---
title: "Dividends, Borrow and Forwards"
book: "Derivatives and Volatility"
subject: quant
language: en
chapter: 5
exercises: 8
source: https://one-course.com/books/quant/5/en/chapter/5-dividends-borrow-and-forwards
---

# Chapter 5 — Dividends, Borrow and Forwards

A share at 50 is on every broker’s hard-to-borrow list. Its one-month options look broken: at the strike of 50 the put trades at 4.04 and the call at 2.78, and a screen that computes implied volatilities from the spot and the interest rate shows the put at 72 volatility and the call at 47. A new trader sees 25 points of arbitrage: sell the dear put, buy the cheap call, hedge. There is none. Put–call parity (One Quant Book 1, chapter 25) says the gap between the call and the put is a forward, and the forward the options are pricing is 48.74, not the 50.16 that the spot and the rate suggest: the market is charging 35% a year to borrow the share, and anyone who wants to be short it through the options pays that rate either way. This chapter is about the forward: what goes into it for a share (dividends and the cost of borrowing), how dividends enter an option model, how the chain itself reveals the forward, and who carries the risk that dividends change.

## 5.1 Forwards with discrete dividends and borrow

A share pays dividends on known ex-dividend dates (One Quant Book 1, chapter 8), and a short seller pays a borrow fee to the lender (One Quant Book 1, chapter 6). Both reduce the forward below the financed spot: the buyer of a forward does not receive the dividends paid before delivery, and the seller who holds the share as a hedge can lend it out and earn the fee.

**Proposition 5.1 (The equity forward).**

With cash dividends $D_i$ paid at $t_i$, a flat financing rate $r$ and a borrow fee $\ell$ (continuously compounded, paid on the value lent), the forward for delivery at $T$ is

$$
F_{0,T}=\Bigl(S_0-\sum_{t_i\le T}D_iP(0,t_i)\Bigr)\frac{e^{-\ell T}}{P(0,T)} .
$$

**Proof.** Buy the share for $S_0$, borrow the present value of the dividends against them, lend the share out and reinvest the fee, sell the forward. At $T$ the position is flat and the dividends have repaid their loan: the cost of the share net of what it earns must equal the present value of the delivery price. ∎

**Definition 5.2 (Proportional dividend).**

A *proportional dividend* is modelled as a fixed fraction $y_j$ of the share price at the ex-date rather than a fixed amount: the share drops by $y_jS_{t_j-}$, and the forward is multiplied by $(1-y_j)$ for every ex-date before $T$. A continuous dividend yield $q$ is its limit when payments are spread evenly over time.

The forward curve of a share is a sawtooth ([Figure 5.1](#fig-dv-dividends-borrow-and-forwards-curve)): it grows at the financing rate less the borrow between ex-dates and drops by each dividend. A continuous yield with the same two-year forward is a straight line through it, wrong at every intermediate date by up to half a quarter’s dividend. For options whose expiry straddles an ex-date the difference is the whole dividend.

![Forward curve of a share at 100 paying 0.60 a quarter, with a 3% financing rate and a 0.5% borrow fee. Between ex-dates the forward grows at 2.5% a year; on each ex-date it drops by the dividend. Illustrative parameters.](https://one-course.com/images/onecourse/chapters/quant-5/dv-dividends-borrow-and-forwards/fig-e08365241332.svg)

***Figure 5.1.** Forward curve of a share at 100 paying 0.60 a quarter, with a 3% financing rate and a 0.5% borrow fee. Between ex-dates the forward grows at 2.5% a year; on each ex-date it drops by the dividend. Illustrative parameters.*

## 5.2 Dividends in the option model

For a European option, only the forward and the distribution around it matter; the Black formula of chapter 3 prices it on the forward. The choice of dividend model decides what the quoted volatility is the volatility of, and how the forward moves when the spot moves.

**Definition 5.3 (Escrowed dividend model).**

In the *escrowed dividend model* the share is split into the present value of the dividends to be paid before expiry, treated as riskless, and the rest, $S^*_t=S_t-\sum_{t<t_i\le T}D_iP(t,t_i)$, which follows a geometric Brownian motion. European options are priced by Black–Scholes with $S^*_0$ in place of $S_0$.

