back to all

Derivatives Pricing

Derivatives Pricing

The objective of the paper is to analyze related terms and pricing parameters, and finally price a CPN issued by UBS. In this paper, we use binomial trees; neglect the difference between discretely and continuously compounded interest rates and dividend yields; the interest rate is considered as constant over time; we neglect the default risk of the buyer not being repaid by UBS until the last question. 

1. Payoff Structure of This Product 

According to the description of the product, the Investor is entitled to receive from the Issuer on the Redemption Date an amount in the Settlement Currency per Product, according to the following scenarios:

  • Where:
  • CF: Cash amount investor received on the Redemption Date 
  • N: Nominal 
  • CP: Capital Protection 
  • Part: Participation 
  • K: Strike Level 

2. Volatility Estimate 

 The data MSCIEmergingMarketsIndex.xls contains daily price levels for the MSCI emerging markets index, starting from Jan 2016. We choose an appropriate part of data from the sheet, estimate the volatility parameter, and interpret this number. 

Firstly, we chose an appropriate part of data n. Secondly, a compromise that seems to work reasonably well is to use closing prices. From daily data over the most recent 90 to 180 days. Thirdly, alternatively, as a rule of thumb, n can be set equal to the number of days to which the volatility is to be applied. 

Therefore, we run a F-test on daily returns in the most recent 90 days. And the second recent 90 days (which is 90-180 days before 2020.3.13). (They are both relatively large sample size (CLT) and independent from each other.) (see in the sheet “volatility, data choosing”) 

Through the F-test we reject the null hypothesis and conclude that the volatilities in those days are significantly different. Hence, daily data over the most recent 90 to 180 days will lead to different volatilities. Which are very sensitive to the period we choose. So, we use the alternative method that picks data from 2017.3.13 to 2020.3.13 to calculate the volatility, since this CPN’s maturity is 3 years. 

Additionally, we ran a F-test on samples during 2014.3.13-2017.3.13 vs 2017.3.14-2020.3.13 to check there is no statistically significant difference between these two 3-years-data-generated volatilities.

The volatility of a stock is a measure of our uncertainty about the returns provided by the stock. During the period we choose, this volatility, 13.90%, represents the variation of the stock price relative to the average stock price on a yearly basis. The higher volatility we observe the more uncertainty in this stock price. 

3. Dividend Yield Estimates 

The underlying index’s dividend yield is equal to 2.8%. This is the dividend yield over the last 15 years. And can be used as a first estimate of the dividend yield parameter, q. Alternatively, we can obtain a second estimate of the dividend yield parameter q for the MSCI emerging markets price index. By applying the standard futures pricing formula, with the price information of an MSCI emerging markets index futures. Additionally, we could use the current Euribor rate of the appropriate maturity. As the relevant interest rate for the futures price formula.  

Standard futures pricing formula: F0=S0*e(r-q)T 

  • Where, 
  • S0: close price of underlying, which is MSCI Emerging Markets Index in this case. F0: the price of forward contract on this underlying. 
  • r: risk-free rate. 
  • q: known yield, which is dividend yield in this case. 

We choose 2020.3.13 as the 0 period. Then the spot price and forward price in this period should cover all the past information according to the weak form of EMH. Hence, S0 is $883.126.  Forward contracts that have the expiration date close to Dolphin CPN’s (February 2021) will be ideal. So, we pick forward expiry in Mar 2021, which settlement price in 2020.3.13 is $851.900. 

( see S0 and F0 in the appendix) 

As for r, the Euribor rates are considered to be the most important reference rates in the European money market. The interest rates do provide the basis for the price and interest rates of all kinds of financial products like interest rate swaps, interest rate futures, saving accounts and mortgages. So, we use a 12-month Euribor rate in 0 period, which is -0.287%. 

(From: https://www.euribor-rates.eu/en/euribor-charts/) 

  • T=1 year because this forward we choose will expire in one year. 
  • By plugging in all the data that we chose: 
  • q=-0.287%-ln(851.9/883.13)=3.31% 

4. Estimate Price with Different Values for Dividend Yield 

(1). Data and Estimates 

Now having collected all necessary parameters to set up a pricing model for the CPN product, we can start by pricing the product at the pricing date, 13 Feb 2018. We have a volatility estimate, two estimates for the dividend yield. Moreover, for the risk-free rate, r, we use the 6-month Euribor rate 

on 13 Feb 2018, which is −0.369% per annum. 

