Derivatives Analysis and Valuation — Practice

Question 1

A stock's current spot price is ₹500, and it pays no dividend over the next 6 months. The risk-free rate is 8% per annum. Compute the theoretical 6-month forward price, and explain the arbitrage available if the actual quoted forward price is ₹530.

Accepted Answer

Theoretical forward price = S₀ × (1+r)^T = ₹500 × (1.08)^0.5 = ₹500 × 1.0392 = ₹519.62 (approximately). Since the actual quoted forward price of ₹530 exceeds this theoretical price, an arbitrageur can execute a cash-and-carry strategy: borrow ₹500 at the risk-free rate today, buy the stock at the spot price of ₹500, and simultaneously sell (go short) the overpriced forward contract at ₹530; at expiry in 6 months, the arbitrageur delivers the stock against the forward contract, receiving ₹530, while owing only ₹500 × 1.0392 = ₹519.62 to repay the borrowing, locking in a risk-free arbitrage profit of ₹530 − ₹519.62 = ₹10.38 per share, with no net investment and no risk, which is precisely why such a mispricing would not persist in a genuinely liquid, efficient market — arbitrage activity of exactly this kind would push the forward price back down toward its theoretical value.

Question 2

A stock index currently stands at 18,000, with a continuous dividend yield of 1.5% per annum. The risk-free rate is 7% per annum (continuous compounding), and the futures contract expires in 3 months. Compute the theoretical futures price.

Accepted Answer

Using the continuous compounding cost-of-carry formula adjusted for a dividend yield: F = S₀ × e^((r−q)×T) = 18,000 × e^((0.07−0.015)×0.25) = 18,000 × e^(0.055×0.25) = 18,000 × e^0.01375 = 18,000 × 1.01384 (approximately) = 18,249 (approximately). The dividend yield is netted against the risk-free rate because holding the index directly earns the 1.5% dividend yield over the 3 months, an income the futures holder does not receive (since the futures contract itself pays no dividend), and the theoretical futures price must therefore be lower than it would be for a non-dividend-paying asset with the identical risk-free rate and time to expiry, reflecting this foregone income the index holder, but not the futures holder, actually earns.

Question 3

A jewellery manufacturer expects to purchase a large quantity of gold in 3 months to fulfil a festival-season order book, and is concerned that gold prices may rise before then. Separately, a gold mining company expects to sell a large quantity of gold in 3 months and is concerned prices may fall. State the correct futures hedge position for each, and explain the exposure each is protecting against.

Accepted Answer

The jewellery manufacturer, expecting to purchase gold in the future and concerned about a price rise, should take a long hedge, buying gold futures today; if gold prices do rise by the time of actual purchase, the gain on the long futures position offsets the higher price the manufacturer must pay in the physical market, locking in an effective purchase price close to today's futures price regardless of how the spot price actually moves. The gold mining company, expecting to sell gold in the future and concerned about a price fall, should take a short hedge, selling gold futures today; if gold prices do fall by the time of actual sale, the gain on the short futures position offsets the lower price the company receives in the physical market, locking in an effective selling price close to today's futures price. The two hedgers face opposite exposures (a future buyer worried about rising prices versus a future seller worried about falling prices) and therefore require opposite futures positions (long versus short) to protect themselves, illustrating precisely why settling the exposure and hedge direction explicitly, before any computation, is the discipline the method chapter emphasises for this entire cluster of chapters.

Question 4

A call option on a stock has a strike price of ₹250. The stock's current spot price is ₹270, and the call option's market premium is ₹28. Compute the option's intrinsic value and time value, and interpret what the time value figure represents.

Accepted Answer

Intrinsic value of the call = Spot price − Strike price (if positive) = ₹270 − ₹250 = ₹20; since this is positive, the option is 'in the money' and has genuine intrinsic value if exercised immediately. Time value = Market premium − Intrinsic value = ₹28 − ₹20 = ₹8. This ₹8 of time value represents the additional amount option buyers are willing to pay beyond the option's immediate exercise value, reflecting the possibility that the option's intrinsic value could increase further before expiry (the stock price could rise even higher, increasing the call's eventual payoff), and this time value component will progressively shrink toward zero as expiry approaches, a phenomenon known as time decay, since less time remains for the stock price to move favourably the closer the option gets to its expiry date, with the time value reaching exactly zero at expiry itself, when the option's value equals only its intrinsic value with no remaining time for further favourable movement.

Question 5

A European call option with a strike price of ₹400 and 6 months to expiry trades at a premium of ₹35. The underlying stock's spot price is ₹390, and the risk-free rate is 8% per annum. Using put-call parity, compute the theoretical price of the corresponding European put option with the same strike and expiry.