Three models are in use. In the escrowed model the volatility is that of $S^*$. In the proportional model it is that of the whole share, which stays lognormal through its proportional drops. In the *spot model* the whole share is lognormal between ex-dates and drops by the cash amount on each: the most realistic, and the only one with no closed form. For a European option with the same forward, the escrowed and proportional models give the same price at the same volatility, since both make the forward lognormal with that volatility; the spot model gives a higher price, because before the ex-date the part of the share that will be paid out as a dividend also moves.

**Proposition 5.4 (The volatility of the escrowed share).**

If the share has volatility $\sigma$ in the spot model and pays one dividend of present value $\bar D$ at $t_1<T$, the escrowed model reproduces its at-the-money price to first order with the effective volatility

$$
\sigma_{\mathrm{eff}}^2\,T=\sigma^2\Bigl(\frac{S_0}{S_0-\bar D}\Bigr)^2t_1+\sigma^2(T-t_1).
$$

**Proof.** Before $t_1$ the escrowed share $S^*=S-\bar D e^{rt}$ moves by the whole share’s moves, $dS^*=dS=\sigma S\,dW$, so its relative volatility is $\sigma S/S^*$; after $t_1$ the two coincide. Adding the variances of the two periods with $S/S^*$ frozen at its initial value gives the formula. ∎

**Example 5.5 (One dividend of 4).**

Share 100, one-year options, $r=3\%$, a dividend of 4 in six months, spot-model volatility 25%. The escrowed and proportional models at 25% imply a flat 25.00; the spot model’s prices imply 25.57 at the 70 strike and 25.49 at the 130; the effective volatility of the proposition is 25.52 ([Figure 5.2](#fig-dv-dividends-borrow-and-forwards-models)). Half a volatility point is 0.2 of premium at the money: a desk that mixes models across systems books it as profit or loss.

![One-year calls on a share at 100 with a dividend of 4 in six months, priced by three dividend models with the same 25% volatility and read back as Black implied volatilities on the common forward. The spot model is half a point above the others and slightly skewed; the effective volatility of matches it at the money. Data: the tutorial.](https://one-course.com/images/onecourse/chapters/quant-5/dv-dividends-borrow-and-forwards/fig-739a74a07836.svg)

***Figure 5.2.** One-year calls on a share at 100 with a dividend of 4 in six months, priced by three dividend models with the same 25% volatility and read back as Black implied volatilities on the common forward. The spot model is half a point above the others and slightly skewed; the effective volatility of [Proposition 5.4](#prop-dv-dividends-borrow-and-forwards-adj) matches it at the money. Data: the tutorial.*

**Definition 5.6 (Mixed dividend model).**

A *mixed dividend model* treats near dividends as cash amounts, announced or forecast with confidence, and dividends beyond a horizon (one to two years) as proportional to the share price, blending the two over a transition period.

The blend is about dynamics. With cash dividends the forward moves one for one with the spot, $\partial F/\partial S=1/P(0,T)$; with [proportional dividends](#def-dv-dividends-borrow-and-forwards-prop) it moves in proportion, $\partial F/\partial S=F/S$. For a five-year option on a share yielding 4% the two deltas differ by almost a fifth, and so does the hedge. A company whose share halves rarely keeps paying the same cash dividend, which is why far dividends are modelled as proportional; near dividends are already declared.

## 5.3 Implying forwards, dividends and borrow from parity

For European options put–call parity, $C-P=P(0,T)(F-K)$, holds at every strike. It is linear in the strike, with slope $-P(0,T)$ and intercept $P(0,T)F$, so a regression across strikes reads both off the chain without any model.

**Method 5.7 (Reading the forward curve off a chain).**

For each expiry:

1. regress $C_K-P_K$ on $K$ over the strikes near the money: the slope is $-P(0,T)$ , the intercept $P(0,T)F$ ;
2. the implied carry $S-FP(0,T)$ is the present value of dividends plus borrow cost to $T$ ; its increments between expiries give the dividend term structure;
3. with a dividend forecast, what remains of the carry is the borrow, the next definition; with the borrow known (general collateral), what remains is the market’s dividend forecast.

For American single-stock options parity is only a pair of inequalities; use European index options, or remove the early-exercise premium first (chapter 6).

