Financial Derivatives

True. An at-the-money put has $S=X$, so its intrinsic value is $\max(X-S,0)=0$. Its price consists only of time value.

True. A put is in the money when the exercise price is above the spot price ($X > S$). Its intrinsic value is $X - S > 0$.

False. Arbitrage is a transaction designed to capture riskless profit from market inefficiency, that is, from price discrepancies for the same or equivalent assets in different markets or forms. Its activity pushes prices back into line and so makes markets more efficient.

False. The bid price is the price at which the market maker is willing to buy the option, so it is the price an investor receives when selling. The investor buys at the higher ask (offer) price, and the difference is the bid-ask spread, the market maker's compensation.

False. A call with a lower exercise price pays more at expiration, but it also costs a higher premium. The premium difference is never more than the difference in exercise prices, so net profit is not necessarily greater, and in percentage terms the higher-exercise-price call offers more leverage and gain (with a larger chance of total loss). The lower-strike call is safer, not clearly more profitable.

True. A call gives the right to buy the stock by paying the exercise price, so its value cannot exceed the stock price $S_0$. If it did, buying the stock directly would be cheaper and arbitrage would be possible. The call value would approach $S_0$ only when the exercise price is zero or the option is very deep in the money.

True. Higher volatility widens the range of possible future stock prices. Option holders gain from favourable moves but their loss is limited to the premium, so the extra upside raises the expected payoff. Hence both call and put values increase with volatility (positive vega).

False. Either party to a futures contract may lose, so the party on the losing side might default. In practice the clearinghouse becomes the counterparty to both sides, and margin requirements with daily marking to market keep default risk very low, but the credit risk is managed rather than absent for one party. This differs from forwards, where each party bears the other's credit risk directly.

True. The Securities Board of Nepal (SEBON), set up under the Securities Act, 2063, regulates the securities market, including exchange-traded derivative instruments. Nepal's derivatives market is still at a very early stage, and foreign-exchange related hedging contracts also fall under Nepal Rastra Bank's rules.

False. With annual compounding for 3 months (0.25 year):

$F_0 = S_0(1+r)^T = 40(1.06)^{0.25} = \text{Rs } 40.59$

The futures price should be about Rs 40.59, not Rs 40.7487.

True (conceptually). A currency swap exchanges interest payments on two different currencies. If both currencies are the same, the exchange rate is irrelevant and the contract becomes an exchange of fixed and floating interest on the same currency, which is an interest rate swap. The notional principals are then equal and are normally not exchanged.

True. Legal risk is the risk that a change in law or regulation, or a court ruling, makes a derivative contract illegal, unenforceable or ultra vires (for example, the government declaring derivatives illegal). The counterparty then cannot be made to perform.

Features of financial derivatives

1. Derived value: the value depends on an underlying asset, rate or index (shares, bonds, currency, interest rate, index).
2. Contract between two parties: a contract on a future date and price (forwards, futures, options, swaps).
3. Future settlement: it is settled at a specified future date, so it has a fixed maturity.
4. Leverage: only a small margin or premium is needed, so gains and losses are magnified.
5. Zero-sum nature: one party's gain is the other party's loss.
6. Risk management: they are used for hedging, speculation and arbitrage.
7. Market type: they are traded on exchanges (futures, options) or over the counter (forwards, swaps).
8. Price discovery: they provide information about expected future prices.

Issues of the derivatives market in Nepal

• No full derivatives exchange, and trading in futures and options has not started.
• The legal framework is weak: the Securities Act, Nepal Stock Exchange rules and Foreign Exchange Act do not clearly provide for derivative contracts.
• The spot capital market is small, shallow and illiquid, and few securities are actively traded.
• Investors and bankers lack awareness and trained manpower, as well as the technology for clearing, settlement and margin management.
• There is no clearing house to manage counterparty risk.
• Capital controls, a fixed peg to the Indian rupee and limited need for currency hedging.
• Fear of speculation and weak regulatory capacity at SEBON and NRB.

(a) The first column is the contract symbol. It identifies the underlying stock (MSFT), the expiration month and year, the option type (put), and the strike price of 25.

(b) The put price on the quotation day (last trade) is 1.63. A change of 0.07 means the previous day's price was 1.63 - 0.07 = 1.56, assuming the price rose.

(c) "Interest" is open interest, meaning the number of option contracts still outstanding (not closed, exercised or expired). Here it is 43,856 contracts.

(d) An investor buying the put pays the ask price, which is 1.64 per share. The buyer pays the dealer's selling price (ask).

(e) The volume of trading is 40 contracts. The bid-ask spread is $1.64 - 1.62 = 0.02$ per option.

Black-Scholes put value

$S=K=100,\ r=0.12,\ \sigma=0.16,\ T=1$

$d_1=\frac{\ln(100/100)+(0.12+0.16^2/2)(1)}{0.16\sqrt1}=\frac{0.1328}{0.16}=0.83$

$d_2=d_1-\sigma\sqrt T=0.83-0.16=0.67$

$N(d_1)=0.7967,\ N(d_2)=0.7486,\ N(-d_1)=0.2033,\ N(-d_2)=0.2514$

Call: $C = 100(0.7967) - 100e^{-0.12}(0.7486) = 79.67 - 66.40 = 13.28$ (approx.)

