_S=-(-500+5,000\times 0.5)=-2,000$ units.
• Hedge: Buy 5,000 traded options; short 2,000 units of spot.

An OTC option portfolio has $\mathrm{\Delta }=-200,\mathrm{\Gamma }=-100,\mathcal{V}=-120$. Two traded options are available:

• Option 1: ${\mathrm{\Delta }}_1=0.5,{\mathrm{\Gamma }}_1=0.02,{\mathcal{V}}_1=0.02$
• Option 2: ${\mathrm{\Delta }}_2=0.1,{\mathrm{\Gamma }}_2=0.01,{\mathcal{V}}_2=0.02$

Let $w_1$ be the number of option 1 added, $w_2$ of option 2, and $w_S$ the units of spot. Full neutrality in all three dimensions requires:

Key logic: first solve the two traded options simultaneously to achieve Gamma and Vega neutrality, then use the underlying spot asset to eliminate the remaining Delta!

Solve the simultaneous linear equations:

• Second minus first: $0.01w_2=20$; back-substitute: $0.02w_1=100-20=80$: $$w_2=2,000\text{ of Option 2},w_1=4,000\text{ of Option 1}$$
• Delta equation (options contribute $2,200$): $$-200+2,200+w_S=0⟹w_S=-2,000$$
• Hedge: Buy 4,000 Option 1 and 2,000 Option 2; short 2,000 units.
• Delta neutrality provides protection against small price moves, while gamma neutrality provides protection against large price moves between hedge rebalancing.

• Daily Delta Hedging: Traders usually ensure that their books and portfolios are delta-neutral at least once a day (often dynamically rebalanced in real time using algorithmic execution).
• Opportunistic Gamma and Vega Management: Whenever the opportunity arises and transaction costs allow, traders trade exchange-traded options to improve gamma and vega neutrality.
• Economies of Scale: As a portfolio becomes larger, hedging becomes relatively less expensive because fixed transaction costs and bid-ask overheads are spread over a much larger notional volume.

A financial institution sold 1,000 European calls on a non-dividend-paying stock at $2.50 each (1 share per option), receiving $2,500.

• Parameters: $S_0=100,K=100,r=0,\sigma =20\mathrm{\%},T=5\text{ weeks}$.
• Black-Scholes price: $2.47 each, $2,470 total; initial theoretical profit: $30.
• How to hedge the short exposure?
  1. Naked position: Do nothing.
  2. Covered position: Buy 1,000 shares immediately.
  3. Stop-loss strategy: Buy stock above $100; sell below $100.
  4. Dynamic delta hedging: Rebalance the stock holding to stay delta neutral.

• With a naked short call, the institution takes no hedging action and simply keeps the initial premium of $2,500.
• P&L Profile at Expiration:
  • If $S_T\le 100$: the option is not exercised and the institution keeps the entire premium $=+\mathrm{\$}2,500$.
  • If $S_T>100$: the option is exercised by the counterparty and the institution faces unlimited upside loss potential!
• Numerical Example: Suppose $S_T=\mathrm{\$}110$:
  $$\text{P/L}=(K-S_T)\times 1,000+\text{option premium}$$ $$=(\mathrm{\$}100-\mathrm{\$}110)\times 1,000+\mathrm{\$}2,500=-\mathrm{\$}7,500$$
• Conclusion: Writing naked calls exposes the institution to uncontrollable, large upside market risk.

• Covered Call Strategy (see Lecture 7): Sell 1,000 call options + buy 1,000 shares at $S_0=\mathrm{\$}100$ (initial cash outlay $=\mathrm{\$}100,000$).
• P&L Profile:
  • If $S_T\ge 100$: the stock is delivered and sold at the strike $K=100$, P/L $=(100-100)\times 1,000+\mathrm{\$}2,500=+\mathrm{\$}2,500$.
  • If $S_T<100$: the calls expire worthless, but the stock held suffers a severe price loss!
• Numerical Example: Suppose $S_T=\mathrm{\$}90$:
  $$\text{P/L}=(S_T-S_0)\times 1,000+\text{option premium}$$ $$\text{P/L}=(-\mathrm{\$}10)\times 1,000+\mathrm{\$}2,500=-\mathrm{\$}7,500$$
• Conclusion: Writing a covered call merely converts upside risk into severe downside risk – by put-call parity, this is essentially equivalent to writing a naked put!