**Definition 5.8 (Implied dividend and implied borrow rate).**

The *implied dividend* of an expiry is the present value of dividends to that expiry that makes the forward read from options, together with the known borrow cost, consistent with the spot. The *implied borrow rate* is the borrow fee $\ell$ that makes the forward consistent with the spot and a dividend forecast: $\ell=-\ln\bigl(FP(0,T)/(S-\bar D)\bigr)/T$.

**Example 5.9 (The hard-to-borrow share).**

The one-month chain of the opening, five strikes from 45 to 55 quoted to the cent, regressed as in [Method 5.7](#met-dv-dividends-borrow-and-forwards-read), gives a forward of 48.74 and a discount factor of 0.99680 (a rate of 3.90%, against 4% true: the cents of rounding). With no dividend due, the carry of 1.42 is all borrow: 35.0% a year. Priced from the naive forward $50e^{0.04\times30/365}=50.16$ at the chain’s 60% volatility, the call would be worth 3.50 instead of 2.78, and the put 3.34 instead of 4.04. The “skew” of 72 against 47 is the wrong forward, not the market’s view of volatility.

Selling the put, buying the call and shorting the share (the reversal of One Quant Book 1, chapter 25) appears to earn 1.42 in a month; it requires borrowing the share for a month at 35%, which costs 1.44. Options on hard-to-borrow shares are where parity fails most visibly when measured from the spot, and academic studies find that the size of the apparent violation tracks the stock-lending market’s fee (Ofek, Richardson and Whitelaw).

![Present value of the carry (dividend plus borrow cost) between consecutive quarterly expiries, recovered by the parity regression from a synthetic chain whose mid prices carry up to two cents of noise at nine strikes. The cumulative forward is recovered to about a cent; each quarter’s increment only to about 0.08. Data: the tutorial.](https://one-course.com/images/onecourse/chapters/quant-5/dv-dividends-borrow-and-forwards/fig-c28e45ce043f.svg)

***Figure 5.3.** Present value of the carry (dividend plus borrow cost) between consecutive quarterly expiries, recovered by the parity regression from a synthetic chain whose mid prices carry up to two cents of noise at nine strikes. The cumulative forward is recovered to about a cent; each quarter’s increment only to about 0.08. Data: the tutorial.*

## 5.4 Dividend risk and dividend swaps

**Definition 5.10 (Dividend risk and dividend swap).**

*Dividend risk* is the sensitivity of a position to a change in the dividends expected before its maturity: a forward, a long call or a short put loses when expected dividends rise. A *dividend swap* is an over-the-counter contract that exchanges, at maturity, the dividends actually paid on a share or index over a period for a fixed amount agreed at inception; the listed version is the dividend future of One Quant Book 1, chapter 22.

[Dividend risk](#def-dv-dividends-borrow-and-forwards-risk) is small for a one-month option and dominant for a five-year one. It accumulates on the books of structured-product issuers: a bank that sells notes paying a function of an index hedges by holding the index, collects its dividends, and has priced those dividends into the notes; if dividends fall, the hedge earns less than was promised. Practitioners describe such issuers as structurally long dividends and as sellers of [dividend swaps](#def-dv-dividends-borrow-and-forwards-risk) and futures to lay the risk off (chapter 18). The buyers are investors who want dividend exposure without equity exposure.

**As of September 2026 — Index dividend futures and 2020.**

The euro-area benchmark index dividend futures listed on Eurex have a contract value of EUR 100 per index point and a tick of 0.1 point (EUR 10), settle in cash on the cumulative gross dividends paid by the index members over the contract year, and list annual maturities up to about ten years out. On 27 March 2020 the European Central Bank recommended that the banks it supervises pay no dividends until at least 1 October 2020, later extended to January 2021. A practitioner account of April 2020 reported 2020 index dividends trading at 55% of what models had predicted and 2021 dividends at 70%, as issuers sold dividends into a falling market.

