1. Stochastic Finance (Hedging with Black-Scholes-Merton Model)
Student’s Name: Ben Hoerlein
Department of MTH, Florida Institute of Technology
Faculty Advisor: Dr. Eugene Dshalalow
The following project was an example of how the hedging of the Black-Scholes-Merton model
was used for the Ameriprise Financial stock during the time frame of September 2015. The
European Call option was used, meaning that the buyer of the stock has a right to sell the stock,
but only at the maturity. The seller used a hedging strategy to invest the premium value of the
portfolio in order to pay the seller the payoff at maturity.
The following formulas were used to calculate the hedging price option. logreturn formula:
𝑙𝑛
𝑆(𝑡)
𝑆(𝑡−1)
,where S(t)is the value for the certain closing price of a stock on a given day. The
volatility(σ) of the stock was calculated from the logreturn formula. The call price of the stock
was calculated with the Black-Scholes-Merton formula: C(t,S(t))=𝑆( 𝑡)Ф((
1
σ√T
) [𝑙𝑛
𝑆( 𝑡)
𝐾
+
(𝑟 +
σ2
2
) 𝑇]) − 𝐾𝑒−𝑟𝑇Ф((
1
σ√T
)[𝑙𝑛
𝑆( 𝑡)
𝐾
+ (𝑟 −
σ2
2
) 𝑇]), where
Ф 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑛𝑜𝑟𝑚𝑎𝑙 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛, K was the strike price at day zero, r was the interest
rate per day, and T is the maturity at 21 days for the stock. Then delta was calculated with the
following formula: Δ(t)=𝐶 𝑠(𝑡, 𝑆( 𝑡) = Ф (𝑑+( 𝑇 − 𝑡, 𝑆( 𝑡)))= Ф ((
1
σ√T
)[𝑙𝑛
𝑆( 𝑡)
𝐾
+ (𝑟 +
σ2
2
)(𝑇 − 𝑡)]).
The seller’s portfolio was calculated by :X(t+1)=X(t)+ Δ(t)[S(t+1)-S(t)]+rD(t). The debt that the
seller owed to the bank after each day was calculated by: D(t)=X(t)- Δ(t)S(t). The option price
worth after the stock maturities was calculated by: payoff= S(20)-K. The final transaction was
calculated by: FT= payoff – X(20), where FT means the final transaction
The seller now has one full share(Δ(t)=1) of the stock its value would be $14.60. When
calculating how much money the seller made, the payoff would be $13.99, which is the exact
payoff the seller has to deliver.The final transaction results showed that the seller ended up
gaining $0.61, if the seller didn’t hedge the option position, the seller would have loss $9.93 and
not gained $0.61.The seller owes the bank $106.38 and he must deliver $13.99 to the buyer.
Also, he will receive $107 for his stock and the portfolio is valued at $14.60.The seller borrowed
$53.9385 from the bank and he has done so until expiration, which created his debt, but since
the seller owned one full share of the stock (Δ(t)=1) for the price of $120.99, which was sold to
breakeven with the bank. The buyer made a net profit of $10.54.
In conclusion, if the seller didn’t hedge the portfolio, he would have collected the premium from
the buyer. If in the event the stock would fall, the buyer and seller would lose the same amount
of money. If the stock would have rose, then the seller would have lose $9.93 while the buyer
would gain $10.54. Hedging with Black-Scholes-Merton model has allowed for the seller to
maximizies his expectation for a profit while minimizing his chances of a net loss. This option
creates a fairness to the premium value that would benefited both the buyer and seller.