• Strategy Mechanics:
  • As soon as the stock price rises above the strike $K=\mathrm{\$}100$, immediately buy 1,000 shares.
  • As soon as the stock price falls below the strike $K=\mathrm{\$}100$, immediately sell 1,000 shares.
• Theoretical Intuition: Hold the stock when the option is in-the-money (a covered position), and hold cash when it is out-of-the-money (avoiding stock price declines).
• Why It Is Bound to Fail in Practice:
  • Whipsaw Losses: When the stock price oscillates around $100, the trader keeps buying high ($100.10) and selling low ($99.90) — each round trip loses $0.20 per share, i.e. $200 on 1,000 shares, rapidly bleeding cash!
  • Gapping & Slippage: Market prices jump discretely; purchases and sales cannot be executed at exactly $100.00.
  • Time Value of Money: Cash flows occur at different points in time and must be discounted.

• Theoretical Cornerstone of the BSM Model: A portfolio of $\mathrm{\Delta }$ units of the underlying asset and 1 short unit of the derivative is locally, instantaneously risk-free:
  $$\mathrm{\Pi }=\mathrm{\Delta }S-f⟹d\mathrm{\Pi }=r\mathrm{\Pi }dt$$
• Here $\mathrm{\Delta }$ is the optimal hedge ratio. It is calculated as the change in value of the derivative divided by the change in stock price (the delta of the option).
• The institution hedges the short exposure from selling 1,000 calls by holding $1,000\times \mathrm{\Delta }$ shares (e.g. 500 shares when $\mathrm{\Delta }=0.5$).
• Futures contracts can also be used for delta hedging, see slide 16: $w_F=e^{-(r-q)T^{\ast }}{w}_S$.
• Why Must the Hedge Be “Dynamic”?
  • The option delta $\mathrm{\Delta }$ is a continuous function of the underlying price $S$ and time $t$.
  • As the stock price moves and time passes, $\mathrm{\Delta }$ changes continually.
  • The position must be rebalanced periodically or continuously to maintain delta neutrality.

In week 5, the option expires in-the-money ($S_5=105>100$). The buyer exercises: 1,000 shares are delivered at $K=100$, and $\mathrm{\$}100,000$ is received.
Net total cost of dynamic hedging $=\mathrm{\$}102,500-\mathrm{\$}100,000=\mathrm{\$}2,500\approx \mathrm{\$}2,470$ (theoretical BS price).

In week 5, the option expires out-of-the-money and worthless ($S_5=96<100$). No delivery is required; the stock position had already been fully sold off before expiration as Delta went to zero.
Net total cost of dynamic hedging $=\mathrm{\$}2,300\approx \mathrm{\$}2,470$ (theoretical BS price).

• The present value of the total cost of dynamically hedging the option is approximately equal to its Black-Scholes theoretical value (the two simulations above gave $2,500 and $2,300, both scattered around $2,470; averaged over many simulations the cost converges to the theoretical value).
• The dynamic hedge portfolio creates a synthetic long position in the option, exactly offsetting the option writer's short exposure.
• Rebalancing Frequency Trade-off: Rebalancing more frequently reduces the tracking error and the variability of hedging cost, but significantly increases cumulative transaction costs.
• Static Hedging with Forward Contracts:
  • Since a forward contract has a constant delta of 1 ($\mathrm{\Delta }=1$), a portfolio of long 1 share of the underlying stock and short 1 forward contract requires no rebalancing at all!
  • Forward contracts allow “hedge and forget” (static hedging). Options, with their inherent nonlinear curvature, inherently rely on dynamic delta hedging.