![Where dividend risk goes. The issuer of equity-linked notes hedges with the underlying, is paid its dividends and has promised their expected value to the note holders; it lays the risk off by selling dividends forward to investors who want them.](https://one-course.com/images/onecourse/chapters/quant-5/dv-dividends-borrow-and-forwards/fig-1ad1cc28f22a.svg)

***Figure 5.4.** Where [dividend risk](#def-dv-dividends-borrow-and-forwards-risk) goes. The issuer of equity-linked notes hedges with the underlying, is paid its dividends and has promised their expected value to the note holders; it lays the risk off by selling dividends forward to investors who want them.*

## 5.5 Tutorial: a forward curve from a chain

**Goal.** Read forwards, discount factors, carry and borrow off a synthetic chain, and compare the three dividend models on one option. **End state:** Figures [5.2](#fig-dv-dividends-borrow-and-forwards-models) and [5.3](#fig-dv-dividends-borrow-and-forwards-carry) and the numbers of [Example 5.9](#ex-dv-dividends-borrow-and-forwards-htb).

1. **Parity by regression**, then carry, borrow and the dividend strip: `def implied_forward (strikes, calls, puts) -> tuple [float , float ]: """Least squares of C - P on K across strikes: slope -P(T), intercept P(T) F. Returns (F, P(T)).""" k = np.asarray(strikes, float ) y = np.asarray(calls, float ) - np.asarray(puts, float ) a = np.column_stack([np.ones_like(k), k]) (intercept, slope), *_ = np.linalg.lstsq(a, y, rcond=None ) df = -slope return float (intercept / df), float (df) def implied_carry (spot: float , fwd: float , df: float ) -> float : """PV of everything the forward removes from the spot: dividends plus the cost of borrow (Book 1, Chapter 25's D_imp).""" return spot - fwd * df def implied_borrow (spot: float , fwd: float , df: float , t: float , pv_dividends: float = 0.0 ) -> float : """Borrow fee l such that F = (S - PV divs) exp(-l T) / P(T), given a dividend forecast.""" return -math.log(fwd * df / (spot - pv_dividends)) / t def strip_dividends (spot: float , expiries, forwards, dfs) -> list [float ]: """PV (today) of the carry implied between consecutive expiries: the dividend term structure.""" carry = [implied_carry(spot, f, d) for f, d in zip (forwards, dfs, strict=True )] return [carry[0 ]] + [b - a for a, b in zip (carry, carry[1 :], strict=False )]` **Listing 5.1.** Forward and discount factor from a chain; implied carry, borrow and dividend strip. code/firm/divfwd/firm_divfwd.py
2. **The spot model** by Gauss–Hermite quadrature over the share just before the ex-date, and the effective volatility of the escrowed share: `def spot_call (k: float , c: dict = ONE, n: int = 96 ) -> float : """Spot model: the share is lognormal with the quoted volatility and drops by the cash amount on the ex-date. Condition on the share just before the ex-date (Gauss-Hermite in its log) and price the remaining period with Black-Scholes.""" x, w = np.polynomial.hermite_e.hermegauss(n) # nodes and weights for exp(-x^2/2) w = w / w.sum() t1, s0, r, v = c[" t_div " ], c[" spot " ], c[" r " ], c[" vol " ] s1 = s0 * np.exp((r - 0.5 * v * v) * t1 + v * math.sqrt(t1) * x) after = np.maximum(s1 - c[" div " ], 1e-12 ) vals = np.array([bs(s, k, c[" t " ] - t1, r, 0.0 , v, " C " ) for s in after]) return float (math.exp(-r * t1) * (w * vals).sum()) def effective_vol (c: dict = ONE) -> float : """Volatility to give the escrowed share S* = S - PV(D) so that it matches the spot model: before the ex-date S* carries the whole share's moves, so its volatility is sigma S / S*.""" pv = c[" div " ] * math.exp(-c[" r " ] * c[" t_div " ]) ratio = c[" spot " ] / (c[" spot " ] - pv) return c[" vol " ] * math.sqrt((ratio ** 2 * c[" t_div " ] + (c[" t " ] - c[" t_div " ])) / c[" t " ])` **Listing 5.2.** The spot model, and the escrowed model’s effective volatility. code/derivatives/05-dividends-borrow-and-forwards/python/dv_dividends.py
3. **Run** `dv_dividends.curve_from_chain()` , `htb_results()` and `fig_dividends.py` .

**What to change next.** Regress on the five strikes nearest the forward only, and on all strikes weighted by the inverse of the quote width, and compare the noise of the quarterly carry; then give the share two dividends before expiry and extend the effective volatility to both.