Accepted Answer

Put-call parity: Call premium + PV(Strike) = Put premium + Spot price. PV(Strike) = ₹400 ÷ (1.08)^0.5 = ₹400 ÷ 1.0392 = ₹384.91. Rearranging: Put premium = Call premium + PV(Strike) − Spot price = ₹35 + ₹384.91 − ₹390 = ₹29.91. The theoretical put premium is approximately ₹29.91. If the actual traded put premium differs materially from this figure, an arbitrage opportunity exists: constructing a long call plus a risk-free bond paying the strike at expiry, and comparing its cost against a long put plus the underlying stock, whichever combination is cheaper can be bought while the more expensive, equivalent-payoff combination is sold, locking in a risk-free profit equal to the mispricing, since both combinations produce identical payoffs at every possible future stock price at expiry and must therefore cost the same today under no-arbitrage.

Question 6

A stock currently trades at ₹100. In one period, it will either rise to ₹120 (u = 1.20) or fall to ₹90 (d = 0.90). The risk-free rate for the period is 5%. Compute the value of a call option with a strike price of ₹100 using the one-period binomial model.

Accepted Answer

Payoff in the up state, Cu = Max(120 − 100, 0) = ₹20. Payoff in the down state, Cd = Max(90 − 100, 0) = ₹0. Risk-neutral probability, p = [(1+r) − d] ÷ (u − d) = [(1.05) − 0.90] ÷ (1.20 − 0.90) = 0.15 ÷ 0.30 = 0.50. Option value = [p × Cu + (1−p) × Cd] ÷ (1+r) = [0.50 × 20 + 0.50 × 0] ÷ 1.05 = 10 ÷ 1.05 = ₹9.52 (approximately). Note that the risk-neutral probability of 0.50 used here is a purely mathematical construct derived from the no-arbitrage condition relating u, d, and the risk-free rate; it is not necessarily, and generally will not be, the actual real-world probability an investor might separately assess for the stock genuinely rising versus falling, which is precisely why this valuation approach is described as risk-neutral pricing — it produces the correct, no-arbitrage option value regardless of what the true, real-world probability of the up or down move might actually be, since the option's value is derived entirely from the replicating-portfolio, no-arbitrage argument rather than from any actual probability assessment of the underlying stock's future movement.

Question 7

A call option has a delta of 0.65. Explain what this figure means, and how many shares an option writer who has sold 1,000 call options (each on one share) should hold to maintain a delta-neutral hedge.

Accepted Answer

A delta of 0.65 means that for a small change in the underlying stock's price, the call option's price is expected to change by approximately 0.65 times that amount; a ₹1 increase in the stock price would be expected to increase the call option's premium by approximately ₹0.65, all else held constant. An option writer who has sold 1,000 call options, each on one underlying share, has effectively taken on a negative delta exposure equal to −1,000 × 0.65 = −650 (since the writer's position moves opposite to the direction a long call holder's position would move); to maintain a delta-neutral hedge, offsetting this exposure, the writer should hold a long position in 650 shares of the underlying stock, since a long position in 650 shares has a delta of exactly +650 (each share having a delta of 1 with respect to its own price), precisely offsetting the −650 delta from the short call position, so that the combined portfolio's value is, for a small price movement, approximately insensitive to changes in the underlying stock's price, at least until the option's own delta itself changes as the stock price moves (a change captured by gamma), requiring the hedge to be periodically rebalanced.

Question 8

An investor holds 1,000 shares of a stock currently trading at ₹200 each, and is concerned about a near-term price decline but does not want to sell the shares outright. Explain how a protective put strategy would address this concern, and its cost.

Accepted Answer

A protective put strategy involves the investor purchasing put options on the same stock (typically one put per share held, so 1,000 puts here) at a chosen strike price, while continuing to hold the underlying shares; this establishes a floor on the value of the combined position, since if the stock price falls below the put's strike price, the investor can exercise the puts and sell the shares at the strike price regardless of how far the market price has fallen, effectively insuring the position against losses below that strike level, while still retaining full upside participation if the stock price rises instead, since the puts would simply expire worthless in that scenario and the investor keeps the appreciated shares. The cost of this strategy is the put premium paid upfront, which represents the price of this downside insurance; this cost is incurred regardless of whether the stock price ultimately falls (in which case the insurance proves valuable) or rises (in which case the premium paid is simply a cost with no offsetting protective benefit realised), analogous to how an ordinary insurance premium is paid regardless of whether the insured event actually occurs, and the investor should weigh this certain cost against the value of the downside protection the strategy provides given their specific concern about a near-term price decline.

Question 9

An investor believes a stock, currently trading at ₹150, will experience a large price movement following an upcoming major corporate announcement, but is uncertain whether the movement will be upward or downward. Explain how a long straddle strategy would let the investor profit from this view, and the specific risk this strategy carries.

