Alternative Financing Instruments for African Economies

This paper investigates the alternative financing instruments that can be used to hedge sovereign risks and finance development in African countries. Many heavily indebted countries are exposed to external risks especially the exchange rate shocks due to limited use of hedging instruments. We propose alternative financing instruments to minimize sovereign risks and the cost of debt. Our paper uses the standard model for pricing options, the Black-Scholes model to determine the fair value of options. The findings show that barrier options have an added advantage over plain vanilla options because of its knock-ins and knock-outs features hence they are the most affordable to use. An important aspect of the effective debt management policies should be on developing local bond market to access alternative financing instruments in the world capital market.

minimize sovereign risks. According to [12] Mu et al. (2013), African bond markets are still small and underdeveloped to access hedging instruments that lower the sovereign default risks.
Another strand of studies show that sub-Saharan Africa's infrastructure needs are approximately USD 130 -170 billion and a financing gap of $67.6 -$107.5 billion annually. This increasing infrastructure gap has prompted African countries to reconsider the role of private sector participation in bridging the gap. Although, most of the sub-Saharan African countries with the exception of South Africa and Nigeria have not been able to attract significant private sector investment in different economic sectors. Moreover, the capacity to effectively construct, design and manage public private partnerships is a significant challenge across Africa. Only a few countries such as South Africa, Nigeria, Zambia, and Kenya have implemented PPP frameworks, however, these countries are still experiencing challenges in terms of resources and expertise. Importantly, for PPPs to be successfully across Sub-Saharan Africa, a sound policy, legal and institutional frameworks with clear guidelines and procedures for development and implementation of PPPs should be enforced. For instance, [13] Leigland (2018), noted that developing countries lag behind in the development of Public Private Partnerships because of the lack of a sound legal and institutional frameworks with which to implement such projects.
Some studies have investigated the role of indexed bonds in hedging risks as a useful tool for debt management. The government can hedge risks by issuing indexed bonds or shortening the maturity of domestic debt. Indexed bonds as pointed out by [7] Price (1997) are cost-effective in reducing government borrowing costs. For instance, the inflation-linked bonds which are financial instruments that tie their interest payment to price changes. Inflation-indexed bonds issued during periods of high inflation offset the debt burden by paying the inflation risk premium. If issuers are exposed to inflation risk, it might lead to a negative risk premium. If the issuer is risk neutral, the risk premium is likely to be positive. Inflation risks can be hedged by shortening the maturity of bonds denominated in foreign currencies.
A number of studies have investigated the issuing of bonds tied to their main export commodities. The issuing of commodity-linked bonds would reduce the debt burden along with declines in export prices. Developing countries should prefer issuing commodity-linked bonds than conventional debt to protect their export commodities from price volatilities. Commodity linked bonds such as gold-linked bonds as shown by [14] Atta-Mensah (2004) could be used as a financing means and as a hedge because it has the option feature. Since developing countries are exposed to substantial commodity price risks, [15] Myers (1992), note that they can use commodity-linked securities as a tool for risk management. Therefore, commodity-linked bonds are used for hedging commodity price shocks.
A related strand of literature focus on financing instruments in which the coupon payments are tied to the GDP of the issuing country. GDP-linked bonds Another strand of literature show that developing countries can use options whose value depend on the value of the underlying asset to hedge commodity price risks. The simplest financial option as shown by [6] Lu and Neftci (2008) is the European plain vanilla options which can be exercised at any time before maturity. Barrier option behaves like a plain vanilla option as long as the underlying asset price does not fall below or predefined barrier. The cost of barrier option as shown by [19] Hull (2003) falls below that of the plain vanilla option. Knock-in barrier options become plain vanilla option when a barrier is hit whereas knock-out barrier options is when the underlying is higher than the barrier. In this case, up-and-out barrier options are the simplest to calibrate.
Other studies focus on credit default swap which is a contract that provides insurance against a default by a sovereign entity and enhance financial stability. Credit derivatives are financial instruments managing default risk which occurs when there is a decline in the ability of the borrower to repay the debt. Credit default swap as suggested by [20] Hull and Allan (2003) allow sovereigns to manage credit risks by insuring against the default of borrowers. Furthermore, [21] Terzi and Ulucay (2011), [22] Duffee (1996), points out that credit default swaps enable the issuing country to transfer or redistribute its credit risks. Empirical findings suggest that the use of sovereign credit default swap spreads increase the level of liquidity hence reducing default risks.
A commodity derivative in the form of a plain vanilla put option should be attached to sovereign Eurobond to hedge the occurrence of a sovereign default risk. When commodity prices are increasing, [6] Lu and Neftci (2008), note that the default risk and borrowing cost decreases. In general, [22] Duffee (1996), finds that the increase in commodity price movements, reduces the default probability hence a bondholder will be compensated by paying for the default swap spread. Credit default swaps are actually put options and provide insurance against price decrease. However, the market for credit default swap in African countries is limited.
Modern studies have shown that Artificial Intelligence can be used in forecasting the applicability of innovative financing in the capital market using intelligent algorithms such as Artificial Neural Networks. The human intelligence includes learning, reasoning and problem solving, which is accomplished by studying how human brain thinks, and how humans learn, decide and work while trying to solve a problem. The ANNs gather their knowledge by detecting the not from programming.