## 5.6 Build: the equity forward curve

**Purpose.** Every equity option of the miniature firm is priced on a forward that comes from this curve; in the pricing library of chapter 28 it is what `MarketData.forward` computes.

**Interface.** `ForwardCurve(spot, rate, cash, proportional, borrow).forward(t)`; `implied_forward(strikes, calls, puts) -> (F, df)`; `implied_carry`; `implied_borrow(spot, fwd, df, t, pv_dividends)`; `strip_dividends(spot, expiries, forwards, dfs)`.

**Rules.** Dividends strictly after today and on or before the delivery date count; [proportional dividends](#def-dv-dividends-borrow-and-forwards-prop) multiply the forward; the regression uses European quotes (or de-Americanised ones).

**Acceptance tests.** `code/firm/divfwd/tests/`: the forward replicates by the cash-and-carry argument; the regression recovers a known forward and discount factor exactly from noiseless quotes; the implied borrow of a clean chain equals the fee used to build it; strips sum to the total carry.

**Stretch.** Fit a smooth borrow curve and a dividend curve jointly across expiries with a penalty on roughness, and flag expiries whose parity residuals are outliers.

Sources and further reading

- E. G. Haug, J. Haug and A. Lewis, “Back to basics: a new approach to the discrete dividend problem”, *Wilmott* (September 2003) 37–47.
- E. Ofek, M. Richardson and R. F. Whitelaw, “Limited arbitrage and short sales restrictions: evidence from the options markets”, *Journal of Financial Economics* 74 (2004) 305–342.
- European Central Bank, Recommendation ECB/2020/19 of 27 March 2020 on dividend distributions during the COVID-19 pandemic.
- Eurex, EURO STOXX 50 Index Dividend Futures, contract specifications.
- E. Barthe, “Hedging structured products during the corona crisis: where did it go wrong?”, Structured Retail Products, 30 April 2020.

## 5.7 Exercises

**Exercise 5.1 ★.**

A share at 100 pays dividends of 1 in three and in nine months; $r=3\%$, no borrow fee. Give the one-year forward.

**Solution of Exercise 5.1.**

$(100-e^{-0.0075}-e^{-0.0225})e^{0.03}=101.0152$.

**Exercise 5.2 ★.**

At one expiry, $C-P$ is 7.02 at the strike 95 and $-2.83$ at the strike 105. Give the discount factor and the forward.

**Solution of Exercise 5.2.**

Slope $(-2.83-7.02)/10=-0.985$: the discount factor is 0.985. Then $7.02=0.985(F-95)$ gives $F=102.1269$.

**Exercise 5.3 ★.**

Check the effective volatility 25.52% of [Example 5.5](#ex-dv-dividends-borrow-and-forwards-adj).

**Solution of Exercise 5.3.**

$\bar D=4e^{-0.015}=3.9404$, $S/(S-\bar D)=1.0410$; $\sigma_{\mathrm{eff}}=0.25\sqrt{1.0410^2\times0.5+0.5}=0.2552$.

**Exercise 5.4 ★★.**

Why does put–call parity give only bounds for American single-stock options, and in which direction is the error when the European relation is used near an ex-date?

**Solution of Exercise 5.4.**

Each side can be exercised early, so $C-P$ is only bounded: $S-D-K\le C-P\le S-Ke^{-rT}$ for American options on a dividend payer. Before an ex-date the call carries an early-exercise premium (it may be exercised to capture the dividend), so $C-P$ is too high and the forward read from the European relation is too high: the [implied dividend](#def-dv-dividends-borrow-and-forwards-implied) comes out too low.

**Exercise 5.5 ★★.**

A share at 100 pays a dividend worth 4% of its price in six months; $r=3\%$, one year. Give $\partial F/\partial S$ when the dividend is modelled as the cash amount 4 and when it is modelled as proportional.

**Solution of Exercise 5.5.**

Cash: $F=(S-4e^{-0.015})e^{0.03}$, so $\partial F/\partial S=e^{0.03}=1.0305$. Proportional: $F=0.96\,Se^{0.03}$, so $\partial F/\partial S=0.9892$. The same dividend gives two hedges 4% apart.

**Exercise 5.6 ★★.**

A three-year at-the-money call on the share of [Figure 5.1](#fig-dv-dividends-borrow-and-forwards-curve) (quarterly dividends of 0.60, $r=3\%$, borrow 0.5%, volatility 20%) is repriced after a 10% cut in every future dividend. By how much does its value change?

