Pricing and Hedging the Guarantee in an Equity-Linked Policy
An Indian insurer has launched a five-year single-premium equity-linked policy. The ₹100,000 premium is notionally invested in the Nifty 50 index, and the maturity benefit is the greater of the premium and 100% of the index return applied to the premium; there is no other benefit and no early surrender value. The insurer intends to hedge the guarantee by buying a five-year European put on the index struck at the index level at issue. At launch the index is 19,500, its dividend yield is 1.2% p.a., the five-year risk-free rate is 6.5% p.a. continuously compounded, and the pricing model uses a volatility of 18% p.a. Five-year puts trade in the market at an implied volatility of 22%. The insurer plans to delta-hedge by rebalancing monthly and has assumed zero transaction costs.
1. Show that the guarantee is a put option, compute its value per policy and the initial delta hedge, and state the cost as a percentage of premium.
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Decomposition. The payoff per premium P is P × max(1, S_T/S₀). Write max(1, S_T/S₀) = 1 + max(0, 1 − S_T/S₀), so the benefit is P + (P/S₀) × max(0, S₀ − S_T). The first term is a certain return of premium; the second is (P/S₀) = 100,000/19,500 = 5.1282 units of a European put struck at S₀.
Black–Scholes with a dividend yield. d₁ = [ln(S/K) + (r − q + ½σ²)T]/(σ√T) = [(0.065 − 0.012 + 0.0162) × 5]/(0.18 × 2.2361) = 0.346/0.4025 = 0.8596; d₂ = 0.8596 − 0.4025 = 0.4571.
Put = K e^(−rT) N(−d₂) − S e^(−qT) N(−d₁) = 19,500 × 0.72253 × 0.3238 − 19,500 × 0.94176 × 0.1950 = 4,562 − 3,581 = ₹981 per index unit, i.e. 5.03% of the index.
Per policy: 5.1282 × 981 = ₹5,032, or 5.0% of the premium. That is the economic cost of the guarantee, and it should be recoverable from the product's charges.
Delta. Put delta = −e^(−qT) N(−d₁) = −0.94176 × 0.1950 = −0.1836 per index unit. Per policy the exposure is 5.1282 × (−0.1836) = −0.9418 index units, worth about ₹18,365. The insurer owes the put, so it is long the hedge: it sells roughly 0.94 index units (₹18,400) for each policy sold, not buys.
2. Recompute at the market implied volatility and assess whether the guarantee is adequately priced. What does the difference tell you?
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At σ = 22%: d₁ = 0.346/(0.22 × 2.2361) = 0.346/0.4919 = 0.7033; d₂ = 0.7033 − 0.4919 = 0.2114. N(−d₁) = 0.2409, N(−d₂) = 0.4163.
Put = 19,500 × 0.72253 × 0.4163 − 19,500 × 0.94176 × 0.2409 = 5,865 − 4,424 = ₹1,441 per index unit, i.e. 7.39% of the index.
Per policy: 5.1282 × 1,441 = ₹7,392, or 7.4% of premium - some 47% more than the ₹5,032 priced at 18%.
Assessment. If the market's implied volatility is the right price of the risk, the insurer has under-charged by about ₹2,360 per policy, roughly 2.4% of premium, on every contract sold. Over a book of 50,000 policies that is nearly ₹120 million of expected loss before any hedging inefficiency.
What the difference means. A 4-point volatility change moves a five-year at-the-money put by half. The insurer's 18% is a long-run realised estimate; the market's 22% reflects the price of bearing the risk plus a volatility risk premium and, for a five-year option on an equity index, a pronounced skew - the strike at 100% of spot is not the point on the smile where the index actually trades. Using a single volatility for a five-year option calibrated to short-dated data is a serious error: the relevant input is the implied volatility for that strike and maturity, not a historical estimate. The pricing model must be recalibrated to market before launch, and the charge reviewed.
3. Explain the risks that remain after the put is bought and delta-hedged monthly. Include the measure and the practical frictions.
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Discrete rebalancing. Black–Scholes assumes continuous trading. With monthly rebalancing the hedger bears a P&L variance approximately equal to ½ Σ Γᵢ² Sᵢ² σ² Δt, i.e. it is a gamma exposure, not a delta exposure. It is small when the option is far in or out of the money, largest near the strike and near maturity, and it is not a risk the insurer can diversify away within one contract. For a five-year option on a book of policies, the hedger should estimate this cost explicitly and add it to the charge.
Path and jump risk. If the index gaps - a market halt, a political event, a trading holiday - the hedge cannot be adjusted and the loss is unbounded relative to the model. The historical data used to calibrate σ understate the tail; a jump-diffusion or a stressed-volatility scenario should inform the capital held.
Basis and frictions. The Nifty 50 is a price index in some series and a total-return index in others; the hedge must be on the same dividend basis as q = 1.2%. Futures must be rolled quarterly, incurring roll and margin costs. Transaction costs, securities transaction tax, stamp duty and the impossibility of shorting single stocks make the index hedge approximate.
Model risk. The price is a discounted risk-neutral expectation and does not depend on the real-world expected index return; the insurer should not 'expect' the fund to grow into the guarantee. Finally, the guarantee is a long-dated liability on the insurer's balance sheet: even a perfectly hedged option position leaves the insurer exposed to the failure of the hedge counterparty, to changes in the implied volatility used to value the liability, and to policyholder behaviour (lapse and paid-up terms) that the option hedge does not track.