Put:

$P = Ke^{-rT}N(-d_2) - S\,N(-d_1) = 88.69(0.2514) - 100(0.2033) = 22.30 - 20.33 \approx \text{Rs } 1.97$

(Check with put-call parity: $P = C - S + Ke^{-rT} = 13.28 - 100 + 88.69 = 1.97$.)

Decision: The market price of Rs 2 is slightly above the model value of Rs 1.97, so the put is marginally overpriced. An investor should not buy it. The difference is only about Rs 0.03, so after transaction costs the put is about fairly priced. A rational investor could write (sell) the put if there is any mispricing.

Short position: 10 contracts $\times$ 100 oz = 1,000 oz at USD 276.50. A short position gains when the price falls.

Initial margin = $10 \times 2{,}500 = USD 25{,}000$; maintenance margin = $10 \times 2{,}000 = USD 20{,}000$.

(a) Daily gain/loss: -1,500; -3,000; +5,000; -4,000; +5,000.

(b) Cumulative: -1,500; -4,500; +500; -3,500; +1,500.

(c) Margin balance: 23,500; 20,500; 25,500; 21,500; 26,500.

(d) The balance never falls below the maintenance margin of USD 20,000 (lowest is USD 20,500 on day 2), so no margin call arises.

(e) If a margin call is made and the investor does not deposit, the broker closes the position at the market price, takes the loss from the margin account and returns any balance. The investor may also lose the account.

$S=956.49,\ F=960.50,\ T=73/365=0.2$

(a) Continuous compounding:

$F^*=Se^{(r-\delta)T}=956.49\,e^{(0.0596-0.0275)(0.2)}=956.49\,e^{0.00642}=962.65$

Actual futures price 960.50 < 962.65, so the futures is underpriced. Buy futures and short the index (arbitrage profit about 2.15).

(b) Annual compounding:

$F^*=S(1+r)^T - FV(D)=956.49(1.0596)^{0.2}-5.27=967.63-5.27\approx 962.36$

Since 960.50 < 962.36, the futures is again underpriced, by about 1.86.

$B(180)=\dfrac1{1+0.05(180/360)}=0.97561,\quad B(360)=\dfrac1{1+0.06}=0.94340$

(a) Fixed rate (per period):

$R=\frac{1-B(360)}{B(180)+B(360)}=\frac{1-0.94340}{0.97561+0.94340}=\frac{0.05660}{1.91900}=0.02950$

Annualised: $0.02950\times2=5.90\%$.

The fixed rate is calculated so that the swap has zero value at initiation (the PV of the fixed payments equals the PV of the floating payments), so that neither party pays to enter it.

(b) Value to the fixed payer:

PV of floating = $(100m)(1+0.025)(0.97561)=\text{Rs }100.00m$

PV of fixed = $2.950m(0.97561)+102.950m(0.94340)=2.878+97.122=\text{Rs }100.00m$

Swap value $=100.00-100.00\approx 0$ (zero at initiation).

(c) First period:

• Fixed payment $=100m\times0.0295=$ Rs 2.950m
• Floating payment $=100m\times0.05\times180/360=$ Rs 2.500m
• Net payment: the fixed payer pays Rs $2.950-2.500=$ Rs 0.450m to the floating payer.

$S=500,\ X=500,\ S_u=600,\ S_d=400,\ r=10\%$. Put payoffs: $P_u=0,\ P_d=100$.

(a) $u=1.2,\ d=0.8$. Risk-neutral probability: $p=\dfrac{1.10-0.80}{1.20-0.80}=0.75$.

$P=\frac{0.75(0)+0.25(100)}{1.10}=\text{Rs }22.73$

This is the fair price because the hedge portfolio of stock and puts is riskless and must earn the risk-free rate; any other price allows arbitrage.

(b) A hedge portfolio combines the stock and the option so that its end value is the same in all states (riskless). Yes, a long stock with long puts works. The hedge ratio is $h=\dfrac{600-400}{0-100}$, so 2 puts per share are bought.

(c) Portfolio: 1 share + 2 puts.

• At Rs 600: $600+2(0)=600$
• At Rs 400: $400+2(100)=600$

The value is Rs 600 in both cases.

(d) Cost today $=500+2(22.73)=545.45$. The return is $600/545.45-1=10\%$, the risk-free rate, because the portfolio is riskless.

(e) A put at Rs 20 is underpriced (fair Rs 22.73). Buy 1 share and 2 puts (cost $500+40=540$) by borrowing Rs 540 at 10%. At expiry, the portfolio is worth Rs 600 in either case and the loan is $540(1.10)=594$.

Profit $=600-594=\text{Rs }6$ per share held, risk-free.

Contract = 100 shares, but profits are shown per share (multiply by 100 for a contract).

For the figures below: the Oct 165 put is not given in the table, so its premium is found from put-call parity, $P=C-S+Ke^{-rT}$, and the same method gives the Aug 165 put.

(a) Long Aug 165 call (premium 5.25). Profit $=\max(S_T-165,0)-5.25$

Breakeven $=165+5.25=170.25$. Max loss $=5.25$ per share (Rs 525 per contract); max gain is unlimited. The graph rises from -5.25 flat until 165, then rises along a 45-degree line.