We build a 12-step binomial tree with 3 months for one step, using all these parameters. And estimate our launch price of the CPN product, to compare the calculated value to the issuance price of UBS. We estimate the price with the two values for the dividend yield, respectively. And interpret the results while applying different dividend yields. 

Figure Explanation Reference 

volatility estimate 13.90% Estimation in 2 

2.80% dividend yield over the  last 15 years
3.31% Estimation in 3

Using the parameters and 12-step estimation, we estimate the launch price of the CPN product to be $980.70 and $983.30 (The dividend yield parameters are 2.80% and 3.31% respectively). It’s a little bit lower than the issuance price of UBS ($995.50 on 21 Feb 2018.). 

5. Interpretation 

The first estimate is the dividend yield over the last 15 years, this super long time period may not be suitable for our valuation. As we can see, the r in the last 15 years is quite different from the recent interest rate. The price return in MSCI Emerging Markets Index is decreasing as time went by. (The most recent one-year price return is negative while the past 3, 5 or 10 years annualized price return is positive.) 

We use different methods to estimate q. The first estimate is calculated directly from the stock in the MSCI Emerging Markets Index for the last 15 years, which is relatively a positive market and leads to a lower price. While the second estimate is calculated by the forward price nearly, people in the market have lower expectations and lead to a higher price. It is based on the rule of one price and people use the true r and q to value the forward.

6. Analysis of the Result 

As the market moves on, the price of the underlying on spot market changes. This will affect the CPN price in the secondary market. On 13th March 2020, about 11 months to expiration, the CPN product was trading at 98.15, recorded on the website of the product. So, we should adjust our estimate of volatility with the price information of the underlying, choose an appropriate time step with a retrievable Euribor rate, and use the binomial tree model to price the product. Finally, we can compare our estimate with the secondary market price.  

The new estimation of the volatility uses 11 months information of underlying, from 2019.4.12 to 2020.3.13, the 11-month volatility is about 15.42%. Because there is about 11-month time to expiration, the historical data’s time interval used to estimate the volatility should be the same as the time to expiration. As for the retrievable Euribor rate, we use the 12-month Euribor rate in 2020.3.13, which is -0.287%. Because the number 11 cannot be equally divided to other integers, we build a 11-step binomial tree with 1 month for one step.  

The underlying price on 13th March 2020 is $891.19, from which we build a 11-step binomial tree to estimate the expiration price, which are all below the barrier level. Then from the expiration price, we can get each payoff for each expiration price, add the denomination, and reverse deduction to the 13th of March 2020 price.  

Through the process, the numbers are all below the barrier level. Since the estimated price on maturity day is lower than in problem 4, the levels of payoff are lower, which causes the real trading price on 13th March 2020 $981.50 to be lower. The estimation on 13th March 2020 is about $1003.71. The difference between $981.50 and 1003.71 may be because the sharp changes in the market cause the index to change faster than the selling price. However, the two numbers just aren’t very different, so the little difference may come from the calculation error, the new parameters we use, Euribor rate, volatility and the step size.

7. The Limitations of The Assumptions 

The binomial tree model is set up with a number of assumptions, and it can be restricted in practice. Assumptions of the binary tree option pricing model are: (1) There is no transaction cost and tax for market investment; (2) Investors are all price takers; (3) It is allowed to fully use short-sale proceeds; (4) Allow borrowing or lending of money at a risk-free rate; (5) The future stock price will be one of two possible values; (6) Risk-free rate and volatility is constant. 

Through the empirical estimation, we find these limitations are:
(1) Volatility is not constant. From the first F-test, we can see that the volatility is not constant. Which violates the assumption of our model that volatility should be constant. And based on the volatility smile we learn in the class, there is a negative correlation between equity prices and volatility.
(2) Risk-free rate is not constant. Risk-free rate is not constant as Euribor is changing over time. And in reality, there is a borrowing constraint that people cannot borrow or lend money at a risk-free rate.
(3) The option will also be taxed with transaction cost. For tax purposes this product is qualified as a combination of an option and bond. In addition, for private investors with tax domicile in Switzerland. Furthermore, the increase of the value of the bond part at sale or maturity is subject to Swiss federal. As well as cantonal and communal income tax.
(4) Taking into account the default risk. According to the traditional estimation which is based on the risk-neutral world, however when we take into account the default risk, maybe we should change the estimation of volatility. An alternative way could be introducing the probability of default and enlarging the trees. Lastly, we can similarly use the risk structure of interest rate and rank the different levels of stock index to add some risk premium.

Back To News

Derivatives Pricing

Recent Articles