Homework 6-Option Pricing Exercise-Revised

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Homework 6-Option Pricing Exercise University/ College Or School Yale SOM MGT805 Fixed Income Professor Cynthia Butterman December 2, 2023

Homework 6-Option Pricing Exercise Midmarket price for 180 bp strike ($) 1,641.48 Midmarket price for 200 bp strike ($) 0.00 Midmarket price for the call spread ($) 1,641.48 Offered price for the call spread ($) =$3,504.2886 Initial hedge Buy or sell? Notional amount =$ 1,000,000 1. The prices of the individual options comprising the call spread described below. In this case, we need to compute the prices of the individual call options based on the information provided. Based on the provided information, the options are as follows: Call with a strike at 1.80% Call with a strike at 2.00% In order to compute the prices, we have to use the Black-Scholes formula for European call options (Zhu & He, 2018): where: C = call option price N = CDF of the normal distribution S t = spot price of an asset K = strike price r = risk-free interest rate t = time to maturity

σ = volatility of the asset We then calculate for each strike prices as follows: Call with a strike at 1.80% T= 1 day K = 1.80% σ = 2.5% (this information is already given; The one-day standard deviation of daily returns was 2.5% (or 40.0% annualized using 256 days)) r = 0.3% (overnight repo) Given the overnight repo=0.30, we can use this value to get the daily rate as follows: Number of trading days=256 Daily rate= overnight repo rate/256 (r/days) Daily rate=0.3%/256 = 0.000011718 Daily rate= 0.000011718 We can then calculate the forward rates (F), for the 10-year treasury yield one day forward. To calculate this, we use the overnight repo and the 10-year treasury yield observed on 3/3/2016 using the formular: (1+K) treasury yield = (1+r) 1/days in a year * (1+F) treasury yield -1/days in a year =(1+1.80%) 10 = (1+0.30%) 1/365 * (1+F) 10-1/365 = 1.80041% F=1.80041% Using the calculator and the above values, the Call price (%) is: Call price (%)= 0.01817 Therefore, using this information, we can calculate the bid price of 100 million notional of strike price of 1.8% as follows: Bid price = (Call price) x (DV01 ) x ( notional amount)

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DV01 = 0.09034 (Information given) = 0.01817%*0.09034*100,000,000 = 1,641.48 Bid price= 1,641.48 Call with a strike at 2.00% T= 1 day K = 2.00% σ = 2.5% (this information is already given) r (Daily rate)= 0.000011718 (calculated in above F=1.80041% (Calculated above) Using the calculator and the above values, the Call price (%) is: Call price (%)= 0.0000 Therefore, using this information, we can calculate the bid price of 100 million notional of strike price of 2.00% as follows: Bid price = (Call price) x (DV01 ) x ( notional amount) DV01 = 0.09034 (Information given) = 0.0000%*0.09034*100,000,000 = 0 Bid price= 0 2. The midmarket “breakeven” price of the call spread. The midmarket breakeven price refers to the average of the bid and ask prices for each option (Keenan et al., 2018). Breakeven price =Midmarket price for 180 bp strike ($)- Midmarket price for 200 bp strike ($) 1,641.48-0= 1,641.48

Therefore, the midmarket “breakeven” price of the call spread= $1,641.48 3. Given the midmarket “breakeven” price you calculated in question (1) above, what will be your offered price to the client? Explain your reasoning. The offered price has to be competitive and profitable. It can be a markup on the breakeven price. For instance, we might offer at a price that provides us a reasonable profit margin. Therefore in this case, we know that the shorted-dated options on 10-year treasury bonds have been trading with a 4-vega bid-offer spread in volatility, therefore, our offer price can be traded at daily volatility of 2.5%+4*vega Which in this case will be: For call at 1.8%: σ/day = 2.5%+4*0.0072=5.37% F (%)= 1.80041 K (%)= 1.80 T (day)=1 r/day= 0.000011718 σ/day= 5.37% Using Calculator: Call (%)= 0.03879 Since the offer price of 100 million notional of strick price of 1.8%, then the Call price * DV01 * notional amt = 0.03879%*0.09034*100,000,000 = 3,504.2886 Offer price=$3,504.2886

For call at 2%: σ/day = 2.5%+4*0=2.5% But the call price (%) is zero(0), calculated above already, therefore, the offer price =0 This means; The offered price for the call spread will be: =$3,504.2886-0==$3,504.2886 4. The notional amount of the 1.625% of 3/3/2026 you need initially to hedge your bank’s position in the call spread, assuming you can transact your hedge at the 5:00 pm price on Thursday March 3rd. Do you need to buy or sell bonds to hedge? In this case, we need to sell the bonds in order to hedge. This is because, holding the short position in call spread, would result in losing money once the rates rise. Therefore, this means that in order to hedge this risk, it would requires us to sell bond considering that this would result in profit once the rates rise while the bond price decreases. 5. What happens to your risk exposure if yields move up a lot? If yields move up a lot, the risk exposure of the call spread decrease. For instance, if we assume the yields move up significantly, beyond the strike prices of the call spread (1.80% for the lower strike and 2.00% for the higher strike), then the risk exposure of the call spread will reduce because in such a case, the loss from selling call-spread is capped after yield increases beyond 2% but the hedging position of selling treasury note results in a profit when yield shoots up (Wu et al., 2023). Therefore, if yields move up a lot, the risk exposure of the call spread decrease.

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References Keenan, R., Lane, R., Matti, J., Gong, H., & Affiliation. (2018). Estimating the Volatility in the Black-Scholes Formula . https://www.valpo.edu/mathematics-statistics/files/2015/07/Estimating-the-Volatility-in- the-Black-Scholes-Formula.pdf Wu, J., Wang, Y., Zhu, M., Zheng, H., & Li, L. (2023). Exotic option pricing model of the Black–Scholes formula: a proactive investment strategy . 11 . https://doi.org/10.3389/fphy.2023.1201383 Zhu, S.-P., & He, X.-J. (2018). A modified Black–Scholes pricing formula for European options with bounded underlying prices. Computers & Mathematics with Applications , 75 (5), 1635–1647. https://doi.org/10.1016/j.camwa.2017.11.023

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