MFM2213 - Financial Mathematics III

3. Greek Letters & Elasticity

Aneesha D. W. Yapa

University of Ruhuna
Department of Mathematics

June 10, 2024

3.4.1 Delta Cont...

Dynamic Aspects of Delta Hedging

Delta hedging involves buying or selling the underlying asset in the same
proportion as the delta to eliminate the risk of price changes.
This strategy is particularly useful for traders who want to maintain a
neutral position in the market.
Consider 2-year European put option with the strike price $50. Current
stock price is $50 and the continuously compounding risk free interest
rate is 10 %. u = 1.5 and d = 0.5.
Construct a trading strategy that lets you hedge the risk of this put using
the stock.
——— Binomial Tree ———–

Aneesha D. W. Yapa MFM2213 - Financial Mathematics III June 10, 2024 2 / 14

3.4.1 Delta Cont...
Our objective is to hedge the put option’s price risk using the underlying
stock.
First, we will define a number referred to as Delta, which is useful when
hedging option risk.
Delta is a number that measures the change in the value of the option
when the stock price changes.
Creating a Riskless Position
At the initial node, ∆ is −0.3155.
This means that to replicate a long (i.e., +1) put option we should sell
0.3155 stocks and lend the proceeds to the money market.
But recall our goal is to create a riskless position when we already own
one put option.
How do we do this?
Aneesha D. W. Yapa MFM2213 - Financial Mathematics III June 10, 2024 3 / 14
3.4.1 Delta Cont...
The answer to is to replicate a short (i.e., −1 put option) position.
To do this requires that we change the signs in a long synthetic put
option position.
That is, from a trading perspective we should borrow from risk free
money market and buy 0.3155 stocks to replicate the short put option.
If we do this we can create a position that is completely riskless.
That is, we are long one put option and simultaneously short one
synthetic put option.
The net effect eliminates all underlying asset price risk from the position.
You can check, at the end of the first period, that your portfolio is worth
$10.69 whether the stock ticks up or down.
In other words, you have created synthetically a risk free bond.
Let us calculate the portfolio values to verify these numbers.
Aneesha D. W. Yapa MFM2213 - Financial Mathematics III June 10, 2024 4 / 14
3.4.1 Delta Cont...
In these calculations, we will assume that you started with one put option
and no cash, so that if you buy stocks, you have to borrow money at the
risk free interest rate (which is 10 % or 0.10).
This actually does not make any difference because you can assume you
have some initial amount of cash, and conduct similar calculations.
At the initial node (A), portfolio value is:
Quantity Value
Put Option 1 9.6769
Stock 0.3155 15.775
Cash -15.775 -15.775
Total 9.6769
Notice that the value of your stocks cancel the borrowings, so that your
portfolio value equals the value of the put option you started with.
Aneesha D. W. Yapa MFM2213 - Financial Mathematics III June 10, 2024 5 / 14
3.4.1 Delta Cont...
If the stock price going up in the next period (at A), the net position is:
Quantity Value
Put Option 1 4.4657
Stock 0.3155 23.6625
Cash -15.775 -17.4341
Total 10.6941
If the stock price going down in the next period (at A), the net position is:
Quantity Value
Put Option 1 20.2419
Stock 0.3155 7.8875
Cash -15.775 -17.4341
Total 10.6953
At the end of the first period, that your portfolio is worth $10.69 whether
the stock price going up or down.
Aneesha D. W. Yapa MFM2213 - Financial Mathematics III June 10, 2024 6 / 14
3.4.1 Delta Cont...
At the node B, since the delta is -0.1667, you need to reduce your stock
holdings.
Portfolio value is:
Quantity Value
Put Option 1 4.4657
Stock 0.1667 12.5025
Cash -6.2741 -6.2741
Total 9.6769
If the stock price going up in the next period (at B), the net position is:
Quantity Value
Put Option 1 0
Stock 0.1667 18.7538
Cash -6.2741 -6.934
Total 11.8198
Aneesha D. W. Yapa MFM2213 - Financial Mathematics III June 10, 2024 7 / 14
3.4.1 Delta Cont...
If the stock price going down in the next period (at B), the net position is:
Quantity Value
Put Option 1 12.5
Stock 0.1667 6.2513
Cash -6.2741 -6.934
Total 11.8173
This confirms that your portfolio will be worth $11.82 along this path if
you follow the trading strategy.
At the node C, since the delta is -1, you need to hold one stock when you
leave this node. Portfolio value is:
Quantity Value
Put Option 1 20.2419
Stock 1 25
Cash -34.5466 -34.5466
Total 10.6953
Aneesha D. W. Yapa MFM2213 - Financial Mathematics III June 10, 2024 8 / 14
3.4.1 Delta Cont...
If the stock price going up in the next period (at C), the net position is:
Quantity Value
Put Option 1 12.5
Stock 1 37.5
Cash -34.5466 -38.1799
Total 11.8201
If the stock price going down in the next period (at C), the net position is:
Quantity Value
Put Option 1 37.5
Stock 1 12.5
Cash -34.5466 -38.1799
Total 11.8201
This confirms that your portfolio will be worth $11.82 along this path if
you follow the trading strategy.
Aneesha D. W. Yapa MFM2213 - Financial Mathematics III June 10, 2024 9 / 14
3.4.1 Delta Cont...