(b) Long Oct 165 put. Parity: $P=8.10-165.13+165e^{-0.0571(102/365)}=8.10-165.13+162.39=5.36$. Profit $=\max(165-S_T,0)-5.36$

Breakeven $=165-5.36=159.64$. Max loss $=5.36$ per share. Max gain $=159.64$ per share (if the stock falls to zero). The graph falls as a line at 45 degrees until 165, then is flat at -5.36.

(c) Covered call: buy stock at 165.13 and write the Oct 165 call for 8.10. Net cost $=157.03$. Profit $=S_T-165.13+8.10-\max(S_T-165,0)$

Breakeven $=157.03$. Max profit $=7.97$ per share (Rs 797 for 100 shares), reached at $S_T\ge165$. Max loss $=157.03$ per share (if stock falls to zero). The graph rises at 45 degrees up to 165, then is flat.

(d) Protective put: buy stock at 165.13 and the Aug 165 put. Parity: $P=5.25-165.13+165e^{-0.0535(46/365)}=4.01$. Cost $=165.13+4.01=169.14$. Profit $=\max(S_T,165)-169.14$

Breakeven $=169.14$. Max loss $=4.14$ per share (Rs 414 for 100 shares). Max profit is unlimited. The graph is flat at -4.14 until 165, then rises at 45 degrees.

False. Derivatives are used for hedging (reducing risk), arbitrage and speculation. Leverage is true for many derivatives, but they are not purely speculative. Hedgers such as farmers, exporters and banks use them to reduce risk.

False. In a protective put (long stock plus long put) the loss is limited, but the gain is unlimited because the stock can rise without limit:

$\text{Profit}=\max(S_T,X)-S_0-P$

The maximum loss is $S_0-X+P$, which is the expression in the statement.

False. Investors who ignore risk are said to be risk-neutral (they care only about expected return). Short selling is selling borrowed securities in the hope of buying them back at a lower price.

True. A short call earns only the premium received (limited gain). If the stock price rises there is no upper limit to the loss, so the strategy is bearish (or neutral).

False. There is a positive relation between the interest rate and the call value. A higher rate reduces the present value of the exercise price, making the call more valuable. (Puts have an inverse relation.)

False. In a 2-for-1 split, the exercise price is halved to Rs 12.50 and the number of options is doubled. The adjusted exercise price becomes Rs 12.50; it is the stock price that is halved, not doubled.

True. Put delta is $N(d_1)-1$ and lies between -1 and 0. A negative delta means that when the stock price rises, the put price falls, and vice versa, so the two are inversely related.

True. Under certainty the futures price equals the spot price plus the cost of carry, $F=S+\text{cost of carry}=S(1+r)^T$, where the cost of carry is the interest and storage cost, less any benefit from holding the asset.

False. Swaps are custom-made contracts between counterparties and are traded in the over-the-counter market, not on organized exchanges. Futures are standardized and sold on exchanges.

True. In a currency swap the notional principals in the two currencies are usually exchanged at the start and re-exchanged at the end at the same exchange rate. In interest rate swaps there is no such exchange.

Current issues of the derivatives market in Nepal

1. No organized market: there are no exchange-traded futures or options, so there is no price discovery for derivatives.
2. Weak legal framework: the Securities Act, the Foreign Exchange Act and the Company Act do not clearly provide for derivative contracts. Their enforceability is uncertain.
3. Underdeveloped spot market: the NEPSE market is small, illiquid and dominated by a few sectors, and there are few underlying assets.
4. No clearing house or margin system to control counterparty risk and settlement.
5. Regulatory and capacity gaps: SEBON and Nepal Rastra Bank lack skills and rules for derivative products. Technology and trained manpower are limited.
6. Currency peg and capital controls: the fixed peg to the Indian rupee and foreign exchange restrictions reduce the need for and the permission to use currency derivatives.
7. Lack of awareness and fear of speculation among investors, banks and corporations.

Way forward: amend the laws, establish a clearing corporation, begin with simple products such as index futures, train market participants, and encourage banks to offer hedging products.

(a) The call (S = 120 > K = 100) is in-the-money. The put (S = 105 > K = 100) is out-of-the-money.

(b) Intrinsic value:

• Call: $\max(120-100,0)=20$
• Put: $\max(100-105,0)=0$

(c) Time value = option price - intrinsic value:

• Call: $26-20=6$
• Put: $3.5-0=3.5$

(d) $T=90/365=0.2466$ and $PV(K)=100/(1.06)^{0.2466}=98.57$.

• European call lower bound: $\max(0,\ S-PV(K))=120-98.57=\text{Rs }21.43$ (the actual price of 26 is above this).
• European put lower bound: $\max(0,\ PV(K)-S)=\max(0,98.57-105)=0$ (the actual price of 3.5 is above this).

$S=140,\ X_1=125\ (C_1=20),\ X_2=130\ (C_2=12)$.

Rule: For American calls that differ only in exercise price, $C(X_1)\ge C(X_2)$, and $C(X_1)-C(X_2)\le X_2-X_1$.

