The quick and dirty on H3

What on Earth is Delta-Hedging?

Warning: the next set of slides will be a mathy hell for y'all as I'm doing the professor thing...

• Stock can go
  • up to \(u\cdot S\) or
  • down to \(d\cdot S\).
• Call payoffs: 
  • ​up: \(C_u=\max(0,uS-X)\)
  • down: \(C_d=\max(0,dS-X)\)
• What's the price \(C\) of a call option at strike price \(X\) today?

Big idea:

• Can we replicate the payoff from the call with other securities?
• If so, use the law of one price!

What on Earth is Delta-Hedging?

• Let's try replicating payoffs with the stock and the risk-free bond.
• Build a portfolio:
  • \(H\) units of the stock
  • \(B\) units of the bond.

lazy notation:
\(1+r_f=R_f\)

• What are payoffs of this portfolio?
  • Case 1: Stock price goes up:
    • \(H\times uS+(1+r_f)\times B\)
  • Case 2: Stock price goes down:
    • ​\(H\times dS+(1+r_f)\times B\)

Task: find \(H\) and \(B\) such that you match the payoffs of the call!

What on Earth is Delta-Hedging?

\[H\cdot u\cdot S+B\cdot R_f  =  C_u\]
\[H\cdot d\cdot S+B\cdot R_f  =  C_d\]

\[H=\frac{C_u-C_d}{(u-d)S},~~B=\frac{uC_d-dC_u}{(u-d)R_f}.\]

The Solution:

Economic Interpretation

• Option payoff is replicated by a portfolio of
  • investment in stocks \[H=\frac{C_u-C_d}{(u-d)S}=\frac{\Delta C}{\Delta S}\]

• risk free investment \[B= \frac{uC_d-dC_u}{(u-d)R_f}.\]

    \(H=\text{the Delta}=\text{change in call values per change in underlying stock}.\)

How does this help us find the price of the call?

\[C=HS+B=\frac{C_u-C_d}{(u-d)S}\cdot S +\frac{uC_d-dC_u}{(u-d)R_f}\]

Knowledge piece:

• \(q\) is the risk neutral probability

• Idea: if \(q\) would be the true probability of \(u\), then investors behave as if they are risk-neutral.

• The law of one price:
  • price of call = \(H\times\) stock price + \(B\)
  • (we normalize the price of the bond to 1)

After simplifications:

\(C=\frac{1}{R_f}\left(q\cdot C_u+(1-q)\cdot C_d\right)\) where \(q=\frac{R_f-d}{u-d}.\)

Is that it?

• Oh no. Now let's do it for another period!

• Price can go to

  • \(u^2S\)

  • \(d^2S\)

  • \(udS=duS\)

• Now we have option values
  • After period 1: \(C_d\) and \(C_u\)
  • After period 2: \(C_{uu},C_{ud},C_{dd}\)
• How to price? \(\to\) go backwards!

\(u^2S\)

\(d^2S\)

\(udS\)

Is that it?

• One period from the end: \[C_u=\frac{1}{R_f}(q\cdot C_{uu}+(1-q)\cdot C_{ud})\] \[C_d=\frac{1}{R_f}(q\cdot C_{ud}+(1-q)\cdot C_{dd})\]
• Two periods from the end \[C=\frac{1}{{R_f}^2}(q^2\cdot C_{uu}+2q(1-q)\cdot C_{ud}+(1-q)^2C_{dd})\]

Let's iterate this further!

• risk-neutral probability \(q\) is constant
• \(R_f\) is per period and not annualized
• The replicating portfolio (\(H,B\)) changes as you move through the tree.
• Values of u and d should be derived from stock's \(\sigma\)

\[u =\exp(\sigma\sqrt{\Delta t}),\] 

\[d =\exp(-\sigma\sqrt{\Delta t}).\]

\[C=\frac{1}{{R_f}^2}\sum^N_{j=0}\underbrace{\left(N \atop j\right)q^j(1-q)^{N-j}}_{\text{prob of \(j\)ups}\atop  \text{in \(N\)rounds}}\cdot \underbrace{\max(0,u^jd^{N-j}S-X)}_{\text{option payoff of \(j\)ups}\atop \text{and \(N-j\)downs}}.\]

Let's iterate this further!

With more rounds/periods, the probability distribution of the final stock prices looks smoother and smoother.

... and that eventually gets us to The Black-Scholes Formula

Price of a European call:

\[C=S\cdot N(d_{1})-X\cdot e^{-rT}\cdot N(d_{2})\]

\(C=\)Current call option value

\(S=\)Current stock price

\(N(d)\)= probability that a random draw from a normal distribution will be less than \(d\),

\(X =\)Exercise price

\(r=\)Risk-free interest rate (annualized, continuously  compounded with the same maturity as the option)

\(T =\)time to maturity of the option in years

\(\sigma = \)Standard deviation of the stock.

where

\(d_{1}  =  \frac{ \ln\left( \frac{S}{X}\right) + T\left(r + \frac{ \sigma^2}{2}\right)}{ \sigma \sqrt{T}}, \)

\(d_{2}  =  d_{1} -  \sigma \sqrt{T}.\)

... and now full cycle to the \(\Delta-\)Hedge

Initial replicating portfolio:    

• \(H=\frac{\partial C}{\partial S}=N(d_1)\) in stocks \(\to\)Delta hedge ratio,
• \(B=-Xe^{-rT}N(d_2)\) in the risk free asset

• Depending on where you "are" currently on the price curve, your hedge ratio differs.
• As the price of underlying asset \(S\) and time-to-expiration \(T\) change, so does the replicating portfolio.

Black-Scholes "hugs" the curve of the payoff at maturity and gets closer has \(t\to\) maturity

steep slope = high hedge ratio

flat slope = low hedge ratio

Last item: volatility

• Unobservable input: volatility.

• Option's market price implies volatility!
• Quotes are often by implied volatilities.
• true vol \(>\) implied vol \(\to\) underpriced
• Lingo: delta-neutral \(=\) hedge against stock-price fluctuations through position in stock s.t. total delta of position \(=0\).
• Often: same underlying,different vol ("volatility smile") \(\to\) inconsistent with BS.
• Estimating "volatility surface" is a big task and now done with machine learning tools.
• Option value is positively related to volatility.

XIU Options from the Montreal Exchange

Source: Bloom (2009). The Impact of Uncertainty Shocks. Econometrica 77(3), 623-685.

A Common Application

Portfolio insurance

• when things go bad, you want your investment loss limited,

• but you also want to benefit from the upside

An Alternative way to create the same payoff!

Compare costs of the two portfolios "today"

Cost of strategy 1: stock + put = \(S+P\)

Cost of strategy 2: bond + call = \(PV(S)+C\)

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