Theoretical Framework
The model builds on the works of [6] Lu and Neftci (2008) who employ Black-Scholes model (1973) to price plain vanilla European options. The option price is a function of asset's price volatility, risk free rate and strike price. The Black-Scholes model continues to be the standard option pricing model. The well-known Black-Scholes formula for option pricing is only valid under the assumption of complete markets. Intuitively, Black-Scholes model is only valid under the assumption of complete markets where volatility is constant. The assumptions made when valuing options include the asset price being exogenous implying that price movements are not affected by actions of markets. The model has been criticized on its strict assumptions such as the constant volatility which is not realistic. Findings by [23] Bakshi et al. (1997), show that the Black-Scholes prediction of zero risk cannot be achieved. The value of call or put option depends on time to maturity, risk free interest rate, volatility, strike price and underlying asset price. The general Black-Scholes model for pricing European call option is given by: where: C t is the price of a European call option at time t, S t is the stock price, K is the strike price, r is the risk free interest rate, t is the time to option expiration, σ is the standard deviation of the option. The simplest financial option, is the European call option. If stock price is greater than the strike price at expiry, then it pays off to exercise call option. The option prices are derived using the general [9] Black-Scholes (1973) formula, where the spot price equals the present value of the striking price, as shown by [6] Lu and Neftci (2008), is given by: where N(d 1 ) and N(d 2 ) are the cumulative distribution function of a standard normal random variable.
where C t denote the price of a European call option at time t, S is the stock price, K is the option striking price, r is the risk free interest rate, t is the time to option expiration, σ is the standard deviation of the option and N(.) is the cumulative distribution function of a standard normal random variable N(0,1). For the special case of a European call or put option, Black-Scholes model indicate that a hedged position can be created consisting of a short position in the option and a long position in the underlying where option's value will not depend on the price of the underlying. In addition, [22] Duffee (1996), note that put call parity can also consist of a long position in a credit default swap combined with a long position in the underlying. Other studies define put-call parity as a relationship between the price of a European call and put option with identical strike prices and maturity time in a frictionless market is given by where C t denote the price of a European call option at time t, S is the stock price, K is the option striking price, r is the risk free interest rate, t is the time to option expiration, σ is the standard deviation of the option and N(.) is the cumulative distribution function of a standard normal random variable N(0,1). The price P t of a European put option at time t with the same expiry date T and strike price K can be obtained by the following put-call parity relationship P t is the current stock price.
Solving partial differentiation equations in closed form is difficult but in the case of Black Scholes model, the European call option at time t = 0 is European put option is given by: When a financial security is traded, the buyer is said to take a long position in the security and the seller is said to take the short position in the security. The derivative pricing problem is solved by determining a fair value for a derivative.
Two boundaries on s(t) are 0 and ∞ representing maximum and minimum price of the underlying asset. Black-Scholes model is also used to price barrier options which behave like a plain vanilla option as long as the underlying asset price does not fall below or predefined barrier . As noted by [19] Hull (2003), the cost of barrier option falls below that of the plain vanilla option. Knock-in barrier options become plain vanilla option when a barrier is hit whereas knock-out barrier options is when the underlying is higher than the barrier. Thus, knock-ins and knock-outs features make options cheaper and fills shortcomings of risk reversals hence, barrier options are suitable for hedging commodity price risk.
However, up-and-out barrier options are the simplest to calibrate. Under suitable assumptions, the value of the contract really depends only on t and S, and it satisfies the following partial differential equation: Journal of Mathematical Finance where r is the risk-free interest rate, σ is the volatility of the underlying asset, S(t) is the current price at time t. The value of a European option has three components, the intrinsic value, the striking price of the option and the insurance value. The intrinsic value is the difference between the price of the underlying asset, S. The striking or exercise price, K is the payoff from exercising option only on the expiration date and the option's premium is the payment for option upfront.
The barrier option behaves like a plain vanilla option as long as the underlying asset price does not fall below or predefined barrier, S b . In up-and-out options, they are active when S b > S as long as the underlying asset price crosses and falls below a predefined barrier, S b . The rationale for barrier options is that by putting a barrier, the payoff is limited. If both down-and-in and down-and-out are held then the effect of the barrier is cancelled and the two barrier options are equivalent to a vanilla put option.