Accepted Answer

A long straddle involves simultaneously buying a call option and a put option on the same stock, at the same strike price (typically at or close to the current spot price of ₹150) and the same expiry date; this strategy profits if the stock price moves significantly in either direction, since a large upward move produces a large profit on the call (which the investor also holds) while the put simply expires worthless, and a large downward move produces a large profit on the put while the call expires worthless, meaning the investor genuinely does not need to correctly predict the direction of the move, only that a sufficiently large move will occur in one direction or the other. The specific risk this strategy carries is that if the stock price does not move significantly in either direction, remaining close to the strike price through expiry, both the call and the put will expire worthless or with minimal value, and the investor loses the entire combined premium paid for both options, meaning a straddle is a bet specifically on the magnitude of price movement (volatility) rather than its direction, and is a losing strategy precisely in the scenario where the anticipated large move fails to materialise, regardless of which direction the price might have otherwise been expected to move.

Question 10

A company has entered into a 2-year interest rate swap as the fixed-rate payer, with a notional principal of ₹100 crore, paying a fixed rate of 8% annually and receiving a floating rate, reset annually, with the next floating payment already determined at 8.5% for the coming year and a further payment expected one year later. The relevant discount rate for both years is 9%. Explain how this swap should be valued by decomposing it into a fixed-rate and a floating-rate bond, and compute its value to the fixed-rate payer.

Accepted Answer

An interest rate swap can be valued as the difference between the value of a floating-rate bond and the value of a fixed-rate bond, both with the same notional principal, since the fixed-rate payer is, in effect, long a floating-rate bond (receiving floating payments) and short a fixed-rate bond (paying fixed payments). Value of the fixed-rate bond (payer's obligation) = PV of fixed coupons + PV of notional at maturity = [₹8 crore ÷ 1.09] + [₹8 crore ÷ (1.09)²] + [₹100 crore ÷ (1.09)²] = ₹7.34 crore + ₹6.73 crore + ₹84.17 crore = ₹98.24 crore. Value of the floating-rate bond (payer's receipt) = the next floating payment (already fixed at 8.5%, since it was set at the last reset date) discounted for one year, plus the bond's own value at that reset date (which equals its face value immediately after a reset, since a floating-rate bond is always worth par immediately after each rate reset) also discounted back one year = [₹8.5 crore + ₹100 crore] ÷ 1.09 = ₹108.5 crore ÷ 1.09 = ₹99.54 crore. Value of the swap to the fixed-rate payer = Value of floating-rate bond received − Value of fixed-rate bond paid = ₹99.54 crore − ₹98.24 crore = ₹1.30 crore, a positive value to the fixed-rate payer, reflecting that the floating rate currently expected (8.5% for the next payment) exceeds the fixed rate the payer is locked into paying (8%), making the swap currently favourable to the party paying fixed and receiving floating.

Question 11

Explain why daily mark-to-market margining on a futures contract substantially reduces counterparty (credit) risk compared to an equivalent forward contract that settles only at maturity, using a scenario where a party's position moves sharply against them.

Accepted Answer

Under daily mark-to-market margining, gains and losses on a futures position are calculated and settled in cash at the end of every trading day, meaning if a party's position moves sharply against them, that loss is realised and must be paid into their margin account immediately, with the clearing house typically issuing a margin call requiring the losing party to top up their account if the balance falls below a maintenance threshold; if the party cannot meet this margin call, their position can be closed out promptly, limiting the accumulated loss to a relatively small, recent period rather than allowing it to build up over the contract's entire remaining life. Under an equivalent forward contract with no daily settlement, losses accumulate silently and are only settled at the single maturity date, meaning a party whose position has moved sharply against them over the contract's life could accumulate a very large unrealised loss with no interim mechanism forcing them to post additional collateral or face an early close-out, and by the time settlement is due at maturity, that party's ability to actually pay the accumulated loss may have deteriorated substantially, or they may simply default entirely, exposing the counterparty to the full accumulated loss with essentially no advance warning or intermediate protection, which is precisely the counterparty risk daily margining in futures markets is specifically designed to contain.

Question 12

An investor moderately bullish on a stock currently trading at ₹300 buys a call option with a strike price of ₹300 for a premium of ₹15, and simultaneously sells a call option with a strike price of ₹340 for a premium of ₹4. Explain the resulting payoff profile of this bull call spread, including the maximum profit, maximum loss, and breakeven point.