Here $C_1-C_2=20-12=8$, which exceeds $X_2-X_1=130-125=5$. So the second part of the rule is violated: the Rs 125 call is overpriced relative to the Rs 130 call. (The first part holds because 20 > 12.)

Arbitrage: Sell the 125 call at Rs 20 and buy the 130 call at Rs 12, for a net inflow of Rs 8.

The minimum profit is Rs 3 per share, with no investment. If the short call is exercised early, the strategy still gives at least Rs 3, because the long 130 call can be exercised to cover the delivery (loss of at most 5 against 8 received).

$S=500,\ X=400,\ S_u=800,\ S_d=200,\ r=8\%$. Call payoffs: $C_u=400,\ C_d=0$.

(a) $u=1.6,\ d=0.4$; $p=\dfrac{1.08-0.4}{1.6-0.4}=0.5667$.

$C=\frac{0.5667(400)+0.4333(0)}{1.08}=\text{Rs }209.88$

(Check by hedge ratio: $h=\dfrac{400-0}{800-200}=\dfrac23$ share; $C=\dfrac23(500)-\dfrac{\frac23(200)}{1.08}=333.33-123.46=209.88$.)

(b) At Rs 220 the call is overpriced (fair 209.88). Arbitrage:

• Sell the call: receive 220
• Buy $\tfrac23$ share: pay 333.33
• Borrow $333.33-220=113.33$ at 8%; repay $113.33(1.08)=122.40$

At expiry:

• If $S=800$: stock worth $533.33$, pay the call holder 400, net $133.33$; less the loan 122.40 gives Rs 10.93.
• If $S=200$: stock worth $133.33$, call expires worthless; less loan gives Rs 10.93.

Riskless profit is Rs 10.93 at expiry (Rs 10.12 today).

$S=230,\ X=215,\ T=0.25,\ r=6\%,\ \sigma=0.50$

$d_1=\frac{\ln(230/215)+(0.06+0.5^2/2)(0.25)}{0.5\sqrt{0.25}}=\frac{0.0675+0.04625}{0.25}=0.4548$

$d_2=d_1-0.5\sqrt{0.25}=0.2048$

$N(d_1)=0.6754,\ N(d_2)=0.5811$

$C=SN(d_1)-Xe^{-rT}N(d_2)=230(0.6754)-215e^{-0.015}(0.5811)$

$=155.34-211.80(0.5811)=155.34-123.08\approx \text{Rs }32.25$

The fair price of the European call is about Rs 32.25.

(a) Buy the stock at Rs 2,500 and at the same time buy a 6-month put with strike Rs 2,500 for Rs 500. This sets a floor on the sale price of Rs 2,500.

(b) Cost $=2{,}500+500=\text{Rs }3{,}000$.

(c) Breakeven stock price $=S_0+P=2{,}500+500=\text{Rs }3{,}000$.

(d) Profit $=\max(S_T,2500)-3000$:

Graph: the profit line is flat at -Rs 500 up to Rs 2,500, crosses zero at Rs 3,000, and then rises at 45 degrees (unlimited upside). The maximum loss is Rs 500 (the put premium) and the upside is unlimited.

Notional $N=$ Rs 60 million. $B(90)=0.9780,\ B(180)=0.9524$.

(a) Swap 1 fixed rate (per period):

$R=\frac{1-B(180)}{B(90)+B(180)}=\frac{1-0.9524}{0.9780+0.9524}=0.02467$

Annualised: $0.02467\times4=9.87\%$. Fixed payment per period $=60m\times0.02467=$ Rs 1.480m.

(b) Day 90 payments (fund receives index 1 return):

Index 1 return $=\dfrac{5609.81}{5514.67}-1=1.725\%$, giving $60m\times0.01725=$ Rs 1.035m received.

• Swap 1 (pay fixed): pays 1.480m. Net: fund pays Rs 0.445m.
• Swap 2 (pay floating): pays $60m\times0.09\times90/360=$ Rs 1.350m. Net: fund pays Rs 0.315m.
• Swap 3 (pay index 2): return $=\dfrac{1231.94}{1212.98}-1=1.563\%$, so pays Rs 0.938m. Net: fund receives Rs 0.097m.

(c) Day 30. Equity leg value $=60m\times\dfrac{5499.62}{5514.67}=$ Rs 59.836m.

• Swap 1: PV of fixed $=1.480(0.9888)+61.480(0.9715)=1.464+59.728=61.191$. Value $=59.836-61.191=$ -Rs 1.355m.
• Swap 2: PV of floating $=60(1.0225)(0.9888)=60.663$. Value $=59.836-60.663=$ -Rs 0.827m.
• Swap 3: PV of index 2 leg $=60\times\dfrac{1201.45}{1212.98}=59.430$. Value $=59.836-59.430=$ +Rs 0.407m.

Swaps 1 and 2 have a negative value to the fund; swap 3 has a positive value.

$S=1500,\ r=8\%,\ T=1$, storage Rs 25 at 6 months and Rs 25 at 12 months, contract size 100 Tola.

(a) Forward price:

$F_0=S(1+r)^T+FV(\text{storage})=1500(1.08)+25(1.08)^{0.5}+25=1620+25.98+25=\text{Rs }1{,}670.98\text{ per Tola}$

Per contract: $1{,}670.98\times100=$ Rs 167,098.