Example 3.2
The price of a certain stock follow a three-period binomial tree model with
u = 1.3 and d = 0.7. A period for the tree is three months. The price of the
stock is currently 80. The stock pays dividends continuously at a rate of 2 %.
The continuously compounded risk-free rate is 5 %. A market-maker writes
100 nine-month at-the-money European call options on the stock.
a) Construct the three period binomial model.
b) Calculate the Delta for each node.
c) Determine the number of shares of the stock the market-maker must buy
or sell to delta-hedge her portfolio.

Aneesha D. W. Yapa MFM2213 - Financial Mathematics III June 10, 2024 10 / 14

3.4.1 Delta Cont...
Delta of a Portfolio
The delta of a portfolio of options or other derivatives dependent on a
single asset whose price is S is
∂
Q

∂S
where is the value of the portfolio.
Q

The delta of the portfolio can be calculated from the deltas of the
individual options in the portfolio.
If a portfolio consists of a quantity wi of option i (1 ≤ i ≤ n), the delta
of the portfolio is given by
n
X
∆= wi ∆i
i=1

where i is the delta of the ith option.

Aneesha D. W. Yapa MFM2213 - Financial Mathematics III June 10, 2024 11 / 14
3.4.1 Delta Cont...
The formula can be used to calculate the position in the underlying asset
necessary to make the delta of the portfolio zero.
When this position has been taken, the portfolio is delta neutral.
Suppose a financial institution has the following three positions in options
on a stock:
1. long position in 100,000 call options with strike price $55 and an expiration date in
3 months. The delta of each option is 0.533.
2. A short position in 200,000 call options with strike price $56 and an expiration date
in 5 months. The delta of each option is 0.468.
3. A short position in 50,000 put options with strike price $56 and an expiration date
in 2 months. The delta of each option is -0.508.
The delta of the whole portfolio is -14,900.
This means that the portfolio can be made delta neutral by buying 14,900
shares.
Aneesha D. W. Yapa MFM2213 - Financial Mathematics III June 10, 2024 12 / 14
3.4.1 Delta Cont...

Example 3.3
Assume the two Binomial tree model applies. The continuously compounded
risk-free rate of interest is 4 %. A share of non dividend-paying stock is
currently worth 120. The volatility of the stock is 20 %. A portfolio consists of
the following European options:
100 one-year, 130-strike written calls
60 one-year, 130-strike written puts
80 two-year at-the-money purchased calls
Determine the number of shares that must be bought or sold in order to delta
hedge this portfolio.

Aneesha D. W. Yapa MFM2213 - Financial Mathematics III June 10, 2024 13 / 14

 - End -

Aneesha D. W. Yapa MFM2213 - Financial Mathematics III June 10, 2024 14 / 14