**Solution of Exercise 5.6.**

The forward rises from 100.3611 to 101.1038 and the call from 12.7558 to 13.1475: $+0.3917$, about 3% of its value, for a 10% cut in dividends.

**Exercise 5.7 ★★★.**

*Coding.* From `synthetic_chain()`, compute the implied forward and discount factor of the one-year expiry and the implied borrow given the known dividends, and compare with the 0.5% used to build the chain.

**Solution of Exercise 5.7.**

Forward 100.1132, discount factor 0.97033; with the known dividends (present value 2.3661) the implied borrow is 0.504%, against 0.5%: the regression error, about a cent on the forward, is 0.004% of borrow.

**Exercise 5.8 ★★★.**

*Find the flaw.* “On this hard-to-borrow share the one-month 50 puts trade at 72 volatility and the calls at 47. Selling puts and buying calls, delta-hedged, captures 25 volatility points.”

**Solution of Exercise 5.8.**

Both volatilities are computed on the wrong forward. On the forward the options price, 48.74, the call and the put at 50 both imply 60%: there is no volatility difference to capture. The trade is a reversal in disguise: selling puts and buying calls, hedged by shorting shares, needs a borrow at 35%, which costs what the “edge” appears to pay.

## 5.8 Problem: The Hard-to-Borrow Share

**Problem 5.1.**

Weekend problem — reading the borrow off the options

The share of the opening trades at 50 and pays no dividend in the next month; the financing rate is 4%. Its one-month European options are quoted (mid): 45 call 5.41, put 1.68; 47.5 call 3.94, put 2.71; 50 call 2.78, put 4.04; 52.5 call 1.90, put 5.65; 55 call 1.26, put 7.50. The chain’s at-the-money volatility is 60%.

**Part I — The forward.**

1. Compute $C-P$ at each strike.
2. Regress it on the strike: give the discount factor and the forward.
3. Give the implied rate, and explain the gap to 4%.
4. Give the implied carry.
5. Give the [implied borrow rate](#def-dv-dividends-borrow-and-forwards-implied) .

**Part II — The wrong forward.**

6. Give the naive forward from the spot and the rate.
7. Price the 50 call and put at 60% on the naive forward.
8. By how much are they mispriced?
9. Give their implied volatilities computed on the naive forward.
10. Why do the two implied volatilities differ, and which forward makes them equal?

**Part III — The apparent arbitrage.**

11. What does the reversal (short share, short put, long call) appear to earn?
12. What does it cost to borrow the share for the month?
13. Is there an arbitrage once borrow is counted?
14. What can a holder of the share do with the options instead of lending it, and what does that earn?
15. How would a desk that is not able to borrow the share express a view that it will fall?

**Part IV — Judgement.**

16. What happens to the options if the borrow fee falls to 5% overnight?
17. Why are listed options on such shares American, and what does that change?
18. What should an options system use as the forward for this share?
19. State the *named result* : the [implied borrow rate](#def-dv-dividends-borrow-and-forwards-implied) and the naive pricing error on the 50 call.
20. In one sentence: what does put–call parity measure on a hard-to-borrow share?

**Solution of Problem 5.1.**

**1.** 3.73, 1.23, $-1.26$, $-3.75$, $-6.24$. **2.** Discount factor 0.99680; forward 48.74. **3.** 3.90%; the quotes are rounded to the cent, and a cent of error on a slope measured over ten points of strike moves the discount factor by $10^{-3}$. **4.** 1.42. **5.** 35.0% a year. **6.** 50.16. **7.** Call 3.50, put 3.34. **8.** The call by $+0.72$ (26% of its price), the put by $-0.70$. **9.** Call 47.3%, put 72.3%. **10.** Parity ties $C-P$ to the forward; with the naive forward the call’s and the put’s volatilities must differ to fit it. On the forward 48.74 both are 60%. **11.** $51.26-50e^{-0.04\times30/365}$: 1.42 today. **12.** $50\times0.35\times30/365=1.44$. **13.** No: the apparent profit is the borrow cost, to within the rounding of the quotes. **14.** Sell the share and buy it back synthetically (buy the call, sell the put): this brings in 1.42 for the month, the same as lending the share at 35%. Holders who cannot lend (a fund whose custodian does not lend) earn the fee this way. **15.** Buy puts or put spreads, or sell calls: the options carry the borrow cost in their prices, so the view is expensive but expressible. **16.** The forward jumps from 48.74 to 49.96: the 50 call rises and the put falls by about 0.6 each, a move unrelated to volatility or spot. **17.** The borrow fee acts like a dividend yield: an American call may be worth exercising early to own the share and lend it at the fee, which adds an early-exercise premium to the calls and blurs parity further. **18.** The forward implied by the options themselves, updated with the borrow market, not spot times the financing rate. **19.** 35.0% a year; the naive price of the 50 call is 0.72 too high. **20.** The forward, and through it the cost of being short the share.