(b) The market forward price of Rs 1,625 is below Rs 1,670.98, so the forward is underpriced. Strategy:

• Buy (go long) the forward at Rs 1,625.
• Sell the silver (or short it) at Rs 1,500 per Tola, and invest the proceeds at 8%, which grows to Rs 1,620. The storage cost that would have been paid (value Rs 50.98 at the end of the year) is saved.
• At expiry, buy the silver under the forward for Rs 1,625 and return it.

Riskless profit $=1{,}670.98-1{,}625=$ Rs 45.98 per Tola, or Rs 4,598 per contract.

(c) The initial value of the forward contract is zero, because the forward price is set equal to the fair price (so that the contract costs nothing at entry).

(d) After 3 months: $S_t=1350$, 9 months remain, storage payments are due in 3 months and 9 months.

$F_t=1350(1.08)^{0.75}+25(1.08)^{0.25}+25=1{,}430.22+25.48+25\approx\text{Rs }1{,}480.70$

Value of the short forward $=PV(F_0-F_t)=\dfrac{1670.98-1480.70}{(1.08)^{0.75}}\approx\text{Rs }179.6$ per Tola, or about Rs 17,961 per contract.

The short position gains because the silver price has fallen.

True. A derivative gets its value from an underlying asset, rate or index, such as a stock, bond, commodity, currency or interest rate. That is why it is called a derivative.

True. An American put can be exercised at any time, and the most it can pay is $X$ (when the stock price falls to zero). So the maximum value of an American put is the exercise price $X$ (the European put maximum is $X/(1+r)^T$).

True. A riskless hedge (as in the binomial model) involves buying the stock and buying puts. The gain on the puts offsets losses on the stock when the price falls, and the portfolio is the same in all states (a protective put).

True. The breakeven of a protective put is $S_0+P$. A put with a higher exercise price costs more, so the premium $P$ is higher, which raises the breakeven price (although the floor is also higher).

False. There is a margin call because the balance (Rs 3,100) is below the maintenance margin (Rs 3,500). But the investor must deposit enough to bring the balance back to the initial margin:

$5{,}000-3{,}100=\text{Rs }1{,}900,$

not Rs 400.

True. For commodities, the futures price is the spot price compounded at the risk-free rate, plus the future value of storage costs (less any convenience yield): $F=S(1+r)^T+FV(\text{storage})$.

False. Unlike interest rate and currency swaps, whose payments are generally positive, equity swap payments can be negative. If the equity index falls, the return is negative, so the equity payer must pay the counterparty (in addition to any other leg).

True. A forward contract usually has one delivery (settlement) date, when the underlying is delivered and paid for. Swaps are a series of forward contracts with multiple settlement dates.

True. With a dividend, the lower bound of the European call is

$c\ge S-PV(D)-\frac{X}{(1+r)^T}=100-1-\frac{100}{1.05}=99-95.24=\text{Rs }3.76$

Reasons for using financial derivatives

• Risk management (hedging): A firm or investor can transfer price risk (of shares, interest rates, currencies, commodities) to someone willing to bear it, by taking an offsetting derivative position.
• Price discovery: Futures and option prices reflect the market's expectations of future spot prices, helping producers and investors plan.
• Low transaction cost and leverage: A small margin or premium gives exposure to a large underlying value, so positions are cheaper and quicker to set up than trading the asset itself.
• Market efficiency and liquidity: Arbitrageurs remove mispricing between spot and derivative markets, which keeps prices aligned.
• Access to otherwise unavailable positions: Derivatives allow short exposure, customised cash flows (swaps) and exposure to assets that are hard to trade directly.
• Speculation: Traders take views on price movements at low cost, which adds liquidity to the market.

Three ways in which derivatives can be misused

1. Excessive speculation instead of hedging: Because of leverage, a small adverse price movement creates very large losses, so a firm that was meant to hedge ends up gambling (Barings Bank, 1995).
2. Taking positions without understanding them: Complex products (exotic options, structured swaps) may carry hidden risks. Poor valuation and weak controls lead to unexpected losses (Orange County, 1994).
3. Concealing risk or manipulating accounts: Off-balance-sheet positions, rogue traders and mis-reporting can hide losses and raise systemic risk, as in the build-up to the 2008 crisis with credit derivatives.

Given: put on 100 shares, exercise price X = Rs 300, premium = Re 4 per share (total Rs 400).

(a) Stock falls to Rs 260: Yes, exercise. The put lets you sell at Rs 300 when the market price is only Rs 260.

Gain from exercising = $(300 - 260) \times 100 = \text{Rs } 4{,}000$

Net gain after premium = $4{,}000 - 400 = \text{Rs } 3{,}600$

(b) Exercise and immediately buy the stock back: You sell at Rs 300 and buy at Rs 260, so the profit is Rs 40 per share, i.e. Rs 4,000. After the Rs 400 premium the net gain is Rs 3,600 (Rs 36 per share).

(c) Stock rises to Rs 360: No, do not exercise. Exercising would mean selling at Rs 300 shares worth Rs 360, a loss of Rs 60 per share. Let the put expire worthless. The only loss is the premium paid, Rs 400, which is the maximum loss of a put buyer.