Accepted Answer

This is a bull call spread, constructed by buying a lower-strike call (₹300 strike, ₹15 premium paid) and selling a higher-strike call (₹340 strike, ₹4 premium received), for a net premium cost of ₹15 − ₹4 = ₹11. Maximum loss is limited to this net premium paid, ₹11, occurring if the stock price at expiry is at or below ₹300, since both options would expire worthless. Maximum profit is capped at the difference between the two strike prices minus the net premium paid: (₹340 − ₹300) − ₹11 = ₹40 − ₹11 = ₹29, occurring if the stock price at expiry is at or above ₹340, since above this level, the gain on the long ₹300-strike call is exactly offset, dollar for dollar, by the increasing loss on the short ₹340-strike call, capping further profit beyond this point. The breakeven point is the lower strike price plus the net premium paid: ₹300 + ₹11 = ₹311, the stock price at which the intrinsic value of the long call exactly offsets the net premium cost of the spread. This strategy trades away some of the unlimited upside a simple long call alone would offer (capped at ₹29 maximum profit here, rather than unlimited) in exchange for a materially lower net cost (₹11 instead of the full ₹15 premium for the long call alone), appropriate for an investor who is moderately, rather than strongly, bullish and does not expect the stock to rise dramatically far beyond the ₹340 level.

Question 13

An Indian company has borrowed in US dollars but earns its revenue primarily in rupees, creating a currency mismatch it wishes to eliminate without repaying and refinancing the original dollar loan. Explain how a currency swap could address this concern.

Accepted Answer

The Indian company could enter a currency swap with a counterparty (typically a bank) under which it agrees to pay rupee-denominated interest and principal to the swap counterparty, in exchange for receiving US dollar-denominated interest and principal from the counterparty, with the dollar amounts received from the swap counterparty used specifically to service the company's original dollar loan obligations to its actual lender. This effectively converts the company's economic exposure from a dollar-denominated obligation to a rupee-denominated one, since the company's own net cash outflow, after netting the swap's rupee payments against its dollar receipts (used to service the original loan), is now in rupees, the same currency as its revenue, eliminating the currency mismatch without needing to actually repay and refinance the original dollar loan itself, which could involve prepayment penalties, transaction costs, or simply not being practically or contractually possible; the swap achieves the desired economic effect of currency conversion synthetically, alongside the unchanged original loan, rather than through an actual, potentially costly or impractical refinancing of that original obligation.

Question 14

An option has a vega of 0.20. If the underlying asset's implied volatility rises from 25% to 28%, estimate the resulting change in the option's premium, and explain why vega matters specifically for option positions but has no equivalent concept for a position purely in the underlying asset itself.

Accepted Answer

Vega measures the sensitivity of an option's price to a change in the underlying's volatility, so a vega of 0.20 means the option's premium is expected to change by approximately 0.20 for each one percentage point change in implied volatility; with implied volatility rising by 3 percentage points (from 25% to 28%), the option's premium would be expected to rise by approximately 0.20 × 3 = 0.60. Vega matters specifically for option positions, and has no equivalent for a position purely in the underlying asset, because an option's value depends directly on the expected magnitude of the underlying's future price movements (higher volatility increases the chance of the option finishing significantly in the money, increasing its value, while the option holder's downside is limited to the premium paid regardless of how large an adverse move occurs), whereas a straightforward long or short position in the underlying asset itself has a value that depends only linearly on the asset's actual price level, with no direct sensitivity to how volatile that price has been or is expected to be, since a plain position gains or loses proportionately with the price change itself, not with the magnitude of uncertainty surrounding future price changes.

Question 15

An airline wants to hedge its exposure to rising jet fuel prices but no jet fuel futures contract is available in the relevant market; it instead uses crude oil futures as a proxy hedge. Explain the specific type of risk this cross-hedge introduces beyond ordinary basis risk, and why the hedge is unlikely to be perfect even if correctly sized.

Accepted Answer

Using crude oil futures to hedge jet fuel exposure is a cross-hedge, since the futures contract's underlying asset (crude oil) is not identical to the airline's actual exposure (jet fuel), and this introduces a specific risk beyond the ordinary basis risk that exists even when hedging with a futures contract on the identical underlying asset: the risk that the price relationship between crude oil and jet fuel itself changes over the life of the hedge, since jet fuel prices are influenced by refining margins, region-specific supply and demand for refined jet fuel specifically, and other factors distinct from, though correlated with, crude oil prices; if this crude-to-jet-fuel price relationship shifts unexpectedly, the crude oil futures position may not move in the same proportion as the airline's actual jet fuel cost, leaving a portion of the exposure unhedged, or even, in an extreme case, moving in the wrong direction relative to the airline's actual cost exposure. Even with a carefully calculated hedge ratio designed to minimise this cross-hedge risk based on historical correlation between crude oil and jet fuel prices, the hedge is unlikely to be perfect precisely because this historical relationship is not guaranteed to hold with the same strength going forward, meaning some residual, unhedgeable risk specific to the imperfect substitutability of the two commodities remains, on top of whatever ordinary basis risk would exist even if a genuine jet fuel futures contract had been available and used instead.

Practice — Derivatives Analysis and Valuation

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