## 5.9 Interview questions

**Interview question 5.1 ★ trader, researcher.**

How do you get the forward of a share from its option prices, without a model?

**Solution of Interview question 5.1.**

Put–call parity: $C-P=P(0,T)(F-K)$. Regress $C-P$ on $K$ over European quotes near the money; the slope is $-P(0,T)$, the intercept $P(0,T)F$. With American options, use out-of-the-money quotes where early exercise is negligible or remove its premium first.

*What the interviewer is looking for: parity as a linear regression across strikes.*

**Interview question 5.2 ★ trader.**

The at-the-money put of a share trades well above the call at the same strike. Give two reasons.

**Solution of Interview question 5.2.**

The forward is below the spot: a large dividend before expiry, or a high borrow fee (a hard-to-borrow share). Both appear in parity as carry; at the forward the two options are worth the same.

*What the interviewer is looking for: dividends and borrow through the forward.*

**Interview question 5.3 ★★ researcher, bank.**

Cash or [proportional dividends](#def-dv-dividends-borrow-and-forwards-prop): which do you use for a one-month option and for a five-year one, and what changes in the hedge?

**Solution of Interview question 5.3.**

Cash for the near dividends, already declared; proportional (or a blend) for far ones, because a company’s dividend falls when its share falls. The hedge changes because the forward’s sensitivity to spot is $1/P(0,T)$ with cash dividends and $F/S$ with proportional ones; for a five-year option the gap is of the order of the dividend yield times five years.

*What the interviewer is looking for: the dynamics, not only the forward level.*

**Interview question 5.4 ★★ trader, risk.**

What is [dividend risk](#def-dv-dividends-borrow-and-forwards-risk), who is naturally long it, and what happened to it in 2020?

**Solution of Interview question 5.4.**

The sensitivity of positions to expected dividends. Issuers of equity-linked structured products, whose hedges collect dividends they have promised to note holders, are long; they sell [dividend swaps](#def-dv-dividends-borrow-and-forwards-risk) and futures. In 2020 supervisors asked banks to suspend dividends and index dividends traded far below models, and issuers’ hedging sales added to the fall.

*What the interviewer is looking for: who carries it, and why the 2020 fall was amplified.*

**Interview question 5.5 ★★ developer, researcher.**

How do you put a discrete cash dividend into a tree pricer for American options, and what goes wrong with the obvious approaches?

**Solution of Interview question 5.5.**

Options: escrowed (a tree on $S-\mathrm{PV}(D)$, adding the PV back for exercise), or a drop of the node values by $D$ at the ex-date. The escrowed tree understates the volatility before the ex-date unless its volatility is adjusted; dropping by $D$ breaks recombination (the tree must be rebuilt after the ex-date or interpolated), and a proportional drop keeps recombination but is wrong for a cash dividend. Test against a European reference with the same forward.

*What the interviewer is looking for: the recombination problem and the volatility mismatch.*

**Interview question 5.6 ★★★ researcher.**

The same volatility of 25% is fed to the escrowed model in one system and the spot model in another. Which gives the higher call price, by roughly how much, and why?

**Solution of Interview question 5.6.**

The spot model: its volatility applies to the whole share including the part paid out, so before the ex-date the forward moves more. For a one-year option with a 4% dividend at six months, about half a volatility point, 0.2 of premium at the money; the adjustment is $\sigma^2(S/(S-\bar D))^2t_1+\sigma^2(T-t_1)$.

*What the interviewer is looking for: what the volatility is the volatility of.*