Data Description
This study employs monthly data in which the commodity prices for oil, gold and silver are from the IMF IFS for Tanzania (Gold), Copper (Zambia) and Oil

Black Scholes Results
In vanilla options, a put option and a call option provides insurance against commodity price shocks. A put option acts as a price floor by providing insurance against price decrease. [6] Lu and Neftci (2008), argue that a call option acts as a price ceiling by providing insurance against price increase so they both set minimum and maximum price. Table 1  As shown in Table 1, prices of at the money put options are relatively higher than out of the money put options. If the reference entity, in this case, the government wants to hedge commodity price risks, they will use the less costly options. Hence, OTM put options will be the alternative instrument to ATM put options which is in line with findings by [ shocks than long-term options. Thus short term options increase more than long-term options. Table 2 lists the prices of out-the-money (OTM) options on the three selected commodities. These prices, OTM put options are less expensive that the ATM put options. OTM put options occur when the strike price of a put option is greater than the spot price of the underlying asset. However, risk reversals being part of the plain vanilla options, are still inefficient to reduce option costs. [25] Carry and Wu (2003), examine the behavior of both the prices of at-the-money and out-of-the-money options as the option's time-to-maturity tends to zero. They conclude that as the maturity tends to zero for OTM options, they exhibit a jump component implying that OTM put options are more liquid than OTM call options. Standard call (put) option contracts promise to pay nothing if they are out of the money at maturity. Since plain vanilla put options and risk reversals are inefficient as suggested above, up-and-out barrier options are considered to hedge commodity price risks. The prices for up-and-out put barrier options at 1 year and 3 year maturity are given in Table 3.  Source: IMF IFS and Bloomberg for the month of December 2012. Note that out of the money put option is when the underlying is higher than the strike price at a given period of time.

The Partial Derivatives
The traditional Black-Scholes model uses partial derivatives to analyze the sensitivity of option premium to small changes in the model's parameters known as the Greeks. These hedging parameters are considered to be useful descriptive statistics to option traders for a portfolio. Within the Black-Scholes model, these sensitivities are obtained by taking the partial derivatives of the option-pricing formula below. Table 4 reports the sensitivities of the options to changes in the model's parameters. The delta is the most important Greek because it has the largest risk. The delta of a call option on a stock is the rate of change of the option premium to change in stock price. Above result show that delta is the most volatile parameter implying that a 1% change in stock price, the option price changes by 90%. Delta hedging involves the buying and selling the underlying asset to eliminate risk. The second Greek parameter is the Ґ gamma which is the rate of change in delta to change in the stock price.
If gamma is small, delta becomes less sensitive whereas if gamma is large, then delta becomes more sensitive to variations in the underlying. It implies that a 1% change in stock price, delta changes by 10%. Delta and gamma hedging are both based on the assumption that the volatility of the underlying is constant. The ρ rho is highly volatile that is a small change in interest rate leads to a significant change in the option price. Our results show that a 1% change in interest rate, the call price will vary by 8%. When interest rates increase, call prices increase Table 4. The Greeks. Similarly, the vega, ϒ, which measures the sensitivity of option premium to changes in volatility shows that a 1% change in volatility leads to a change in option price by 39%. According to [19] Hull (2003) and [27] Wilmott (2000), options are viewed as instruments of volatility in the underlying price, which leads to gamma gains and vega gains. An increase in volatility will increase the option price and a decrease in volatility, will decrease the option price in the same rate.
Therefore, the Greeks provide traders a descriptive summary of determining the sensitivity of options with respect to fluctuations in stock prices, volatility, interest rate and time to maturity.

The Fluctuation of Commodity Prices (1971-2012)
Developing countries depend heavily on primary commodities for their export earnings which exposes them to commodity price shocks. The commodity price shocks are persistent and highly volatile which is the most challenging issues facing policymakers in heavily indebted countries. [28] Senhadji (2003), note that the borrowing behaviour of a developing country that relies heavily on primary commodities for its export earnings is faced with a uncertainty about the longevity of external shocks. Uncertainty concerning the longevity of shocks generates substantial debt accumulation. However, [2] Muhanji and Ojah (2011), note that external shocks significantly influence debt accumulation in African countries.
Over-borrowing during the 1970s was the result of developing countries' favourable commodity prices which in the long-run, deteriorated.       They propose to transfer risks to countries which can bear the commodity risks.
However, [33] Borensztein and Panizza (2009), demonstrate that the welfare gains associated with hedging against commodity price risks are significant in influencing export earnings. Increase in export earnings will reduce external debt burden hence achieving macroeconomic stability.

Conclusions
This paper investigates the alternative financing instruments that can be used in the selected African countries to minimise the default risks and finance development. Hedging instruments minimize shocks that are hitting the economy while minimizing the likelihood of sovereign risks and defaults. Most of the developing countries are exposed to interest rate shocks and commodity price fluctuations lead to debt accumulation [34] (Demirguc-Kunt and Detrigiache, 1994), markets are still small and illiquid to access hedging instruments that lower the Journal of Mathematical Finance sovereign default risks. Thus, it is important for policymakers to design sound financial instrument policies that will hedge sovereign risks and finance sustainable development. Secondly, African countries should use linked-bonds such as commodity linked bonds to minimize commodity price shocks. Third, African countries should be discouraged to borrow in foreign currency debt which is exposed to foreign exchange rate risks. In practice, foreign exchange rate risks are hedged through currency derivatives in the forex market. From the policy perspective, innovative financing instruments in African countries will assist policymakers in designing policies that will minimize the cost of debt and hedge sovereign risks.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.