Given: $S_0 = 230$, $X = 200$, $T = 0.25$ year, $\sigma = 0.5$, market price of call = Rs 45. The risk-free rate is not stated, so 5% per annum is assumed.

$d_1 = \frac{\ln(230/200) + (0.05 + 0.5^2/2)(0.25)}{0.5\sqrt{0.25}} = \frac{0.1398 + 0.04375}{0.25} = 0.734$

$d_2 = d_1 - \sigma\sqrt{T} = 0.734 - 0.25 = 0.484$

$N(d_1) = 0.7685$, $N(d_2) = 0.6858$

$C = S_0 N(d_1) - X e^{-rT} N(d_2) = 230(0.7685) - 200 e^{-0.0125}(0.6858)$

$C = 176.76 - 135.46 \approx \text{Rs } 41.3$

Strategy: The market price (Rs 45) is higher than the model value (about Rs 41.3), so the call is overpriced. Sell (write) the call at Rs 45 and hedge by buying $N(d_1) \approx 0.77$ share per option, financed by borrowing. The gap of about Rs 3.7 per call is the arbitrage profit as the market price converges to the model value.

Given: futures price Rs 414, 10 contracts long. Initial margin = $20 \times 10 =$ Rs 200, maintenance margin = $15 \times 10 =$ Rs 150. Contracts are priced at Rs 414 on the starting day.

(Daily gain = price change × 10 contracts.) On day 5 the balance of Rs 140 falls below Rs 150, so a call is issued for Rs 60 to bring the account back up to the initial margin of Rs 200.

Price triggering a margin call: The balance falls below Rs 150 when the cumulative loss exceeds Rs 50, i.e. Rs 5 per contract. From a purchase price of Rs 414 (with no earlier gains) this means a futures price below Rs 409. Day 5 (Rs 408) is the first price that crosses this level.

If the margin call is not met: The broker will close out (liquidate) the position at the prevailing price, and the investor bears the loss. The remaining margin is returned after settling losses, and the broker may restrict the account.

(a) Buying the stock and writing a call on it is a covered call position.

(b) Profit per share = (stock price at expiration − 77) − call payoff + 6, where call payoff = $\max(S_T - 80, 0)$.

(c)

• Maximum profit $= (80 - 77) + 6 =$ Rs 9 per share, i.e. Rs 900, for any $S_T \ge 80$.
• Break-even $= 77 - 6 =$ Rs 71 per share.
• Maximum loss occurs if the stock falls to zero: $77 - 6 =$ Rs 71 per share, i.e. Rs 7,100.

Given: $S_0 = 500$, $r = 8\%$ (continuous), $T = 0.5$ year. Dividends of Rs 10 at 2 and 5 months.

(a) PV of dividends:

$I = 10e^{-0.08(2/12)} + 10e^{-0.08(5/12)} = 9.868 + 9.672 = 19.54$

$F_0 = (S_0 - I)e^{rT} = (500 - 19.54)e^{0.04} = \text{Rs } 500.07$

The forward price is set so the contract has zero value at the start, so the initial value of the forward is zero.

(b) After 3 months: $S = 480$, remaining time = 3 months, and only the 5-month dividend remains (due in 2 months).

$I = 10e^{-0.08(2/12)} = 9.868$

$F = (480 - 9.868)e^{0.08(0.25)} = 470.13 \times 1.0202 = \text{Rs } 479.63$

Value of the short forward with delivery price $K = 500.07$:

$f_{short} = (K - F)e^{-rT} = (500.07 - 479.63)e^{-0.02} = 20.44 \times 0.9802 \approx \text{Rs } 20.03$

The short position gains about Rs 20.03 per share because the price has fallen.

Given: $S_0 = 200$, $X = 200$, two periods of six months, $1 + r = 1.0408$.

$u = 223.6/200 = 1.118$ and $d = 178.88/200 = 0.8944$

$p = \frac{(1+r) - d}{u - d} = \frac{1.0408 - 0.8944}{1.118 - 0.8944} = 0.6547$

Terminal call values (X = 200): uu: 250 → 50; ud: 200 → 0; dd: 160 → 0.

Step 1 (after six months):

$C_u = \frac{0.6547(50) + 0.3453(0)}{1.0408} = \text{Rs } 31.45, \qquad C_d = 0$

Step 2 (today):

$C_0 = \frac{0.6547(31.45) + 0.3453(0)}{1.0408} = \text{Rs } 19.79$

The fair value of the call is about Rs 19.79.

Strategy if the call sells at Rs 10: The call is underpriced (Rs 10 < Rs 19.79), so buy the call and hedge by short selling the stock. The hedge ratio at time 0 is

$\Delta = \frac{31.45 - 0}{223.6 - 178.88} = 0.703 \text{ share per call}$

Short 0.703 share (proceeds ≈ Rs 140.7), pay Rs 10 for the call and lend the balance at 4.08% per period. The position is riskless in each period and revalued at each node. Expected arbitrage profit is about Rs 9.79 per call.

