Options Theory Dashboard

Frankfurt School - Trading & Sales

Dashboard
Black-Scholes
Greeks
Butterfly
Hedging

Options Chain Dashboard

Live pricing and Greeks for multiple strikes - similar to trading terminal views

Strike Call Price Put Price Delta (Call) Gamma Vega Theta (Call)
Spot Price
$100
ATM Call
$0.00
ATM Put
$0.00
Days to Expiry
91

Black-Scholes-Merton Formula

Implied Volatility (Slide 34)

What is Implied Volatility?
Implied volatility measures the market's expectation of how much an asset will move over a given period. It is the level of volatility σ that, when inserted into the Black-Scholes-Merton formula, matches the value of a traded option.

Uses of Implied Volatility:
• Gauge general market uncertainty about future returns
• Higher uncertainty ⇒ higher option prices ⇒ higher implied volatility
• Forward-looking measure (augments historical volatility)
• Compare relative pricing of options with different strikes/maturities

Implied Volatility Surface:
The 3D surface below shows how implied volatility varies across different strike prices and spot prices, helping traders identify relatively expensive or cheap options.

Call Price
$0.00
Put Price
$0.00
d₁
0.0000
d₂
0.0000

Delta (Δ)

∂C/∂S = e^(-qT) × N(d₁)
Call Delta = 0.0000
Put Delta = 0.0000

Gamma (Γ)

∂²C/∂S² = φ(d₁)e^(-qT) / (S×σ×√T)
Gamma = 0.0000

Vega (ν)

∂C/∂σ = S×φ(d₁)×√T×e^(-qT)
Vega = 0.0000 per 1%

Theta (θ)

∂C/∂t = -S×φ(d₁)×σ×e^(-qT)/(2√T) - r×K×e^(-rT)×N(d₂)
Call Theta = 0.0000 per day

Butterfly Strategy

Buy 1 lower strike, Sell 2 middle strikes, Buy 1 upper strike

Long Call

Strike: 95
Qty: +1

Short Calls

Strike: 100
Qty: -2

Long Call

Strike: 105
Qty: +1

Net Premium

$0.00

Delta Hedging (Slides 51-52)

Portfolio: Long 1 call option, Short Δ shares of stock
Goal: Maintain $\Pi = V - \Delta S \approx 0$ through continuous rebalancing

Hedging Example (Slide 52)

Setting:
We hold a short call option position with strike K=100
Current stock price: S=100$
Δ of Call: +0.6

Without Delta Hedge:
Stock rises by ↑1$. Since we are short the call, our position suffers a mark-to-market loss of $0.60.

With Delta Hedge:
We took on the opposite Δ, thereby shorted 0.6 shares of the stock.
Our short call loses: -0.60$
Our short stock gain: +0.60$
Net effect = 0