(a) Fixed rate. Discount factors $B_0(t) = \dfrac{1}{1 + L_0(t)\,(t/360)}$:

$\sum B_0 = 3.72448$

$\text{Fixed rate per period} = \frac{1 - B_0(720)}{\sum B_0} = \frac{1 - 0.88261}{3.72448} = 0.03152$

$\text{Annualized fixed rate} = 0.03152 \times 2 = 6.30\% \ (\approx 6.304\%)$

(b) Market value later. The new curve gives the remaining payment dates at 20, 200, 380 and 560 days. Discount factors:

$\sum B = 3.79881$

PV of the fixed leg per unit of notional:

$0.03152(3.79881) + 0.90218 = 0.11974 + 0.90218 = 1.02192$

Value of the equity leg per unit is the index ratio: $\dfrac{1204.10}{1150.89} = 1.04623$.

$\text{Value to the equity receiver} = (1.04623 - 1.02192) \times 10{,}000{,}000 \approx \text{Rs } 243{,}000$

The swap has a positive value of about Rs 2.43 lakh to the asset manager, who receives the equity return and pays fixed.

(c) Equity swap. An equity swap is a contract in which two parties exchange periodic payments, one based on the return of a stock or stock index (dividends plus price change) and the other on a fixed or floating interest rate (e.g. LIBOR), applied to an agreed notional principal. No principal is exchanged. It is used to gain or reduce exposure to an equity market without buying or selling the shares, to avoid transaction costs, taxes and restrictions, and to shift a portfolio between equity and fixed income quickly.

True. A derivative gets its value from an underlying asset (a stock, index, commodity, currency or interest rate), so its price moves with the value of that asset.

True. A longer time to expiration gives the stock more time to rise above the exercise price. This raises the time value, so a call's value generally increases with time to expiration.

False. The same risk-neutral probability of an up move ($p = \frac{1+r-d}{u-d}$) and of a down move ($1-p$) is used for puts and calls. Only the payoff changes, to $\max(X - S, 0)$.

False. The put buyer's loss is limited to the premium paid. The put seller's loss is large but limited, because the stock price cannot fall below zero (maximum loss = $X$ − premium). Unlimited loss applies to the writer of an uncovered call.

False. A margin call is issued only when the balance falls below the maintenance margin. Rs 4,000 is above Rs 3,500, so there is no margin call.

True. A futures contract has zero value at initiation. Afterwards, with daily mark-to-market, a position can show a negative value when the price moves against it (for example, a long position after prices fall).

True. A swap where both parties pay floating rates tied to different indices (e.g. LIBOR against a T-bill rate) is called a basis swap.

False. Both the buyer and the seller of a futures contract must deposit the initial margin with the clearing house through their brokers, because both can lose.

True. Intrinsic value of the put = $X - S = 250 - 235 =$ Rs 15. Time value = premium − intrinsic value = $20 - 15 =$ Rs 5.

Derivative market: A derivative market is a financial market where contracts such as forwards, futures, options and swaps are traded, whose value is derived from an underlying asset (stocks, bonds, commodities, currencies, interest rates, indices).

Major functions in an economy

1. Price discovery: Futures and option prices show the market's expectations of future spot prices, which helps in production and investment decisions.
2. Risk transfer (hedging): Hedgers pass price, interest rate and exchange rate risk to speculators who are willing to bear it.
3. Market efficiency: Arbitrage between spot and derivative markets removes mispricing and brings prices in line.
4. Liquidity and market depth: Derivatives attract hedgers, speculators and arbitrageurs, which makes the underlying markets more liquid.
5. Lower transaction cost and leverage: Positions are taken with small margins or premiums, so large exposures are cheaper to build.
6. Capital allocation and financial innovation: Stable, predictable cash flows encourage investment and new products. Derivatives also make markets more complete by allowing short positions and tailored payoffs.

Given: long put, X = Rs 130, premium = Rs 12. Profit = $\max(130 - S_T, 0) - 12$.

(a)

(b) Breakeven price $= X - \text{premium} = 130 - 12 =$ Rs 118. This agrees with part (a): the profit is −2 at Rs 120 and +8 at Rs 110, so it is zero at Rs 118.

(c) Maximum loss = premium = Rs 12 per share (if $S_T \ge 130$). Maximum profit = $130 - 12 =$ Rs 118 per share (if the stock falls to zero).

Given: $S = 260$, $X = 250$, $T = 0.5$, $r = 6\%$, and the standard deviation for the six months is 0.30, so $\sigma\sqrt{T} = 0.30$.

$d_1 = \frac{\ln(260/250) + 0.06(0.5) + \frac{1}{2}(0.30)^2}{0.30} = \frac{0.03922 + 0.03 + 0.045}{0.30} = 0.381$

$d_2 = 0.381 - 0.30 = 0.081$

$N(d_1) = 0.6483$, $N(d_2) = 0.5322$

$C = 260(0.6483) - 250e^{-0.03}(0.5322)$

$C = 168.56 - 250(0.97045)(0.5322) = 168.56 - 129.12 \approx \text{Rs } 39.45$

The value of the call option is about Rs 39.4.

Given: $S_0 = 900$, $T = 30/365$, $r = 5\%$.

(a) Futures price with no carrying cost (continuous compounding):

$F_0 = S_0 e^{rT} = 900\,e^{0.05(30/365)} = 900 \times 1.004119 \approx \text{Rs } 903.71$

(b) With storage cost of Rs 30 payable at expiration:

$F_0 = S_0 e^{rT} + \text{storage} = 903.71 + 30 = \text{Rs } 933.71$

(c) Similarities of futures and forward contracts

• Both are agreements to buy or sell an asset at a fixed price on a future date.
• Both are used for hedging and speculation, and both have zero value at initiation.
• Both are priced by the cost-of-carry model, and their prices converge to the spot price at maturity.
• Both can be settled by delivery or in cash, and both carry a long and a short position.
• If interest rates are constant, futures and forward prices for the same maturity are the same.

Given: long 1 contract (1 kg) at Rs 8,400,000. Initial margin = Rs 200,000, maintenance margin = Rs 150,000.

On day 2 the balance of Rs 104,000 is below the maintenance margin of Rs 150,000, so the investor must deposit Rs 96,000 to bring it back to the initial margin of Rs 200,000. Excess margin on day 4 is not withdrawn. Check for day 5: $200{,}000 + 96{,}000 - 50{,}000 = 246{,}000$.

Price that triggers a margin call: A call occurs when the balance falls below Rs 150,000, i.e. after a loss of more than Rs 50,000. The trigger price is $8{,}400{,}000 - 50{,}000 =$ Rs 8,350,000 (a margin call follows if the price falls below this level, as with Rs 8,304,000 on day 2).

If the margin call is not met: The broker closes out the position at the current price, the loss is deducted from the margin, and any balance is returned to the investor.

Given: the manager pays the small-cap index return and receives fixed 5.5%. Notional = Rs 50,000,000.

Fixed payment per period $= 50{,}000{,}000 \times 0.055 \times \frac{180}{360} = \text{Rs } 1{,}375{,}000$ (received by the manager).

(a) Index return $= \dfrac{238.41}{234.10} - 1 = 1.841\%$

Equity payment $= 50{,}000{,}000 \times 0.01841 = \text{Rs } 920{,}547$ (paid by the manager).

Net $= 1{,}375{,}000 - 920{,}547 =$ Rs 454,453, paid by the dealer to the manager.

(b) Index return $= \dfrac{241.27}{234.10} - 1 = 3.063\%$

Equity payment $= 50{,}000{,}000 \times 0.03063 = \text{Rs } 1{,}531{,}397$ (paid by the manager).

Net $= 1{,}375{,}000 - 1{,}531{,}397 = -$ Rs 156,397, so the manager pays Rs 156,397 to the dealer.

Given: $S_0 = 400$, $X = 350$, $u = 1.2$, $d = 0.8$, $1 + r = 1.05$ per period.

$p = \frac{1.05 - 0.8}{1.2 - 0.8} = 0.625, \qquad 1 - p = 0.375$

Stock prices: $S_u = 480$, $S_d = 320$; $S_{uu} = 576$, $S_{ud} = 384$, $S_{dd} = 256$.

Terminal call values: $C_{uu} = 226$, $C_{ud} = 34$, $C_{dd} = 0$.

One period before expiration:

$C_u = \frac{0.625(226) + 0.375(34)}{1.05} = \frac{141.25 + 12.75}{1.05} = 146.67$

Early exercise would give $480 - 350 = 130 < 146.67$, so hold.

$C_d = \frac{0.625(34) + 0.375(0)}{1.05} = 20.24$

Exercise value is 0, so hold.

Today:

$C_0 = \frac{0.625(146.67) + 0.375(20.24)}{1.05} = \frac{91.67 + 7.59}{1.05} = \text{Rs } 94.53$

Early exercise today would give only $400 - 350 = 50$. The American call is worth the same as a European call here because no dividend is paid. The maximum price to pay is about Rs 94.53.

Strategy if the call sells for Rs 30: The call is underpriced by about Rs 64.53. Buy the call and short sell the stock to hedge. The hedge ratio is

$\Delta = \frac{C_u - C_d}{S_u - S_d} = \frac{146.67 - 20.24}{480 - 320} = 0.79 \text{ share}$

Short 0.79 share per call, invest the net proceeds at the risk-free rate, and rebalance each period. The hedge removes price risk and locks in a profit of about Rs 64.5 per call.

Given: $S_0 = 170$, call premium = Rs 20, put premium = Rs 10, $X = 170$. Figures are per share; for a contract of 100 shares multiply by 100.

(a) Long call: Profit $= \max(S_T - 170, 0) - 20$

Breakeven $= 170 + 20 =$ Rs 190. Maximum loss = Rs 20 per share (Rs 2,000 per contract). Maximum gain is unlimited.

(b) Long put: Profit $= \max(170 - S_T, 0) - 10$

Breakeven $= 170 - 10 =$ Rs 160. Maximum loss = Rs 10 per share (Rs 1,000 per contract). Maximum gain $= 170 - 10 =$ Rs 160 per share (Rs 16,000), if the price falls to zero.

(c) Covered call (buy stock at 170, write call at 20): Profit $= (S_T - 170) - \max(S_T - 170, 0) + 20$

Breakeven $= 170 - 20 =$ Rs 150. Maximum profit = Rs 20 per share (Rs 2,000), for any price of Rs 170 or more. Maximum loss $= 170 - 20 =$ Rs 150 per share (Rs 15,000), if the stock falls to zero.